Download - Integrable Systems in Symplectic Geometryjansa/thesis.pdffriends in Eindhoven, among them are, Amir Hossein Gharamarian, Kambiz Pournazeri and his wife Sima Nasr, Parsa Beigi, Neda

Integrable Systemsin

Symplectic Geometry

Integrable Systems in Symplectic GeometryEsmaeel AsadiISBN:978-90-9023000-9

THOMAS STIELTJES INSTITUTE

FOR MATHEMATICS

Copyright c© 2008, E. Asadi, Amsterdam

All rights reserved. No part of this publication may be reproduced in anyform or by any electronic or mechanical means (including photocopying,recording, or information storage and retrival system) without permissionin writing from the author.

Printed by PrintPartners Ipskamp, the Netherlands.

VRIJE UNIVERSITEIT

Integrable Systems in Symplectic Geometry

ACADEMISCH PROEFSCHRIFT

ter verkrijging van de graad Doctor aande Vrije Universiteit Amsterdam,op gezag van de rector magnificus

prof.dr. L.M. Bouter,in het openbaar te verdedigen

ten overstaan van de promotiecommissievan de faculteit der Exacte Wetenschappen

op maandag 26 mei 2008 om 13.45 uurin de aula van de universiteit,

De Boelelaan 1105

door

Esmaeel Asadi

geboren te Arak, Iran

promotor: prof.dr. J. Hulshofcopromotoren: dr. J.A. Sanders

dr. J.P. Wang

research supervisor: dr. J.A. Sandersdr. J.P. Wang

reading committee: prof.dr. A. V. Mikhailovdr. J. Draismadr. F. Pasquottodr. J. van de Leur

To my wife, my mother andthe soul of my father

Acknowledgments

We could say that everyone’s life has several layers. The surface layer is constantlychanging and sometimes it transmutes into a storm. This layer, usually deals with dailylife happenings, tasks, and responsibilities. The exterior layer, tends to draw great dealof attention towards itself: the semester is starting, deadlines are pushing, tasks arelining up, moving to a new apartment is on the way, and many more. However, I havelearned to focus, and see beyond the surface layer, deeper into my inner life and whatI consider to be of more importance to me. For the last four years, I have been ableto track the inner progress of my personality and the changes that happen in the stateof my mind. Every now and then, while sitting behind my desk and sipping coffee, Ilook into the progress and the new challenges on my way, which provide me with freshperspective. The droplets of formulas and equations open their way through this aridsoil of technical jargon. I wait for a spark of life, for a sprout of passion to appear. I didas Albert Einstein says :“Life is like riding a bicycle. To keep your balance you mustkeep moving.”

Now here is the place to thank several people who have been pushing me to move.First of all there is dr. Jan Sanders, thank you very much for being a great supervisor,for all discussion we had in your room, for pointing me always in the right direction tolearn things and get the results and for all your help and dedication. I would also liketo thank your wife, Wil for being so kind to me.

I am specially grateful to my second advisor dr. Jing Ping Wang for reading thedrafts and thesis and giving bright comments which were so helpful.

I would also like to thank my promotor, prof.dr. Joost Hulshof and my thesis com-mittee for their support: prof.dr. A. V. Mikhailov (University of Leeds), dr. Jan Draisma(Technische Universiteit Eindhoven), dr. Federica Pasquotto (Vrije Universiteit) and dr.Johan van de Leur (Universiteit Utrecht).

I would also like to thank dr. Sara Lombardo for her kind support. Thanks alsogo to dr. Guido Carlet and dr. Vladimir Novikov and staff members of the Universityof Kent in the UK where I was a visitor for four months. I am grateful particularlyto the mathematics department of VU, staff members of Mathematical analysis and inparticular I should thank Maryke Titawano and drs. S.J. Chedi for their help withaccomodations and my visa procedure. Thanks also go to prof. Ron Perline for his

vii

permission and exchanged e-mails for his home-grown soliton surface picture which Iused to design the cover of my thesis. I would also should thank my cousin, Mr. RezaAkbari for his help to make that cover.

I am so happy that during the time I had been studying my bachelor in IUST andmaster in Sharif university of Technology, I met many people with whom I am still intouch. Among them are dr. B. Mirzaii, dr. F. Behrouzi, dr. K. Malakpoor and Mr. S.Mohammadian. I had their valuable support. These are friends who I am so grateful tohave. I never forget their kindness. I should also thanks Mrs. A. Kamali Sarvestani forher hospitality and her lovely sons Ilia and Aria.

During these years in Netherlands I had a great time specially in Eindhoven with myfriends, Ali Etaati, Lusine Hakobyan and Kamyar Malakpoor. We all together had lotsof fun, went to many nice places specially our trip to Germany was perfect having fun,singing Armenian songs in the car and so on. I had also happy moments with my otherfriends in Eindhoven, among them are, Amir Hossein Gharamarian, Kambiz Pournazeriand his wife Sima Nasr, Parsa Beigi, Neda Sepacian and many other friends.

In Amsterdam many people deserve thanks for their contribution. My dear friend, dr.Mohammad Abry, was my roommate for four years and I am so grateful to get to knowsuch a friend, my thank also goes to his wife, Monireh and his two lovely sons, Moeenand Soroush. My other officemate, including Dave Visser, Kirsten Valkenburg, DesireeBasile and Adi Setiawan are all deserve to thank of. I also should thank Hamid Abbasi,Hamid Falah and Hu Hai for all those moments I had with them playing table tennis.Thanks go to dr. S. Eslami, his wife, Mr. M. Botshekan and dr. Reza Bakhtiari forhis lovely hospitality in Italy, Pisa and to Saeed Torkzaban and his wife Shiva, as wellas Behnam Norouzizadeh, Leila Mohammadi, Marjan Asgari, Mathijs Louws, ArezooBahramirad. I am so happy as well to have met Hala Elrofai and became very closefriend, she pushed me to move forward significantely in my life.

I never forget how I missed my mother Mrs Ghamartaj Nadjafi, my sister MissMarhamat and my brother Mr. Abolfazl, his wife and my dear nephews, Ali and Amir-mohammad. Thank you all for your unconditional supports.

Most of all , I must express my heartfelt to my wonderful wife, Masoumeh Robatmeili.Having her with me was my on time move. Thank you for all your patience and yourkindness and understanding to be with me in a house with a light, window and happiness.I should also thank my mother-in-law and father-in-law for their support specially whenwe were at home last few months. I would also like to thank my sisters-in-law particularlyMiss Maryam and Mahboobeh Robatmeili who were so nice to me and I hope one daywe start to watch the rest of “Daii jan Napelon”.

Amsterdam, March 2008 Esmaeel Asadi

viii

Contents

Introduction 1

Some history . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1 Quaternionic algebra 7

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2 Inner product on quaternionic vectors . . . . . . . . . . . . . . . . . . . . 12

1.3 Symplectic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.4 Killing form in Symplectic Lie algebra . . . . . . . . . . . . . . . . . . . . 14

2 Variational complex and geometric operators 17

2.1 Algebraic approach to the concept of geometric operator . . . . . . . . . . 17

2.2 Formal Variational Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2.1 Classical formal variational calculus . . . . . . . . . . . . . . . . . 20

2.2.2 Variational calculus for geometric dynamical symbols . . . . . . . . 24

2.3 Schouten bracket . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Symplectic operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Nijenhuis operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Preliminaries of Geometry 35

3.1 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Cartan’s moving frame method . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Ehresmann connection and Cartan geometry . . . . . . . . . . . . . . . . 41

3.4 Homogeneous space, symmetric space . . . . . . . . . . . . . . . . . . . . 43

4 Geometric method of Integrable system 45

4.1 Integrable system in Riemannian geometry . . . . . . . . . . . . . . . . . 45

4.2 Integrable system in (p,q)-Orthogonal geometry . . . . . . . . . . . . . . 54

4.3 Integrable system in symplectic geometry . . . . . . . . . . . . . . . . . . 59

4.4 Geometric operators in the form of Lie algebraic objects... . . . . . . . . . 71

ix

5 Lax representation of an integrable system 775.1 Theory of Lax method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Lax method in Symplectic geometry . . . . . . . . . . . . . . . . . . . . . 885.3 Higher symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6 Computation of Geometric operator 996.1 Hamiltonian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.2 Symplectic operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

A Computations in Hamiltonian operators 123

B Some Killing form identities 125

C Computation of Killing form 131

Summary 141

x

Introduction

Some history

Many of the equations and systems which now are called integrable have been known indifferential geometry. One of them is the famous sine-Gordon equation (SG), which wasderived to describe surfaces with constant negative Gaussian curvature. Another oneis the Liouville equation describing minimal surfaces in 3−dimensional Euclidean space.For physicists, the prototype examples of integrable systems are the Korteweg-De Vriesequation (KDV) [15] and the nonlinear Schrodinger equation (NLS) [19].

“what is an integrable system?”

This is a question with many answers of varying degree of precision, generality andplausibility. We will try briefly to list few of these answers.

Newton’s equations of motion are those three famous equations which are taught toevery student of elementary Classical Mechanics. The Kepler two-body problem and afew other equations turned out to have “exact solutions”.

In fact these equations are particular cases of more general mechanical systems,known as (finite dimensional) Hamiltonian system, with a Hamiltonian function and aPoisson bracket.

A Hamiltonian system is called “completely integrable” if it has as many independentfunctions in involution with the Hamiltonian function and themselves as it has degreesof freedom.

In the nineteenth century, Liouville provided a general framework characterizing thecases where completely integrable Hamiltonian system are “solvable by quadratures”,i.e., the general solution is found by integration and algebraic operations only, see [3].

The discovery of the physical soliton is attributed to John Russell’s observation in1834 as he described it in his “Report on Waves” [55]. Much later in 1895 Korteweg and

1

2

de Vries derived the equation for water waves in shallow channels, which confirms theexistence of solitary waves. The equation which now bears their names is of the form

ut = u3 + uu1 (KDV equation).

In 1965, Kruskal and Zabusky, following a computationally numerical study of theBoussinesq anharmonic lattice of equal masses which was done by Fermi, Pasta and Ulam(FPU), rederived the KDV equation and found its stable pulse-like waves. They namedsuch waves solitons. These are solitary waves in the form of pulses whose behavior hasmany particle-like features. During their evolution, solitons propagate without changeof shape and with no energy loss. When two or more solitons with different propagationspeed collide, after a highly nonlinear interaction the pulses emerge with the same initialform and no energy is lost in radiation in the course of the interaction.

The stability and particle-like behavior of the solitons could only be explained by theexistence of many conservation laws : DtU +DxF = 0; in which U is called conserveddensity and F conserved flux. Zabusky and Kruskal started to find more of them.Later, it was proved by Miura, Gardner and Kruskal in 1968 that there was indeed aconserved density of each order [50].

Gardner was the first to notice that the KDV equation could be written in a Hamil-tonian framework. Later Zakharov and Fadeev showed how this could be interpretedas a completely integrable Hamiltonian system in a same sense as finite dimensionalintegrable systems [77] where one finds for each degree of freedom a conserved density.

Perhaps the richest group of equations known to be integrable are pseudosphericalsurfaces. They are surfaces in R3 with constant negative Gaussian curvature. Bianchi[18] discovered a beautiful relation between iterated Backlund transformations: thepermutability theorem. This theorem assert that for two Backlund transformationsf1 = Bσ1f and f2 = Bσ2f of a pseudospherical surface f corresponding to angles σ1, σ2

between the normals, there exists a fourth pseudospherical surface f which is simulta-neously a Backlund transformation of f1 and f2 :

f = Bσ1f2 = Bσ2f1.

Moreover f can be computed algebraically from f, f1, f2. In this way, the Backlundtransformation generates an infinite-dimensional ‘symmetry group’ acting on the set ofpseudospherical surfaces and the permutability theorem shows the possibility of writingdown explicit solutions starting with a simple f.We might say that the symmetry of an equation is the conserved geometric feature ofsolitons. The symmetry groups of differential equations were first studied by SophusLie. In his framework, these consist of geometric transformations of independent anddependent variables of the system. In the case of KDV, there are four such symmetries,namely arbitrary translation in x and t, Galilean boost and scaling. In the contextof pseudospherical surfaces, see [6], Backlund transformation Bσ is the transform of aBianchi transformation by means of a Lie transformation Lσ, symbolically

Bσ = L−1σ Bπ/2Lσ. (0.0.1)

3

As we explained, we get the following features of an integrable system:

1. infinitely many generalized symmetries;

2. infinitely many conservation laws;

3. explicit solutions;

4. complete integrability in the sense of Liouville.

Where does integrability come from?

A starting point from which all this rich structure can be derived is a zero-curvatureformulation of the underlying problem. The Lax (or Zakharov-Shabat (ZS)) representa-tion of nonlinear equation can be given in a form of compatibility condition

Ut(λ)− Vx(λ) + [U(λ), V (λ)] = 0 (0.0.2)

of two linear equations

Ψx = U(λ)Ψ, Ψt = V (λ)Ψ.

See [22] for the reduction of this construction in symmetric space. In that case, U =λA + Q(x, t) in which A is constant element of the underlying Lie algebra and Q ispotential function. In this way, Zakharov and Mikhailov [76] use a pole expansionU =

∑U(q)(λ− λi)−1, while others [34, 37] favor polynomial expansions.

The zero curvature representation (0.0.2) has a transparent geometrical origin. Indifferential geometry, the embedded surface is the Gauss-Codazzi equation represented asa compatibility condition of linear equations for the moving frame (the Gauss-Weingartenequations), see Lund and Regge [46]. The spectral parameter λ in this representationdescribes deformation of surfaces preserving their properties.

The connection between geometry and integrable systems is clarified by Hasimoto[31] in 1972. He found the transformation between the equations governing the curvatureand torsion of a thin vortex filament (FM) moving in an incompressible inviscid fluidand the NLS equation. The equation FM can be modeled as

γt = γs × γss

in which γ(s, t) is a curve evolving in 3−dimensional space R3. In fact Hasimoto con-structed the complex function ψ = κ exp(i

∫ s0 τds) of the curvature and torsion of the

curve γ, and showed that if the curve evolves according to the vortex filament equation,then ψ solves the cubic nonlinear Schrodinger equation

iψt + ψss +1

2|ψ|2ψ = 0.

Lamb [40] used the Hasimoto transformation to connect other motion of curves to theintegrable equations like modified KDV (mKDV) and SG equations. Balakrishnan et

4

al. [5] have investigated another aspect of space curve formulation: the geometric phaseassociated with the time evolution of the curve and its connection to integrability.

Sasaki [62] gave a geometric interpretation of the ZS spectral problem in terms ofpseudospherical surfaces. Chern and Tenenblat [10] characterized the mKDV hierar-chy as a relation between local invariants of a certain foliation on a surface of constantnonzero Gauss curvature. Terng, Tenenblat, Sattinger and Uhlenbeck in a series of pa-pers [69, 69, 70, 68, 67], studied the symplectic, Lie theoretic, and differential geometricproperties of soliton theory. They construct a pencil of connections depending on thedeformation parameter λ, and prove that the pencil is flat for all λ ∈ R if and onlyif the dynamical variables or the invariance of the one parameter of surfaces follows aHamiltonian flow. See also [9, 8].

Langer and Perline [42] showed in 1991 that the dynamics of a nonstretching vortexfilament in R3 gives rise, through the Hasimoto transformation, to the recursion oper-ator of the NLS hierarchy. The appearance of the recursion operator can be explainedobserving that the Frenet equations for the curve in R2 and R3 are equivalent to the ZSspectral problem without the spectral parameter.

Doliwa and Santini [12] showed that certain elementary geometric properties of themotion of a curve select the hierarchy of integrable dynamics. The motion should be non-stretching and occur in a N−dimensional sphere of radius R and the dynamics indepen-dent of the radius of the sphere. They give a simple geometric meaning of the Hasimototransformation: Hasimoto transformation is induced by a gauge transformation from theFrenet frame to the parallel or natural frame. Wang [72] uses this interpretation to findthe generalized (from R3 to RN ) Hasimoto transformation.

Generalizing Doliwa and Santini’s approach, Sanders, Wang and Beffa showed thatmotion of a curve in a 3−dimensional Riemannian manifold with constant curvaturefollows an arc-length preserving geometric evolution and the evolution of its curvatureand torsion is always a Hamiltonian flow.

Cartan’s Lemma leads us to use the Lie algebra valued 1−form instead of the Levi-Civita connection defined on Riemannian manifold, so that having a frame on the curveembedded in the Riemannian manifold is equivalent to specifying the Cartan connectionapplied on the γs. Indeed Sanders and Wang [58] showed that choosing a natural frameand having the Cartan connection specified according to the natural frame, the Cartanstructure equation leads to the recursion equation of integrable equation. In this waythey found the Hamiltonian operator out of curvature part and symplectic operatorresulted from solving the free torsion tensor. Authors applied a similar method to thecase of conformal geometry [59], in which case making the proper choice of ”naturalframe”, leads to the Hamiltonian and symplectic operator.

Outline and Summary of results

In this thesis, we generalized the former idea to other geometries, such as o(p, q)−orthogonal geometry and mainly to symplectic geometry.

Chapter 1 is an introduction to the algebra of quaternions and the symplectic Lie

5

algebra using quaternions.Chapter 2 explains the variational calculus and in particular defines Hamiltonian and

symplectic operators suitable for Lie algebraic domains.Chapter 3 is an introduction to differential geometry.Chapter 4 is the core of the thesis. Starting from Riemannian geometry in Section

4.1 we prove that if we choose the natural moving frame for a flow of a curve preserv-ing arclength and embedded in the Riemannian manifold with Levi-Civita connectioncompatible with its metric, then evolution of the differential invariants of the curve fol-lows the vector mKDV equation and gives us the recursion operator to produce highersymmetries as well as the Hamiltonian and symplectic operator. Using Cartan’s Lemmawe see that these objects can be obtained by just writing down the Cartan structureequation for Euclidean geometry.

We then proceed with o(p, q)-orthogonal geometry in Section 4.2, generalizing theEuclidean geometry, choosing the natural moving frame for the connection matrix . TheCartan structure equations lead to the evolution equation

Dtu = −I1p−1,qu3x −

3

2ux < u,u >, I1

p−1,q =

(Ip−1 0

0 −Iq

)

which is an mKDV type equation with recursion operator R = HI where the operatorH and I are proved to be the Hamiltonian and symplectic operator, respectively.

Next in Section 4.3, we consider the symplectic geometry defined by the homogeneousspace Sp(n)/Sp(1) × Sp(n− 1), which indeed is identified with projective quaternionicspace HPn. We study the Cartan structure equation, and see that choosing natural orparallel frame

u =: ω(Dx) =

0 1 0T

−1 u −uT

0 u 0

,

one can find the time evolution of invariants of a family of curves embedded in thehomogeneous space. That is,

(uu

)

t

= HI

(vv

)+ A

(vv

).

Replacing v by trivial symmetry ux, we obtain a noncommutative scalar-vectormKDV equation:

ut =1

4u3 +

3

8(−uu1u− uu2 + u2u) +

3

2〈u,u〉u1 + 〈u,u1〉u+ 1

2u〈u,u1〉

+2u〈u1,u〉 − 12 〈u1,u〉u+

3

2Cuu2,

ut = u3 +3

2u2u+

3

4u1(u1 +

1

2u2 + 2〈u,u〉).

This is equation 4.3.10. The reduction u = 0 leads to the second version of the non-commutative mKDV scalar equation and the reduction u = 0 yields the vector mKDVequation.

6

Then in Section 4.4 one rewrites this equation using the Lie bracket, Killing formand projections. In the symplectic case, we will see that

ut = HIv + Av,

in which

H = Dx − π1adu − aduD−1x π0adu,

I = −1

2uD−1

x K(u, .)− (1

2ρ1 + ρ0)π1ada(Dx − adu)adaπ1(

1

2ρ1 + ρ0),

A = ρ0 + 2ρ1 − aduD−1x ρ1.

This way of writing the geometric operator can be generalized to any other Cartangeometry.

We prove that the operator A is Nijenhuis operator, that is, the Nijenhuis tensor van-ishes. Furthermore we claim that H and HA∗ are Hamiltonian operators and A−1∗IA−1

is symplectic. The proofs can be found in Chapter 6Generalizing the Drinfel’d-Sokolov method to symplectic geometry, we find the Lax

representation of the symplectic case in Chapter 5. The Lax operator is indeed

L = Dx + λA+ q,

in which A is the projection of ω(Dx) to the vector space sp(n)/sp(1) × sp(n− 1) as aconstant element of the symplectic Lie algebra and q is the projection of ω(Dx) to thesubalgebra sp(1)× sp(n− 1).

Chapter 1

Quaternionic algebra

1.1 Introduction

In this chapter we consider quaternionic numbers and vectors and see how an innerproduct can be defined on the space of quaternionic vectors. We introduce the Liealgebra of quaternions and compute the Killing form of two elements of this Lie algebra.We then find what can be the relation between the inner product of two vectors inquaternionic space and the Killing form of two specific elements of the Lie algebra.

Let us first briefly recall some basic facts about quaternions. The quaternions werediscovered on 1843 by Sir William Rowan Hamilton. They form a noncommutative,associative algebra over R :

H = a+ bi+ cj + dk | a, b, c, d ∈ R,

where algebra multiplication is defined as

i2 = j2 = k2 = −1, ij = k = −ji, jk = i = −kj, ki = j = −ik,

and scalar multiplication defined as

α(a+ bi+ cj + dk) = (αa) + (αb)i + (αc)j + (αd)k for α ∈ R.

In general, by associative algebra over R, we mean a vector space over the field R witha multiplication on it which is associative, distributive over addition and satisfies

α(q1q2) = q1(αq2) = (αq1)q2 for α ∈ R and q1, q2 ∈ H.

Moreover H is an involutive algebra, i.e, there is a map ∗ : H→ H such that (uv)∗ = v∗u∗

and u∗∗ = u, for u, v ∈ H. The involution of q ∈ H is conjugation of quaternionic numberq = a+ bi+ cj + dk, defined by

q = a− bi− cj − dk.

Clearly for all u, v ∈ H we have that uv = v u. We define the modulus of a quaternion qby

|q| = (qq)1/2 = (a2 + b2 + c2 + d2)1/2.

The principal properties of the modulus of H are as follows. If q, q1, q2 ∈ H, then

7

8 Chapter 1. Quaternionic algebra

1. |q| = 0 if and only if q = 0;

2. |q1 + q2| ≤ |q1|+ |q2| and |q1q2| = |q1||q2|;

3. |q| = |q|; and

4. If q 6= 0, then q−1q = qq−1 = 1, where q−1 =q

|q|2 .

Observe that the set C of complex numbers appears as a real subalgebra of H,meaning that C can be embedded in H as an associative algebra over R. More preciselyC can be seen as SpanR1, i residing inside H where SpanR1, i = a1 + bi | a, b ∈ Ris spanned by 1, i with coefficients in R. The following result gives some of the propertiesof the subalgebra C in H.(The proofs are elementary and omitted.)

Lemma 1.1.1. Consider C as the real subalgebra SpanR1.i of H. Then:

1. SpanR1, i = q ∈ H | qi = iq;

2. SpanRj, k = q ∈ H | qi = −iq;

3. λq = qλ for every λ ∈ SpanR1, i = C and q ∈ SpanRj, k.

Remark 1.1.2. 1. The first point to be emphasized in the third item of this lemmais that if λ ∈ C, then λ can be obtained from λ by an similarity transformationλ→ q−1λq in H.(This is impossible in C except for the trivial case where λ ∈ R.)

2. It follows that H is not an algebra over C, since for λ ∈ C and q1, q2 ∈ H, twoquantities q1(λq2) and (λq1)q2 are not necessarily equal.

We now derive two representation of quaternions by complex vector and matrices aswell as real matrices. The identification of C within H affords a useful linear represen-tation of quaternions, as well as n−tuples of quaternions, by complex vectors. Namely,if q = a0 + α1i+ α2j + α3k ∈ H, then

q = (α0 + α1i) + (α2 + α3i)j = γ1 + γ2j (1.1.1)

where γ1 = α0 + α1i and γ2 = α2 + α3i belong to C. Thus, each quaternion q isuniquely represented by a pair of complex numbers γ1, γ1 via (1.1). In fact the functionρ : H→ C2 defined by

ρ(q) =

(γ1

−γ2

)

is a linear one-to-one map between H and C2 as vector spaces over R. This representationextends to the real vector space Hn of n−tuples of quaternions. Indeed, define thefunction

ρ : Hn → C2n by ρ(ξ) =

(ξ1

−ξ2

),

1.1. Introduction 9

where ξ = ξ1 + ξ2j ∈ Hn and ξ1, ξ2 ∈ Cn. A second useful representation of quaternionsis via M(2,C) as the set of all 2 × 2 complex matrices. Consider the function φ : H →M(2,C) defined as follows: if q = γ1 + γ2j, as in (1.1), then

φ(q) =

(γ1 γ2

−γ2 γ1

).

The function φ is simply injective and satisfies, for all q, q1, q2 ∈ H and α ∈ R, theequations

φ(q1 + q2) = φ(q1) + φ(q2), φ(q1q2) = φ(q1)φ(q2), φ(αq) = αφ(q).

Moreover,

φ(q) = φ(γ1 + (−γ2)j) =

(γ1 −γ2

γ2 γ1

)= φ(q)∗

where φ(q)∗ denotes the conjugate transpose of the 2 × 2 complex matrix φ(q). Thusφ preserves the linear, multiplicative, and involutive structure of H; that is, φ is aninjective ∗-homomorphism from the real involutive algebra H into the real involutivealgebra M(2,C).

A homomorphic embedding of M(n,H) into M(2n,C) is similarly constructed. First,note that by applying (1.1) entry wise to a matrix Q ∈ M(n,H), one can write Q asQ = Γ1 +Γ2j, where Γ1 and Γ2 are n×n complex matrices. The function Φ : M(n,H)→M(2n,C), defined for Q = Γ1 + Γ2j ∈M(n,H) by

Φ(Q) =

(Γ1 Γ2

−Γ2 Γ1

)

is an injective ∗− homomorphism. That is, Φ satisfies, for all Q,Q1, Q2 ∈M(n,H) andα ∈ R, the equations:

1. Φ(Q1 +Q2) = Φ(Q1) + Φ(Q2) and Φ(Q1Q2) = Φ(Q1)Φ(Q2),

2. Φ(αQ) = αΦ(Q), and

3. Φ(Q∗) = Φ(Q)∗ where Q∗ is conjugate transpose of matrix Q ∈ M(n,H), that is

Q∗ = Qt.

Observe that for Q1, Q2 ∈M(n,H), we have (Q1Q2)∗ = Q∗2Q∗1 either by direct reasoning

or using similar identity for complex matrices and properties above of the function Φ.

Proposition 1.1.3. Left quaternionic inverse is also right inverse for quaternionic ma-trices.

Proof. If Q1Q2 = I, then Φ(Q1)Φ(Q2) = I. Hence also I = Φ(Q2)Φ(Q1) = Φ(Q2Q1).Now Φ is injective and it follows that Q2Q1 = I.


Proposition 1.1.4. The image of the function Φ is

Φ(M(n,H)) = P ∈M(2n,C) | JP = P J

where J = Jj, with

J =

(0 −σInσIn 0

).

Proof. We see that Φ(Inj) = J, and if Φ(Q) = P where Q = Γ1 + Γ2j, then

JP = Φ(Inj)Φ(Q) = Φ(InjQ) = Φ(jQ).

Using Lemma 1.1.1, part 3, we see that

jQ = jΓ1 + jΓ2j = Γ1j + (Γ2j)j = Γ1j − Γ2.

Hence JP = Φ(Γ1j − Γ2). On the other hand, from the definition we obtain that

P =

(Γ1 −σΓ2

σΓ2 Γ1

)=

(Γ1 −σΓ2

σΓ2 Γ1

)= Φ(Γ1 + Γ2j).

Consequently we get that

PJ = Φ(Γ1 + Γ2j)Φ(Inj) = Φ((Γ1 + Γ2j)Inj) = Φ(Γ1j − Γ2).

Hence JP = PJ and hence JP = JjP = JPj = PJj = P J . On the other hand, if

JP = P J, then JP = PJ. Hence let P be P =

(Γ1 Γ2

Γ3 Γ4

), then we obtain that

Γ4 = Γ1, Γ3 = −Γ2.

This indeed, by definition, means that Φ(Γ1 + Γ2j) = P.

The following Lemma shows how the function ρ : Hn → C2n is related to the functionΦ : M(n,H)→M(2n,C).

Lemma 1.1.5. Assume that Q ∈M(n,H), ξ ∈ Hn, and λ ∈ C. Then:

1. ρ(Qξ) = Φ(Q)ρ(ξ),

2. ρ(ξλ) = λρ(ξ), and

3. Φ(Q)ρ(ξ) = λρ(ξ) if and only if Qξ = ξλ.

Proof. Express Q and ξ in linear form: Q = Γ1 + Γ2j and ξ = ξ1 + ξ2j, for someΓ1,Γ2 ∈M(n,C) and ξ1, ξ2 ∈ Cn. Thus,

Qξ = (Γ1 + Γ2j)(ξ1 + ξ2j)

= Γ1ξ1 + Γ1ξ2j + Γ2(jξ1) + Γ2j(ξ2j)

= Γ1ξ1 + Γ1ξ2j + Γ2ξ1j + Γ2ξ2j2

= (Γ1ξ1 − Γ2ξ2) + (Γ1ξ2 + Γ2ξ1)j,

1.1. Introduction 11

using part 3 of Lemma 1.1.1. Hence by definition we obtain that

ρ(Qξ) =

Γ1ξ1 − Γ2ξ2

−Γ1ξ2 + Γ2ξ1

=

Γ1ξ1 − Γ2ξ2

−Γ1ξ2 − Γ2ξ1

=

Γ1 Γ2

−Γ2 Γ1

ξ1

−ξ2

= Φ(Q)ρ(ξ),

which completes the proof of (1).To prove (2), note that ξλ = (ξ1 + ξ2)λ = ξ1λ + ξ2jλ = (ξ1λ) + (ξ2λ)j (by Lemma

1.1.1), and so

ρ(ξλ) =

(ξ1λ

−ξ2λ

)= λ

(ξ1

−ξ2

)= λρ(ξ),

which proves (2). It might be remarked that λξ = λξ1 + (αξ2)j, hence ρ(λξ) =

(λξ1

−λ ξ2

)

which is not equal to λρ(ξ) nor ρ(ξ)λ. To prove (3), first assume that Qξ = ξλ. Applyρ and obtain ρ(Qξ) = ρ(ξλ). But this equation is, by (1) and (2), precisely Φ(Q)ρ(ξ) =λρ(ξ). Conversely, assume that Φ(Q)ρ(ξ) = λρ(ξ). Then, by (1) and (2), ρ(Qξ) = ρ(ξλ).Because ρ is injective, we have that Qξ = ξλ, thereby proving (3).

For further study about quaternions, see [20].Now we will discuss the identification of quaternionic matrices with real matrices.

Since we can write Q ∈M(n,H) uniquely as

Q = Γ0 + Γ1i+ Γ2j + Γ3k

where Γ0,Γ1,Γ2,Γ3 are real matrices, The map µ : M(n,H)→M(4n,R) given by

µ(Q) =

Γ0 −Γ1 −Γ2 −Γ3

Γ1 Γ0 −Γ3 Γ2

Γ2 Γ3 Γ0 −Γ1

Γ3 −Γ2 Γ1 Γ0

is an injective homomorphism of two real associative algebras over real numbers. Thatmeans that if Q,Q1, Q2 ∈M(n,H) and α ∈ R, then:

µ(Q1 +Q2) = µ(Q1) + µ(Q2), µ(Q1Q2) = µ(Q1)µ(Q2), µ(αQ) = αµ(Q).

It is also clear that

µ(Q∗) = µ(Γt0 − Γt1i− Γt2j − Γt3k) = µ(Q)t.


1.2 Inner product on quaternionic vectors

The set of all n−tuples of quaternions Hn is a right module over the division ring H. wecall Hn a right H−module. That means, for all q, q1, q2 ∈ H and ξ, ξ1, ξ2 ∈ Hn we havethat

1. (ξ1 + ξ2)q = ξ1q + ξ2q,

2. ξ(q1 + q2) = ξq1 + ξq2,

3. ξ(q1q2) = (ξq1)q2, and

4. ξ1 = ξ.

Definition 1.2.1. A Hermitian inner product on a right H−module is a quaternionic-valued bilinear form on it, which is right-linear in the second slot, and is positive definite.That is, it satisfies the following properties.

1. < ξ, ξ1 + ξ2 >=< ξ, ξ1 > + < ξ, ξ2 >,

2. < ξ1, ξ2q >=< ξ1, ξ2 > q,

3. < ξ2, ξ1 > =< ξ1, ξ2 >, and

4. < ξ, ξ >≥ 0 and < ξ, ξ >= 0 if and only if ξ = 0.

The basic example is the form

< ξ, η >= ξ∗η =

n∑

i=1

ξiηi, (1.2.1)

(by considering ξ as column matrix) on the right H−module Hn.Using (2) and (3) of the definition above, we find that

< ξ1q, ξ2 >= q < ξ1, ξ2 > .

Indeed

〈ξ1q, ξ2〉 = 〈ξ2, ξ1q〉 = 〈ξ2, ξ1〉q= q〈ξ2, ξ1〉 = q〈ξ1, ξ2〉.

On the complex vector space C2n, we can also define an Hermitian inner product as in(1.2.1). The following proposition relates the Hermitian inner product on quaternionicvectors as above with that on the corresponding complex vectors defined by the map ρ.Notice that we will use the same notation for both inner products.

Lemma 1.2.1. If ξ, η ∈ Hn, then

〈ξ, η〉 = 〈ρ(ξ), ρ(η)〉 − 〈ρ(ξ), J ρ(η)〉

1.3. Symplectic group 13

Proof. Let ξ and η be the vectors ξ = ξ1 + ξ2j and η = η1 + η2j where ξ1, ξ2, η1, η2 ∈ C.Then we see that

〈ξ, η〉 = ξ∗η = (ξ1t − ξt2j)(η1 + η2j)

= ξ1tη1 + ξt2η2 + (ξ1

tη2 − ξt2η1)j

= 〈ρ(ξ), ρ(η)〉 − 〈ρ(ξ), Jρ(η)〉j= 〈ρ(ξ), ρ(η)〉 − 〈ρ(ξ), J ρ(η)〉.

1.3 Symplectic group

We will denote the set of invertible n× n matrices over H by GL(n,H). Now we definethe symplectic Lie group over quaternions as

Sp(n) = Q ∈ Gl(n,H) |< Qξ,Qη >=< ξ, η > for all ξ, η ∈ Hn.

Observe that, by definition,

< Qξ,Qη >= (Qξ)∗Qη = (ξ∗Q∗)(Qη) = ξ∗(Q∗Qη) =< ξ,Q∗Qη > .

Hence Q ∈ Sp(n) if and only if Q∗Q = I. Thus by Proposition 1.1.3, we have QQ∗ = Ias well. Such s matrix is called a unitary quaternionic matrix.

Proposition 1.3.1. Φ(Sp(n)) = N ∈M(2n,C) | N ∗N = I and JN = NJ.

Proof. Observe that Φ is ∗-homomorphism and use Proposition 1.1.4.

Proposition 1.3.2. Φ(Sp(n)) consist of N ∈M(2n,C) such that

〈Nρ(ξ), Nρ(η)〉 = 〈ξ, η〉, 〈Nρ(ξ), JNρ(η)〉 = 〈ξ, Jη〉, (1.3.1)

for all ξ, η ∈ Hn. In other word, two inner products 〈ξ, η〉 and 〈ξ, Jη〉 will be left un-changed if we substitute ξ, η with Nρ(ξ), Nρ(η), respectively.

Proof. Let Q ∈ M(n,H) and ξ, η ∈ Hn. Then applying Lemmas 1.1.5 and 1.2.1, weobtain

〈ξ, η〉 = 〈ρ(ξ), ρ(η)〉 − 〈ρ(ξ), Jρ(η)〉and

〈Qξ,Qη〉 = 〈Φ(Q)ρ(ξ),Φ(Q)ρ(η)〉 − 〈Φ(Q)ρ(ξ), JΦ(Q) ρ(η)〉.Now it is clear that if Q ∈ Sp(n) then Φ(Q) ∈ M(2n,C) satisfies (1.3.1). On theother hand, assume that N ∈ M(2n,C) satisfies in (1.3.1), then from the first equationwe obtain that N ∗N = I and from the second one, N ∗JN = J . This is nothing butJN = NJ. Now the proof of Proposition 1.3.2 is done using Proposition 1.3.1.


See also [11], p.16-24.The Lie algebra of symplectic group Sp(n) is

sp(n) = A ∈M(n,H) | A∗ +A = 0.

A Lie bracket is an anti-commutative bracket [Q,P ] = QP − PQ. The dimension ofsp(n) as vector space over R is 2n2 + n. Notice that mentioned Lie group is compact,connected, semisimple, and simple connected Lie group. See the references [73, 11] forexact definitions.

1.4 Killing form in Symplectic Lie algebra

Now we compute the Killing form of the two general elements of symplectic group andthen for two specific ones we see that Killing form can be expressed in term of theHermitian inner product. To do so, let us for a while represent a quaternionic numberq by q = q1 + qii + qjj + qkk. The componentwise product of two quaternions p and qis defined by < p, q >r= p1q1 + piqi + pjqj + pkqk and so is the componentwise innerproduct of vectors ξ, η, by < ξ, η >r=

∑ni=1 < ξi, ηi >r . It is easy to see that

< ξ, η >r=1

2(< ξ, η > + < η, ξ >),

which shows that componentwise inner product is just symmetrization of the Hermitianinner product.

Lemma 1.4.1. The Killing form of two elements A and B of the symplectic Lie algebrais following:

K(A,B) = −4(n+ 1)

n∑

s=1

< Ass, Bss >r −8(n+ 1)∑

p<q

< Apq, Bpq >r . (1.4.1)

The proof can be found in Appendix C. Let us represent a general element A of thesymplectic Lie algebra as

A =

(p −pt

p P

),

where p is a pure quaternionic number, p is quaternionic vector and P ∈ sp(n− 1). Wedefine a projection of sp(n) by

π(A) =

(0 −0t

0 P

).

This is an orthogonal projection with respect to the Killing form, that is,

K(π(A), (1 − π)(B)) = 0.

1.4. Killing form in Symplectic Lie algebra 15

Notice that neether π nor 1− π is a Lie algebra homomorphism. Indeed

(1− π)([A,B]) − [(1− π)(A), (1 − π)(B)] =

= [A,B]− [π(A), π(B)] − [(1− π)(A), (1 − π)(B)]

= [A, π(B)] + [π(A), B]

= 2[π(A), π(B)].

Lemma 1.4.2.

K(A,B) = K(π(A), π(B)) +K((1− π)(A), (1 − π)(B)).

Concerning the Killing form and Lie bracket of matrices of the type in the last lemma,we can easily prove the following lemma.

Lemma 1.4.3. For pure quaternionic numbers p, q, r, vectors p,q and matrices P,Qand R of dimension n− 1 in sp(n− 1), we have that

[(p 00 0

),

(q 00 0

)]=

(Cpq 0

0 0

), (1.4.2a)

[(0 −pT

p 0

),

(0 −qT

q 0

)]=

(Cqp 0

0 qpT − pqT

), (1.4.2b)

[(0 00 Q

),

(0 −pT

p 0

)]=

(0 −Qp

T

Qp 0

), (1.4.2c)

K

([(0 −pT

p 0

),

(0 −qT

q 0

)],

(q 00 0

))= K

((Cqp 0

0 0

),

(q 00 0

)), (1.4.2d)

K

((p 00 P

),

[(r 00 R

),

(q 00 0

)])= K

((p 00 0

),

[(r 00 0

),

(q 00 0

)]). (1.4.2e)

Proof. First and second equalities are trivial. For the second and third ones, we needjust prove that

K

((p 00 P

),

(q 00 0

))= K

((p 00 0

),

(q 00 0

)).

But this follows from

K

((0 00 P

),

(q 00 0

))= 0,

by using the formula for the Killing form of two elements

(0 00 P

)and

(q 00 0

)as in the

Lemma 1.4.1.


Consequently, though the matrix[(0 −pT

p 0

),

(0 −qT

q 0

)]is not of the type of

one of the matrices inside the bracket, but the Killing form of that with the matrix of

the type

(q 00 0

), would give rise to the Killing form of two matrices of former type.

Chapter 2

Variational complex and geometric

operators

2.1 Algebraic approach to the concept of geometric oper-

ator

Definition 2.1.1. Let A be any Lie algebra over a ring containing the real numbers.Then we say that the real vector space M is left A−module if a bilinear operation isgiven which assign to each pair a ∈ A,m ∈M an element a ·m ∈M such that

a1 · a2 ·m− a2 · a1 ·m = [a1, a2] ·m.This is called a representation of A on M .

Example 2.1.1. Let X be a smooth finite-dimensional manifold. The elements of theLie algebra A are vector fields on X and M is C∞(X,R). The operation [a1, a2] is thecommutator of vector fields and a ·m is the result of the action of the vector field a onthe function m ∈ M. Notice that in this example, M has a ring structure. But this isnot essential in the theory of Hamiltonian formalism, as in the next section which willbe devoted to the variational calculus, we would have left A−module M, without havingring structure on it.

Our objective is to construct a Hamiltonian structure on the pair (A,M) using onlythe structures present in A and M. There are thus two basic operation: the commutator[a1, a2] and the action a ·m. We shall attempt to express all operations in terms of thesebasic operations.

We first define forms and differentials of forms, analogous to what in the case ofExample 2.1.1 is called the de Rham complex.

Definition 2.1.2. A q-form is a multilinear function ω on A with values in M, that isω : A× . . .A→M. A 0-form is by definition a fixed element m ∈M. The differential orcoboundary operator dq is given by the formula

dqω(a1, . . . , aq+1) =∑

i

(−1)i+1ai · ω(a1, . . . , ai, . . . , aq+1) (2.1.1)

+∑

i<j

(−1)iω(a1, . . . , ai, . . . , [ai, aj ], . . . , aq+1),

where the notation . means the argument under the hat sign will be removed.

17

18 Chapter 2. Variational complex and geometric operators

Remark 2.1.1. In the de Rham complex one restricts attention to antisymmetric forms,and this has become the norm in Lie algebra cohomology. As Loday [45] pointed outin his study of Leibniz algebras, there is no need to do so. The only reason for theantisymmetry assumption is that if one works with Lie algebras on which a ring R actsby left multiplication (as in vector fields multiplied by functions) and where R itself is anontrivial A-module, with rules like (cf. Palais [52])

[fx, y] = f [x, y]− y(f)x, x, y ∈ A, f ∈ R,

then to show if ω is R-linear, which is the usual tensor assumption, one also has thatdqω is R-linear, one needs to assume that ω is antisymmetric.

In particular, d0m : A→M is a linear map with values in M ; the value of the 1-formd0m on an element a ∈ A is given by the formula

d0m(a) = a ·m.

If ξ is a 1-form, i.e., a linear mapping ξ : A→M, then

d1ξ(a1, a2) = a1 · ξ(a2)− a2 · ξ(a1)− ξ([a1, a2]).

We can simply see that d1d0m = 0 is equivalent to the principle axiom of A−module M,that is:

d1d0m(a1, a2) = a1 · d0m(a2)− a2 · d0m(a1)− d0m([a1, a2])

= a1 · a2 ·m− a2 · a1 ·m− [a1, a2] ·m = 0.

One can verify that dq+1dq = 0 in general.

Definition 2.1.3. A q-form ω is called closed if dqω = 0.

We denote by Cq(A,M) the space of q-forms. We observe that C q(A,M) is also aleft A-module; the action of a is called the Lie derivative which is given by

L0am = a ·m, m ∈ C0(A,M) = M, (2.1.2)

(L1aξ)(b) = a · ξ(b)− ξ([a, b]), ξ ∈ C1(A,M). (2.1.3)

The general formula is, with q > 0,

ιqbLqa = Lq−1

a ιqb − ιq[a,b],

where

ιqaω(b1, . . . , bq−1) = ω(a, b1, . . . , bq−1) for ω ∈ Cq(A,M).

We can show that

Lqa1Lqa2− Lqa1

Lqa2= Lq[a1,a2]

2.2. Formal Variational Calculus 19

and

Lqa = ιq+1a dq + dq−1ιqa. (2.1.4)

One can use these Cartan rules as axioms to derive formula (2.1.1).Now we suppose that in the space of 1-forms, a subspace E∗ ⊂ C1(A,M) is fixed

which contains the differential of all 0-forms (i.e., of elements of M .) In concrete situa-tions considered in the next section, E∗ will be specially described.

Let H : E∗ → A be a linear operator.

Definition 2.1.4. The operator H is called anti-symmetric if for any ξ1, ξ2 ∈ E∗ wehave

ξ1(Hξ2) = −ξ2(Hξ1).

Notation 2.1.1. Sometimes we use the notation (ξ, a) for ξ(a).

Definition 2.1.5. With any anti-symmetric operator H : E∗ → A, we connect a 2-formωH defined on the image im(H) by

ωH(a1, a2) = (H−1a2)(a1), a1, a2 ∈ im(H), (2.1.5)

where by H−1a2 we denote any element ξ2 such that Hξ2 = a2.

Lemma 2.1.2. The bilinear map ωH is indeed a well-defined 2-form, that is, independentof the choice of ξ2 in Definition 2.1.5.

Proof. Suppose a1, a2 ∈ im(H), i.e., they have the form ai = Hξi. In addition assumethat a2 = Hξ3. Since H is anti-symmetric, we see that

ξ3(a1) = ξ3(Hξ1) = −ξ1(Hξ3) = −ξ1(a2).

Likewise we have that ξ2(a1) = −ξ1(a2). Hence there holds ξ3(a1) = ξ2(a1) for any twomembers of ξ2, ξ3 ∈ H−1a2 and a1 ∈ im(H).

Definition 2.1.6. We say that an anti-symmetric operator H : E∗ → A is Hamiltonianif

1. The image of the operator H is a subalgebra of the Lie algebra A and

2. d2ωH = 0 on this subalgebra.

In order to indicate criteria that an operator be Hamiltonian, we must obviouslyreformulate the condition that the form ωH defined by formula 2.1.5 be closed directlyin terms of the form generating the operator H.

2.2 Formal Variational Calculus

This section is devoted to the background and set up for geometric operators such asHamiltonian, symplectic operator. In particular we describe noncommutative variationalcalculus and explain how this can be set up in the framework of a matrix Lie algebra.


2.2.1 Classical formal variational calculus

In the formal variational calculus introduced by Gel’fand and Dikii, [27], [26], one startswith coordinates uα, α = 1, . . . , N , which take their values in a field or division ring andthen one defines A to be the algebra of sums and products in the uα and its derivativeswith respect to the independent variable x. In other words, A is a free associativealgebra over the real numbers. One then defines Der(A), the space of derivations on Aas a ∈ DerA satisfying

aab = a(a)b+ aa(b).

That means a act on A according to the Leibniz rule.

Remark 2.2.1. If one takes A to be the space of continuous functions on the real linewith pointwise multiplication, then Der(A) = 0.

The space Der(A) is a Lie algebra with the bracket defined as

[a1, a2] = a1a2 − a2a1.

We can express a ∈ Der(A) as

a =N∑

α=1

∞∑

k=0

aα(k)

∂

∂uαk, aα(k) ∈ A.

Here the term aα(k)

∂

∂uαkstands for the replacement of uαk by aα(k), using the Leibniz rule.

Example 2.2.1. Let A be generated by uk, k = 0, . . .. Then aukul = a(k)ul + uka(l).

Example 2.2.2. Let

Dx =

N∑

α=1

∞∑

k=0

uαk+1

∂

∂uαk.

Then Dx ∈ Der(A).

Definition 2.2.1. LetE = a ∈ Der(A) | [a, Dx] = 0,

be the space of all derivations on A commuting with Dx. It is clear that E is a Liesubalgebra of Der(A) due to the Jacobi identity. Moreover, since every a ∈ Der(A) canbe determined by its action on uαk , hence every a ∈ E can be determined by its actionon only uα. Indeed if a ∈ E, then

aα(k) = auαk = aDkxu

α = Dkxau

α = Dkxa

α(0) =: aαk ,

so that a ∈ E can be identified with some (a1, . . . , αN ) ∈ AN , and we can write a ∈ E as

a =N∑

α=1

∞∑

i=0

aαi∂

∂uαi,


with the replacement rule as before. We call elements of the Lie subalgebra E evolu-tionary vector fields. The map p : AN → E assigning (a1, . . . , aN ) to a is called theprolongation map.

Definition 2.2.2. Let W be an arbitrary module over E. Then the Frechet derivativeof S ∈W in the direction a ∈ E, is defined as

DS [a] =d

dεS(u+ εau)|ε=0 ∈W

Example 2.2.3. One has DDx [a] = 0, since Dx is not u-dependent.

Lemma 2.2.2. The Lie bracket on E induces a Lie bracket on AN :

[a, b]A = Db[p(a)] −Da[p(b)], a, b ∈ AN .

One has [p(a), p(b)]E = p([a, b]A), that is, p is a Lie algebra homomorphism.

Proof. Assume that a, b ∈ AN and c ∈ A are given. Then

[p(a), p(b)]E(c) = p(a)(p(b)c) − p(b)(p(a)c)

=

N∑

α=1,β=1

∞∑

i=0,j=0

aβj∂

∂uβj

(bαi

∂c

∂uαi

)−

N∑

α=1,β=1

∞∑

i=0,j=0

bβj∂

∂uβj

(aαi

∂c

∂uαi

)

=

N∑

α=1,β=1

∞∑

i=0,j=0

(aβj∂bαi

∂uβj

)( ∂c

∂uαi

)+

N∑

α=1,β=1

∞∑

i=0,j=0

(aβj b

αi

)( ∂2c

∂uβj ∂uαi

)

−N∑

α=1,β=1

∞∑

i=0,j=0

(bβj∂aαi

∂uβj

)( ∂c

∂uαi

)−

N∑

α=1,β=1

∞∑

i=0,j=0

(bβj a

αi

)( ∂2c

∂uβj ∂uαi

)

=

N∑

α=1,β=1

∞∑

i=0,j=0

(aβj∂bαi

∂uβj

)( ∂c

∂uαi

)−

N∑

α=1,β=1

∞∑

i=0,j=0

(bβj∂aαi

∂uβj

)( ∂c

∂uαi

)

= p(p(a)b)(c) − p(p(b)a)(c)

= p(p(a)b − p(b)a)(c)

= p([a, b]A)(c).

Let us explain here some of terms we used. The term

(aβj∂bαi

∂uβj

)( ∂c

∂uαi

),

is computed as follows. The monomial uαi in c will be replaced by the expression aβj∂bαi

∂uβj,

which is itself the result of replacing aβj with uβj in bαi according to the Leibniz rule inboth process.


Also the term (aβj b

αi

)( ∂2c

∂uβj ∂uαi

)

is understood similarly. That is just replacing uαi and uβj with bαi and aβj respectively inc.

Notice that p(a)b =(p(a)b1, . . . , p(a)bN

)∈ AN . Also we see that

p(a)(c) =

N∑

α=1

∞∑

i=0

aαi∂c

∂uαi

=N∑

α=1

∞∑

i=0

d

dεc[uαi + εaαi ]|ε=0

=

N∑

α=1

∞∑

i=0

d

dεc[uαi + εp(a)(uαi )]|ε=0

=d

dεc[u+ εp(a)(u)]|ε=0

= Dc[p(a)].

Hence Db[p(a)] = p(a)(b) ∈ AN .

Definition 2.2.3. Let V be a E-module. We define an equivalence relation on V by

a1 ∼ a2 if and only if a1 − a2 = Dx.a3 for a1, a2, a3 ∈ V.Notice that Dx ∈ E, so this is well defined. We denote equivalence classes by V = V/DxVand its elements by

∫a for a ∈ V ; these are called functionals.

Definition 2.2.4. We define an action of E on V by a ·∫a =

∫a · a for a ∈ E, a ∈ V.

The action is well defined: Let b ∈ V. Then we have that

a ·∫Dx.b =

∫a · (Dx.b) =

∫[a, Dx].b+

∫Dx.(a · b) = 0.

So evolutionary vector fields and functionals go well together.

Lemma 2.2.3. The space V is an E−module under the action just defined.

Proof. In fact

a1 · a2 ·∫a− a2 · a1 ·

∫a = a1 ·

∫a2 · a− a2 ·

∫a1 · a

=

∫a1 · a2 · a− a2 · a1 · a

=

∫[a1, a2] · a

= [a1, a2] ·∫a.


In the case V = A, one has, due to the Leibniz rule, integration by part as∫

(Dxa)b =−∫a(Dxb). Now if a = p(a), a ∈ AN , then for b ∈ A, we have that

a ·∫b =

∫a · b

=

N∑

α=1

∫ ∞∑

i=0

aαi∂b

∂uαi

=

N∑

α=1

∫ ∞∑

i=0

(Dixaα)

∂b

∂uαi

=

∫ N∑

α=1

aα∞∑

i=0

(−Dx)i(∂b

∂uαi) (2.2.1)

Definition 2.2.5. The partial variational derivation δδuα : A→ A is defined by

δ

δuα=

∞∑

i=0

(−Dx)i∂

∂uαi.

Lemma 2.2.4 (Gel’fand and Dikii[26]). We have that δδuα Dx = 0.

Proof. One has [Dx,∂

∂uαi] =

∂

∂uαi−1

. Therefore

δ

δuαDx =

∞∑

i=0

(−Dx)i∂

∂uαiDx

=∞∑

i=0

(−Dx)i+1 ∂

∂uαi+∞∑

i=1

(−Dx)i∂

∂uαi−1

= 0,

and the result follows.

Thus one can define δδuα on A, that is, δ

δuα : A→ A. This is usually called in literature

the gradient of functional. The operator δδuα is called Euler operator.

According to (2.2.1) we can write the action of a on A in terms of variational deriva-tives:

a ·∫b =

N∑

α=1

∫aα

δb

δuα.

Remark 2.2.5. Some Hamiltonian operators associated with (a, A) have been shownto be connected with certain very interesting nonlinear partial differential equations (cf[27],[26] and [28]).


2.2.2 Variational calculus for geometric dynamical symbols

As we have seen in Chapter 1, we derived the operator acting on a certain subspaceconsisting of the functions with values in a Lie algebra g. For instance, this subspace inthe symplectic geometry is the space of functions with values of the form

0 0 00 p −pt

0 p 0

∈ spn+2,

depending on the matrix

U = ω(Dx) =

0 0 00 u −ut

0 u 0

and its derivatives Ui = DixU for i = 0, . . . .

Definition 2.2.6. Let K ∈ C2(spn+2,R). A space E∗ ⊂ C1(E, V ) is given by theformula

b?(a) =

∫K(b, a), b ∈ E.

Here we assume that K is nondegenerate, that is, if b∗ = 0 it follows that b = 0. Werecall that here V is the associated algebra generated by single dynamical variables u,uand their derivatives and furthermore V is defined as before.

Definition 2.2.7. The inner product on E defined by

< a, b >=

∫K(a, b),

induces an inner product on E∗ as follows:

< a∗, b∗ >∗=< a, b > .

Definition 2.2.8. If we have nondegenerate pairing, then for the linear operator H :E∗ → E, we can define the adjoint operator H∗ : E→ E∗ by

< a,Hb∗ >=< H∗a, b∗ >∗ .

Now if H : E∗ → E is anti-symmetric, then by definition a∗(Hb∗) = −b∗(Ha∗).Hence

− < H∗b, a∗ >∗ = − < b,Ha∗ >= −b∗(Ha∗)

= a∗(Hb∗) =< a,Hb∗ >=< a∗, (Hb∗)∗ >∗ .

Thus (Hb∗)∗ = −H∗b.


Remark 2.2.6. Following the definition of adjoint operator H ∗ of H : E∗ → E we seethat

c∗(Hb∗) = (H∗c)b. (2.2.2)

Indeed

c∗(Hb∗) =< c,Hb > = < H∗c, b∗ >∗= < (H∗c)∗, b >= ((H∗c)∗)∗(b) = (H∗c)b

Lemma 2.2.7. We have that Da∗ [c] = (Dac)∗. Moreover, given the anti-symmetric

operator H : E∗ → E we have that

DH∗ [a](c) = −(DH [a](c∗))∗, where H∗ : E→ E∗. (2.2.3)

Proof.

Da∗ [c](b) =d

dεa∗[u+ εc](b)

=d

dε

∫K(a[u+ εc], b)

=

∫K(

d

dεa[u+ εc], b)

=

∫K(Dac, b)

= (Dac)∗(b).

Now for the anti-symmetric operator H : E∗ → E we have that

DH∗ [a](c) =( ddεH∗[u+ εa]

)(c)

=d

dε

(H∗[u+ εa](c)

)

= − d

dε

(H[u+ εa](c∗)

)∗

= −( ddεH[u+ εa](c∗)

)∗

= −(DH [a](c∗)

)∗.

Lemma 2.2.8. We do have that

1. DH [a](b∗) = DHb∗ [a]−H(Db∗ [a]),

2. DH∗ [a](b) = DH∗b[a] −H∗(Db[a])


3. Dc∗ [a](b) = Dc∗(b)[a]− c∗(Db[a]).

4. DS [a](a) = DSb[a]− S(Dba). For the operator S : E→ E∗.

Proof.

Dc∗(b)[a] =d

dεc∗(u+ εau)(b(u+ εau))|ε=0

= Dc∗ [a](b) + c∗(Db[a])

Lemma 2.2.9. For the operator H : E∗ → E, the following identity holds:

c∗(DH [a]b∗

)= b∗

((DH∗ [a](c))∗

).

Moreover if H is anti-symmetric we have that

c∗(DH [a]b∗

)= −b∗(DH [a](c∗)).

Proof. Firstly we have that

Dc∗(Hb∗)[a] = c∗(DHb∗ [a]) +Dc∗ [a](Hb∗)

= c∗(H(Db∗ [a])) + c∗(DH [a](b∗)) +Dc∗ [a](Hb∗) (2.2.4)

Similarly we get that

D(H∗c)b[a] = (H∗c)(Db[a]) +DH∗c[a](b)

= (H∗c)(Db[a]) +H∗(Dc[a])(b) + (DH∗ [a](c))(b) (2.2.5)

Applying (2.2.2) we see that

c∗(Hb∗) = (H∗c)b, Dc∗ [a](Hb∗) = (Dc[a])∗(Hb∗) = (H∗Dc[a])b,

andc∗(H(Db∗ [a])) = c∗(H(Db[a])∗) = (H∗c)Db[a].

Hence the two equations (2.2.4) and (2.2.5) yield the following identity.

c∗(DH [a](b∗)) = (DH∗ [a](c))(b) = b∗(

(DH∗ [a](c))∗).

Now if H is anti-symmetric, then (DH∗ [a](c))∗ = −DH [a](c∗) by Lemma 2.2.7.

Remark 2.2.10. Likewise we can define the adjoint of differential operator Da wherea ∈ E as an operator D∗a : E→ E. Indeed we define it as

< D∗ab, c >=< b, Dac > .


Remark 2.2.11. Assuming that K is nondegenerate, then we can identify E∗ with E.Hence if H : E∗ → E is anti-symmetric, then the map H : E→ E satisfies H ∗ = −H.

Lemma 2.2.12. Let m =∫K(b, c) and a, b, c ∈ E. Then

a ·m =

∫K(a, D∗bc +D∗c b).

Proof. We compute

a ·m =

∫K(Dab, c) +K(b, Dac)

=

∫K(Dba, c) +K(b, Dca) +

∫K([a, b], c) +K(b, [a, c])

=

∫K(a, D?

bc) +K(D?c b, a)

=

∫K(a, D?

bc +D?c b),

where we used the invariance of the Killing form.

Corollary 2.2.13. d0m ∈ E∗ for each m ∈ A.

We can define the Lie derivative of the operators between E and E∗ similarly to thatof the q-forms. For instance the Lie derivative of H : E∗ → E is the map LaH : E∗ → E

defined by LaH(b∗) = La(Hb∗)−H(Lab∗). Using the last lemma, we can write the Lie

derivative of all differential object in terms of the Frechet derivatives. In order to dothis we need the following definition.

Definition 2.2.9. Given the operator H : E→ F, the conjugate operator H † : F∗ → E∗

is defined as

(H†(f∗))(e) = f∗(He).

Lemma 2.2.14. The Lie derivative of the basic objects with respect to b ∈ E are givenby the following formulas:

a ∈ E, Lba = Dab−Dba, (2.2.6a)

ξ ∈ E∗, Lbξ = Dξh+D†bξ. (2.2.6b)

H : E∗ → E, LbH = DHb−DbH −HD†b, (2.2.6c)

I : E→ E∗, LbI = DIb + IDb +D†bI, (2.2.6d)

S : E→ E, LbS = DSb−DbS + SDb, (2.2.6e)

T : E∗ :→ E∗, LbT = DTb +D†bT − TD†b, (2.2.6f)

Proof. 1. The first one is trivial.


2. For the second one, indeed we have

Lbξ(a) = Lb(ξ(a)) − ξ(Lba)

= (Dξb)(a) + ξ(Dab)− ξ(Dab−Dba)

= (Dξb)(a) + ξ(Dba)

= (Dξb)(a) + (D†bξ)(a)

Hence we have thatLbξ = Dξb +D†bξ.

3. To prove third one we compute

LbH(ξ) = Lb(Hξ)−H(Lbξ)

= DHξb−DbHξ −H(Dξb +D†bξ)

= DHb(ξ) +H(Dξb)−DbHξ −H(Dξb +D†bξ)

= DHb(ξ)−DbHξ −H(D†bξ)

Hence we have that LbH(ξ) = DHb−DbH −HD†b.4.

LbI(a) = Lb(Ia) − I(Lba)

= DIab +D†bIa− I(Dab−Dba)

= DIb(a) + I(Dab) +D†bIa− I(Dab−Dba)

= DIb(a) +D†bIa− I(Dba)

Hence LbI = DIb +D†bI + IDb.

5.

(LbS)(a) = Lb(Sa) − S(Lba)

= DSab−DbSa− S(Dab−Dba)

= DSb(a) + S(Dab)−DbSa− S(Dab−Dba)

= DSb(a)−DbSa + S(Dba)

Thus LbS = DSb−DbS + SDb.

6.

LbT (ξ) = Lb(Tξ)− T (Lbξ)

= DTξb +D†bTξ − T (Dξb +D†bξ)

= DT b(ξ) + T (Dξb) +D†bTξ − T (Dξb +D†bξ)

= DT b(ξ) +D†bTξ − T (D†bξ)

Therefore LbT = DTb +D†bT − TD†b.

2.3. Schouten bracket 29

2.3 Schouten bracket

In this section an effective algebraic characterization of a Hamiltonian operator will begiven.

Definition 2.3.1. The Schouten bracket [H,H] for the operator H : E∗ → E is definedas

[H,H](a∗, b∗, c∗) = (LHa∗b∗)(Hc∗) + cycl.,

for a∗, b∗, c∗ ∈ E∗.

Notice that the Schouten bracket can be expressed in terms of Frechet derivatives:

Proposition 2.3.1. For the antisymmetric operator H : E∗ → E we have that

[H,H](a∗, b∗, c∗)

= b∗(DH [Ha∗](c∗)) + c∗(DH [Hb∗](a∗)) + a∗(DH [Hc∗](b∗))

= < DH [Ha∗](c∗), b > + < DH [Hb∗](a∗), c > + < DH [Hc∗](b∗), a > .

Proof. First notice that for a∗, b∗, c∗ ∈ E∗, by definition we have

(LHa∗b∗)(Hc∗) = Ha∗.b∗(Hc∗)− b∗([Ha∗,Hc∗])

= < Ha∗, D∗bHc∗ +D∗Hc∗b > − < b, DHc∗Ha∗ −DHa∗Hc∗ >

= < DbHa∗,Hc∗ > + < DHc∗Ha∗, b > − < b, DHc∗Ha∗ −DHa∗Hc∗ >

= < DbHa∗,Hc∗ > + < b, DHa∗Hc∗ >

= < DbHa∗,Hc∗ > + < b, DH [Hc∗](a∗) > + < b,H(Da∗ [Hc∗]) >

= < DbHa∗,Hc∗ > + < b, DH [Hc∗](a∗) > − < Hb∗, Da[Hc∗] > .

Now simply if we take cyclic permutation, then we get that

(LHa∗b∗)(Hc∗) + cycl. = < b, DH [Hc∗](a∗) > +cycl.

= b∗(DH [Hc∗](a∗)) + cycl.

= [H,H](a∗, b∗, c∗),

which proves the proposition.

Definition 2.3.2. For operator H : E∗ → E and b∗ ∈ E∗ we define linear operator

DHb∗ : E→ E, by DHb∗(a) = DH [a](b∗).

The operator DHb∗ is dual to the Frechet derivative in some sense.

Lemma 2.3.2. For an antisymmetric operator H : E∗ → E, the Schouten bracket canbe expressed as image of 1−form of its own argument.

[H,H](a∗, b∗, c∗) = c∗(− (DHb∗)(Ha∗) + (DHa∗)(Hb∗)−H((DHb∗)†a∗)

).


Proof. According to Definitions 2.2.9 and 2.3.2, we have that

[H,H](a∗, b∗, c∗)

= b∗(DH [Ha∗](c∗)) + c∗(DH [Hb∗](a∗)) + a∗(DH [Hc∗](b∗))

= b∗(DH [Ha∗](c∗)) + c∗(

(DHa∗)(Hb∗))

+ a∗(

(DHb∗)(Hc∗))

= b∗(DH [Ha∗](c∗)) + c∗(

(DHa∗)(Hb∗))

+(

(DHb∗)†a∗)

(Hc∗).

Now applying Lemma 2.2.9, we obtain that

[H,H](a∗, b∗, c∗)

= −c∗(DH [Ha∗](b∗)) + c∗(

(DHa∗)(Hb∗))

+(

(DHb∗)†a∗)

(Hc∗).

From the fact that H is antisymmetric, the Schouten bracket will take following form:

[H,H](a∗, b∗, c∗)

= −c∗((DHb∗)(Ha∗)) + c∗(

(DHa∗)(Hb∗))− c∗(H((DHb∗)†a∗))

= c∗(− (DHb∗)(Ha∗) + (DHa∗)(Hb∗)−H((DHb∗)†a∗)

),

which is the desired result.

In the following theorem, we give explicit criteria to decide whether an operator isHamiltonian in terms of Frechet derivatives.

Theorem 2.3.3. The anti-symmetric operator H : E∗ → E is Hamiltonian according toDefinition 2.1.6 if and only if [H,H] = 0.

Proof. First we prove that if [H,H] = 0 then im(H) is closed, or im(H) is a subalgebraof E. From the vanishing of the Schouten bracket, the last lemma and nondegeneracy,we have that

−(DHb∗)(Ha∗) + (DHa∗)(Hb∗)−H((DHb∗)†a∗) = 0.

Then we can compute the bracket of two elements of im(H) as two elements of the Liealgebra E.

[Ha∗,Hb∗]

= DHb∗Ha∗ −DHa∗Hb∗

= DH [Ha∗](b∗) +H(Db∗ [Ha∗])−DH [Hb∗](a∗)−H(Da∗ [Hb∗])

= H(Db∗ [Ha∗]−Da∗ [Hb∗]

)+DHb∗(Ha∗)−DHa∗(Hb∗)

= H(Db∗ [Ha∗]−Da∗ [Hb∗]

)−H((DHb∗)†a∗)

= H(Db∗ [Ha∗]−Da∗ [Hb∗]− (DHb∗)†a∗

).

2.4. Symplectic operator 31

Hence im(H) is subalgebra of the Lie algebra E. Thus the bilinear map ωH is indeed a2−form and we can compute its exterior derivative. Using the definition of action of E

on A, Lemmas 2.2.2 and 2.2.8, we can prove that (see Appendix A for the proof)

d2ωH(a1, a2, a3) = [H,H](b∗1, b∗2, b∗3), ai = Hb∗i , i = 1, 2, 3.

Now the proof is complete.

Remark 2.3.4. In Chapter 4 we come up with the operator H : E → E. In fact H isdefined on a subspace of E, but simply we can just define the value of H on remainingspace, which is nothing but spn, to be zero. This operator can be lifted to the operatorH : E∗ → E, since we have identified E∗ with E. In the other words, we have takena specific subspace of C1(E, A). Hence we say that H : E → E is Hamiltonian if therelated lifted operator H : E∗ → E is Hamiltonian, that is the Schouten operator [H,H]described in terms of Frechet derivative as in Proposition 2.3.1 would vanish. We willdiscuss this in the next sections.

2.4 Symplectic operator

Let us now suppose that we have A−module M, and so we can build a complex (Ω, d)defined by the pair (A,M). An specific example of which we will work later on would bethe pair (E, A). In this section we will consider linear operators of the form S : E→ E∗

and define the symplectic operator by defining a symplectic structure as for Hamiltonianoperator and then we will give a criteria in term of Frechet derivatives for an operatorto be symplectic in the formal variational complex.

Definition 2.4.1. The operator S : E→ E∗ is anti-symmetric if

(Sa)b = −(Sb)a for a, b ∈ E.

Let us assume, throughout this section, that we have the nondegenerate pairingbetween vector space E∗ and the Lie algebra E as defined in the previous section.

Definition 2.4.2. The adjoint operator S∗ : E → E∗ for the operator S : E → E∗ isdefined due to the nondegeneracy of pairing as follows:

< S∗b∗, a >=< b∗, Sa >∗ .

If operator S is anti-symmetric, then (Sa)∗ = −S∗a∗. This is easy to prove using non-degeneracy. Indeed

< S∗b∗, a >=< b∗, Sa >∗ = < b, (Sa)∗ >= (Sa)b

= −(Sb)a =< −(Sb)∗, a > .

Definition 2.4.3. Suppose that S : E→ E∗ is a linear operator. Then we define 2-formωS on E by

ωS(a, b) = (Sb)(a). (2.4.1)

It is clear that if S is anti-symmetric, so is ωS.


Definition 2.4.4. The anti-symmetric linear operator S : E → E∗ is called symplecticif the 2−form ωS is closed.

Lemma 2.4.1. For an anti-symmetric operator S : E→ E∗, we have that

d2ωS(a, b, c) = La(Sc)(b) + (cycl.) (2.4.2)

where (cycl.) means terms with arguments cyclically permuted.

Proof.

d2ωS(a, b, c) = a.ωS(b, c)− b.ωS(a, c) + c.ωS(a, b)

−ωS([a, b], c) + ωS([b, c], b) − ωS([b, c], a)

= a.Sc(b) + b.Sa(c) + c.Sb(a)

−ωS([a, b], c) + ωS([b, c], b) − ωS([b, c], a)

= LaSc(b) + LbSa(c) + LcSb(a).

As in the setup of the Hamiltonian operator, the right hand side of (2.4.2) can beformulated by defining a similar Schouten bracket [S, S] for the operator S : E → E∗.That is

[S, S](a, b, c) = (LaSc)(b) + cycl.

Corollary 2.4.2. If the pairing is nondegenerate, then the anti-symmetric operator Sis symplectic if and only if the Schouten bracket [S, S](a, b, c), vanishes for all a, b and c

in E.

The Schouten bracket for the operator S : E → E∗ can be expressed in terms ofFrechet derivatives.

Lemma 2.4.3. For an anti-symmetric operator S : E→ E∗, we have that

[S, S](a, b, c) = LaSc(b) + cycl. = −∫K(DS∗ [a]c∗, b

)+ cycl.

Proof. By Definition 2.1.3 we get that

LaSc(b) = a.(Sc(b)) − (Sc)([a, b]) = a. < (Sc)∗, b > −(Sc)([a, b])

=

∫K(a, D∗(Sc)∗b +D∗b(Sc)∗)−

∫K((Sc)∗, Dba−Dab)

=

∫K(D(Sc)∗a, b) +

∫K(Dba, (Sc)∗)−

∫K((Sc)∗, Dba−Dab)

=

∫K(D(Sc)∗a, b) +

∫K((Sc)∗, Dab)

= −∫K(DS∗c∗a, b) −

∫K(S∗c∗, Dab)

= −∫K(DS∗ [a]c∗ + S∗(Dc∗ [a]), b

)−∫K(S∗c∗, Dab)

= −∫K(DS∗ [a]c∗, b

)+

∫K(Dca, S

∗b∗)−∫K(S∗c∗, Dab)

2.5. Nijenhuis operator 33

Hence by taking cyclic permutation, one can get rid of two last terms of last line, so thatwe get the equality as stated.

Remark 2.4.4. There is a similar observation as we explained in Remark 2.3.4. In factto prove that the operator S : E→ E is symplectic, we simply prove that the Schoutenbracket [S, S] described as in the last lemma vanishes using the properties of the Killingform and the Jacobi identity.

2.5 Nijenhuis operator

The concept of Nijenhuis operator has been introduced into the theory of integrablesystem in the work of Magri, Gelfand and Dorfman(see [14]) and under the name ofhereditary operators, in that of Fuchssteiner and Fokas, see [24],[23]. The defining rela-tion for this operator was originally found as a necessary condition for an almost complexstructure to be complex, i.e., as an integrability condition. Its important property is toconstruct an abelian Lie algebras.

Definition 2.5.1. The linear map N : E→ E is a Nijenhuis operator if

[Nx, Ny] −N [Nx, y] −N [x, Ny] +N 2[x, y] = 0 (2.5.1)

for any pair x, y ∈ E.

In the local version, if u is dynamical symbol, then using the Frechet derivatives, thecondition (2.5.1) takes the form as in the following proposition.

Proposition 2.5.1. In local version, linear map N : E→ E is Nijenhuis operator if andonly if

DN [Ny](x) −DN [Nx](y) +N(DN [x](y) −DN [y](x)) = 0. (2.5.2)

for any pair x, y ∈ E.

Proof. For any pair x, y ∈ E. we have that

[Nx, Ny] −N [Nx, y] −N [x, Ny] +N 2[x, y]

= DNyNx−DNxNy−N(DyNx−DNxy)−N(DNyx−DxNy)

+N2(Dyx−Dxy)

= DN [Nx](y) +N(DyNx)−DN [Ny](x) −N(DxNy)

−N(DyNx−DN [y](x) −N(Dxy))−N(DN [x](y) +N(Dyx)−DxNy)

+N2(Dyx−Dxy)

= DN [Nx](y) −DN [Ny](x) +N(DN [y](x) −DN [x](y)).


The following theorem explain how one can construct an abelian Lie algebra out ofNijenhuis operator. The proof can be found in [14], theorem 3.4.

Theorem 2.5.2. Let a ∈ E be the symmetry of a Nijenhuis operator N, i.e., LaN = 0.Then all elements of N ja ∈ E are symmetries of N for all j, k > 0. If N is invertible, thesame is true for all j, k ∈ Z. These symmetries commute.

An example of such an operator is given in Chapter 4. For more information, see[51]. Good applications are [48], [38] and [47].

Chapter 3

Preliminaries of Geometry

3.1 Riemannian geometry

In this section we present all concepts about Riemannian manifolds that we need lateron. A manifold will be C∞−manifold. A good reference for these matters would be [53].

Definition 3.1.1. Let U be a neighborhood of a point p on a C∞-manifold M. Wedenote the ring of C∞-functions on U by C∞(U) and the space of derivations on C∞(U)by X∞(U).

Lemma 3.1.1. The space X∞(U) is a Lie algebra.

Proof. Let Xi, i = 1, 2 be derivations, that is, Xi(fg) = (Xif)g + f(Xig). The spaceX∞(U) is an associative algebra, where the composition is given by (X · Y )(f) =X(Y (f)). This implies antisymmetry and the Jacobi identity, when we define [X,Y ] =X · Y − Y ·X. One now has to show that the commutator is indeed a derivation. Onehas

[X,Y ]fg = (X · Y − Y ·X)(fg)

= X(Y (f)g + fY (g)) − Y (X(f)g + fX(g))

= XY (f)g + Y (f)X(g) +X(f)Y (g) + fXY (g)

−Y X(f)g −X(f)Y (g) − Y (f)X(g) − fY X(g)

= XY (f)g + fXY (g) − Y X(f)g − fY X(g)

= [X,Y ](f)g + f [X,Y ](g),

and this implies that [X,Y ] ∈ X∞(U).

Definition 3.1.2. A connection on a manifold M is an operator ∇ : X∞(U)×X∞(U)→X∞(U) which assigns to two C∞ vector fields X and Y with domain U, a third C∞

vector field denoted by ∇XY with the same domain U, in such a way that the followingproperties are satisfied:

1. ∇X(Y + Z) = ∇XY +∇XZ,

2. ∇X+WY = ∇XY +∇WY,

3. ∇fXY = f∇XY,

35

36 Chapter 3. Preliminaries of Geometry

4. ∇XfY = X(f)Y + f∇XY,

for any X,W vectors at p ∈ M, Y, Z smooth fields and f a smooth function definedon a neighborhood of p.

Definition 3.1.3. We say an n−dimensional manifold M is a Riemannian manifold if Mis endowed with a symmetric and positive definite 2-covariant tensor field <,>, that is,it is C∞(U)-bilinear. The tensor <,> is called the Riemannian metric of the manifold,and it allows us to define distances, length, angles, orthogonality, etc., in the naturalway. In particular, the length of a vector X is defined as

‖X‖ =√〈X,X〉.

Definition 3.1.4. A Riemannian connection on a Riemannian manifold M is a connec-tion ∇ on M such that

1. ∇XY −∇YX = [X,Y ](the connection is torsion free ),

2. Z〈X,Y 〉 = 〈∇ZX,Y 〉+ 〈X,∇ZY 〉,for all fields X,Y and Z with the common domain.

Definition 3.1.5. The curvature tensor of a connection ∇ is a tensor R that assigns toeach pair of vectors X,Y at a point p a linear transformation R(X,Y ) of the tangentspace to p, as TpM, into itself. After extending X,Y and Z to smooth vector fields onU , R(X,Y )Z is defined via the relation

R(X,Y )Z = ∇X∇Y Z −∇X∇Y Z −∇[X,Y ]Z. (3.1.1)

That this defines a tensor has to be proved. The value of this expression is independentof the way the vector fields were extended.

Definition 3.1.6. The torsion tensor of a connection ∇ is defined by

T (X,Y ) = ∇XY −∇YX − [X,Y ],

where X and Y are smooth vector fields on U.

Definition 3.1.7. The Riemann-Christoffel curvature tensor (of type (0, 4)) is the4−covariant tensor

K(X,Y,Z,W ) = 〈R(Z,W )Y,X〉for any X,Y,Z, and W tangent vectors at p.

Riemannian curvature tensors have the following properties:

Theorem 3.1.2. The following relations are true:

1. K(X,Y,Z,W ) = −K(Y,X,Z,W ) = −K(X,Y,W,Z),

3.2. Cartan’s moving frame method 37

2. K(X,Y,Z,W ) = K(Z,W,X, Y ).

Definition 3.1.8. Given two independent vectors X,Y ∈ TpM, the normalized quadraticform,

sec(X,Y ) =K(X,Y,X, Y )

〈X,X〉〈Y, Y 〉 − 〈X,Y 〉2 ,

is called sectional curvature of X,Y. It can easily be checked that sec(X,Y ) dependsonly on the plane π spanned by X and Y, and so the sectional curvature is also calledK(π), the Riemannian curvature of the plane section π.

Definition 3.1.9. A Riemannian manifold M is said to have constant Riemannian cur-vature κ if the Riemannian curvature of all plane sections is the constant κ.

Proposition 3.1.3. The following properties are equivalent:

1. K(π) = κ for all 2-planes in TpM.

2. R(X,Y )Z = κ(〈Y,Z〉X − 〈X,Z〉Y ) for any X,Y and Z in TpM.

Corollary 3.1.4. Assume the manifold M has certain constant Riemannian curvature.Then

1. ∇XR = 0 along any direction determined by the vector field X. That is, theRiemannian curvature tensor is parallel.

2. If Z is orthogonal to X and Y, then R(X,Y )Z = 0.

3. If W is orthogonal to X and Y, then K(W,Z,X, Y ) = 0 for any Z.

3.2 Cartan’s moving frame method

This method was introduced by Elie Cartan at the beginning of last century. Cartan’s in-sight was that the local properties of a manifold equipped with a geometric structure canbe very well understood if one knows how the frames of the tangent bundle (compatiblewith the geometric structure) vary from one point of the manifold to another.

Let M be an arbitrary Riemannian manifold with metric 〈, 〉. We choose a localorthonormal moving frame X = ei | i = 1, . . . , n. Denote by (θi) the dual coframe, i.e.,

θi(ej) = δij .

Remark 3.2.1. Notice that the notion of “dual of frame” which is another frame isdefined as another frame Y = fi | i = 1, . . . , n so that 〈ei, fj〉 = δij . See [41] for morediscussion on this issue.

Lemma 3.2.2 (E. Cartan). On the Riemannian manifold M as above, there exists acollection of 1-forms ωij uniquely defined by the requirements

(a) ωij = −ωji .


(b) dθi = θj ∧ ωij.

Proof. Uniqueness: Suppose ωij satisfy the conditions mentioned above. Since

θk | k = 1, . . . , n

form a basic for 1-forms and

θj ∧ θk | j, k = 1, . . . , n and j < k

a basic for 2-forms, there exist functions f ijk and gijk such that

ωij =∑

k

f ijkθk, and dθi =

∑

j<k

gijkθj ∧ θk, gijk = −gikj,

so that we have then

n∑

j

θj ∧ ωij =

n∑

j,k=1

θj ∧ f ijkθk

= −n∑

j,k=1

f ijkθk ∧ θj

=n∑

j,k=1,j<k

(f ijk − fkj)θj ∧ θk

Then the condition (a) is equivalent to

(a1) f ijk = −f jikwhile (b) gives

(b1) f ijk − f ikj = gijk.

The above two relations uniquely determine the f ′s in terms of the g′s via a cyclicpermutation of the indices i, j, k as

f ijk =1

2(gijk + gjki − gkij). (3.2.1)

Existence: Consider the functions gijk defined by

dθi =∑

j<k

gjjkθi ∧ θk, gijk = −gikj.

Next define ωij = f ijkθk where f ′s are given by (3.2.1). Then the forms ωij satisfy both

(a) and (b).

3.2. Cartan’s moving frame method 39

Let the matrix a be invertible, so that X.a is another moving frame and the dualframe associated to this moving frame is simply

θX.a = a−1θX ,

in which θX is the dual 1-forms as above associated to X. In the following lemma wedetermine how 1-form ω behaves under the change of moving frame.

Lemma 3.2.3. We have that

ωX.a = a−1da+ a−1ωXa, (3.2.2)

where ωX and ωX.a are unique 1-forms as in Lemma 3.2.2 associated to moving framesX and X.a, respectively.

Proof. We compute

d(aθX.a) = da ∧ θX.a + a.dθX.a

= da ∧ (a−1.θX) + a.(θX.a ∧ ωX.a)= (da.a−1) ∧ θX − a.(ωX.a ∧ θX.a)= (da.a−1) ∧ θX − a.(ωX.a ∧ a−1θX)

= (da.a−1) ∧ θX − (a.ωX.a.a−1) ∧ θX

= (da.a−1 − a.ωX.a.a−1) ∧ θX .

On the other hand from the fact aθX.a = θX , we get that

d(aθX.a) = dθX = θX ∧ ωX = −ωX ∧ θX .

This implies that

da.a−1 − a.ωX.a.a−1 = −ωX , or ωX.a = a−1.da+ a−1.ωX .a.

We see that the 1-form ω corresponding to the moving frame does not behave like atensor when we change the coordinate system or moving frame.

Definition 3.2.1. Cartan connection on a manifold M is an assignment of a matrixvalued 1-form ω to every moving frame such that (3.2.2) holds.

Lemma 3.2.4. Let ∇ be the Levi Civita connection on Riemannian manifold M com-patible with its metric and e′is and θ′s are as mentioned above and also ωij be the 1-formsas in Cartan’s Lemma. Then

∇ekej = ωij(ek)ei.


Proof. Define ωij by

∇ekej = ωij(ek)ei.

Since the connection is compatible, thus

0 = ∇ek〈ej , el〉 (3.2.3)

= 〈∇ekej , el〉+ 〈ej,∇ekel〉 (3.2.4)

= 〈ωij(ek)ei, el〉+ 〈ej , ωil(ek)ei〉 (3.2.5)

= ωlj(ek) + ωjl (ek), (3.2.6)

Hence ωlj = −ωjl .The differential of θi can be computed in terms of the Levi-Civita connection and

we have that

dθi(ej , ek) = ejθi(ek)− ekθ

i(ej)− θi([ej , ek])= −θi([ej , ek])= −θi(∇ejek) + θi(∇ekej)

= −θi(ωlk(ej)el) + θi(ωlj(ek)el)

= ωij(ek)− ωik(ej)

where the first equality follows from the fact that θi(ek)′s are constant and second

equality from the fact that ∇ is torsion free connection. But we see that

(θ ∧ ω)i(ej , ek) = θl ∧ ωil(ej , ek)= θl(ej)ω

il(ek)− θl(ek)ωil(ej)

= ωij(ek)− ωik(ej).

So

dθ = θ ∧ ω.Thus the ω′s satisfy both condition (a) and (b) of Cartan lemma, so that, by uniqueness,we must have ωij = ωij.

In other word, the lemma shows that o(n)−valued 1−form ω = (ω ij) is the 1 formassociated to moving frame X via Levi-Civita connection ∇. In particular, as we showin the following lemma, the 2-forms

Ω = dω − ω ∧ ω, Θ = dθ − ω ∧ θ, (3.2.7)

which are called Cartan curvature form and torsion form, respectively are the Rieman-nian curvature and torsion tensor on the Riemannian manifold. Together the equations(3.2.7) form the Cartan structure equation.

Lemma 3.2.5. 1. Ωik(el, ej) = 〈R(el, ej)ek, ei〉.

3.3. Ehresmann connection and Cartan geometry 41

2. Θl(ej, ek) = 〈T (ej, ek), el〉.where R and T are curvature and torsion tensors defined in (3.1.1) and (3.1.6).

Proof. By definition we have

R(el, ej)ek = ∇el∇ejek −∇ej∇elek −∇[el,ej ]ek

= ∇elωik(ej)ei −∇ejω

ik(el)ei −∇∇el

ej−∇ej elek

= +el(ωik(ej))ei + ωik(ej)ω

mi (el)em

−ej(ωik(el))ei − ωik(el)ωmi (ej)em

−(ωij(el)ωmk (ei)em − ωil(ej)ωmk (ei)em)

Hence

〈R(el, ej)ek, ei〉= +el(ω

ik(ej)) + ωmk (ej)ω

im(el)

−ej(ωik(el))− ωmk (el)ω

im(ej)

−(ωmj (el)ωik(em)− ωml (ej)ω

ik(em))

On the other hand, we have that

Ωik(el, ej) = (dω − ω ∧ ω)ik(el, ej)

= el(ωik(ej))− ej(ω

ik(el))− ωik([el, ej ])

−ωmk ∧ ωim(el, ej)


ik(el))− ωik(ωmj (el)em − ωml (ej)em)

−(ωmk (el)ωim(ej)− ωmk (ej)ω

im(el))


ik(el))− (ωmj (el)ω

ik(em)− ωml (ej)ω

ik(em))

+ωmk (ej)ωim(el)− ωmk (el)ω

im(ej)

= 〈R(el, ej)ek, ei〉

Thus we obtain that

〈R(el, ej)ek, ei〉 = (dω − ω ∧ ω)ik(el, ej).

For the proof of the second part, see the comprehensive book written by Michael Spivak[64] volume II.

3.3 Ehresmann connection and Cartan geometry

We generalize the idea of classical Cartan connection. The content of this section can befound in various references, Spivak’s [64] and Kobayashi’s [35] are comprehensive books,Sharpe [63] describes the Cartan generalization of Klein’s Erlangen program.


As in Euclidean space there is a natural way to parallel-translate and compare vectorsat different points, likewise in a general manifold a choice of a connection prescribes away of translating tangent vectors “parallel to themselves” and to intrinsically define adirectional derivative.

In the case of a principal bundle P with structure group G over a manifold M :

G // P

π

M

We explain the rule of a connection when thinking of lifting a vector field v ∈ TM to avector field v ∈ TP in a unique way. For each p ∈ P, let Gp be the vector subspace of TpPconsisting of all the vectors tangent to the vertical fiber. That is Gp = ker(dπ(p)) ⊂ TpPin which dπ(p) : TpP → TπpM.

The lifting of v will be unique if we require v(p) to lie in a subspace of TpP comple-mentary to Gp. A smooth and G−invariant choice of such a complementary subspace iscalled a Ehresmann connection (Cf. [17]) on P. This leads to the following definition.

Definition 3.3.1. A connection on a principal bundle P is a smooth assignment of asubspace Hp ⊂ TpP, for each p ∈ P such that:

1. TpP = Gp ⊕Hp,

2. Hgp = Tp(Lg)Hp for each g ∈ G, where Lg is the left-translation in G and conse-quently Tp(Lg) : TpP → TgpP.

Given a connection, the horizontal subspace Hp is mapped isomorphically by dπ ontoTπpM. Therefore the lifting of v is the unique horizontal v which projects onto v. Anequivalent way of assigning a connection is by means of a Lie algebra valued 1−form ω(Cartan connection). If X ∈ g, let X † be the vector field on P induced by the action ofthe 1−parameter subgroup etX . Since the action of G maps each fiber into itself, thenX† is tangent to the vertical fiber at each point, i.e., X ∈ Gp. For each v ∈ TpP, wedefine ω(v) as the unique X ∈ g such that X † is equal to the vertical component of v.It follows that ω(v) = 0 if and only if v is horizontal.

Proposition 3.3.1. A Cartan connection 1−form ω has the following properties:

1. ω(X†) = X,

2. L∗gω = Adgω for each g ∈ G, in which Ad is adjoint representation of G.

The proof can be found in [35] and appendix A of [63].Now we define the Cartan geometry based on the Ehresmann connection. Assume

here that H is a group with the Lie algebra h as subalgebra of g.

Definition 3.3.2. A Cartan geometry ξ = (P, ω) on M modeled on (g, h) with groupH consist of the following data:

3.4. Homogeneous space, symmetric space 43

1. a smooth manifold M ;

2. a principal left H bundle P over M ;

3. a g−valued 1−form ω on P satisfies the following conditions:

(a) for each point p ∈ P, the linear map ωp : TpP → g is an isomorphism;

(b) (Lh)∗ω = Ad(h)ω for all h ∈ H;

(c) ω(X†) = X for all X ∈ h.

The g−valued form on P given by

Ω = dω +1

2[ω, ω]

is called the curvature. If ρ : g → g/h is the canonical projection, then ρ(Ω) is calledthe torsion. If Ω takes values in the subalgebra h, we say that the geometry is torsionfree.

Definition 3.3.3. Let M is a connected manifold. Then Cartan geometry ξ = (P, ξ)has constant curvature if Ωp(Xp, Yp) is independent of p ∈ P whenever the vector fieldsX and Y are ω−constant vector fields.

That may also be expressed by saying that the curvature function

K : P → Hom(C2(g/h), h), K(p) = Ωp(ω−1p (u), ω−1

p (v))

is constant.

Definition 3.3.4. A Cartan geometry whose curvature vanishes at every point is calledflat.

Notice that while structure equation always holds for a Lie group, meaning that thecurvature of Maurer-Cartan form vanishes, not all Cartan geometry are flat.

3.4 Homogeneous space, symmetric space

The material of this section is taken from [4] and [32]. A homogeneous space of aLie group G is any differentiable manifold P on which G acts transitively, that is, forp1, p2 ∈M, there is g ∈ G so that g.P1 = p2. The subgroup

H = Hp0 = g ∈ G : g.p0 = p0

is called the isotropy group at p0. It is a theorem that each such P can be identifiedwith a coset space G/H for some subgroup H and that this H plays the rule of isotropygroup of some point.


Let g, h be the Lie algebras of G and H respectively, and let m be the vector spacecomplement of h in g. Then

g = h⊕m, [h, h] ⊂ h,

and m is identified with the tangent space Tp0M of M = G/H at point p0. At themoment we know nothing of [h,m] and [m,m].

Definition 3.4.1. When g satisfies the more stringent conditions:

g = h⊕m, [h, h] ⊂ h, [h,m] ⊂ m,

then M = G/H is called a reductive homogeneous space.

These spaces possess canonically defined connection with curvature and torsion.Evaluated at fixed point p0, the curvature and torsion tensors are given purely in termsof the Lie bracket operation on m :

(R(X,Y )Z)p0 = −[[X,Y ]h, Z], X, Y, Z ∈ m,

T (X,Y )p0 = −[X,Y ]m, X, Y ∈ m,

where subscript h and m refer to the component of [X,Y ] in those vector subspaces.

Definition 3.4.2. When g satisfies the conditions:

g = h⊕m, [h, h] ⊂ h, [h,m] ⊂ m,

[m,m] ⊂ h,

then g is called a symmetric algebra and G/H is a symmetric space.

For these spaces the above mentioned canonical connection is derived from a metric,which is itself given by the restriction of the Killing form to m. Clearly this connectionis torsion free and curvature tensor is given as

(R(X,Y )Z)p0 = −[[X,Y ], Z], X, Y, Z ∈ m.

For the symmetric spaces with constant curvature, we do have that

κ =K(R(X,Y )Y,X)

K(X,X)K(Y, Y )−K(X,Y )2= − K([[X,Y ], Y ], X)

K(X,X)K(Y, Y )−K(X,Y )2,

for X,Y ∈ m.

Remark 3.4.1. There also can be defined a Levi-Civita connection.

Remark 3.4.2. On p.518 of Helgason’s book [32] there is a table of symmetric spaces.Directly beneath this table those spaces which are Hermitian are listed.

Remark 3.4.3. The space Sp(n + 1)/Sp(n) × Sp(1) is homogeneous space. There wehave that m = g/h in which g = sp(n+ 1) and h = sp(n)× sp(1). Moreover this space isnaturally reductive space. For definition see [4].

Chapter 4

Geometric method of Integrable system

4.1 Integrable system in Riemannian geometry

What follows is the description of a natural frame and natural formulae (in comparisonwith the Frenet frame and the Frenet formulae) for any smooth curve on a Riemannianmanifold under some nondegeneracy conditions that follow from the construction.

Let γ : U ⊂ R→ M be a smooth curve on a Riemannian manifold M with Riemannianconnection ∇. From now on we will assume that all vector fields are defined on somecommon open subset U. Let V (x) be the tangent field at x obtained by differentiationwith respect to x (also called the velocity vector). We will naturally say that γ isparametrized by arclength whenever |V (x)| = 1 for all x in the domain of γ. Assumethat γ is nondegenerate, that is, V (x) 6= 0 for all x ∈ U. We can then define the firstvector in the natural frame, the unit tangent vector, as

e1(x) =V (x)

|V (x)| .

Now as is well known in differential geometry, we can construct the Frenet frame. Moreexplicitly, let the orthonormal basis e1, e2, e3 along the curve be Frenet frame, that is,

e1

e2

e3

x

=

0 k1 0−k1 0 k2

0 −k2 0

e1

e2

e3

.

The matrix in this equation is called the Cartan matrix.The construction of natural frame is due to the work of Bishop and Hasimoto. In

1975, Bishop, cf. [7] discovered the same transformation as Hasimoto [31] when hestudied the relation between two different frames to frame a curve in 3−dimensionalEuclidean space. In order to build up the natural frame, let us introduce the followingnew basis

e1

f2

f3

=

1 0 00 cos(θ) −sin(θ)0 sin(θ) cos(θ)

e1

e2

e3

, θ =

∫k2.

Its frame equation is

e1

f2

f3

x

=

0 k1cos(θ) k1sin(θ)−k1cos(θ) 0 0−k1sin(θ) 0 0

e1

f2

f3

.

45

46 Chapter 4. Geometric method of Integrable system

We call the basis e, f1, f2 the natural frame or parallel frame.As a matter of fact, Hasimoto transformation is the one which converts the Frenet

frame into such a natural or parallel frame. This transformation is induced by a gaugetransformation, cf. [12, 72] where the Hasimoto transformation has been generalizedto arbitrary n−dimensional space. In the following definition we have formulated theparallel frame for a curve embedded in a n−dimensional Riemannian manifold.

Definition 4.1.1. The natural orthonormal frame ei, i = 1, . . . , n is the frame satis-fying the following equation which are called natural formulae.

∇e1e1 = −∑n

i=2 kiei

∇e1ei = kie1, i ≥ 2,(4.1.1)

In the matrix form we have

∇e1e1

∇e1e2...

∇e1en

=

0 −k2 . . . −knk2 0 . . . 0...

......

...kn 0 . . . 0

e1

e2...

en

.

Definition 4.1.2. Suppose γ(s, t) is any one parameter family of arclength parametrizedcurves embedded in a Riemannian manifold, and let X be the vector filed X = γt. ThenX is locally arclength preserving if ∇γsX has no tangential component, i.e.,

〈∇γsX, γs〉 = 0.

Remark 4.1.1. The vector field X in the above definition is called Sym-Pohlmeyer fieldin the literature, See for instance [43], [66] and [54].

In the following theorem, using the natural moving frame for the curves, we analyzethe correspondence between curve evolution and natural curvature evolution by meansof a geometric recursion operator. See [42, lemma in page 81] and [49, theorem 2] in thecase of Frenet moving frame.

Theorem 4.1.2. Let M be n−dimensional Riemannian manifold with constant curva-ture κ, and let γ(x, t) be family of curves on M satisfying a geometric evolution systemof equations of the form

γt = h1e1 + . . .+ hnen, (4.1.2)

where e1, . . . , en is the natural frame of γ, and h1, . . . , hn are arbitrary smooth func-tions of the curvatures k2, . . . kn and their derivatives with respect to x. Assume thatx is the arclength parameter and that evolution (4.1.2) is arclength preserving, that is,〈e1, e1〉 = 1. Then the curvatures k = (k2, . . . , kn) satisfy the evolution

kt = HIh− κh,

4.1. Integrable system in Riemannian geometry 47

where, if we denote by Dx the total differentiation operator with respect to x,

I = −Dx −D−1x (< k, . >)k, H = Dx + H1

in which

H1g = D−1x (kgt − gkt)k, for g = (g2, . . . , gn)t.

Proof. A short comment on the calculations to follow: Let us denote e1 = γx, assumingx to be arclength. These vectors are defined as the push-forward vectors of the vectors∂

∂t,∂

∂x, tangent to the domain of γ, through γ. That is, if γ : U ⊂ R2 → M, then

γt = γ∗∂

∂tacting on functions as γt(f) =

∂

∂tf(γ(t, x)); likewise for x : meaning that

e1(f) = γx(f) =∂

∂xf(γ(t, x)) in which f is a function f : M→ R. Thus, by applying γt

or γx to functions defined along γ, we are indeed taking their derivatives with respect tot or x, respectively. If (4.1.2) is arclength preserving, these two vectors will commute:

[γx, γt] = 0,

since∂

∂xand

∂

∂tcommute and γ∗ preserves Lie bracket.

Now it follows from Definition 3.1.4 of the Riemannian connection and the naturalformulae, that

γt〈∇e1ei,∇e1ei〉 = 2〈∇γt∇e1ei,∇e1ei〉= 2〈∇γt∇e1ei, kie1〉= 2ki〈∇γt∇e1ei, e1〉, (4.1.3)

for i = 2, . . . , n. On the other hand we can write:

γt〈∇e1ei,∇e1ei〉 = γt〈kie1, kie1〉= γt(k

2i )

= 2kiki,t, (4.1.4)

where we have denoted ki,t to γt(ki). Observe that here is the first place we have usedthe fact that curve is arclength-preserving, i.e., 〈e1, e1〉 = 1.

Hence, using the definition of Riemannian curvature tensor, the Riemann-Christoffelcurvature tensor and the fact that e1 = γx and γt commute, the evolution of ki can be


derived from (4.1.3) and (4.1.4) as follows.

ki,t = 〈∇γt∇e1ei, e1〉= 〈∇e1∇γtei +∇[e1,γt]ei +R(γt, e1)ei, e1〉= 〈∇e1∇γtei, e1〉+K(e1, ei, γt, e1)

= e1〈∇γtei, e1〉 − 〈∇γtei,∇e1e1〉+K(e1, ei, γt, e1)

= e1〈∇γtei, e1〉 − 〈∇γtei,−n∑

l=2

klel〉+K(e1, ei, γt, e1)

= e1〈∇γtei, e1〉+n∑

l=2

kl〈∇γtei, el〉+K(e1, ei, γt, e1).

Let us denote 〈∇γtei, ej〉 by ωji (γt) for i, j = 1, . . . , n according to the Lemma 3.2.4and Cartan’s Lemma 3.2.2. Also let us denote by Dx the total differentiation withrespect to x, that is, Dx = e1. Then the evolution for ki takes the form:

ki,t = Dxω1i (γt) +

n∑

l=2

klωli(γt) +K(e1, ei, γt, e1) (4.1.5)

In order to compute ωli(γt), we use first the fact that the connection is torsion free:

∇γte1 = [γt, e1] +∇e1γt

= ∇e1γt

= ∇e1(

n∑

i=1

hiei)

=

n∑

i=1

e1(hi)ei +

n∑

i=1

hi∇e1ei

= (Dxh1 +n∑

i=2

hiki)e1 +n∑

i=2

(Dxhi − h1ki)ei (4.1.6)

Hence since 〈ei, ej〉 = δij , so

0 = ω11(γt) = 〈∇γte1, e1〉 = Dxh1 +

n∑

i=2

hiki,

and

〈∇γte1, ei〉 = ωi1(γt) = Dxhi − h1ki for i = 2, . . . , n.

Hence we can solve the first equation and find h1 in term of h2, . . . , hn, as follows:

h1 = −D−1x (< k,h >), where h = (h2, . . . , hn)t. (4.1.7)


Notice that the inner product here is just the natural inner product in Rn−1.Also we obtain that

〈∇γte1, ei〉 = ωi1(γt) = Dxhi − h1ki for i = 2, . . . , n.

Hence ω1i (γt) = −ωi1(γt) = −Dxhi −D−1

x (< k,h >)ki. We denote by I the operator

I := −Dx −D−1x (< k, . >)k (4.1.8)

Now using again the properties of Riemannian connection and Riemannian curvaturetensor we can compute the following expression for i, l ≥ 2:

Dxωii(γt) = e1〈∇γtei, el〉

= 〈∇e1∇γtei, el〉+ 〈∇γtei,∇e1el〉= 〈∇γt∇e1ei +R(e1, γt)ei, el〉+ 〈∇γtei, kle1〉= 〈∇γt(kie1), el〉+K(el, ei, e1, γt) + kl〈∇γtei, e1〉= ki〈∇γte1, el〉+ kl〈∇γtei, e1〉+K(el, ei, e1, γt)

= kiωl1(γt)− klωi1(γt) +K(el, ei, e1, γt).

Hence we simply have that

ω1i (γt) = D−1

x (kiωl1(γt)− klωi1(γt)) +D−1

x (K(el, ei, e1, γt)).

Thus, finally, (4.1.5) leads to

ki,t = Dxω1i (γt) +

n∑

l=2

D−1x (kiω

l1(γt)− klωi1(γt))kl (4.1.9)

+n∑

l=2

D−1x (K(el, ei, e1, γt))kl +K(e1, ei, Dt, e1). (4.1.10)

We use the symbol H for the operator

H := Dx + H1 (4.1.11)

whereH1g = D−1

x (kgt − gkt)k, for g = (g2, . . . , gn)t.

Now we simply write the equation for kt as follows.

kt = HIh +

n∑

l=2

klD−1x (K(el, e., e1, γt)) +K(e1, e., γt, e1). (4.1.12)

Since the Riemann-Christoffel curvature tensor K in (4.1.12) is multilinear with re-spect to C∞(M) in all its components), it is enough to compute K(e1, ei, ej, e1) andK(el, ei, e1, ej) for all i, l = 2, . . . , n and j = 1, . . . , n. We see that

K(e1, ei, ej , e1) = K(ej , e1, e1, ei) = 0, for j 6= 1, i,


since ej then is perpendicular to e1 and ei. On the other hand

K(e1, ei, ei, e1) = −K(ei, e1, ei, e1) = −sec(ei, e1) = −κ

andK(e1, ei, e1, e1) = −K(e1, ei, e1, e1) = 0.

Thus we have that

K(e1, ei, ej , e1) = −κδij, for i ≥ 2, and j ≥ 1.

Similarly for all i, l 6= 1 and j ≥ 1, we have

K(el, ei, e1, ej) = K(e1, ej, el, ei) = 0.

Therefore the equation (4.1.12) becomes as follows:

kt = HIh− κh = Rh− κh. (4.1.13)

where the operator R is R = HI, in which the operators I and H have been defined in(4.1.8) and (4.1.11), respectively.

Remark 4.1.3. The operator

P(h1e1 + h2e2 + . . . + h3e3) = (−D−1x < k,h >)e1 + h2e2 + . . . + h3e3

is called the renormalization operator in [43].

Remark 4.1.4. For related topics and similar constructions, see [74, 75, 61, 42].

Remark 4.1.5. Let us now take h = kx in Theorem 4.1.2. Then we obtain

kt = −k3 +3

2〈k,k〉kx − κk.

Indeed we have

Ih = −k2x +1

2〈k,k〉k, and H1Ih = kx〈k,k〉 − k〈kx,k〉.

Theorem 4.1.6. The operators H is Hamiltonian, that is, the Schouten bracket [H,H] =0 and I is a symplectic operator. The operator R is thus a hereditary operator.

Proof. See [58].

Remark 4.1.7. For a curve γ embedded in a Riemannian manifold whose tangent spaceembedded in a Lie algebra it sufficse to be arclength-preserving, that

K(γs, γs) = 1,

in which K is the Killing form of the Lie algebra.


Remark 4.1.8. One could ask what sort of geometric operators and evolution equationswill come up if we use the Frenet frame? The answer can be found in [49] for nonzerocurvature which would be a generalization of [42] in the case of zero curvature. Brieflyit was found that evolution of κ, τ (taken as k1, k2 here) with respect to time, takes theform (

κτ

)

t

= P

(h3

h1

)

where

P =

−τDx −Dxτ D2x

1

κDx −

τ2

κDx +Dxκ

Dx1

κD2x −Dx

τ2

κ+ κDx Dx(

τ

κ2+Dx

τ

κ2)Dx + τDx +Dxτ

+κ

01

κDx

Dx1

κ0

.

Notice the difference between the curvature of the curve κ and sectional curvature κ.The division in 1/κ is consequence of having torsion free connection, see (4.1.6), andsolving the corresponding equations dividing by κ. Instead we used nonlocal operatorD−1x as in (4.1.7). Then one can split the operator P into the anti-symmetric operators

as below.

P = B + D + E + κC ,

in which

B =

−τDx −Dxτ −τ

2

κDx

−Dxτ2

κ0

, D =

(0 Dxκ

κDx τDx +Dxτ

),

and

E =

0 D2x

1

κDx

Dx1

κD2x Dx(

τ

κ2+Dx

τ

κ2)Dx

, C =

01

κDx

Dx1

κ0

.

These operators form a quadruplet of compatible Hamiltonians and the Hamiltonianpair P and C gives a hereditary operator R1 = PC−1.

Now let us back to Theorem 4.1.2 and evolution of the curve γ in Riemannian man-ifold M. Corresponding to the natural frame e1, . . . , en, let us denote Rn-valued dual1-form θ to θ = (θ1, . . . , θn). Then we have that

θ(Dx) = θ(e1) = (1, 0, . . . , 0), θ(γt) = (h1, . . . , hn), (4.1.14)


where γt =∑n

i=1 hiei. Let us denote by Dt the total differentiation with respect to t,that is, Dt = γt. According to the Cartan Lemma 3.2.2, we do have that

∇e1ej = ωkj (e1)ek.

This implies that if we let ω = (ωji )i,j , then in the natural frame (4.1.1), we have that

ω(e1) =

0 −k2 . . . −knk2 0 . . . 0...

......

...kn 0 . . . 0

. (4.1.15)

As in the proof of Theorem 4.1.2, we could find ω(Dt) = ω(γt) from the the curvatureof the Levi-Civita connection ∇ and the fact that the connection is torsion free. Thuswe expect to obtain the same equations and geometric operator if we use Cartan struc-ture equations (3.2.7) as we discussed above on the Riemannian manifold modeled onEuclidean geometry. The Cartan structure equation applied to (e1, γt) = (Dx, Dt) givesthe following equations:

Ω(Dx, Dt) = (dω − ω ∧ ω)(Dx, Dt)

= Dxω(Dt)−Dtω(Dx)− ω([Dx, Dt])

−ω(Dx)ω(Dt) + ω(Dt)ω(Dx)

= Dxω(Dt)−Dtω(Dx)− ω(Dx)ω(Dt) + ω(Dt)ω(Dx),

since [Dx, Dt] = 0. The torsion form Θ is as follows:

Θ(Dx, Dt) = (dθ − ω ∧ θ)(Dx, Dt)

= Dxθ(Dt)−Dtθ(Dx)− T ([Dx, Dt])

−ω(Dx)θ(Dt) + ω(Dt)θ(Dx)

= Dxθ(Dt)−Dtθ(Dx)− ω(Dx)θ(Dt) + ω(Dt)θ(Dx).

We use the Cartan structure equations:

Ω(Dx, Dt) = Dxω(Dt)−Dtω(Dx)− ω(Dx)ω(Dt) + ω(Dt)ω(Dx)

Θ(Dx, Dt) = Dxθ(Dt)−Dtθ(Dx)− ω(Dx)θ(Dt) + ω(Dt)θ(Dx).(4.1.16)

In the case of Riemannian manifold, we have Θ(Dx, Dt) = 0.Now we replace the instance of θ(Dx), θ(Dt) ∈ Rn and ω(Dx) ∈ on as in (4.1.14) and

(4.1.15) into the Cartan structure (4.1.16) and keep the matrix ω(Dt) as general elementof on which is to be computed. Let us take the matrix ω(Dt) as a general element of onas

ω(Dt) =

(0 −mt

m M

), M ∈ on−1 and mt = (m2, . . . ,mn).


With respect to this representation, we write ω(Dx) as

ω(Dx) =

(0 −kt

k 0

), kt = (k2, . . . , kn).

Similarly we write θ(Dx) and θ(Dt) as

θ(Dx) =

(10

), θ(Dt) =

(h1

h

)ht = (h2, . . . , hn).

Also we write

Ω(Dx, Dt) =

(0 −rt

r R

)R ∈ on−1 and rt = (r2, . . . , rn).

Then the Cartan structure equations (4.1.16) in components will be as follows.

Curvature part :

r = Dxm−Dtk + Mk,R = DxM + kmt −mkt

(4.1.17a)

Torsion part :

0 = Dxh1 + 〈k,h〉0 = Dxh− kh1 + m

(4.1.17b)

where 〈, 〉 is standard inner product on Rn−1.Now from the torsion part, we can solve the first equation 4.1.17b and find that

h1 = −D−1〈k,h〉.

Then the second one can be solved for m as m = −Dxh− kD−1〈k,h〉 = Ih, where theoperator I = −Dx − kD−1〈k, .〉 is exactly the operator described in (4.1.8) and provedto be symplectic.

From the curvature part, we see that

M = D−1(mkt − kmt) +D−1(R).

This implies that

Dtk = Dxm + Mk− r

= Dxm +D−1x (mkt − kmt)k +D−1(R)k− r

= Hm +D−1(R)k− r

where H = Dx+ H1 is operator (4.1.11) which is proved to be Hamiltonian. Now similarto the discussion at the end of the proof of Theorem 4.1.2, we obtain R = 0 and r = κh.Therefore

Dtk = Rh− κh,

in whichR = HI.


Remark 4.1.9. If one compares the computational efforts of setting up a moving Frenetframe and structure equations (Levi-Civita connection) just using the metric with theCartan formulation in terms of connection, the difference is striking: not only one cannot see what one is doing, but there is no need to see it, since every thing goes right byconstruction, and the entities one writes down are automatically differential invariants.See [59].

Remark 4.1.10. One can start with the Euclidean Lie algebra g = eucn(R) = on(R)nRnand h = on and assume that ω is the Cartan connection in Euclidean geometry as aCartan geometry. Let us choose ω(Dx) and ω(Dt) as follows.

ω(Dx) =

0 −kt 1k 0 0t

0 0 0

, ω(Dt) =

0 −mt h1

m M ht

0 0 0

.

Now if we write down the Cartan structure equation

Ω(Dx, Dt) = Dxω(Dt)−Dtω(Dx) + [ω(Dt), ω(Dx)],

then we will find exactly the same formula for evolution of k.

4.2 Integrable system in (p,q)-Orthogonal geometry

We extend the idea of the last section in finding the interaction between geometry in asense of Cartan and integrable system. The direct generalization of Euclidean geometryas an example of a Cartan geometry to the Riemannian geometry of signature p, q willbe defined as follows.

Definition 4.2.1. Let

Σp,q =

(Ip 00 −εIq

).

Then define the orthogonal group of signature p, q by

Op,q(R) = A ∈ Glp+q(R)|AΣp,qAt = Σp,q.

The Lie algebra of this Lie group is

op,q(R) = A ∈Mp+q(R)|AΣp,q + Σp,qAt = 0.

Any elements of op,q(R) has the form(A BC D

),

where A ∈M(p, p), B ∈M(p, q), C ∈M(q, p) and D ∈M(q, q) and that

A+At = D +Dt = 0, Bt = εC.

The reason we use ε in the definition of this Lie group is that we can keep trace ofthe Riemannian case and see how is consistent with that of last section by taking ε = −1and of course to standard geometry of orthogonal group of signature p, q by taking ε = 1.

4.2. Integrable system in (p,q)-Orthogonal geometry 55

Definition 4.2.2. The Riemannian geometry of signature p, q in a sense of Klein ge-ometry, see Sharpe [63] definition 3.2., is described as a pair of Lie groups (G,H) inwhich

G = Op,q(R)nRp+q, H = Op,q(R).

Similar to the Riemannian case, we first choose a moving frame namely parallel frameand write the Cartan structure equation having Cartan connection ω with free torsion.Notice that the Riemannian manifold M = G/H is assumed to have zero constantcurvature. The parallel moving frame is described as

ω(Dx) =

(A′ B′

C ′ D′

)=

0 u1 . . . up−1 up . . . up+q−1

−u1 0 . . . 0 0 . . . 0...

......

......

......

−up−1 0 . . . 0 0 . . . 0εup 0 . . . 0 0 . . . 0

......

......

......

...εup+q−1 0 . . . 0 0 . . . 0

,

as an element of op,q(R).

Remark 4.2.1. As in the Riemannian case, we consider x as arclength parameter, sothat the number of invariants are p+ q − 1. In general this number is just dim(M)− 1.In the sequel, we take this consideration into account.

Now let ω(Dt) be the matrix

ω(Dt) =

(A BC D

),

in which the matrices A ∈ M(p, p), B ∈ M(p, q), C ∈ M(q, p) and D ∈ M(q, q), possessthe following properties:

A+At = D +Dt = 0, Bt = εC.

As in the Riemannian case, let the torsion 1-form at Dx and Dt be

τ(Dx) = (1, 0, . . . , 0, 0, . . . , 0)t,

τ(Dt) = (h1, h2, . . . , hp−1, hp, . . . , hp+q)t,

respectively.

Theorem 4.2.2. With the assumptions above, the Cartan struction equations as in(4.1.16) gives the evolution of ut = (u1, . . . , up+q−1) as below.

Dtu = R(Iεp−1,qh), R = HI, (4.2.1)


in which

ht = (h2, . . . , hp+q), Iεp−1,q =

(Ip−1 0

0 −εIq

),

andH = Dx + H1, I = −Dx − uD−1

x < u, . >pq,

where

H1w = (D−1x (w(Iεp−1,qu)t − u(Iεp−1,qw)t)u, for wt = (w1, . . . , wp+q−1),

and< u,h >pq=< u, Iεp−1,qh > .

The inner product on the right hand side is the normal inner product on Rp+q−1.

Proof. Let us first compute the multiplication type terms in the Cartan structure equa-tion (4.1.16) and find that ω(Dt)τ(Dx) = (A11, A21, . . . , Ap1, C11, . . . , Cq1)t and

ω(Dx)τ(Dt) = (

p+q−1∑

k=1

ukhk+1,−u1h1,−u2h1, . . . ,−up−1h1, εuph1, . . . , εup+q−1h1)t.

Now we use the Cartan structure equation applied to (Dx, Dt) as described in (4.1.16)and obtain the following equations of which the first group is concerned with the curva-ture part, and the second one with the free torsion part.

DxAi1 +Dtui−1 −p∑

k=2

Aikuk−1 +

q∑

k=1

Bikεuk+p−1 = 0, (4.2.2a)

for 1 < i ≤ p,DxAij +Ai1uj−1 −Aj1ui−1 = 0, for 1 < i, j ≤ p (4.2.2b)

DxB1j −Dtup+j−1 −p∑

k=2

uk−1Bkj −q∑

k=1

uk+p−1Dkj = 0, (4.2.2c)

for 1 ≤ j ≤ q,DxBij +Ai1up+j−1 + ui−1B1j = 0, (4.2.2d)

for 1 < i ≤ p, 1 ≤ j ≤ q,DxDij + Ci1up+j−1 − Cj1up+i−1 = 0, for 1 ≤ i, j ≤ q. (4.2.2e)

Dxh1 −p+q−1∑

k=1

ukhk+1 = 0 (4.2.3a)

Dxhi + ui−1h1 +Ai1 = 0 for 1 < i ≤ p (4.2.3b)

Dxhi − εui−1h1 + C(i−p)1 = 0 for p+ 1 ≤ i ≤ q + p (4.2.3c)

Now from (4.2.2d) and the fact that B1j = εCj1, we find that

Bij = −D−1x (Ai1up+j−1 + εui−1Cj1).

4.2. Integrable system in (p,q)-Orthogonal geometry 57

Also from Equations (4.2.2b) and (4.2.2e) we obtain

Aij = D−1x (Aj1ui−1 −Ai1uj−1), Dij = D−1

x (Cj1up+i−1 − Ci1up+j−1).

Hence plugging Aij , Bij and Dij into (4.2.2a) and (4.2.2c), one can derive the evolutionof ui−1 for i = 2, . . . p and up+j−1 for j = 1, . . . q respectively as below:

Dtui−1 = −DxAi1 +

p∑

k=2

Aikuk−1 −q∑

k=1

Bikεuk+p−1

= −DxAi1 +

p∑

k=2

uk−1D−1x (Ak1ui−1 −Ai1uk−1)

+

q∑

k=1

εuk+p−1D−1x (Ai1up+k−1 + εui−1Ck1), (4.2.4)

Dtup+j−1 = DxB1j −p∑

k=2

uk−1Bkj −q∑

k=1

uk+p−1Dkj

= εDxCj1 +

p∑

k=2

uk−1D−1x (Ak1up+j−1 + εuk−1Cj1)

−q∑

k=1

uk+p−1D−1x (Cj1up+k−1 − Ck1up+j−1). (4.2.5)

Hence the evolution equation can be rewritten as follows:

Dtu = −(Dx + H1)Iεp−1,qw = −HIεp−1,qw, (4.2.6)

where w = (A21, . . . , Ap1, C11, . . . , Cq1)t.Now (4.2.3a) of the torsion part gives us

h1 = D−1x (

p+q−1∑

k=1

ukhk+1). (4.2.7)

Using the inner product defined on Rp+q−1 we can write h1 as h1 = D−1x < u,h >

in which h = (h2, . . . hp+q)t. Also we can derive Ai1 for 1 < i ≤ p and C(i−p)1 for

p+ 1 ≤ i ≤ q+ p from the equations (4.2.3a) and (4.2.3a) of the torsion part as follows:

Ai1 = −Dxhi − ui−1h1 for 1 < i ≤ p,C(i−p)1 = εui−1h1 −Dxhi for p+ 1 ≤ i ≤ q + p.

These equations together with (4.2.7) give us the formulae for the vector w in termsof h = (h2, . . . hp+q)

t as below:

w = −Dxh− Iεp−1,quD−1x < u,h >,


from which we simply find that

Iεp−1,qw = −DxIεp−1,qh− uD−1

x < u,h >

= (−Dx − uD−1x < u, . >pq)(I

εp−1,qh)

= I(Iεp−1,qh)

Hence we can write the evolution (4.2.6) as follows:

Dtu = −(Dx + H1)Iεp−1,qw

= −HIεp−1,qw

= HI(Iεp−1,qh).

Let us replace h by the trivial symmetry ux, and compute the evolution equation(4.2.1). First we see that

I(Iεp−1,qux) = −Iεp−1,qu2x −1

2u < u,u > .

Then we find that

H1I(Iεp−1,qux) = u < ux,u > −ux < u,u > .

Hence we find the evolution equation:

Dtu = −Iεp−1,qu3x −1

2ux < u,u > −u < ux,u >

u < ux,u > −ux < u,u >

= −Iεp−1,qu3x −3

2ux < u,u > .

This is well known mKDV equation type.

Theorem 4.2.3. The operators H, I and R are Hamiltonian, symplectic and hereditaryoperator respectively.

Remark 4.2.4. Notice that the inner product < u,h >pq is nothing but the Killingform of two specific matrices in the Lie algebra op,q(R), namely

0 ut ut

−u 0 0εu 0 0

,

0 ht

ht

−h 0 0εh 0 0

,

where for instance ut = (u1, . . . , up−1) and ut = (up, . . . , up+q−1), likewise for ht

and ht.

Remark 4.2.5. One can compare the operator H presented here and the one derivedin [58] in the case of Riemannian geometry. In fact we do have that

Hw =∑

i<j

J ′ijuD−1 < J ′iju,w >pq,

in which J ′ij = JijIεp−1,q and (Jij)kl = δikδjl − δilδjk.

4.3. Integrable system in symplectic geometry 59

4.3 Integrable system in symplectic geometry

We consider the group G = Sp(n + 2) over the division ring of quaternions H of allinvertible matrix in GL(n+ 2,H) as defined in Chapter 1.

Now let H be the closed subgroup SP (1)×SP (n+ 1) of G. In fact SP (1) sits in thefirst diagonal entry of G and SP (n + 1) sits in the diagonal block left. Let us denotethe Lie algebra of G by g and that of H by h. As is known in the literature, M = G/His then a smooth manifold. In fact M is the quaternionic projective space HPn−1. For acomprehensive reference, see [63].

Similar to the situation in Riemannian manifold [58], given a curve in M, we knowits tangent vectors Dx and want to compute all possible Dt. Let ω be Cartan 1−formwith its values in the Lie algebra g. We make a specific choice of ω(Dx) and leave ω(Dt)as a general element of g. We see that the dimension of M is equal to the dimension ofg/h which is easily computed to be 4n. With x taken to be the arc length parameter,the dimension of the space of differential invariants in ω(Dx) describing the curve mustbe one less the dimension of the manifold, that is, 4n− 1, see Remark 4.2.1.

Instead of working on curvature and torsion separately, we can put them in onepicture. That means we can increase the dimension by one and put τ(Dx) in first rowof ω(Dx). Following this schema, we will have ω(Dt) as general element of sp(n+ 2).

Now let us choose a Cartan matrix ω(Dx) similar to that of the parallel coframe inRiemannian geometry with proper dimension counting as follows:

ω(Dx) =

0 1 0t

−1 u −ut

0 u 0

,

where ω(Dx) is taken as an element of sp(n+ 2).

Remark 4.3.1. Important notice should be taken into consideration that u is purelyimaginary, and u ∈ Hn−1 following the fact that ω(Dx) is in sp(n+ 2).

Remark 4.3.2. Other choices of coframe tend to destroy the scalar-vector character ofthe analysis and complicate matters tremendously, which is one of the main reasons whythe n-dimensional analysis using Frenet frames seems to be out of reach.

We see that this matrix is parametrized by 4n− 1 real parameters. Notice that herewe have taken the curvature and torsion part of Cartan form in one picture.

Now ω(Dt) must be a typical element of g which we write as follows:

ω(Dt) =

m11 m12 −mt

1

m21 m22 −mt2

m1 m2 M

.

In the Riemannian case, if we use a parallel frame and assume constant curvature κ,this can be taken zero and we still can derive all the geometric quantities. Therefore we


have taken the curvature equal to zero. Hence the Cartan structure equation evaluatedat the evolutionary vector fields Dx, Dt is as follows:

Dxω(Dt)−Dtω(Dx) + [ω(Dt), ω(Dx)] = 0.

Before we explore the Cartan structure equation, let us define some notation. Com-mutators of vectors and scalars are defined by

Cum2 := 〈u,m2〉 − 〈m2,u〉, Cum22 := um22 −m22u,

where the inner product 〈·, ·〉 is the Hermitian inner product. Right multiplication byscalar u on vector h and left multiplication by vector u on scalar h are defined respectivelyby

Ruh = hu, Luh = uh.

On the other hand, the anti-commutators on vector and scalar quantities are defined by

Auh = 〈u,h〉+ 〈h,u〉, Auh = uh+ hu.

Now we explicitly write the components of the Cartan structure equation. Amongthese equations, the four first equations are concerned with the curvature and the lastthree with the torsion. These equations lead to evolution of the scalar invariant u and thevector invariant u as combination of geometric operators applied on the proper torsionvariables of ω(Dt) according to the proposition below, in which we have defined H1 asthe operator acting on vectors by

H1h =(D−1x (hu t − uh t)

)u, (4.3.1)

where, for instance, hu t is the outer product of a vector and a covector, that is, a matrix.As we have seen before, this operator appear also in the case of Riemannian geometry ofsignature p, q, see Theorem 4.2.2. Hence, for instance, we can write Mu = H1m2. whenwe see in (4.3.2c) below.

Dxm11 −m12 −m21 = 0 (4.3.2a)

Dxm22 −Dtu− Cum22 + Cum2 +m12 +m21 = 0 (4.3.2b)

Dxm2 −Dtu +Rum2 + H1m2 − Lum22 + m1 = 0 (4.3.2c)

DxM−m2ut + um2

t = 0 (4.3.2d)

Dxm1 −m2 − um21 = 0 (4.3.2e)

Dxm12 +m11 −m22 +m12u− 〈m1,u〉 = 0 (4.3.2f)

Dxm21 +m11 −m22 − um21 + 〈u,m1〉 = 0 (4.3.2g)

Solving these equations we obtain

Theorem 4.3.3. The evolution of differential invariants can be written in the form(DtuDtu

)= HI

(m12 +m21

m1

)+ A

(m12 +m21

m1

), (4.3.3)


where

H =

(Dx − Cu Cu

−Lu Dx +Ru + H1

), A =

((2Dx − Cu)D−1

x 0−LuD

−1x I

),

and

I =

12Dx −

1

4Cu −

1

4AuD

−1x

12Au

12Cu + 1

2uD−1x Au

−12LuD

−1x

12Au − 1

2Lu Dx + 12LuD

−1x Au

.

Proof. We start with (4.3.2b) and (4.3.2c) to find the evolution

(DtuDtu

)as presented

below.

(DtuDtu

)=

(Dx − Cu Cu

−Lu Dx + Ru + H1

)(m22

m2

)+

(m12 +m21

m1

)

= H

(m22

m2

)+

(m12 +m21

m1

). (4.3.4)

Now from (4.3.2f) and (4.3.2g) which are actually the torsion part, we obtain m11−m22

in terms of m12,m21 and m1. Hence in the evolution (4.3.4) we subtract and add m11

in a proper way.

(DtuDtu

)= H

(m22 −m11

m2

)+ H

(m11

0

)+

(m12 +m21

m1

)

= H

(m22 −m11

m2

)+

(Dx − Cu 0−Lu 0

)(m11

m1

)+

(m12 +m21

m1

)

(4.3.5)

If we subtract (4.3.2f) from (4.3.2g), we deduce that

Dx(m21 −m12) = um21 +m12u−Aum1.

Using the fact that 12Au(m12 +m21) = m12u+ um21 we obtain that

m21 −m12 = D−1x (

1

2Au(m12 +m21)−Aum1). (4.3.6)

Taking the difference of (4.3.2f) and (4.3.2g), we derive an expression for m22 −m11:

m22 −m11 =1

2Dx(m12 +m21) +

1

2Cum1 +

1

2(m12u− um21).

Using the fact that m12u−um21 = −12Cu(m12 +m21)+u(m12−m21) and (4.3.6) enable

us to convert last equation to the following one.


m22 −m11 =1

2Dx(m12 +m21) +

1

2Cum1

− 1

4Cu(m12 +m21)− 1

2uD−1

x (1

2Au(m12 +m21)−Aum1).

On the other hand, we can write m21 as below.

m21 =m21 −m12

2+m21 +m12

2

=1

2D−1x (

1

2Au(m12 +m21)−Aum1) +

m21 +m12

2.

Hence by using (4.3.2e), we can express

(m22 −m11

m2

)as follows.

(m22 −m11

m2

)

=

(12Dx −

1

4Cu − 1

2uD−1 1

2Au12Cu + 1

2uD−1x Au

− 12LuD

−1x

12Au − 1

2Lu Dx + 12LuD

−1x Au

)(m12 +m21

m1

)

= I

(m12 +m21

m1

). (4.3.7)

Now from (4.3.2f) we obtain that

(m11

m1

)=

(D−1x 00 I

)(m12 +m21

m1

). (4.3.8)

The last step is to substitute the equations (4.3.7) and (4.3.8) in the evolution (4.3.5)to prove the statement of the theorem:

(DtuDtu

)= HI

(m12 +m21

m1

)

+

(Dx − Cu 0−Lu 0

)(D−1x 00 I

)(m12 +m21

m1

)+

(m12 +m21

m1

)

= HI

(m12 +m21

m1

)

+

(((Dx − Cu)D−1

x 0−LuD

−1x 0

)+

(DxD

−1x 0

0 I

))(m12 +m21

m1

)

= HI

(m12 +m21

m1

)+ A

(m12 +m21

m1

).


Remark 4.3.4. Similar to Remark 4.2.4 in the Riemannian geometry of signature p, q,if we apply the Killing form formula (1.4.1) in symplectic geometry, then we can see that

m21 −m21 = D−1x (

1

2Au(m12 +m21)−Aum1)

= D−1x K(

0 0 00 u −ut

0 u 0

,

0 0 00 m12 +m21 −mt

1

0 m1 0

).

Of course up to some constant coefficient.

Let us put

(hh

)= A

(m12 +m21

m1

). Then we obtain

(m12 +m21

m1

)= A−1

(hh

), A−1 =

(Dx(2Dx − Cu)−1 0Lu(2Dx − Cu)−1 I

).

Hence the evolution in the theorem takes the following form:(DtuDtu

)= R

(hh

)+

(hh

), R = HIA−1. (4.3.9)

If we make the specialization

(hh

)=

(u1

u1

), where u1,u1 are the derivatives of u

and u with respect to x, respectively, then we obtain the noncommutative evolutionequations:

ut =1

4u3 +

3

8(−uu1u− uu2 + u2u) +

3

2〈u,u〉u1 + 〈u,u1〉u+ 1

2u〈u,u1〉

+2u〈u1,u〉 − 12〈u1,u〉u+

3

2Cuu2,

ut = u3 +3

2u2u+

3

4u1(u1 +

1

2u2 + 2〈u,u〉).

(4.3.10)

Remark 4.3.5. It is remarkable to see how the procedure will go if we choose

ω(Dx) =

σ 1 0t

−1 u −ut

0 u 0

, ω(Dt) =

m11 m12 −mt

1

m21 m22 −mt2

m1 m2 M

.

In this case, the Cartan structure equation on the manifold with zero constant curvatureleads to the following evolution:

DtσDtuDtu

=

Dx −Cσ 0 0

0 D − Cu Cu

0 −Lu Dx + H + Ru

m11 −m22

m22 −m11

m2

+

(D − Cσ)m22

(D − Cu)m11

−Lum11

+

−m12 −m21

m12 +m21

m1

.


Then, with little effort, we realize that we can not solve these equations. Therefore thedimension of the gauge matrix is a criterium for the Cartan structure to be solved. Theproblem remains why if we put u in entry (2, 2) of the gauge matrix instead of the entry(1, 1) of this matrix, we succeed to find all the integrable properties, but not in entry(1, 1). In other words, the choice of the frame work is so important that if we change iteven very slightly, we will get either nasty extra nonlocal expression which we can notsolve or trivial equations. Notice that in this sense, facing such problems, the situationwith the Lax method is the same.

Now we give the specific version of Definition 2.2.6 for the pairing in the currentgeometry.

Definition 4.3.1. The pairing between

(hh

)and

(gg

)is defined by

〈(hh

),

(gg

)〉 =

∫K(σ

(hh

), σ

(gg

)),

in which σ is a section of g/h subject to the zero constant curvature condition into thesubspace of the Lie algebra g generated just as the Cartan matrix ω(Dx). For instance,one can take

σ

(hh

)=

0 0 0

0 h −hT

0 h 0

.

The adjoint of the operator P has been defined in (2.2.8). In the current geometry,this definition reduces to:

〈(hh

), P

(gg

)〉 = 〈P ∗

(hh

),

(gg

)〉.

Since the pairing is nondegenerate, P ∗ is well-defined.

Example 4.3.1. We compute the adjoint operator of the operator A. We see that

A∗ =

(D−1x (2Dx − Cu) D−1

x Cu

0 I

). This can be done in a few steps. Let us put A1 =

(Cu 00 0

). Then we compute its adjoint as follows.

〈(hh

),A1

(gg

)〉 =

∫K(

0 0 0

0 h −hT

0 h 0

,

0 0 00 Cug 00 0 0

)

=

∫K(

(h −h

T

h 0

),

(Cug 0

0 0

)) =

∫K(

(h −h

T

h 0

),[(u 0

0 0

),

(g 00 0

)])

= −∫K(

(g 00 0

),[(u 0

0 0

),

(h −h

T

h 0

)]) = −

∫K(

(g 00 0

),

(Cuh 0

0 0

))

= −∫K(

(g −gT

g 0

),

(Cuh 0

0 0

)) = −〈

(gg

),A1

(hh

)〉,


in which we have used Lemmas 1.4.2 and 1.4.3 for some of the equalities. Thus A∗1 = −A1.

Now let us put A2 =

(0 0Lu 0

). Hence we have that

〈(hh

),A1

(gg

)〉 =

∫K(

0 0 0

0 h −hT

0 h 0

,

0 0 0

0 0 −Lugt

0 Lug 0

)

=

∫K(

(h −h

T

h 0

),

(0 −Lug

t

Lug 0

))

=

∫K(

(h −h

T

h 0

),[(0 −uT

u 0

),

(g 00 0

)])

= −∫K(

(g 00 0

),[(0 −uT

u 0

),

(h −h

T

h 0

)])

= −∫K(

(g 00 0

),

(−Cuh −uh

T

uh hut − uht

))

= −∫K(

(g 00 0

),

(−Cuh 0

0 0

))

=

∫K(

(g −gt

g 0

),

(Cuh 0

0 0

))

= 〈(gg

),A∗2

(hh

)〉,

where A∗2 =

(0 Cu

0 0

). Here again we have used the Lemmas 1.4.2 and 1.4.3 and the fact

that the Killing form is invariant under adjoint action. Now it is clear that (D−1x )∗ =

−D−1x . Thus we have that

(CuD

−1x 0

0 0

)∗= (A1D

−1x )∗ = (D−1

x )∗A∗1 = D−1x A1 =

(D−1x Cu 00 0

).

Similarly we do have that

(0 0

LuD−1x 0

)∗= (A2D

−1x )∗ = (D−1

x )∗A∗2 =

(0 −D−1

x Cu

0 0

).

Hence it should be clear that

A∗ =

(D−1x (2Dx −Cu) D−1

x Cu

0 I

).

In the following proposition we show that there is a meaningful link between theoperator H and A.


Proposition 4.3.6. The following equality holds

AH = HA∗.

Proof. It is simple multiplication of two matrix operator as follows.

AH− HA∗ =

((2Dx − Cu)D−1

x 0−LuD

−1x I

)(Dx − Cu Cu

−Lu Dx +Ru + H1

)

−(Dx − Cu Cu

−Lu Dx +Ru + H1

)(D−1x (2Dx − Cu) D−1

x Cu

0 I

)

=

(2Dx − 3Cu +CuD

−1x Cu (2Dx − Cu)D−1

x Cu

−2Lu + LuD−1x Cu −LuD

−1x Cu +Dx +Ru + H1

)

−(

2Dx − 3Cu + CuD−1x Cu 2Cu − CuD−1

x Cu

−2Lu + LuD−1x Cu −LuD

−1x Cu +Dx +Ru + H1

)

= 0

Corollary 4.3.7. The operator H is antisymmetric as well as HA∗, that is H∗ = −H

and (HA∗)∗ = −HA∗. Furthermore the operator I itself is also antisymmetric.

Proof. We decompose the operator H into two operators and seperately prove that each

of them is antisymmetric. Let us denote the operator

(0 00 Ru

)by P1. Then we can

compute its adjoint as follows:

〈(hh

), P1

(gg

)〉 =

∫K(

(h −h

T

h 0

),

(0 −guT

gu 0

))

=

∫K(

(h −h

T

h 0

),[(0 −gT

g 0

),

(u 00 0

)])

=

∫K(

(0 −gT

g 0

),[(u 0

0 0

),

(h −h

T

h 0

)])

=

∫K(

(0 −gT

g 0

),

(Cuh hu

t

−hu 0

)) =

∫K(

(0 −gT

g 0

),

(0 hu

t

−hu 0

))

=

∫K(

(g −gT

g 0

),

(0 hu

t

−hu 0

)) = 〈

(gg

),−P1

(hh

)〉.

This indeed shows that P ∗1 = −P1. Now put P2 =

(0 00 H1

). Then using the notations

defined in section 4.4 and Lemmas 4.4.1, we see that


〈(hh

), P2

(gg

)〉

=

∫K(

0 0 0

0 h −ht

0 h 0

,

0 0 0

0 0 −D−1x (gut − ugt)u

t

0 D−1x (gut − ugt)u 0

)

= −∫K(

0 0 0

0 h −ht

0 h 0

,

[

0 0 00 u −ut

0 u 0

, D−1

x π0

[

0 0 00 u −ut

0 u 0

,

0 0 00 g −gt

0 g 0

]]

)

=

∫K([

0 0 00 u −ut

0 u 0

,

0 0 0

0 h −ht

0 h 0

],

D−1x π0

[

0 0 00 u −ut

0 u 0

,

0 0 00 g −gt

0 g 0

])

= −∫K(D−1

x

[

0 0 00 u −ut

0 u 0

,

0 0 0

0 h −ht

0 h 0

],

π0

[

0 0 00 u −ut

0 u 0

,

0 0 00 g −gt

0 g 0

])

= −∫K(D−1

x π0

[

0 0 00 u −ut

0 u 0

,

0 0 0

0 h −ht

0 h 0

],

[

0 0 00 u −ut

0 u 0

,

0 0 00 g −gt

0 g 0

])

= −∫K([

0 0 00 u −ut

0 u 0

, D−1

x π0

[

0 0 00 u −ut

0 u 0

,

0 0 0

0 h −ht

0 h 0

]],

0 0 00 g −gt

0 g 0

)

= −∫K(

0 0 0

0 0 −D−1x (hut − uh

t)u

t

0 D−1x (hut − uh

t)u 0

,

0 0 00 g −gt

0 g 0

).


The last expression is nothing but 〈(gg

),−P2

(hh

)〉. Hence P ∗2 = −P2. Using the com-

putation in example 4.3.1 together with the two last identities shows that the operatorH is indeed antisymmetric.

In order to prove that the operator I is antisymmetric, having proved part of that,

let us first denote P3 =

(0 00 LuD

−1x Au

). Then we can compute its adjoint as follows:

〈(hh

), P2

(gg

)〉 =

∫K(

(h −h

t

h 0

),

(0 −LuD

−1x Aug

t

LuD−1x Aug 0

))

=

∫K(

(h −h

t

h 0

),

(0 −utD−1

x AuguD−1

x Aug 0

))

=

∫K(

(h −h

t

h 0

),

(0 −ut

u 0

)).D−1

x Aug

= −∫D−1x K(

(h −h

t

h 0

),

(0 −ut

u 0

)).Aug

=

∫(D−1

x Auh).(Aug)

= −∫

(D−1x Auh).K(

(0 −ut

u 0

),

(g −gt

g 0

))

= −∫K(

(0 −utD−1

x AuhuD−1

x Auh 0

),

(g −gt

g 0

))

= 〈(gg

),−P2

(hh

)〉.

Therefore P ∗2 = −P2. Now it should be clear that I is antisymmetric.

Theorem 4.3.8. The operator N = A is indeed a Nijenhuis operator. That is, theNijenhuis tensor vanishes.

Proof. According to Definition 2.5.1, we have to prove that the identity (4.3.11) holds:

DN [Nψ](ϕ) −DN [Nϕ](ψ) +N(DN [ϕ](ψ) −DN [ψ](ϕ)) = 0 (4.3.11)

for any pair of vector fields ϕ,ψ ∈ X(M). In our specific case, the ϕ and ψ take the

following form: ϕ =

(pp

), ψ =

(qq

).

Here we are given Nϕ =

((2Dx −Cu)D−1

x p−LuD

−1x p+ p

). Now we compute the Frechet deriva-

tive of N as follows and use it later on with different arguments

DN [ψ](ϕ) =

(−CqD−1

x 0−LqD

−1x 0

)(pp

)=

(−CqD−1

x p−LqD

−1x p

).


Now we see that

DN [Nψ](ϕ) −DN [Nϕ](ψ)

=

(−C(2Dx−Cu)D−1

x qD−1x p

−L−LuD−1x q+qD

−1x p

)−(−C(2Dx−Cu)D−1

x pD−1x q

−L−LuD−1x p+pD

−1x q

)

=

(−C(2Dx−Cu)D−1

x qD−1x p+ C(2Dx−Cu)D−1

x pD−1x q


−1x p+ L−LuD

−1x p+pD

−1x q

).

On the other hand we obtain

DN [ϕ](ψ) −DN [ψ](ϕ) =

(−CpD−1

x q−LpD

−1x q

)−(−CqD−1

x p−LqD

−1x p

)

=

(−CpD−1

x q + CqD−1x p

−LpD−1x q + LqD

−1x p

).

Hence we get that

N(DN [ϕ](ψ) −DN [ψ](ϕ)

)

=

(2Dx − Cu)D−1

x

(− CpD−1

x q + CqD−1x p)

−LuD−1x

(− CpD−1

x q + CqD−1x p)

+ (−LpD−1x q + LqD

−1x p)

Now first we prove that the scalar part or the first component of expression

DN [Nψ](ϕ) −DN [Nϕ](ψ) +N(DN [ϕ](ψ) −DN [ψ](ϕ))

in (4.3.11) vanishes:

−C(2Dx−Cu)D−1x qD

−1x p+ C(2Dx−Cu)D−1

x pD−1x q

+(2Dx − Cu)D−1x

(− CpD−1

x q + CqD−1x p)

= −C(2Dx−Cu)D−1x qD

−1x p+ C(2Dx−Cu)D−1

x pD−1x q

+2(−CpD−1x q + CqD

−1x p)−CuD−1

x

(− CpD−1

x q + CqD−1x p)

= −2CqD−1x p+ CCuD−1

x qD−1x p+ 2CpD

−1x q − CCuD−1

x pD−1x q

+2(−CpD−1x q + CqD

−1x p)−CuD−1

x

(− CpD−1

x q + CqD−1x p)

= +CCuD−1x qD

−1x p−CCuD−1

x pD−1x q − CuD−1

x

(− CpD−1

x q + CqD−1x p)

Jacobi id= CuCD−1

x qD−1x p− CuD−1

x

(− CpD−1

x q + CqD−1x p)

= CuD−1x DxCD−1

x qD−1x p− CuD−1

x

(− CpD−1

x q + CqD−1x p)

= CuD−1x

(CqD

−1x p− CpD−1

x q)− CuD−1

x

(− CpD−1

x q + CqD−1x p)

= 0,


We have used once the Jacobi identity as

+CCuD−1x qD

−1x p− CCuD−1

x pD−1x q = CuCD−1

x qD−1x p.

We manipulate the second component (vector part) as follows:


−1x p+ L−LuD

−1x p+pD

−1x q

−LuD−1x

(− CpD−1

x q + CqD−1x p)

+ (−LpD−1x q + LqD

−1x p)

= −L−LuD−1x qD

−1x p+ L−LuD

−1x pD

−1x q − LuD

−1x

(− CpD−1

x q + CqD−1x p)

= Lu(CD−1x qD

−1x p)− LuD

−1x

(− CpD−1

x q + CqD−1x p)

= LuD−1x Dx(CD−1

x qD−1x p)− LuD

−1x

(− CpD−1

x q + CqD−1x p)

= LuD−1x

(CqD

−1x p−CpD−1

x q)− LuD

−1x

(− CpD−1

x q + CqD−1x p)

= 0,

In which again we have used the Jacobi identity this time as follows:

−L−LuD−1x qD

−1x p+ L−LuD

−1x pD

−1x q = Lu(CD−1

x qD−1x p).

This is nothing but the Jacobi identity for the following three elements of sp(n+ 2) :

0 0 00 0 −ut

0 u 0

,

0 0 00 D−1

x p 00 0 0

and

0 0 00 D−1

x q 00 0 0

.

Remark 4.3.9. Notice that operator A is not recursion operator, consequently it is notinvariant under the flow.

Remark 4.3.10. Such an operator A appeared as ”Starting operator” in Fokas-Santini’spapers [60] and [21] where they give the recursion operator and bi-Hamiltonian structurein multidimensional equations.

Remark 4.3.11. Since N = A is invertible, so by [14, proposition 3.2], the operatorA−1 itself is Nijenhuis operator.

In the light of Theorem 4.3.8, the operator R can be written as

R = HIA−1

= (HA∗)(A−1∗IA−1).

This decomposition of the operator R is the key to find Hamiltonian and symplecticoperator. In fact the main result of this chapter is the next theorem presenting thesefacts. In the next section we express these operator in terms of the Lie bracket, Killingform and some projections and in the Chapter 6 we will prove the theorem in detail.

4.4. Geometric operators in the form of Lie algebraic objects... 71

Theorem 4.3.12. The operators H and HA∗ are Hamiltonian operators and the operatorA−1∗IA−1 is symplectic.

Remark 4.3.13. Theorems 4.3.8 and 4.3.12 show that the manifold M = G/H is en-dowed with a so called ‘Poisson-Nijenhuis structure’ as is defined in [48]. We will notgo further in this direction. For more information, the reader is referred to the works ofMagri, as presented in for instance [47] and [38].

Remark 4.3.14. Notice that we can decompose the entry in the first column and firstrow of H. One can write

(Dx +Ru)(2Dx − Cu)−1(Dx − Lu)

= (Dx +Au −Cu

2)(2Dx − Cu)−1(Dx −

Au + Cu2

)

= (Au2

+2Dx − Cu

2)(2Dx − Cu)−1(−Au

2+

2Dx − Cu2

)

=1

2Dx −

1

2Au(2Dx − Cu)−1 1

2Au −

1

4Cu

=1

2Dx −

1

2uD−1

x1

2Au −

1

4Cu

Remember that the operator acts on purely imaginary arguments. Then the last equalityfollows from (2Dx − Cu)−1 = 1

2(Dx − 12Cu)−1 and (Dx − 1

2Cu)−1 acting on real valuedfunction gives D−1

x acting on the same function, as well as Au acting on the real functionsgives twice acting u on the same function. Indeed the result of action Au on a imaginaryvalued function is real function. Therefore the operator (2Dx−Cu)−1 = 1

2 (Dx− 12Cu)−1

acting on this real function would yield 12D−1x action on the same real function, for

assume that f is the real function and (Dx − 12Cu)−1f = g. Then f = (Dx − 1

2Cu)g.Since Cug is imaginary, hence we do have that Dxg0 = f where g0 is the real part ofg. Thus (Dx − 1

2Cu)−1f = g0. One can see this from the expansion of (Dx − 12Cu)−1 as

well. Indeed (Dx − 12Cu)−1 = D−1

x + 12D−1x CuD

−1x + .... Hence CuD

−1x f = 0. Therefore

we have that (Dx − 12Cu)−1f = D−1

x f.Also notice that Au acting on the real function f is equal to 2uf.This decomposition indicates that if we put u = 0, then geometric operators, Hamil-

tonian, symplectic and recursion operators, will reduce to the ones that appeared in[30].

4.4 Geometric operators in the form of Lie bracket, Killing

form and projections

In the method we are using, the only tools we have are the Lie algebra and the Cartangeometry, hence we expect to be able to write the geometric operators H and I in termsof the Lie bracket, the Killing form and proper projections. In this sense for instancesee Remark 4.3.4. Also as one can see in the proof of Theorem 4.3.8, we could use the


Jacobi identity for the matrices in the Lie algebra sp(n+2) which in fact are projectionsof bigger matrices.

Let us define the projections π0 and π1 as follows:

π0

m11 0 0

0 m22 −mt2

0 m2 M

=

0 0 00 0 00 0 M

,

and

π1

m11 0 0

0 m22 −mt2

0 m2 M

=

0 0 00 m22 −mt

2

0 m2 0

.

In the following lemma, we give the Lie algebraic form of the operator H1 defined asin (4.3.1).

Lemma 4.4.1.

[

0 0 00 u −ut

0 u 0

, D−1

x π0

[

0 0 00 u −ut

0 u 0

,

0 0 00 m22 −mt

2

0 m2 0

]]

=

0 0 00 0 −utD−1(−umt

2 + m2ut)

0 −D−1(−umt2 + m2u

t)u 0

.

Proof. The proof just follows from computing the Lie bracket of the elements of the Liealgebra sp(n+ 2).

Let u and m2 be the projection of ω(Dx) and ω(Dt) over the Lie subalgebra h,respectively as well as a and m1, the projections of ω(Dx) and ω(Dt) over the vectorspace g/h which itself is indeed the dual orthogonal of h with respect to the Killing form.In other words

u =

0 0 00 u −ut

0 u 0

, a =

0 1 0−1 0 00 0 0

,

and

m2 =

m11 0 0

0 m22 −mt2

0 m2 M

, m1 =

0 m12 −mt1

m21 0 0m1 0 0

.

Now let us define the projections ρ0 and ρ1 on the diagonal and offdiagonal of theimage of π1 as follows:

ρ1

0 0 00 m22 −mt

2

0 m2 0

=

0 0 00 m22 00 0 0


and

ρ0

0 0 00 m22 −mt

2

0 m2 0

=

0 0 00 0 −mt

2

0 m2 0

.

We give some identities which in fact are interactions of projections ρ0, ρ1 and π1.

Proposition 4.4.2. For every matrix q in the image of π1, we have that

1. (12ρ1 + ρ0)π1ad2

aq = −q.

2. π1adaaduadaq = −aduρ1q − π1adρ0uρ0q,

The following theorem describes how we can express the geometric operator in termsof Lie algebraic notions, such as Killing form, adjoint representation and the projections.

Theorem 4.4.3. The evolution of the u following the Cartan structure equation onM = G/H can be expressed as

ut = HIm0 + Am0,

in which the Lie algebra form I of geometric operator I and A of Nijenhuis operator A

appears as

H = Dx − π1adu − aduD−1x π0adu,

I = −1

2uD−1

x K(u, .)− (1


1

2ρ1 + ρ0),

A = ρ0 + 2ρ1 − aduD−1x ρ1.

Proof. From the curvature part or in fact the equations (4.3.2b),(4.3.2c) and (4.3.2d)and the previous lemma, we simply find that

ut = H(π1m2) + m0, m0 = π1adam1,

Now the torsion part gives the following matrix equation:

ada(m2) = (Dx − adu)m1. (4.4.2)

Since ad2a 6= λI for any λ ∈ R, we can not solve equation (4.4.2) in the usual way.

Therefore the existence of the Nijenhuis operator A plays a crucial rule in the symplecticgeometry. Notice that in the Riemannian case we do have ad2

a = −I.


In order to get rid of this difficulty, we do as follows:

ut = H

0 0 00 m22 −m11 −mt

2

0 m2 0

+ H

0 0 00 m11 00 0 0

+ m0

=

0 0 00 m22 −m11 −mt

2

0 m2 0

+ (Dx − π1adu)

0 0 00 m11 00 0 0

+ m0

= H

0 0 00 m22 −m11 −mt

2

0 m2 0

+

((2Dx − adu)D−1

x ρ1 + ρ0

)m0

= H

0 0 00 m22 −m11 −mt

2

0 m2 0

+ Am0. (4.4.3)

Now the matrix

0 0 00 m22 −m11 −mt

2

0 m2 0

can be expressed in terms of m2 and consequencely in terms of m1 using the identity(4.4.2) as follows:

0 0 00 m22 −m11 −mt

2

0 m2 0

= −(

1

2ρ1 + ρ0)π1ad2

am2

= −(1

2ρ1 + ρ0)π1ada(Dx − adu)m1. (4.4.4)

Since m21 −m12 = D−1x K(u, m0), thus the matrix m1 can be written in terms of m0 as

follows.

m1 =

0m12 −m21

20

m21 −m12

20 0

0 0 0

+

0m12 +m21

2−mt

1

m21 +m12

20 0

m1 0 0

= −1

2aD−1

x K(u, m0) + ada(1

2ρ1 + ρ0)m0.


Hence we identify the operator I as follows:

0 0 00 m22 −m11 −mt

2

0 m2 0

= −(

1

2ρ1 + ρ0)π1ada(Dx − adu)m1

= −(1

2ρ1 + ρ0)π1ada(Dx − adu)

(− 1

2aD−1

x K(u, m0) + ada(1

2ρ1 + ρ0)m0

)

= −1

2(1

2ρ1 + ρ0)π1ada(adua)D−1

x K(u, m0)

−(1

2ρ1 + ρ0)π1adA(Dx − adu)adA(

1

2ρ1 + ρ0)m0

= +1

2(1

2ρ1 + ρ0)π1(ad2

au)D−1x K(u, m0)

−(1


1

2ρ1 + ρ0)m0

= −1

2uD−1

x K(u, m0)− (1


1

2ρ1 + ρ0)m0

= −1

2uD−1

x K(u, m0)− (1

2ρ1 + ρ0)π1adA(Dx − adu)adAπ1(

1

2ρ1 + ρ0)m0

= I,

in which we have used the fact that ( 12ρ1 + ρ0)π1(ad2

au) = −u by applying the previouslemma. In the last line, we add π1 at end to have symmetrized expression, since it wouldnot change anything.

Thus replacing the last equation into the evolution (4.4.3), we get that

ut = HIm0 + Am0.

Remark 4.4.4. Similar results have been derived in the general case of Riemanniansymmetric spaces in [2]. The Author has given a definition of parallel frame based onthe choice of a. It seems there is a gap if we compare two result. Indeed if we choose aas we have chosen here, then ω(Dx) will be determined according to his set up and thatis not what we have. This needs further research.

Remark 4.4.5. This is exactly the Poisson operator in [69, 1.13] which in generalis defined on Hermitian symmetric spaces. See also [70, 68]. The equation (4.4.2)corresponds to the λ coefficient of Lax representation, see Chapter 5.

Remark 4.4.6. For the related topics and similar construction, see [36, 74, 75, 61],and also [42]. For instance in [43], the authors apply a method of Sym and Pohlmeyer,[66, 54], to the Fordy-Kulish generalized nonlinear Schrodinger systems associated withHermitian symmetric spaces [22]. Furthermore in [43], the authors gives also an appro-priate specialization in the context of the symmetric space SO(p + 2)/SO(p) × SO(2)


which yields evolution equations for curves in Rp+1 and Sp, with natural curvatures sat-isfying a generalized mKDV system. In fact this example is related to the constructionsof Doliwa and Santini and illuminates certain features of the latter.

Remark 4.4.7. The rule of A operator is very much similar to that of interwiningoperator Z defined as in [43, p. 166]. That is

Z(Y ) = adA(AdφY )

for Y as Sym-Pohlmeyer field and is proved that

ZRY = (R − λ)ZY, RY = −P([T, asY ]), RY = (as − adQa−1s adQ)adA

in which P is defined asP(B) = a−1

s [Q,Bm] +Bm.There R and P are called respectively geometric recursion operator and renormalizationoperator. See the next section for the Lie algebraic form of recursion operator H whichis exactly of the form R.

Chapter 5

Lax representation of an integrable

system

5.1 Theory of Lax method

It is well known that most of the integrable nonlinear partial differential equations,

ut = F (t, x, u, ux, . . . , unx),

admit a Lax representation,Lt = [A,L],

in which L,A are linear differential operators. Originally, the subject is due to thediscovery by Gardner, Greene, Miura and Kruskal in [25] that the eigenvalues of theSchrodinger operator are integrals of the Korteweg-de Vries equation. At the sametime, Lax presented a general principle for associating nonlinear evolution equationswith linear operators so that the eigenvalues of the linear operator are integrals of thenonlinear equation, see [44]. To have a simple picture of the Lax construction, let B besome Hilbert space of functions, for instance a space of smooth functions, chosen so thatthe function u(t) lies in B. We recall from functional analysis that a Hilbert space is avector space with an inner product so that there can be defined a norm on the spaceand in addition it is complete. Suppose that to each function u ∈ B, we can associatea self adjoint operator L = Lu over some Hilbert space,

u→ Lu, (5.1.1)

with the following property: If u changes with t subject to the equation

ut = K(u),

the operators L(t), which also change with t, remain unitary equivalent. If this is the case,then eigenvalues of Lu constitute a set of integrals for the equation under consideration.The unitary equivalence of the operators L(t), mentioned above, means that there is aone-parameter family of unitary operators U(t) such that

U(t)−1L(t)U(t), (5.1.2)

is independent of t. This fact can be expressed by setting the t derivative of (5.1.2) equalto zero:

−U−1UtU−1LU + U−1LtU + U−1LUt = 0. (5.1.3)

77

78 Chapter 5. Lax representation of an integrable system

As a matter of fact, Stone’s theorem [65], says that a one-parameter family of unitaryoperators on a Hilbert space satisfies a differential equation of the form

Ut = AU (5.1.4)

where A(t) is an antisymmetric operator. Conversely, every solution of (5.1.4) withA∗ = −A is a one-parameter family of unitary operators.

Now substituting (5.1.4) into (5.1.3) we obtain

Lt = AL− LA = [A,L]. (5.1.5)

If u satisfies the equation ut = K(u), then Lt can be expressed in terms of u, and allthat remains to verify is that equation (5.1.5) has an antisymmetric solution A. Thatmeans that the unitary equivalence of the operators L(t) is nothing but finding theantisymmetric solution A for (5.1.5), which is called the Lax representation.

The drawback of this method is that it requires one to guess correctly the relation(5.1.1) between the function u and the operator L. What we will do later on is that wetake the operator L depending to specific function u, then we will proceed by choosingproper operator, or ansatz, A, so that the Lax representation (5.1.5) hold from whichwe find the evolution equation.

To put some light on the subject, let us consider the Schrodinger operator

L = D2 +1

6u.

Then Lt is multiplication by1

6ut. Thus we have to find an antisymmetric operator A

whose commutator with L is multiplication. To obtain a nontrivial result, let us chooseA1 = D3 + aD +Da. The coefficient a is to be chosen. Now we have

[A1, L] = (1

2ux − 4ax)D2 + (

1

2uxx − 4axx)D +

1

6uxxx − axxx +

1

3aux.

Clearly, to eliminate all but the zero order terms we have to choose

a =1

8u.

With this choice, [A1, L] is multiplication by

1

24(uxxx + uux).

Setting A = 24A1, we verify that

[A,L] = K(u),

where K(u) = ut is KDV equation

ut + uxxx + uux = 0.

5.1. Theory of Lax method 79

Later on, Drinfel’d and Sokolov in [16], showed that given the operator

L = Dx + aλ+ q(x, t), (5.1.6)

where λ is the so called spectral parameter, q belongs to a Lie algebras g, and a is aconstant element of g, one can construct an operator A =

∑ni=0 piλ

i such that the Laxrepresentation Lt = [A,L] is equivalent to the evolution equation of the form

qt = F (q, qx, qxx, . . .). (5.1.7)

The method given in [16], which is based on bringing the operator L to diagonal form,allows one to constructively build, in addition to the A operator, the higher symmetriesand integrals (conservation laws) for (5.1.7). The construction of [16], can be generalizedto the case of L operators of a more general form:

L = Dx + aλn+1 +

n∑

i=−mqi(x, t)λ

i.

In this chapter, we set up the method of Drinfel’d and Sokolov developed in [16] and[29]. The proofs here will be based on normal form theory for filtered Lie algebras. Thenwe will specialize to the case of the symplectic Lie algebra.

Definition 5.1.1. The Lie algebra g =⊕

i∈Z gi is called Z-graded if gi are vector spacesuch that [gi, gj] ⊂ gi+j where i, j ∈ Z. It is clear that g0 is a subalgebra of g.

Let us consider a Z−graded Lie algebra g =⊕

i∈Z gi and an Lax operator of theform

L = D + α+ q(x, t), q ∈ g0. (5.1.8)

Here α ∈ g1 satisfy the condition

g = ker(adα)⊕ im(adα). (5.1.9)

Indeed the condition 5.1.9 is an assumtion on α.Now let us define Fn =

∏∞i=n gi where gi = g−i. Then Fn is a filtration. Let us denote

α by α−1 which by definition must be in F−1.

Lemma 5.1.1. Any element Z of the Lie algebra can be written as

Z = [α−1, X] + Y, (5.1.10)

where Y ∈ ker(adα−1) and X ∈ im(adα−1). Moreover if Z ∈ Fn then Y ∈ Fn andX ∈ Fn+1.


Proof. By decomposition there exist Z (1) ∈ ker(adα−1) and Z(2) ∈ g such that Z =Z(1) + [α−1, Z

(2)]. Applying decomposition for Z (2), we find Z(2) = X(1) + X(2) withX(1) ∈ ker(adα−1) and X(2) ∈ im(adα−1). Hence [α−1, Z

(2)] = [α−1, X(2)]. Now we just

take X = X(2).Now suppose Z ∈ Fn. Then [α−1, X], Y ∈ Fn. Assume that X 6= 0 and k is lowest

order that X ∈ Fk and Xk ∈ gk is lowest nonzero term of it. Then [α−1, Xk] ∈ gk−1∩Fn.Now if k − 1 < n, Then it follows that [α−1, Xk] must be zero, i.e Xk ∈ ker(adα−1).On the other hand X ∈ im(adα−1) and hence Xk ∈ im(adα−1). Thus Xk ∈ ker(adα−1) ∩im(adα−1) = 0 which is in contradiction with the assumption that Xk is lowest nonzeroterm of X. Hence we conclude that k − 1 ≥ n, and so X ∈ Fk ⊆ Fn+1.

Lemma 5.1.2. Let us define the following maps

Fk/Fk+1

αk−1→ Fk−1/Fkαk−1−1→ Fk−2/Fk−1.

Now we have that

Fk−1/Fk = ker(αk−1−1 )⊕ im(αk−1).

Proof. We see that

ker(αk−1−1 ) = Z + Fk|Z ∈ Fk−1, [α−1, Z] ∈ Fk−1,

and if Zk−1 is first term of Z, then that means that [Zk−1, α−1] = 0. Also we have that

im(αk−1) = Z + Fk|∃X ∈ Fk such that Z − [α−1, X] ∈ Fk.

This means that Zk−1 = [α−1, Xk]. Now suppose Z ∈ Fn, Then by the general decom-

position there exist Y ∈ ker(adα−1) and X ∈ im(adα−1) such that Z = Y + [α−1, X]. Itis clear that Y, [α−1, X] ∈ Fn. Hence

Z + Fn+1 = Y + Fn+1 + [α−1, X] + Fn+1.

Obviously [α−1, Y ] = 0 ∈ Fn, hence Y + Fn+1 ∈ ker(αn−1−1 ). Now suppose X ∈ Fk. If

k ≥ n+ 1 then X ∈ Fn+1. Hence

[α−1, X]− [α−1, X] = 0 ∈ Fn+1.

Thus [α−1, X] + Fn+1 ∈ im(αn−1). Therefore

Z + Fn+1 = Y + Fn+1 + [α−1, X] + Fn+1,

in which Y + Fn+1 ∈ ker(αn−1−1 ) and [α−1, X] + Fn+1 ∈ im(αn−1). Now suppose k < n+ 1

and X = X + X in which X ∈ Fn+1. But then

[α−1, X] + Fn+1 = [α−1, X ] + [α−1, X ] + Fn+1.


Now since [α−1, X], [α−1, X ] ∈ Fn and that [α−1, X] ∈ Fk−1 − Fn−1 so

[α−1, X ] = 0.

Thus[α−1, X] + Fn+1 = [α−1, X ] + Fn+1.

Again [α−1, X]− [α−1, X ] = 0 ∈ Fn+1 in which X ∈ Fn+1. Therefore [α−1, X ] + Fn+1 ∈im(αn−1). Hence

Z + Fn+1 = Y + Fn+1 + [α−1, X] + Fn+1,

in which Y + Fn+1 ∈ ker(αn−1−1 ) and [α−1, X ] + Fn+1 ∈ im(αn−1).

Now we give a second proof of Lemma 5.1.1 using the previous lemma: Let us Supposethat Zk−1 ∈ Fk−1. Then by the previous lemma we have that

Zk−1 + Fk = Yk−1 + Fk +Wk−1 + Fk,

where Yk−1 +Fk ∈ ker(αk−1−1 ) and Wk−1 +Fk ∈ im(αk−1). Therefore W− [α−1, Xk] ∈ Fk in

which Xk ∈ Fk and [α−1, Yk−1] ∈ Fk−1. Here we can suppose that Xk ∈ Fk ∩ im(adα−1).

On the other hand, we have that [α−1, Yk−1k−1 ] = 0. Hence

Zk−1 + Fk = Yk−1 + [α−1, Xk] + Fk = Y k−1k−1 + [α−1, Xk] + Fk.

Thus

Zk−1 = Y k−1k−1 + [α−1, Xk] + Zk.

Similarly we haveZk = Y k

k + [α−1, Xk+1] + Zk+1,

where again [α−1, Ykk ] = 0 and Xk+1 ∈ Fk ∩ im(adα−1). Hence we get that

Zk−1 = Y k−1k−1 + Y k

k + ...+ [α−1, Xk +Xk+1 + ....] = Y + [α−1, X].

So is clear that Y ∈ Fk−1 ∩ ker(adα−1) and X ∈ Fk ∩ im(adα−1).

Remark 5.1.3. In the notation of [56], L0 = D + α−1 + q(0) is in F01 and u1 ∈ F1

0. Ingeneral L = D+α−1 + q ∈ F0

1 and u ∈ F10 and L0 = D+α−1 +h ∈ F0

1 and L0 and L arein the same equivalence class and L0 is normal form for that class. For more informationabout normal form theory, see [56].

The following proposition ( [29]) plays a key rule in constructing an integrable equa-tion using a Lax pair.

Proposition 5.1.4. There exist an element u =∑∞

1 ui, ui ∈ im(adα−1) ∩ gi, and h =∑∞0 hi, hi ∈ ker(adα−1) ∩ gi such that

eaduL = D + α−1 + h.


Proof. The claim is that for each n ∈ N ∪ 0,−1 there exist

un ∈ Fn ∩ im(adα−1), and q(n) ∈ ker(adα−1) ∩ F0 + Fn,

such thateadun (Ln) = Ln+1, where Ln = D + α−1 + q(n).

When n = −1 then we take u−1 = 0 ∈ F0∩ im(adα−1) then L−1 = L0 = D+α−1 + q(−1)

with q(−1) = q(0) = q ∈ F0 ⊆ F0 + ker(adα−1)∩F0. Now suppose the statement holds forn. Then by assumption we can write

q(n) = q(n) + yn,

where q(n) ∈ ker(adad−1) ∩ F0 and yn ∈ Fn. Now by semisimple decomposition

yn = yn + [α−1, un+1],

where yn ∈ keradα−1 ∩ Fn ⊂ keradα−1 ∩ F0 and un+1 ∈ Fn+1 ∩ im(adα−1) by Lemma5.1.1. It follows that

D + α−1 + q(n+1) = eadun+1 (D + α−1 + q(n))

= D + α−1 + q(n) + [un+1, α−1] mod Fn+1

= D + α−1 + q(n) + yn mod Fn+1.

Henceq(n+1) = q(n) + yn mod Fn+1.

This indeed means that q(n+1) ∈ keradα−1 ∩ F0 + Fn+1. Thus the proof of the inductionstatement is complete. By the Campbell-Baker-Hausdorff formula there exist

u(n) ∈ F1 ∩ im(adα−1),

such that

eadeu(n)L0 = eadun eadun−1 . . . eadu0L−1 = Ln = D + α−1 + q(n).

Since in our case⋂

Fn = 0, it follows that in the limit n→∞ one has

eadu∞ (D + α−1 + q) = D + α−1 + q∞.

Notice that according to [29], we are supposed to choose A+,A− in such a way that

g = A+ ⊕ A−, and A− ⊆ ker(adAλ).

Now lets set

g+ =⊕

i>0

gi + A+, g− =⊕

i<0

gi + A−. (5.1.11)

We use grading and not filtering from now on. The next result shows that Lt and [A+β , L]

sit in the same space. We show this in more general form of Lax operator as below. Onecan see that the previous theorem holds in this case as well.


Proposition 5.1.5. Consider the Lax operator L = D + Aλn+1 +∑n

i=−m qiλi and let

β be a constant from the center of subalgebra ker(ad−(αn+1)),

Aβ = e−aduβ, and A+β = (e−aduβ)+,

where “+” denotes the projection onto g+ parallel to g−. Then [A+β , L] ∈∑n

i=−m gi.

Proof. We have

[e−aduβ, L] = [e−aduβ, e−adu(D + αn+1 + h)] = e−adu [β,D + αn+1 + h]

= e−adu(−βx + [β, αn+1] + [β, h]) = e−adu(0) = 0,

using the assumption that β is in the center of Ker(adαn+1) and that α−(n+1), hi ∈Ker(adαn+1). Hence

[(e−aduβ)+, L] = −[(e−aduβ)−, L].

Now [(e−aduβ)+, L] sits in∑∞−m gi and [(e−aduβ)−, L] in

∑n−∞ gi, hence [(e−aduβ)+, L]

sits in intersection space which is∑n−m gi. Indeed if e−aduβ =

∑1−∞Xi+X−+

∑∞1 Xi+

X+ where Xi ∈ gi, X− ∈ A− and X+ ∈ A+, then we have that

[(e−aduβ)+, L] = [∞∑

1

Xi +X+, L]

=

∞∑

1

(−Xi,x + [Xi, αn+1] +

n∑

j=−m[Xi, qj ])−X+,x + [X+, αn+1]

+

n∑

j=−m[X+, qj]

∈∞∑

−mgi,

and

[(e−aduβ)−, L] = [−1∑

−∞Xi +X−, L]

=−1∑

−∞(−Xi,x + [Xi, αn+1] +

n∑

j=−m[Xi, qj ])−X−,x + [X−, αn+1] +

n∑

j=−m[X−, qj ]

=

−1∑

−∞(−Xi,x + [Xi, αn+1] +

n∑

j=−m[Xi, qj ])−X−,x

+

n∑

j=−m[X−, qj]

∈n∑

−∞gi,


using the fact that A− ⊂ ker(adαn+1). Hence [(e−aduβ)+, L] ∈∑n−m gi.

Proof. second proof : Here L = D + α−(n+1) + q where q ∈ F−n/F(m+1). Let us define

a few notations. We denote Fk/Fk+1 by Gk and Gk + ... + Gl by Hlk where k < l. Wewant to prove that

[(e−aduβ)+, L] ∈ Hm−n.

We see that

[e−aduβ, L] = [e−aduβ, e−adu(D + αn+1 + h)]

= e−adu [β,D + αn+1 + h]

= e−adu(−βx + [β, αn+1] + [β, h])

= e−adu(0) = 0,

using the assumption that β is in the center of ker(adα−(n+1)) and that α−(n+1), hi ∈

ker(adαn+1). Hence

[(e−aduβ)+, L] = −[(e−aduβ)−, L].

Let us put (e−aduβ)− = X+X− in which X− ∈ A− and X ∈ F1 = H∞1 . Also (e−aduβ)+ =Y +X+ in which X+ ∈ A+ and Y ∈ H−1

−∞. Then we have that

[(e−aduβ)−, L] = −Xx −X−,x + [X,α−(n+1)] + [X−, α−(n+1)]

+[X, q] + [X−, q]

= −Xx −X−,x + [X,α−(n+1)] + [X, q] + [X−, q].

Here [X,α−(n+1)] ∈ H∞−n, and [X, q] ∈ H∞−n+1. Hence [(e−aduβ)−, L] ∈ H∞−n. Similarly

[(e−aduβ)+, L] = −Yx −X+,x + [Y, α−(n+1)] + [X+, α−(n+1)]

+[Y, q] + [X+, q].

Here [Y, α−(n+1)] ∈ H−1−∞ and [Y, q] ∈ Hm−1

−∞ and also [X+, q] ∈ Hm−n. Hence we have that

[(e−aduβ)+, L] ∈ Hm−∞. Therefore [(e−aduβ)+, L] ∈ Hm

−∞ ∩ H∞−n = Hm−n.

Remark 5.1.6. Instead of Hqp we may use the notation in [56] as f p|q and call it q− jet

of fp.

The claim of proposition above is that relation Lt = [A+β , L] is equivalent to some

evolution system for the unknown q. In the case we will work out, the operator L istaken to be L = D + α+ q with q ∈ G0 and α−1 = Aλ ∈ G−1.

Proposition 5.1.7. Let M ∈ F−n in which n > 0. Suppose that we have

[Dt −M,L0] = 0 or L0,t = [M,L0], (5.1.12)

where L0 = Dx + α−1 + h and h ∈ ker(adα−1) ∩ F0. Then M ∈ ker(adα−1).


Proof. We see thatht = [M−n, α−1] mod F−n.

Thus [M−n, α−1] ∈ Ker(adα−1) + F−n. Hence [M−n, α−1] ∈ ker(adα−1) and so M−n ∈ker(adα−1). Thus M ∈ ker(adα−1) + F−n+1. Now suppose M−n+i ∈ Ker(adα−1) fori = 0, ..., k − 1. Hence

M = M + M,

in which M ∈ Ker(adα−1) and M ∈ F−n+k. From (5.1.12), we have that

ht + Mx − [M, h] = −Mx + [M , α−1] + [M, h].

Since ht+Mx−[M, h] ∈ ker(adα−1). The same reasoning shows that M−n+k = M−n+k ∈ker(adα−1). Hence by the filtering topology in F′ns we get that M ∈ ker(adα−1).

Lemma 5.1.8. 1. Let β and γ be arbitrary elements of the center of subalgebraker(adAλ). Consider the equation

Lt = [A+β , L], (5.1.13)

thendAγdt = [A+

β , Aγ ].

2. [A+β , Aγ ]+ = [Aγ , (Aβ)−]+ = [A+

γ , (Aβ)−]+. Notice that these identities does notdepend on (5.1.13).

Proof. From (5.1.13) we obtain

0 = [dt −A+β , L] = [dt −A+

β , e−aduL0] = e−adu [eadu(dt −A+

β ), L0],

thus0 = [eadu(dt −A+

β ), L0] = [dt − A+β , L0],

where L0 = Dx + α +∑0−∞ hi in which hi ∈ ker(adα) ∩ gi and α ∈ g1. From the last

proposition we know that A+β ∈ ker(adα). Now we want to prove that [dt −A+

β , Aγ ] = 0or

[dt − A+β , γ] = 0,

or[A+

β , γ] = 0,

since γ is a constant. Now since γ is in the center of ker(adα) and A+β in the ker(adα),

therefore this equality is trivial. To prove (2), one notice that

[A+β , Aγ ] = [(e−aduβ)+, e

−aduγ]

= [e−aduβ − (e−aduβ)−, e−aduγ]

= [e−aduβ, e−aduγ] + [e−aduγ, (e−aduβ)−]

= e−adu [β, γ] + [e−aduγ, (e−aduβ)−]

= [e−aduγ, (e−aduβ)−]

= [Aγ , (Aβ)−],

since β, γ are in the center of ker(adα) as well as in ker(adα) itself clearly.


Theorem 5.1.9. Let β and γ be two arbitrary elements of the center of subalgebraker(adAλ). Then the flows

Lt = [A+β , L], and Lτ = [A+

γ , L],

commute with each other, i.e,

D[A+β ,L][A

+γ , L]−D[A+

γ ,L][A+β , L] = 0, or equivalentely, (Lt)τ = (Lτ )t.

In particular the flows Ltn = [A+βλn , L] will commute. One can choose β = α, since α

and αλn = Aλn+1 are naturally in the center of ker(adAλ).

Proof. We have

d

dτ

(dL

dt

)=

d

dτ[A+

β , L] = [d

dτA+β , L] + [A+

β ,d

dτL].

According to Lemma 5.1.8,d

dτAβ = [A+

γ , Aβ ].

Sod

dτA+β = [A+

γ , Aβ ]+. Thus

d

dτ

(dL

dt

)= [[A+

γ , Aβ ]+, L] + [A+β , [A

+γ , L]].

Similarlyd

dtA+γ = [A+

β , Aγ ]+ and that

d

dt

(dL

dτ

)= [[A+

β , Aγ ]+, L] + [A+γ , [A

+β , L]].

Using Jacobi identity, we obtain

d

dτ

(dL

dt

)− d

dt

(dL

dτ

)= [

d

dτA+β −

d

dtA+γ + [A+

β , A+γ ], L]

= [[A+γ , Aβ ]+ − [A+

β , Aγ ]+ + [A+β , A

+γ ], L]

But[A+

γ , Aβ ]+ = [Aβ , (Aγ)−]+ = [A+β , (Aγ)−]+

as in the previous lemma. Therefore

[A+γ , Aβ ]+ − [A+

β , Aγ ]+ + [A+β , A

+γ ]

= [A+β , (Aγ)−]+ − [A+

β , Aγ ]+ + [A+β , A

+γ ]+

= [A+β ,−A+

γ ]+ + [A+β , A

+γ ]+

= [A+β ,−A+

γ +A+γ ]+ = 0.


Henced

dτ

(dL

dt

)− d

dt

(dL

dτ

)= 0.

Now we give some well known examples of Z−graded Lie algebras.

Example 5.1.1. The first example is the Lie algebra of the Laurent series

g = b[[λ, λ−1]] =⊕

i∈Zgi,

where gi consist of elements Aλi in which A belongs to the Lie algebra b. Here g isdecomposed into sum of a polynomial in λ and a series containing only negative powersof λ. The instance of this decomposition in the set up above is (5.1.11). Notice thatthere we have taken A− = 0 and A+ = b. See [16] to get more on this specific Z−gradedLie algebras.

Example 5.1.2. The second example is known as Kac-Moody algebras which indeedassign a Z−graded Lie algebra to any automorphism of finite order on a given Lie algebraand is constructed as follows.

Let σ be any automorphism of finite order m on a finite dimensional Lie algebrab. By bi we denote a subspace of the space b, consist of elements like X ∈ b so thatσ(X) = µiX, where µ is a primitive m−root of unity. We have

b = ⊕i∈Zmbi, (5.1.14)

so that [bi, bj] ⊂ bi+j. We say that b is graduated modulo m. Now, to a Lie algebra(5.1.14), we assign a graduated Lie algebra L(b, σ), by considering Laurent series

L(b, e) = b[[λ, λ−1]] =⊕

i∈Zbλi,

and taking in it a subalgebra

L(b, σ) =⊕

i∈Zλibi,

where bi will be taken as modulo m. We denote gi to biλi, so that L(b, σ) =

⊕i∈Z gi.

For more information, see the work of V. G. Kac in [33].

Proposition 5.1.10. Suppose that the automorphism σ on the Lie algebra b is irre-ducible, i.e., automorphism with respect to which the algebra can not be decomposedinto a direct sum of invariant ideals. Then there exist elements

E0, . . . , Er ∈ g1, F0, . . . , Fr ∈ g−1, and H0, . . . ,Hr ∈ g0,

such that


1. The sets E0, . . . , Er and F0, . . . , Fr form a basis in g1 and g−1 respectively andthe set H0, . . . ,Hr generates g0.

2. The following relations hold:

(a) [Hi,Hj ] = 0,

(b) [Ei, Fj ] = δijHi,

(c) [Hi, Ei] = AiiEj,

(d) [Hi, Fj ] = −AijFj ,

where Aii = 2 for all i.

We use Theorem 5.1.9 to construct a recursion operator for a certain integrableevolution equations. Let us specifically work on Kac-Moody algebras g defined on theLie algebra b. Assume that the the degree of automorphism σ is k. Let us take α = Aλ inwhich A is constant element of b so that the decomposition (5.1.9) holds. Now we choosea sequence of constant elements of center of ker(adα) as Aλnn∈N. Then we would geta sequence of commuting equation or hierarchy of integrable equation. Here

A+Aλn+k = (e−aduAλn+k)+ = λk(e−aduAλn)+ + (λk(e−aduAλn)−)+.

Hence

Ltn+k= [A+

Aλn+k , L] = λkLtn + [Rk, L], (5.1.15)

whereRk = (λk(e−aduAλn)−)+ is clearly of degree k and must be taken as the polynomial

Rk = Nkλk +Nk−1λ

k−1 + . . .+N0, Ni ∈ gi. (5.1.16)

as an element of Kac-Moody algebras g.Equating coefficients of λ powers of both side of (5.1.15) we will get hierarchy of

equations as well as the Recursion operator as we can see in the section below forSymplectic Lie algebras of quaternions.

5.2 Lax method in Symplectic geometry

Let us consider the Lie algebra of the Symplectic group b = sp(n + 1). We first buildup a Kac-Moody algebras g defined on the Lie algebra b. To be specific we define theautomorphism σ as follows:

σ(X) = T ′XT ′−1,

where

T ′ =(−1 00 In

).

Obviously σ2 = Id and eigenvalues of σ are 1 and −1, Indeed in this particular case,µ = −1. Hence

g2i = geλ2i, g2i+1 = goλ2i+1,

5.2. Lax method in Symplectic geometry 89

where

ge =

∗ 0 00 ∗ ∗0 ∗ ∗

, go =

0 ∗ ∗∗ 0 0∗ 0 0

.

We immediately see that if we let a = sp(n)× sp(1), then

ge = a, and go = b/a.

Let us choose

A =

0 1 0−1 0 00 0 0

∈ go.

In the next proposition we show that the Kac-Moody algebra decompose into the kerneland image of adjoint representation as in (5.1.9).

Proposition 5.2.1. Following decomposition holds.

g = ker(adAλ)⊕ im(adAλ).

Moreover

ker(adAλ) =⊕

i

0 t 0−t 0 00 0 0

λ2i+1

⊕⊕

i

p 0 00 p 00 0 P

λ2i,

where t is real number, p a pure quaternionic, and P ∈ sp(n− 1). Also

im(adAλ) =⊕

i

0 p −pT

p 0 0p 0 0

λ2i+1

⊕⊕

i

q 0 00 −q −qT

0 q 0

λ2i.

where p, q are pure quaternionic numbers.

Proof. Let us take general element M of the basic Lie algebra b. Thus M has followingform.

M =

m11 m12 −mT

1

m21 m22 −mT2

m1 m2 M

, (5.2.1)

in which m21 = −mT12 and m11,m22 are pure quaternionic numbers. Hence the bracket

of M and A is

[M,A] =

−m12 −m21 m11 −m22 mT

2

m11 −m22 m21 +m12 −mT1

−m2 m1 0

. (5.2.2)


Since m11 −m22 and m12 + m21 are pure quaternionic number, im(adAα) is exactly asin the proposition. Let us suppose [M,A] = 0. Hence

m1 = 0 = m2, m11 = m22 and m12 +m21 = 0.

Now if M ∈ ge, then m11 = m22 and if M ∈ go, then m12 +m21 = 0, which indeedmeans that m12 = −m21 = t ∈ R. This shows that ker(adAλ) is as in the proposition.Also it is clear that the sum of the elements of the form of coefficients of λ2i+1 inker(adAλ) and im(adAλ) constitute go and those of λ2i form ge. This shows that thedecomposition holds.

Notation 5.2.1. From now on, we denote the set of pure quaternionic numbers byim(H).

Let us take Lax operator L = Dx + Aλ+ q as in (5.1.8), in which

q =

0 0 0T

0 u −uT

0 u 0

∈ sp(n)× sp(1).

Notice that in the case of real numbers instead of quaternions, we see that q is nothingbut natural frame in comparison with the Frenet frame in classical differential geometry.

Before we proceed, it would be useful to compute the bracket of general elementM ∈ b = sp(n+ 1) as in (5.2.1) and the matrix q ∈ b as we do it as below.

[M,U ]

=

0 m12u− < m1,u > −m12uT

−um21+ < u,m1 > Cum2 − Cum22 −m22uT + umT

2 + uTM−um21 m2u+ Mu− um22 −m2u

T + umT2

.

where the inner product involved is the hermitian inner product, as we defined it Chapter1.

Notation 5.2.2. The standard inner product on Rn is denoted by <,>r .

Now the degree of σ is 2. Therefore the sequence of flows associated with Laxoperators in (5.1.15) becomes

Ltn+2 = [A+Aλn+2 , L] = λ2Ltn + [R2, L], (5.2.3)

and R2 becomes

R2 = Nλ2 +Mλ+K, N,K ∈ ge and M ∈ go. (5.2.4)

Now we compute coefficients of λ′s of (5.2.3). First step: the coefficient of λ3. Simplythe coefficient of λ3 vanishes , that is :

[N,A] = 0. (5.2.5)


This leads to following integrability conditions.

n11 = n22, (5.2.6a)

n2 = 0. (5.2.6b)

The next coefficient is of λ2 which gives us

0 = Utm + [N,U ] + [M,A]−DxN. (5.2.7)

We derive the following equations using grading and equations above.

0 = m12 +m21 +Dxn11, (5.2.8a)

0 = m21 +m12 + n11u− un11 −Dxn11 + utm , (5.2.8b)

0 = −m1 + un11 − utm , (5.2.8c)

0 = DxN. (5.2.8d)

From (5.2.8a) and (5.2.8b), we find that (2Dx + Cu)n11 − utm = 0. Hence

n11 = (2Dx + Cu)−1utm .

As a result we obtain(m12 +m21

m1

)=

(−1 00 I

)(Dn11

m1

)

=

(−1 00 I

)(D(Cu + 2D)−1 0Lu(Cu + 2D)−1 −I

)(utmutm

)

=

(−D(Cu + 2D)−1 0Lu(Cu + 2D)−1 −I

)(utmutm

).

Denoting by A the operator

A =:

(−D(Cu + 2D)−1 0Lu(Cu + 2D)−1 −I

),

we simply express last matrix equation as follows:(m12 +m21

m1

)= A

(utmutm

).

Now the coefficient of λ is:

[M,U ] + [K,A] −DxM = 0. (5.2.9)

Hence we derive the following equations:

0 = k2 + um21 +Dxm1, (5.2.10a)

0 = DxM, (5.2.10b)

0 = m12u−Dm12 − k22 + k11− < m1,u >, (5.2.10c)

0 = −um21 −Dm21 − k22 + k11+ < u,m1 >= 0. (5.2.10d)


By adding (5.2.10d) and (5.2.10c) we find that

2k22 − 2k11 = m12u− um21 −Dx(m12 +m21) + Cum1

= −1

2Cu(m12 +m21) + u(m12 −m21)

−Dx(m12 +m21) + Cum1.

Notice that we have used the fact that

m12u− um21 = −1

2Cu(m12 +m21) + u(m12 −m21).

Also by subtracting those two equations we get that

m12u+ um21 −D(m12 −m21)−Aum1 = 0.

On the other hand, 12Au(m12 +m21) = (m12u+ um21). Hence we obtain m12 −m21 as

follows:

m12 −m21 = D−1(1

2Au(m12 +m21)−Aum1).

Remark 5.2.2. We can express this using the Killing form:

m12 −m21 = −1

2D−1K(

(u −uT

u 0

),

(m12 +m21 −mT

1

m1 0

)).

Also we can simply compute m21 as below.

m21 =m21 −m12

2+m21 +m12

2

= −1

2D−1(

1

2Au(m12 +m21)−Aum1) +

m21 +m12

2.

We are able now to compute the following vector.

(k22 − k11

k2

)

=

(−1

4Cu + 1

2uD−1 1

2Au − 12D −1

2uD−1Au + 1

2Cu

12LuD

−1(12Au)− 1

2Lu −12LuD

−1Au −D

)(m12 +m21

m1

).

Let us denote the last operator we derived by I:

I =

(−1

4Cu + 1

2uD−1 1

2Au − 12D −1

2uD−1Au + 1

2Cu

12LuD

−1(12Au)− 1

2Lu −12LuD

−1Au −D

).

Hence we write(k22 − k11

k2

)= I

(m12 +m21

m1

). (5.2.11)


The constant part gives

[K,U ]−DxK − Utm+2 = 0, (5.2.12)

from which one can find the following equation.

utm+2 = −(Cu +D)k22 + Cuk2, (5.2.13a)

utm+2 = −uk22 + Ku + k2u−Dk2, (5.2.13b)

k11 = 0, (5.2.13c)

DKij = uikj2 − ki2u

j . (5.2.13d)

This yields the expression for

(utm+2

utm+2

)in terms of

(k22

k2

)as follows.

(utm+2

utm+2

)=

(−(Cu +D) Cu

−Lu −D +Ru + H1

)(k22

k2

), (5.2.14)

in which the notation H1 is used to be the operator acting as follows:

H1k2 = D−1(uikj2 − ki2u

j)u.

We denote by H the operator that just appeared, i.e.,

H =:

(−(Cu +D) Cu

−Lu −D +Ru + H1

).

Hence the constant coefficient of Lax representation can be written as follows.

(utm+2

utm+2

)= HIA

(utmutm

).

There is a link between the operators A and H in following proposition.

Proposition 5.2.3.

A−1HA∗ = H.

Proof. See Proposition 4.3.6 of Chapter 4.

Remark 5.2.4. This may be related to the Hamiltonian map as in [39, theorem 5.2.9],the map Φ is Hamiltonian if and only if

Φ(B) = D(Φ)B1D(Φ)∗,

where B and B1 are Hamiltonian operator acting on C and C1, respectively.


Remark 5.2.5. We can see that:

(Dx + Lu)(2Dx + Cu)−1(−Dx +Ru) =

= (Dx +Au +Cu

2)(2Dx + Cu)−1(−Dx +

Au − Cu2

)

= (Au2

+2Dx + Cu

2)(2Dx + Cu)−1(

Au2− 2Dx + Cu

2)

= −1

2Dx +

1

2Au(2D + Cu)−1 1

2Au −

1

4Cu

= −1

2Dx +

1

2uD−1

x

1

2Au −

1

4Cu.

Therefore we can write the operator I as below.

I =

((Dx + Lu)(2Dx + Cu)−1(−Dx +Ru) −1

2uD−1x Au + 1

2Cu12LuD

−1(12Au)− 1

2Lu −12LuD

−1Au −D

).

See Remark 4.3.14 for the motivation of this decomposition.

Now if we let utm = u1 and utm = u1, as trivial symmetry, then we get the followingsystem of scalar-vector equation.

ut = −1

4u3 +

3

8(uu1u− uu2 + u2u)− 3

2< u,u > u1

−2 < u,u1 > u+ 12u < u,u1 > −u < u1,u >

−12 < u1,u > u+

3

2Cuu2,

ut = −u3 +3

2u2u+ u1(

3

4u1 −

3

8u2 − 3

2< u,u >).

(5.2.15)

We check that if we apply again

R = HIA,

on the equation itself, the result will commute with the equation itself. This means thatwe can construct the hierarchy of equations starting with this new integrable system.The splitting the operator R to Hamiltonian and symplectic has been worked out inChapter 4.

Remark 5.2.6. We compute the previous equation (5.2.15), explicitly and step by step.Let utm = u1 and utm = u1 then

(Dn11

m1

)=

(12u1

12uu− u1

).

Hence (m12 +m21

m1

)=

(−1

2u112uu− u1

).

5.3. Higher symmetry 95

Thus we obtain

(k22

k2

)according to (5.2.11) as follows.

(k22

k2

)=

1

8Cuu1 +

1

4u2 + (1

2u < u,u > − 12Cuu1) + 1

2u(−1

4u2+ < u,u >)

12u(−1

8u2+ < u,u >) +

1

4uu1 − (1

2u1u+ 12uu1 − u2)

=

1

8Cuu1 +

1

4u2 + (u < u,u > − 1

2Cuu1)− 1

8u3)

12u(−1

8u2+ < u,u >)− ( 1

2u1u+1

4uu1 − u2)

,

and so (5.2.14) yields the following expression for

(utm+2

utm+2

)explicitly in terms of

(uu

)

and its total derivatives.

(utm+2

utm+2

)=

−1

4u3 +

3

8(uu1u− uu2 + u2u)− 3

2< u,u > u1

−2 < u,u1 > u+ 12u < u,u1 > −u < u1,u >

−12 < u1,u > u+

3

2Cuu2

−u3 +3

2u2u+ u1(

3

4u1 −

3

8u2 − 3

2< u,u >)

.

5.3 Higher symmetry

We apply the recursion operator R to the equation 5.2.15 and find fifth order equation

of the hierarchy as follows To have short expression, let us put R

(uu

)=

(S0

S1

). Then

we have that

S = 16u5 + (−40u)u4 + u3

[30u2 − 60u1 + 40 < u,u >

]

+u2

[60 < u1,u > +20 < u,u1 > −24 < u,u > u− 50u2

+10u1u+ 10u1u0 − 36u < u,u > +50uu1 − 5u3]

+u1

[50 < u2,u > +40 < u1,u1 > −7 < u1,u > u− 7 < u1,u > u

+30 < u,u2 > +23 < u,u1 > u+ 30 < u,u >2 −15u3 +5

2u2u

−30u1 < u,u > −5

2uu1u+

45

2uu2 − 23u < u,u1 >

−53u < u1,u > +25

2u1u

2u21 −

5

8u4 − 6u2u1 + 15u2 < u,u >

−5

2uu1u− 23u < u,u1 > −53u < u1,u > +u1u

2 +25

2u2

1

].


and also we obtain that

S0 = u5 +5

2u4u−

5

2uu4 − 5uu3u+ u2u3 +

5

2u3u1 +

3

2u3u

2 − 5

2u1u3

1

2u2u1u− u2uu1 −

3

8u2u

3 − 7

2u1u2u−

3

2u1uu2 − 4uu2u1

−19

8uu2u

2 − 1

2uu1u2 +

13

8u2u2u+

9

8u3u2

−1

4u4u1 +

7

8u3u1u+

7

8u2u1u

2 +7

4u2u2

1 +3

8uu1u

3 +5

4uu1uu1

−3

4uu2

1u+1

4u1u

2u1 −5

4u1uu1u−

5

4u2

1u2 − 5

2u3

1

3u4 < u,u > −3u3 < u,u > u− 7u3 < u,u1 > −2u3 < u1,u >

−6u2 < u,u >2 +5u2 < u,u1 > u+ 3u2 < u,u2 > +5u2 < u1,u > u

+6u2 < u1,u1 > −27u2 < u2,u > u+ 6u < u,u >2 u

+12u < u,u >< u,u1 > +24u < u,u >< u1,u >

−7u < u,u1 > u2 − 25u < u,u2 > u− 13u < u,u3 >

−7u < u1,u > u2 + 12u < u1,u >< u,u > −20 < u1,u1 > u

−9u < u1,u2 > +5u < u2,u > u+ 21u < u2,u1 >

+57u < u3,u1 > −8 < u,u > u < u,u1 > +16 < u,u > u < u1,u >

+26 < u,u >< u,u1 > +2 < u,u >< u1,u > u+ 4 < u,u1 > u3

+24 < u,u1 > u < u,u > +9 < u2,u1 > u− 7 < u3,u > u

−6 < u1,u >< u,u > u+ 14 < u1,u1 > u2 + 63 < u,u3 > u

− < u1,u > u3 + 18 < u,u1 >< u,u > u+ 37 < u,u2 > u2

+12 < u1,u > u < u,u > +7 < u2,u > u2 + 39 < u1,u2 > u

−6uu1u < u,u > −3uu1 < u,u > u− 4uu1 < u,u1 > −24uu1 < u1,u >

−14u < u,u1 > u1 − 12u < u,u1 >< u,u > −14u < u1,u > u1

−3u1u2 < u,u > +3u1u < u,u > u+ 7u1u < u,u1 > −3u1u < u1,u >

+4u1 < u1,u > u+ 12u1 < u1,u1 > +66u1 < u2,u > +6u1 < u,u >2

+4u1 < u,u1 > u+ 6u1 < u,u2 > −6 < u,u > uu1u+ 8 < u,u1 > uu1

+24 < u,u > u1 < u,u > +29 < u,u1 > u1u+ 74 < u,u2 > u1

−2 < u1,u > uu1 + 9 < u1,u > u1u+ 28 < u1,u1 > u1 + 14 < u2,u > u1

−9uu2 < u,u > +3u2u < u,u > +6u2 < u,u > u+ 11u2 < u,u1 >

+31u2 < u1,u > −6 < u,u > uu2 + 6 < u,u > u2u+ 29 < u,u1 > u2

+9 < u1,u > u2 + 6u3 < u,u > +4 < u,u > u3

+18 < u,u >< u,u2 > −12 < u,u >< u1,u1 >

−54 < u,u >< u2,u > −20 < u3,u1 > −30 < u4,u > +30 < u,u4 >

−6 < u2,u >< u,u > +12 < u1,u1 >< u,u >

+42 < u,u2 >< u,u > +60 < u,u1 >2 +20 < u1,u3 > −60 < u1,u >3,

5.3. Higher symmetry 97

The code we used to find out that the equation itself will commute with what we canfind by applying the recursion operator is written in FORM, see [71] and [57].

Chapter 6

Computation of Geometric operator

6.1 Hamiltonian operator

This section is devoted to the computation of the Hamiltonian operator. In Chapter4, Theorem 4.3.12, we claimed that the operator H = HA∗ is indeed Hamiltonian. Weproved also that the operator A is a Nijenhuis operator, see Theorem 4.3.8. As is known,see for instance [69], the operator H is indeed a Hamiltonian operator. We will prove thatthe Lie algebra form of H which is denoted by H, as in Theorem 4.4.3, is Hamiltonianusing the definition and techniques described in Chapter 2. We will prove that

∫K(pi, DH

[Hpi+2](pi+1)) = 0. (6.1.1)

Notation 6.1.1. Here we have taken the sum over index i, but we take into account therule of shifting, that is for instance, we can add to index i, by 1, 2 and use, for instance,the fact that

pi+3 = pi, pi+4 = pi+1. (6.1.2)

Notation 6.1.2. Here and after, we simply use the notation H for the operator H andpi for the matrix pi and likewise for u so that we write as

H = Dx − π1adu − aduD−1π0adu.

The Frechet derivative of H is

DH [q] = −π1adq − adqD−1π0adu − aduD

−1π0adq.

Now we compute the expression on the left of (6.1.1):∫K(pi, DH [Hpi+2](pi+1))

=

∫K(pi, (−π1adHpi+2 − adHpi+2D

−1π0adu − aduD−1π0adHpi+2)pi+1

)

= −∫K(pi, π1adpi+2,x−π1adupi+2−aduD

−1x π0adupi+2

pi+1)

−∫K(pi, adpi+2,x−π1adupi+2−aduD

−1x π0adupi+2

D−1π0adupi+1)

−∫K(pi, aduD

−1π0adpi+2,x−π1adupi+2−aduD−1x π0adupi+2

pi+1). (6.1.3)

99

100 Chapter 6. Computation of Geometric operator

This expression is a combination of the Schouten brackets of three operators constitutingthe operator H, these are, H1 = Dx,H2 = π1adu and H3 = aduD

−1π0adu. In order toprove that H is Hamiltonian, we show that

[Hi,Hj] = 0, for i, j = 1, 2, 3.

We will break the proof of this claim into the several part.

Lemma 6.1.1. The Schouten bracket [H3,H3] vanishes.

Proof. We explain every single step and every single rule we use, so that later on we willjust do it. It is clear from (6.1.3) that

[H3,H3](p1, p2, p3) =

∫K(pi, adaduD

−1x π0adupi+2

D−1π0adupi+1)

+

∫K(pi, aduD

−1π0adaduD−1x π0adupi+2

pi+1).

We simplify the first term as follows:

∫K(pi, adaduD

−1x π0adupi+2

D−1π0adupi+1)

=

∫K(pi+1, adaduD

−1x π0adupi

D−1π0adupi+2).

in which we have used the shifting rule (6.1.2). For the second term, we derive thefollowing:

∫K(pi, aduD


pi+1)

= −∫K(adupi, D


pi+1)

=

∫K(D−1adupi, π0adaduD

−1x π0adupi+2

pi+1)

=

∫K(D−1π0adupi, adaduD

−1x π0adupi+2

pi+1)

= −∫K(adaduD

−1x π0adupi+2

D−1π0adupi, pi+1),

where the first equality follows from the fact that the Killing form is invariant under theadjoint action, that is,

K(adXY,Z) +K(Y, adXZ) = 0, X, Y,X ∈ g. (6.1.4)

The second equality follows the integration by parts, the third equality follows the rulestated in Lemma 1.4.2, the fourth equality again the invariance of the Killing form.

6.1. Hamiltonian operator 101

Hence we obtain

[H3,H3](p1, p2, p3) =

∫K(pi+1, adaduD

−1x π0adupi

D−1π0adupi+2)

−∫K(adaduD

−1x π0adupi+2

D−1π0adupi, pi+1).

Now using the Jacobi identity for the three elements u,D−1π0adupi+2 and D−1π0adupiof the Lie algebra g, we obtain

[H3,H3](p1, p2, p3) =

∫K(pi+1, aduadD−1

x π0adupiD−1π0adupi+2).

Again using the rules mentioned above, we see that

[H3,H3](p1, p2, p3) = −∫K(adupi+1, adD−1

x π0adupiD−1π0adupi+2)

= −∫K(π0adupi+1, adD−1

x π0adupiD−1x π0adupi+2).

The last integrand we obtained is in the image of total derivative, more precisely,

Remark 6.1.2.

DxK(p, adqr) = K(px, adqr) +K(p, adqxr) +K(p, adqrx)

= K(px, adqr) +K(adrp, qx)−K(adqp, rx).

Hence

DxK(pi, adpi+1pi+2) = 3K(pix , adpi+1pi+2).

Therefore K(pix , adpi+1pi+2) = 1/3DxK(pi, adpi+1pi+2).

It follows that

[H3,H3](p1, p2, p3) = −1

3

∫DxK(D−1

x π0adupi+1, adD−1x π0adupi

D−1x π0adupi+2).

Hence by definition [H3,H3](p1, p2, p3) = 0.

Lemma 6.1.3. [H3,H2] = 0.

Proof. From (6.1.3), we obtain that

[H3,H2](p1, p2, p3)

= +

∫K(pi, π1adaduD

−1x π0adupi+2

pi+1) +

∫K(pi, adπ1adupi+2

D−1π0adupi+1)

+

∫K(pi, aduD

−1π0adπ1adupi+2pi+1). (6.1.5)


Now we manipulate the expression on the right. The first term is simplified as follows:∫K(pi , π1adaduD

−1x π0adupi+2

pi+1) =

∫K(pi , adaduD

−1x π0adupi+2

pi+1)

= −∫K(adaduD

−1x π0adupi+2

pi , pi+1) =

∫K(adpiaduD

−1x π0adupi+2 , pi+1)

= −∫K(aduD

−1x π0adupi+2 , adpipi+1) =

∫K(D−1

x π0adupi+2 , aduadpipi+1)

=

∫K(D−1

x π0adupi+2 , π0aduadpipi+1)

Here we have used Lemma 1.4.2, 6.1.4, anti-symmetricity of the Lie bracket, (6.1.4)(twice), and Lemma 1.4.2, respectively.

The second and third terms of (6.1.5) together simplify as follows:∫K(pi, adπ1adupi+2

D−1π0adupi+1) +K(pi, aduD−1x π0adπ1adupi+2

pi+1)

= −∫K(adπ1adupi+2

pi, D−1x π0adupi+1)−K(adupi, D

−1π0adπ1adupi+2pi+1)

= −∫K(adπ1adupi+2

pi, D−1x π0adupi+1)

+

∫K(D−1

x adupi, D−1π0adπ1adupi+2

pi+1)

= −∫K(π0adπ1adupi+2

pi, D−1π0adupi+1)

+

∫K(D−1

x π0adupi+1, π0adπ1adupipi+2)

= −∫K(π0adadupi+2

pi, D−1x π0adupi+1) +K(D−1

x π0adupi+1, π0adadupipi+2)

= −∫K(π0(adadupi+2

pi − adadupipi+2), D−1x π0adupi+1)

=

∫K(π0aduadpipi+2, D

−1x π0adupi+1)

=

∫K(π0aduadpi+1pi, D

−1x π0adupi+2).

Here the first is term modified according to (6.1.4), Lemma 1.4.2, and finally the factthat one has

π0adπ1adupi+2pi = π0adadupi+2

pi. (6.1.6)

Similarly the second term is manipulated using (6.1.4), integration by part, Lemma 1.4.2,and again the equality (6.1.6). Then these two terms together modified using the Jacobiidentity and then shifting rule. Now it should be clear that

[H3,H2](p1, p2, p3)

6.1. Hamiltonian operator 103

vanishes.

Lemma 6.1.4. [H2,H2] = 0.

Proof. We see that

[H2,H2](p1, p2, p3) =

∫K(pi, π1adπ1adupi+2

pi+1)

=

∫K(pi, adπ1adupi+2

pi+1)

= −∫K(pi, adpi+1π1adupi+2)

=

∫K(adpi+1pi, π1adupi+2)

= −∫K(adpi+1pi, π1adpi+2u)

= −∫K(adpi+1pi, adpi+2u− π0adpi+2u)

=

∫K(adpi+2adpi+1pi, u) +

∫K(adpi+1pi, π0adpi+2u)

=

∫K(adpi+1pi, π0adpi+2u)

=

∫K(π0adpi+1pi, π0adpi+2u)

=

∫K(adρ0pi+1ρ0pi, adρ0pi+2ρ0u)

= −∫K(adρ0pi+2adρ0pi+1ρ0pi, ρ0u)

= 0.

Notice that we have used the equality

∫K(adpi+2adpi+1pi, u) = 0,

using the Jacobi identity and also the fact that

π0adpi+1pi = adρ0pi+1ρ0pi. (6.1.7)

This concludes the proof of the lemma.

Lemma 6.1.5. [H1,H2] = 0


Proof.

[H1,H2](p1, p2, p3) = −∫K(pi, π1adpi+2,xpi+1)

= −∫K(pi, adpi+2,xpi+1)

= −1

3

∫DxK(pi, adpi+2pi+2)

= 0.

This concludes the proof of the lemma.

Lemma 6.1.6. [H1,H3] = 0

Proof.

[H1,H3](p1, p2, p3) =

= −∫K(pi, adpi+2,xD

−1x π0adupi+1)−

∫K(pi, aduD

−1x π0adpi+2,xpi+1)

=

∫K(adpi+2,xpi, D

−1x π0adupi+1) +

∫K(adupi, D

−1x π0adpi+2,xpi+1)

= −∫K(D−1

x adpi+2,xpi, π0adupi+1) +

∫K(π0adupi+1, D

−1x adpi,xpi+2)

=

∫K(π0adupi+1, D

−1x adpipi+2,x) +

∫K(π0adupi+1, D

−1adpi,xpi+2,x)

=

∫K(π0adupi+1, adpipi+2)

=

∫K(π0adupi+1, π0adpipi+2)

=

∫K(adρ0uρ0pi+1, adρ0piρ0pi+2)

=

∫K(ρ0u, adρ0pi+1adρ0piρ0pi+2)

= 0.

Here again (6.1.7) plays a key role.

Now we obtain the main result out of these lemmas.

Theorem 6.1.7. The operator

H = Dx − π1adu − aduD−1π0adu

is a Hamiltonian operator.


Now we can simply prove that the operator HA∗ or its systematic form in terms ofthe Lie bracket and projection as HA∗ is also Hamiltonian. In fact we have that

HA∗ = (Dx − π1adu − aduD−1x π0adu)(ρ0 + 2ρ1 − ρ1D

−1x π1adu)

= (Dx − π1adu − aduD−1x π0adu)(ρ0 + 2ρ1)− ρ1π1adu + π1aduρ1D

−1x π1adu

= (Dx − π1adu − aduD−1x π0adu)(ρ0 + 2ρ1)− ρ1π1adu + aduρ1D

−1x π1adu

= (Dx − π1adu − aduD−1x π0adu)(ρ0 + 2ρ1)− ρ1π1adu + aduD

−1x ρ1π1adu

= Dx − π1adu − aduD−1x π0adu +

Dxρ1 − π1aduρ1 + ρ1π1adu + aduD−1x ρ1π1adu.

Let us write HA∗ = C1 + C2 in which

C1 = Dx − π1adu − aduD−1x π0adu,

andC2 = Dxρ1 − π1aduρ1 + ρ1π1adu + aduD

−1x ρ1π1adu.

We already proved that C1 is Hamiltonian, that is, [C1, C1] = 0. Similarly we can provethat [C2, C2] = 0. It is not difficult to show that [C1, C2] = 0. Hence we can simplyconclude the following lemma.

Theorem 6.1.8. HA∗ is also Hamiltonian.

Remark 6.1.9. Indeed this representation of the Hamiltonian operator is a specific caseof typical Hamiltonian operators, see for instance a series of papers [69], [70] and [68]and [13].

6.2 Symplectic operator

As announced in Chapter 4, this section is devoted to the proof of the fact that theoperator A−1∗IA−1 is symplectic. In order to do so, is enough to show that the Liealgebra form A−1∗IA−1 of this operator is symplectic. This will be done in a few steps.First we give some identities which we use later on.

Proposition 6.2.1. Using the notation of Section 4.4, we have that

1. (12ρ1 + ρ0)π1ad2

aq = −q.

2. π1adaaduadaπ1q = −aduρ1q − π1adρ0uρ0q,

where the matrix q and u have the form

q =

0 0 00 q −qt

0 q 0

, u =

0 0 00 u −ut

0 u 0

. (6.2.1)

We have removed here the hat sign on top of the symbols for the matrices to have asimpler looking notation.


Now using this proposition, the operator I can be expressed and simplified as in thefollowing lemma.

Lemma 6.2.2. The operator I can be decomposed as

I = S0 + S1 + S2,

in which

S0 = −1

2uD−1

x K(u, ·), S1 =1

2Dxρ1 −

1

4adρ1uρ1,

and

S2 = Dxρ0 −1

2adρ0uρ1 −

1

2π1adρ0uρ0.

Proof.

I

= −1

2uD−1

x K(u, ·)− (1


1

2ρ1 + ρ0)

= −1

2uD−1

x K(u, ·)− (1

2ρ1 + ρ0)π1ad2

aπ1(1

2ρ1 + ρ0)Dx

+(1

2ρ1 + ρ0)π1adaaduadaπ1(

1

2ρ1 + ρ0)

= −1

2uD−1

x K(u, ·) + (1

2ρ1 + ρ0)Dx

+(1

2ρ1 + ρ0)(−adu

1

2ρ1 − π1adρ0uρ0)

= −1

2uD−1

x K(u, ·) + (1

2ρ1 + ρ0)Dx

−1

4adρ1uρ1 −

1

2adρ0uρ1 −

1

2π1adρ0uρ0

In the following proposition we compute A−1, A−1∗ and also prove that the operatorI and consequently the operator A−1∗IA−1 is anti-symmetric.

Proposition 6.2.3. We do have that

1. A−1 = DxBρ1 + (adρ0uBρ1 + ρ0), in which B = (2Dx − adρ1u)−1.

2. B∗ = −B.

3. A−1∗ = BDxρ1 +Bπ1adρ0uρ0 + ρ0.

4. S∗i = −Si for i = 0, 1, 2.


Proof. 1. We see that

AA−1 = (ρ0 + 2ρ1 − aduD−1x ρ1)(DxBρ1 + (adρ0Bρ1 + ρ0))

= 2DxBρ1 + adρ0uBρ1 + ρ0 − aduBρ1

= (2Dx − adu + adρ0u)Bρ1 + ρ0

= (2Dx − adρ1u)Bρ1 + ρ0

= ρ1 + ρ0.

Similarly we have that

A−1A = (DxBρ1 + (adρ0Bρ1 + ρ0))(ρ0 + 2ρ1 − aduD−1x ρ1)

= 2DxBρ1 −DxBadρ1uD−1x ρ1

+2adρ0uBρ1 − adρ0uBadρ1uD−1x ρ1 + ρ0 − adρ0uD

−1x

= DxB(2Dx − adρ1u)D−1x ρ1

+adρ0uB(2Dx − adρ1u)D−1x ρ1 + ρ0 − adρ0uD

−1x

= ρ1 + ρ0.

2. We can write the operator B as follows.

B =1

2(I − 1

2D−1x adρ1u)−1D−1

x

=1

2(I +

1

2D−1x adρ1u +

1

4D−1x adρ1uD

−1x adρ1u + . . .)D−1

x

=1

2(D−1

x +1

2D−1x adρ1uD

−1x +

1

4D−1x adρ1uD

−1x adρ1uD

−1x + . . .).

Notice that the summation is finite when applied to elements of the Lie algebra E.Now using the integration by parts and (6.1.4), we find that

∫K(Bp, q) =

∫K(

1

2(D−1

x +1

2D−1x adρ1uD

−1x

+1

4D−1x adρ1uD

−1x adρ1uD

−1x + . . .)p, q)

=

∫K(p,−1

2(D−1

x +1

2D−1x adρ1uD

−1x

+1

4D−1x adρ1uD

−1x adρ1uD

−1x + . . .)q)

=

∫K(p,−Bq).

Hence we proved that B∗ = −B.


3. We can compute the adjoint of the operator A−1 as follows.∫K(A−1p, q) =

∫K((DxBρ1 + adρ0uBρ1 + ρ0)p, q)

= −∫K(Bp,Dxρ1q)−K(Bρ1p, π1adρ0uρ0q) +K(p, ρ0q)

= +

∫K(p,BDxρ1q) +K(p,Bπ1adρ0uρ0q) +K(p, ρ0q)

= +

∫K(p, (BDxρ1 +Bπ1adρ0uρ0 + ρ0)q).

Hence by definition A−1∗ = BDxρ1 +Bπ1adρ0uρ0 + ρ0.

4. For this equality, we use Lemma 1.4.2. See Remark 6.2.4 below.∫K(S0p, q) =

∫K(− 1

2uD−1

x K(u, p), q)

= −∫

1

2K(u, q).D−1

x K(u, p)

= +

∫1

2D−1x K(u, q).K(u, p)

= +

∫1

2K(uD−1

x K(u, q), p)

=

∫K(− (

1

2uD−1

x K(u, q)), p)

Hence by the definition of adjoint operator, we see that S∗0 = −S0. Notice thatthis can be considered as an example of how we work with such expressions. Forinstance the technique used here can be seen in the lemmas afterwards proving thatA−1∗IA−1 is symplectic, or applied to the current case, more precisely A−1∗S0A

−1

is symplectic.

Now we compute the adjoint operator of S1 :∫K(S1p, q) =

∫K(

(1

2Dxρ1 −

1

4adρ1uρ1)p, q

)

= +

∫K(

1

2Dxρ1p, ρ1q)−

1

4K(adρ1uρ1p, ρ1q)

= −∫K(ρ1p,

1

2Dxρ1q) +K(ρ1p,

1

4adρ1uρ1q)

= −∫K(p,

1

2Dxρ1q) +K(p,

1

4adρ1uρ1q)

=

∫K(p,−(

1

2Dxρ1 −

1

4adρ1uρ1)q

)

So that S∗1 = −S1.


Similarly we can manipulate S∗2 as follows using the same technics.∫K(S2p, q)

=

∫K(

(Dxρ0 −1

2adρ0uρ1 −

1

2π1adρ0uρ0)p, q

)

=

∫K(Dxρ0p, ρ0q)−

1

2K(adρ0uρ1p, ρ0q)−

1

2K(π1adρ0uρ0p, ρ1q)

= −∫K(ρ0p,Dxρ0q) +

1

2K(ρ1p, π1adρ0uρ0q) +

1

2K(ρ0p, adρ0uρ1q)

= −∫K(p,Dxρ0q) +

1

2K(p, π1adρ0uρ0q) +

1

2K(p, adρ0uρ1q)

=

∫K(p, (−Dxρ0 +

1

2π1adρ0uρ0 +

1

2adρ0uρ1)q

).

Thus S∗2 = −Dxρ0 + 12π1adρ0uρ0 + 1

2adρ0uρ1 = −S2.

Remark 6.2.4. There is a technical point here: using Lemma 1.4.2, if we have ρi in thefirst component of the Killing form, then that would move to the other component. Forexample K(ρ0p, q) = K(p, ρ0q) or K(ρ1p, adρ0uρ0q) = K(ρ1p, π1adρ0uρ0q) and so on.

The next lemma is essential in what follows, in particular in proving that the operatormentioned above is symplectic.

Lemma 6.2.5. The following identity holds for the operator B and arbitrary matricesp, q, r of the form (6.2.1):

∫K(adBρ1pBρ1r, ρ1q) = −

∫K(adρ1pBρ1r − adρ1rBρ1p,Bρ1q).

Proof. By definition and the Leibniz rule we have that

2DxK(adBρ1pBρ1r,Bρ1q)

= K(ad2DxBρ1pBρ1r,Bρ1q)

+ K(adBρ1p2DxBρ1r,Bρ1q)

+ K(adBρ1pBρ1r, 2DxBρ1q)

= K(adρ1pBρ1r,Bρ1q) +K(adadρ1uBρ1pBρ1r,Bρ1q)

+ K(adBρ1pρ1r,Bρ1q) +K(adBρ1padρ1uBρ1r,Bρ1q)

+ K(adBρ1pBρ1r, ρ1q) +K(adBρ1pBρ1r, adρ1uBρ1q)

= K(adρ1pBρ1r,Bρ1q)−K(adBρ1radρ1uBρ1p,Bρ1q)

+ K(adBρ1pρ1r,Bρ1q) +K(adBρ1padρ1uBρ1r,Bρ1q)

+ K(adBρ1pBρ1r, ρ1q)−K(adρ1uadBρ1pBρ1r,Bρ1q).

We have used the fact that 2DxBρ1p = ρ1p+ adρ1uBρ1p.


Now using the Jacobi identity we obtain

2DxK(adBρ1pBρ1r,Bρ1q) = K(adρ1pBρ1r,Bρ1q) +K(adBρ1pρ1r,Bρ1q)

+ K(adBρ1pBρ1r, ρ1q).

Hence by definition∫K(adρ1pBρ1r,Bρ1q) +K(adBρ1pρ1r,Bρ1q) +K(adBρ1pBρ1r, ρ1q) = 0.

The statement is proved.

Now we yield the operators A−1∗SiA−1 explicitly.

Lemma 6.2.6. 1.

A−1∗S2A−1 =

1

2Bπ1adadρ0uDxρ0uBρ1 −

1

2BDxπ1adρ0uρ0

+Bπ1adρ0uDxρ0 +Dxadρ0uBρ1 +Dxρ0 −1

2adρ0uDxBρ1.

2. A−1∗S1A−1 = 1

2BD3xBρ1 −

1

4BDxadρ1uDxBρ1.

3. For the operator S0 we have that

A−1∗S0A−1 = −1

2BDx

((ρ1u)D−1

x K(u,DxBρ1 + ρ0))

−1

2(ρ0u)D−1

x K(u,DxBρ1 + ρ0).

Proof. 1. We have that

A−1∗S2A−1

= −1

2BDxπ1adρ0uadρ0uBρ1 −

1

2BDxπ1adρ0uρ0

+ Bπ1adρ0uDxadρ0uBρ1 +Bπ1adρ0uDxρ0

− 1

2Bπ1adρ0uadρ0uDxBρ1

+ Dxadρ0uBρ1 +Dxρ0 −1

2adρ0uDxBρ1

= −1

2Bπ1adDxρ0uadρ0uBρ1 +

1

2BDxπ1adρ0uadDxρ0uBρ1

− 1

2BDxπ1adρ0uρ0

+ Bπ1adρ0uDxρ0 +Dxadρ0uBρ1 +Dxρ0 −1

2adρ0uDxBρ1

=1

2Bπ1adadρ0uDxρ0uBρ1 −

1

2BDxπ1adρ0uρ0

+ Bπ1adρ0uDxρ0 +Dxadρ0uBρ1 +Dxρ0 −1

2adρ0uDxBρ1.


We use the Jacobi identity in the last expression on the first two terms, so that

−1

2Bπ1adDxρ0uadρ0uBρ1 +

1

2BDxπ1adρ0uadDxρ0uBρ1

=1

2Bπ1adadρ0uDxρ0uBρ1.

2. The second identity is simple.

3. Now for the third one, notice that the image of adρ0uBρ1 is contained in the imageof ρ0, since the image of Bρ1 is contained in the image of ρ1. Hence we do havethat for instance K(u, adρ0uBρ1·) = 0 using the invariance property of the Killingform under adjoint action (6.1.4). Taking these facts into account we see that

A−1∗S0A−1 =

= (BDxρ1 +Bπ1adρ0uρ0 + ρ0)(−1

2uD−1

x K(u, ·))(DxBρ1 + adρ0uBρ1 + ρ0)

= (BDxρ1 +Bπ1adρ0uρ0 + ρ0)(−1

2uD−1

x K(u,DxBρ1 ·+ρ0·))

= −1

2BDx

((ρ1u)D−1


−1

2(ρ0u)D−1

x K(u,DxBρ1 + ρ0).

This concludes the proof of the three statements of the lemma.

As we discussed in Chapter 2, in order to prove that an operator is symplectic, wefirst need to compute its Frechet derivative.

Lemma 6.2.7. The Frechet derivatives of A−1∗S0A−1, A−1∗S1A

−1 and A−1∗S2A−1 are

expressed as follows:

1.

DA−1∗S2A−1 [pi+2]

= +1

2Badρ1pi+2Bπ1adadρ0uDxρ0uBρ1 +

1

2Bπ1adadρ0pi+2Dxρ0uBρ1

+1

2Bπ1adadρ0uDxρ0pi+2

Bρ1 +1

2Bπ1adadρ0uDxρ0uBadρ1pi+2Bρ1

−1

2Badρ1pi+2BDxπ1adρ0uρ0 −

1

2BDxπ1adρ0pi+2ρ0

+Badρ1pi+2Bπ1adρ0uDxρ0 +Bπ1adρ0pi+2Dxρ0

+Dxadρ0pi+2Bρ1 +Dxadρ0uBadρ1pi+2Bρ1

−1

2adρ0pi+2DxBρ1 −

1

2adρ0uDxBadρ1pi+2Bρ1.


2.

DA−1∗S2A−1 [pi+2]

= +1

2Badρ1pi+2BD

3xBρ1 +

1

2BD3

xBadρ1pi+2Bρ1

−1

4Badρ1pi+2BDxadρ1uDxBρ1 −

1

4BDxadρ1uDxBadρ1pi+2Bρ1

−1

4BDxadρ1pi+2DxBρ1.

3.

DA−1∗S0A−1 [pi+2]

= −1

2Badρ1pi+2BDx

((ρ1u)D−1


−1

2BDx

((ρ1pi+2)D−1


−1

2BDx

((ρ1u)D−1

x K(pi+2, DxBρ1 + ρ0))

−1

2BDx

((ρ1u)D−1

x K(u,DxBadρ1pi+2Bρ1))

−1

2(ρ0pi+2)D−1

x K(u,DxBρ1 + ρ0)

−1

2(ρ0u)D−1

x K(pi+2, DxBρ1 + ρ0)

−1

2(ρ0u)D−1

x K(u,DxBadρ1pi+2Bρ1).

Proof. We apply the Leibniz rule many times. Notice that we do have

Dadρ1u[pi+2] = adρ1pi+2 ,

and

DB [pi+2] = Badρ1pi+2B.

To prove that A−1∗IA−1 is symplectic, it is enough to show that the operatorsA−1∗SiA−1 for i = 0, 1, 2 are symplectic. As we have seen these operator are anti-symmetric. We start with the operator involving S0. In order to shorten the proof, letus introduce some notation. We denote ρ1pi = qi and ρ0pi = qi. Also ρ0u = v andρ0u = v.

Theorem 6.2.8. The operator A−1∗S0A−1 is symplectic.


Proof. As we discussed in Chapter 4, we need to prove that

[A−1∗S1A−1,A−1∗S1A

−1] =

∫K(pi, DA−1∗S0A−1 [pi+2](pi+1)) = 0,

where the bracket is the Schouten bracket. But we have that

[A−1∗S0A−1,A−1∗S0A

−1] =

∫K(pi, DA−1∗S0A−1 [pi+2](pi+1))

= −∫K(qi,

1

2Badqi+2BDx

((v)D−1

x K(v,DxBqi+1)))

−∫K(qi,

1

2Badqi+2BDx

((v)D−1

x K(v,qi+1)))

−∫K(qi,

1

2BDx

((qi+2)D−1

x K(v,DxBqi+1)))

−∫K(qi,

1

2BDx

((qi+2)D−1

x K(v,qi+1)))

−∫K(qi,

1

2BDx

((v)D−1

x K(qi+2, DxBqi+1)))

−∫K(qi,

1

2BDx

((v)D−1

x K(qi+2,qi+1)))

−∫K(qi,

1

2BDx

((v)D−1

x K(v,DxBadqi+2Bqi+1)))

−∫K(qi,

1

2(qi+2)D−1

x K(v,DxBqi+1))

−∫K(qi,

1

2(qi+2)D−1

x K(v,qi+1))

−∫K(qi,

1

2(v)D−1

x K(qi+2, DxBqi+1))

−∫K(qi,

1

2(v)D−1

x K(qi+2,qi+1))

−∫K(qi,

1

2(v)D−1

x K(v,DxBadqi+2Bqi+1)). (6.2.2)

We now choose subexpressions and show they are zero.

1. The first expression to be considered is

∫K(qi,−

1

2(qi+2)D−1

x K(v,qi+1))

+

∫K(qi,−

1

2(v)D−1

x K(qi+2,qi+1)).

The first and second terms of this expression become, respectively,

−∫

1

2K(qi,qi+2).D−1

x K(v,qi+1), −∫

1

2K(qi,v).D−1

x K(qi+2,qi+1).


That is because a real valued expression such as D−1x K(v,qi+1) can pulled out of

the integrand

−K(qi,

1

2(qi+2)D−1

x K(v,qi+1)).

Then we apply integration by part and the shifting rule. We obtain the followingterms:

+

∫1

2D−1x K(qi,qi+2).K(v,qi+1), −

∫1

2K(qi+1,v).D−1

x K(qi,qi+2).

Now it is clear that the expression we chose vanishes.

2. Similar to the first one, we see that the expression

∫K(qi,−

1

2(qi+2)D−1

x K(v,DxBqi+1))

+

∫K(qi,−

1

2BDx

((v)D−1

x K(qi+2,qi+1)))

will vanish as well:

+

∫K(qi,−

1

2(qi+2)D−1

x K(v,DxBqi+1))

+

∫K(qi,−

1

2BDx

((v)D−1

x K(qi+2,qi+1)))

= −∫

1

2K(qi,qi+2).D−1

x K(v,DxBqi+1)

−∫

1

2K(DxBqi, v).D−1

x K(qi+2,qi+1)

= +

∫1

2D−1x K(qi,qi+2).K(v,DxBqi+1)

−∫

1

2K(DxBqi+1, v).D−1

x K(qi,qi+2)

= 0

3. The next expression is:

+

∫K(qi,−

1

2(v)D−1

x K(qi+2, DxBqi+1))

+

∫K(qi,−

1

2BDx

((qi+2)D−1

x K(v,qi+1)))

+

∫K(qi,−

1

2(v)D−1

x K(v,DxBadqi+2Bqi+1))

+

∫K(qi,−

1

2Badqi+2BDx

((v)D−1

x K(v,qi+1))),


which can be simplified as follows.

1

4

∫D−1x K(qi+1,v).K(qi, advBqi+2)−K(advBqi, qi+2).D−1

x K(v,qi+1)

−1

4

∫D−1x K

(qi+1,v).K(qi, advBqi+2) +K(advBqi, qi+2).D−1

x K(v,qi+1).

Now it is clear that the expression is zero. See for more details Appendix B.

4. The last expression is:

+

∫K(qi,−

1

2Badqi+2BDx

((v)D−1

x K(v,DxBqi+1)))

+

∫K(qi,−

1

2BDx

((qi+2)D−1

x K(v,DxBqi+1)))

+

∫K(qi,−

1

2BDx

((v)D−1

x K(qi+2, DxBqi+1)))

+

∫K(qi,−

1

2BDx

((v)D−1

x K(v,DxBadqi+2Bqi+1))).

Again this expression is converted to the following one by using the same rule asin third item:

−∫

1

4K(adqi+2Bqi, v).D−1

x K(v,DxBqi+1)

+

∫1

4K(adqi+2Bqi, v).D−1

x K(v,DxBqi+1)

−∫

1

4D−1x K(DxBqi+1, v).K(v, adqiBqi+2)

+

∫1

4D−1x K(DxBqi+1, v).K(v, adqiBqi+2)

Now in the last expression every term is canceled by another, so that the wholeexpression vanishes. See for more detail on this computation Appendix B.

In what follows, we simply use the following identity as a rule.

2DxBρ1q = ρ1q + adρ1uBρ1q. (6.2.3)



Proof. The Schouten bracket is computed and simplified, using (6.2.3), as follows.

[A−1∗S1A−1,A−1∗S1A

−1] =

∫K(pi, DA−1∗S1A−1 [pi+2](pi+1)

= −1

4

∫K(qi, (Badqi+2BDxadvDxB +BDxadvDxBadqi+2B)qi+1

)

− 1

4

∫K(qi, BDxadqi+2DxBqi+1

)

+1

2

∫K(qi, Badqi+2BD

3xBqi+1 +BD3

xBadqi+2Bqi+1

)

=1

4

∫K(Badqi+2Bqi, DxadvDxBqi+1

)− 1

4

∫K(DxadvDxBqi, Badqi+2Bqi+1

)

− 1

4

∫K(DxBqi, adqi+2DxBqi+1

)

+1

2

∫K(DxBadqi+2Bqi, D

2xBqi+1

)− 1

2

∫K(D2xBqi, DxBadqi+2Bqi+1

)

=1

4

∫K(Badqi+2Bqi −BadqiBqi+2, DxadvDxBqi+1

)

− 1

16

∫K(

advBqi+1 + qi+1, adqi(advBqi+2 + qi+2))

+1

2

∫K(DxBadqi+2Bqi −DxBadqiBqi+2, D

2xBqi+1

)

= −1

4

∫K(DxBadqi+2Bqi −DxBadqiBqi+2, advDxBqi+1

)

− 1

16

∫K(

advBqi+1 + qi+1, adqiqi+2

)

− 1

16

∫K(qi+1, adqiadvBqi+2

)

− 1

16

∫K(

advBqi+1, adqiadvBqi+2

)

+1

4

∫K(DxBadqi+2Bqi −DxBadqiBqi+2, DxadvBqi+1 +Dxqi+1

)

= − 1

16

∫K(

advBqi+1 + qi+1, adqiqi+2

)

− 1

16


)

− 1

16

∫K(


)

+1

4

∫K(DxBadqi+2Bqi −DxBadqiBqi+2, adDxvBqi+1 +Dxqi+1

)

In Appendix B we have simplified the terms in the last expression that has been found.There, in the Lemmas B.0.13 and B.0.11, we proved that the terms in the second line


and third line are simplified to:

− 1

16


)=

1

16

∫K(

advqi+1, adqiBqi+2 − adqi+2Bqi

)

and

− 1

16

∫K(


)

= − 1

32

∫K(

advadvBqi+1 +Badvadvqi+1, adqi+2Bqi − adqiBqi+2

).

The last term, as in the Lemma B.0.12, becomes

1

4


)

=1

8

∫K(

adqi+2Bqi − adqiBqi+2, BadvadDxvBqi+1

)

+1

16

∫K(

adqi+2Bqi − adqiBqi+2, Badvadvqi+1 + advqi+1

)

+1

16

∫K(

advBqi+2 + qi+2, adqi+1qi

).

This has been proved again in Appendix B. Hence by replacing these equations into theexpression for the Schouten bracket we obtain:

[A−1∗S1A−1,A−1∗S1A

−1]

= − 1

32

∫K(


)

+1

8

∫K(


)

+1

16

∫K(

adqi+2Bqi − adqiBqi+2, Badvadvqi+1

)

=1

16

∫K(

adqi+2Bqi − adqiBqi+2,

1

2Badvadvqi+1 + 2BadvadDxvBqi+1 −

1

2advadvBqi+1

)

Again by using (6.2.3), we can find that

advadvBqi+1 = B(2Dx − adv)advadvBqi+1

= 2BadDxvadvBqi+1 + 2BadvadDxvBqi+1 + 2BadvadvDxBqi+1

− BadvadvadvBqi+1

= 2BadDxvadvBqi+1 + 2BadvadDxvBqi+1

+ BadvadvadvBqi+1 +Badvadvqi+1 −BadvadvadvBqi+1

= 2BadDxvadvBqi+1 + 2BadvadDxvBqi+1 +Badvadvqi+1.


Using the last identity in the Schouten bracket, we obtain the following equation:

[A−1∗S1A−1,A−1∗S1A

−1]

=1

16

∫K(

adqi+2Bqi − adqiBqi+2, BadvadDxvBqi+1 −BadDxvadvBqi+1

).

Now for the second component of the Killing form in the integrand, we can use theJacobi identity and see that

[A−1∗S1A−1,A−1∗S1A

−1]

= +1

16

∫K(

adqi+2Bqi − adqiBqi+2,−BadBqi+1advDxv).

We apply Lemma 6.2.5 to simplify the Schouten bracket as follows:

[A−1∗S1A−1,A−1∗S1A

−1] =1

16

∫K(

adBqi+2Bqi, adBqi+1advDxv).

Now we take the last step. We use the invariance property of the Killing form under theadjoint action (6.1.4):

[A−1∗S1A−1,A−1∗S1A

−1] = − 1

16

∫K(

adBqi+1adBqi+2Bqi, advDxv).

This expression is simply zero if we apply the Jacobi identity to the first component ofthe Killing form.


Proof. The Schouten bracket of the operator is as follows:

[A−1∗S2A−1,A−1∗S2A

−1] =

∫K(pi, DA−1∗S2A−1 [pi+2]pi+1)

=

∫K(qi, Dxadqi+2

Bqi+1 +DxadvBadqi+2Bqi+1

)

+

∫K(qi,−

1

2adqi+2

DxBqi+1 −1

2advDxBadqi+2Bqi+1

)

+

∫K(qi,

1

2Badqi+2Bπ1adadvDxvBqi+1 +

1

2Bπ1adadqi+2Dxv

Bqi+1

)

+

∫K(qi,

1

2Bπ1adadvDxqi+2

Bqi+1 +1

2Bπ1adadvDxvBadqi+2Bqi+1

)

+

∫K(qi, Badqi+2Bπ1advDxqi+1 +Bπ1adqi+2

Dxqi+1

)

+

∫K(qi,−

1

2Badqi+2BDxπ1advqi+1 −

1

2BDxπ1adqi+2

qi+1

).


Those subexpressions which are compatible are listed below and it is proved that eachof them cancels. The first of these three subexpressions is:

− 1

2

∫K(qi, adqi+2

DxBqi+1

)− 1

2

∫K(qi, BDxπ1adqi+2

qi+1

)

+

∫K(qi, Dxadqi+2

Bqi+1

)+

∫K(qi, Bπ1adqi+2

Dxqi+1

)

= −1

2

∫K(Dxadqi+2

qi, Bqi+1

)+

1

2

∫K(Bqi+1, Dxπ1adqiqi+2

)

+

∫K(

adqi+2Dxqi, Bqi+1

)−∫K(Bqi+1, π1adqiDxqi+2

)

= −∫K(Dxadqi+2

qi, Bqi+1

)

+

∫K(

adqi+2Dxqi, Bqi+1

)+

∫K(Bqi+1, π1adDxqi+2

qi

)

=

∫K(−Dxadqi+2

qi + adqi+2Dxqi + adDxqi+2

qi, Bqi+1

)

= 0.

This has been done by just applying the rules used so far. For instance, in the first linewe used invariance of the Killing form (6.1.4) and integration by parts, and also the factthat the operator B is anti-symmetric.

The second subexpression is:

1

2

∫K(qi, Badqi+2Bπ1adadvDxvBqi+1) +K(qi, Bπ1adadvDxvBadqi+2Bqi+1)

The first term of the expression is simplified to

−1

4

∫K(Badqi+2Bqi, π1adadvDxvBqi+1

)

using B∗ = −B and the invariant property (6.1.4). The second term similarly becomes

1

2

∫K(

adadvDxvBqi, Badqi+2Bqi+1

),

using the same rules. Then if one applies the shifting rule, we obtain

1

2

∫K(

adadvDxvBqi+1, BadqiBqi+2

).

Hence the couple simplifies to:

−1

2

∫K(Badqi+2Bqi −BadqiBqi+2, adadvDxvBqi+1

).


Now if we apply Lemma 6.2.5 we obtain the following expression:

−1

2

∫K(

adBqi+2Bqi, adadvDxvBqi+1

).

Again using the invariance rule, we obtain:

−1

2

∫K(

adBqi+1adBqi+2Bqi, advDxv).

The last term is zero because of the Jacobi identity.The last subexpression is:

1

2

∫K(qi, Bπ1adadqi+2DxvBqi+1

)+

1

2

∫K(qi, Bπ1adadvDxqi+2

Bqi+1

)

− 1

2

∫K(qi, advDxBadqi+2Bqi+1

)− 1

2

∫K(qi, Badqi+2BDxπ1advqi+1

)

+

∫K(qi, DxadvBadqi+2Bqi+1

)+

∫K(qi, Badqi+2Bπ1advDxqi+1

).

Applying the rules just mentioned, the expression becomes as follows:

− 1

2

∫K(

adBqi+1Bqi, adqi+2Dxv

)−∫K(

adBqi+1Bqi, advDxqi+2

)

− 1

2

∫K(Dxadvqi, Badqi+2Bqi+1

)+

1

2

∫K(Badqi+2Bqi, Dxπ1advqi+1

)

+

∫K(

advDxqi, Badqi+2Bqi+1

)−∫K(Badqi+2Bqi, π1advDxqi+1

).

Now, using Lemma 6.2.5, the first line becomes

−1

2

∫K(

adBqi+1Bqi, adqi+2Dxv + advDxqi+2

).

Applying the shifting rule, the first line equals∫K(

advDxqi+1, BadqiBqi+2 −Badqi+2Bqi

),

So that the the whole expression takes the form

− 1

2

∫K(BadqiBqi+2 −Badqi+2Bqi, adqi+1

Dxv + advDxqi+1

)

− 1

2

∫K(Dxadvqi+1, BadqiBqi+2 −Badqi+2Bqi

)

+

∫K(

advDxqi+1, BadqiBqi+2 −Badqi+2Bqi

)

=1

2

∫K(−Dxadvqi+1 + advDxqi+1 + adDxvqi+1, BadqiBqi+2 −Badqi+2Bqi

).


The last expression is zero, since Dx is a derivation. Hence it has been proved thatthe Schouten bracket

[A−1∗S2A−1, A−1∗S2A

−1]

vanishes and this shows that the operator A−1∗S2A−1 is a symplectic operator.

Appendix A

Computations in Hamiltonian operators

In this appendix, we prove the following identity for an anti-symmetric operator H :E∗ → E :

d2ωH(a1, a2, a3) = [H,H](b∗1, b∗3, b∗2),

where [H,H] is the Schouten or Schouten bracket of H and ai = Hb∗i for b∗i ∈ E∗.We compute the left hand side of the identity. Using the formula for the coboundaryoperator d2 as in (2.1.4) or (2.1.1), we have that

d2ωH(a1, a2, a3)

= a1.ωH(a2, a3)− a2.ωH(a1, a3) + a3.ωH(a1, a2)

−ωH([a1, a2], a3) + ωH([a1, a3], a2)− ωH([a2, a3], a1)

= a1.ωH(a2, a3) + a2.ωH(a3, a1) + a3.ωH(a1, a2)

−ωH([a1, a2], a3) + ωH([a1, a3], a2)− ωH([a2, a3], a1).

Hence using Definition 2.1.5 of the 2-form ωH , we find that

d2ωH(a1, a2, a3)

= La1(b∗3(a2)) + La2(b∗1(a3)) + La3(b∗2(a1))

−b∗3([a1, a2]) + b∗2([a1, a3])− b∗1([a2, a3]).

Using (2.1.3), we see that

d2ωH(a1, a2, a3)

= b∗3(La1a2) + (La1b∗3)(a2)

b∗1(La2a3) + (La2b∗1)(a3)

b∗2(La3a1) + (La3b∗2)(a1)

−b∗3([a1, a2]) + b∗2([a1, a3])− b∗1([a2, a3])

= (La1b∗3)(a2) + (La2b

∗1)(a3) + (La3b

∗2)(a1)

= (LHb∗1b∗3)(Hb∗2) + (LHb∗2b

∗1)(Hb∗3) + (LHb∗3b

∗2)(Hb∗1).

According to Definition 2.3.1, this shows that

d2ωH(a1, a2, a3) = [H,H](b∗1, b∗3, b∗2).

123

Appendix B

Some Killing form identities

Justification of some equlities used in Theorem 6.2.9 did not fit there to include. Herewe prove that those equlities indeed holds.

Lemma B.0.11. Let B be the antisymmetric operator B = (2Dx − adv)−1 acting on

purely imaginary (in the quaternion sense) functions. Then

∑

i∈Z/3

∫K(


)

=1

2

∑

i∈Z/3

∫K(


).

Here we explicitly sum over i ∈ Z/3, contrary to the convention of Notation 6.1.1.

Proof. The integral on the left hand side of the equality can be expanded as below usingthe invariance of the Killing form K and Jacobi identity and sorting out the expressionsafterwards.

∫K(


)

=1

2

∫K(


)+

1

2

∫K(


)

=1

2

∫K(


)− 1

2

∫K(

adqiadvBqi+1, advBqi+2

)

=1

2

∫K(

advBqi+1,−advadBqi+2qi − adBqi+2adqiv)

− 1

2

∫K(− advadBqi+1qi − adBqi+1adqiv, advBqi+2

)

=1

2

∫K(

advadBqi+1qi, advBqi+2

)− 1

2

∫K(

advBqi+1, advadBqi+2qi

)

+1

2

∫K(

adBqi+1adqiv, advBqi+2

)− 1

2

∫K(

advBqi+1, adBqi+2adqiv)

125

126 Appendix B. Some Killing form identities

Now the expression becomes as follows using the invariance of the Killing.

∫K(


)

=1

2

∫K(

advadvBqi+1, adBqi+2qi

)− 1

2

∫K(

advadvadBqi+1qi, Bqi+2

)

+1

2

∫K(

adBqi+2advBqi+1, adqiv)− 1

2

∫K(

adBqi+1advBqi+2, adqiv)

= −1

2

∫K(

advadvBqi+1, adqiBqi+2

)− 1

2

∫K(

adBqi+1qi, advadvBqi+2

)

+1

2

∫K(

adBqi+2advBqi+1, adqiv)− 1

2

∫K(

adBqi+1advBqi+2, adqiv)

It follows that

∑

i∈Z/3

∫K(


)

=1

2

∑

i∈Z/3

∫K(

advadvBqi+1, adqi+2Bqi − adqiBqi+2

)−K

(advadBqi+1Bqi+2, adqiv

)

=1

2

∑

i∈Z/3

∫K(


)+K

(adBqi+1Bqi+2, advadqiv

)

It follows from Lemma 6.2.5 that the following identity holds for the operator B:

∫K(

adBqi+1Bqi+2, ·)

=

∫K(Badqi+1Bqi+2 −Badqi+2Bqi+1, ·

).

We find that

∑

i∈Z/3

∫K(


)

=1

2

∑

i∈Z/3

∫K(


)

− 1

2

∑

i∈Z/3

∫K(Badvadqi+1v, adqi+2Bqi − adqiBqi+2

).

=1

2

∑

i∈Z/3

∫K(

advadvBqi+1 −Badvadqi+1v, adqi+2Bqi − adqiBqi+2

)

=1

2

∑

i∈Z/3

∫K(


)

127

Lemma B.0.12.∫K(DxBadqi+2Bqi −DxBadqiBqi+2, adDxvBqi+1 +Dxqi+1

)

=1

2

∫K(


)

+1

4

∫K(

adqi+2Bqi − adqiBqi+2, Badvadvqi+1 + advqi+1

)

+1

4

∫K(


).

Proof. Remember that DxB =1

2+

1

2advB.


)

=1

2

∫K(

advBadqi+2Bqi − advBadqiBqi+2, adDxvBqi+1 +Dxqi+1

)

+1

2

∫K(

adqi+2Bqi − adqiBqi+2, adDxvBqi+1 +Dxqi+1

)

and it follows that

∑

i∈Z/3


)

=1

2

∑

i∈Z/3

∫K(


)

− 1

2

∑

i∈Z/3

∫K(

adDxvadqi+2Bqi, Bqi+1

)+

1

2

∑

i∈Z/3

∫K(Bqi+2, adqiadDxvBqi+1

)

− 1

2

∑

i∈Z/3

∫K(Bqi, adqi+2Dxqi+1

)+

1

2

∑

i∈Z/3

∫K(Bqi+2, adqiDxqi+1)

=1

2

∑

i∈Z/3

∫K(


)

− 1

2

∑

i∈Z/3

∫K(

adDxvadqiBqi+1, Bqi+2

)+

1

2

∑

i∈Z/3

∫K(Bqi+2, adqiadDxvBqi+1

)

− 1

2

∑

i∈Z/3

∫K(Bqi+2, adqi+1Dxqi

)+

1

2

∑

i∈Z/3

∫K(Bqi+2, adqiDxqi+1)

=1

2

∑

i∈Z/3

∫K(


)

− 1

2

∑

i∈Z/3

∫K(

adBqi+1adqiDxv,Bqi+2

)− 1

2

∑

i∈Z/3

∫K(Bqi+2, Dxadqi+1qi

)

128 Appendix B. Some Killing form identities

where the last two terms where obtained by applying the Jacobi identity and the deriva-tion property. We now have, using again Lemma 6.2.5,

∑

i∈Z/3


)

=1

2

∑

i∈Z/3

∫K(


)

+1

2

∑

i∈Z/3

∫K(

adBqi+1Bqi+2, adqiDxv)− 1

2

∑

i∈Z/3

∫K(Bqi+2, Dxadqi+1qi

)

=1

2

∑

i∈Z/3

∫K(

adqi+1Bqi+2 − adqi+2Bqi+1

, BadvadDxvBqi +BadvDxqi −BadqiDxv)

+1

4

∑

i∈Z/3

∫K(


)

=1

2

∑

i∈Z/3

∫K(

adqi+1Bqi+2 − adqi+2Bqi+1, BadvadDxvBqi +BDxadvqi

)

+1

4

∑

i∈Z/3

∫K(


)

=1

2

∑

i∈Z/3

∫K(

adqi+1Bqi+2 − adqi+2Bqi+1, BadvadDxvBqi

)

+1

4

∑

i∈Z/3

∫K(

adqi+1Bqi+2 − adqi+2Bqi+1, advqi +Badvadvqi

)

+1

4

∑

i∈Z/3

∫K(


)

=1

2

∑

i∈Z/3

∫K(


)

+1

4

∑

i∈Z/3

∫K(

adqi+2Bqi − adqiBqi+2, advqi+1 +Badvadvqi+1

)

+1

4

∑

i∈Z/3

∫K(


)

where we used the relation BDx =1

2+

1

2Badv.

129

Lemma B.0.13.

∑

i∈Z/3


)

= −∑

i∈Z/3

∫K(


).

Proof.

∑

i∈Z/3


)

= −∑

i∈Z/3

∫K(qi+1, advadBqi+2qi

)−∑

i∈Z/3

∫K(qi+1, adBi+2adqiv

)

= +∑

i∈Z/3

∫K(

advqi+1, adBqi+2qi

)+∑

i∈Z/3

∫K(

adBqi+2qi+1, adqiv)

= −∑

i∈Z/3

∫K(

advqi+1, adqiBqi+2

)−∑

i∈Z/3

∫K(

adqi+2Bqi, adqi+1v)

= −∑

i∈Z/3

∫K(


).

Appendix C

Computation of Killing form

Here we give a direct proof of Lemma 1.4.1.

Proof. Let X1, X2 be the subsets of spn defined by

X1 = E(q)ss | s = 1, . . . , n, q = i, j, k,

X2 = F (1)rs , F

(i)rs , F

(j)rs , F

(k)rs | r < s with r = 1, . . . , n− 1, s = 2, . . . , n,

respectively, in which, for instance, E(i)ss is a matrix with entry (E

(i)ss )ss = i and zero

elsewhere , F(1)rs with entry (F

(1)rs )rs = 1 and (F

(1)rs )sr = −1 and zero elsewhere, matrix

F(i)rs with entry (F

(i)rs )rs = i = (F

(i)rs )sr. and zero elsewhere. Then X1 ∪X2 is a basis for

sp(n,H).Let us denote linear map adA adB shortly by T and take element E from the first

set X1 in which Ess 6= 0. Then we can compute the matrices below.

(BE)rt =

0 if t 6= sBrsEss if t = s

, (EB)rt =

0 if r 6= sEssBst if r = s

.

Hence

[B,E]rt =

0 if r, t 6= s−EssBst if r = s, t 6= sBrsEss if r 6= s, t = sBssEss −EssBss if r = s, t = s

.

Now we need to compute (TE)ss. To do so we compute following entry of matricesA[B,E] and [B,E]A:

(A[B,E])ss = Ass(BssEss −EssBss) +

n∑

k=1,k 6=sAskBksEss,

([B,E]A)ss = (BssEss −EssBss)Ass +n∑

k=1,k 6=s(−EssBsk)Aks.

131

132 Appendix C. Computation of Killing form

Thus we have

(TE)ss = [A, [B,E]]ss

= (AssBssEss +EssBssAss)− (AssEssBss +BssEssAss)

+

n∑

k=1,k 6=s(AskBksEss +EssBskAks).

Now we take element F from the second subset X2 of basis in which Fpq 6= 0, for p < q.Then

(BF )rt = BrpFpt +BrqFqt =

0 if t 6= p, qBrqFqp if t = pBrpFpq if t = q,

and

(FB)rt = FrpBpt + FrqBqt =

0 if r 6= p, qFpqBqt if r = pFqpBpt if r = q,

Hence

[B,F ]rt =

0 if r 6= p, q−FpqBqt if r = p, t 6= p, q−FqpBpt if r = q, t 6= p, qBrqFqp if t = p, r 6= p, qBrpFpq if t = q, t 6= p, qBpqFqp − FpqBqp if r = p, t = pBppFpq − FpqBqq if r = p, t = qBqqFqp − FqpBpp if r = q, t = pBqpFpq − FqpBpq if r = q, t = q

Thus we have that

(A[B,F ])pq = App(BppFpq − FpqBqq) +Apq(BqpFpq − FqpBpq)

+

n∑

k=1,k 6=p,qApk(BkpFpq),

and

([B,F ]A)pq = (BpqFqp − FpqBqp)Apq + (BppFpq − FpqBqq)Aqq

+n∑

k=1,k 6=p,q(−FpqBqk)Akq.

133

Therefore we obtain

(TF )pq = (AppBpp +ApqBqp)Fpq + Fpq(BqpApq +BqqAqq)

− (AppFpqBqq +BppFpqAqq +ApqFqpBpq +BpqFqpApq)

+∑

k=1,k 6=p,q(ApkBkpFpq + FpqBqkAkq).

Then K(A,B) can be obtained as follows.

K(A,B) =n∑

s=1

(TE(i)ss )i + (TE(j)

ss )j + (TE(k)ss )k

+∑

p<q

(TF (1)pq )1 + (TF (i)

pq )i + (TF (j)pq )j + (TF (k)

pq )k

=

n∑

s=1

[−8 < Ass, Bss >r −6

n∑

k 6=s,k=1

< Ask, Bsk >r]

+∑

p<q

[−4 < App, Bpp >r −4 < Aqq, Bqq >r −12 < Apq, Bpq >r

− 4(∑

k 6=p,q< Apk, Bpk >r + < Aqk, Bqk >r)]

= − 4(n+ 1)

n∑

s=1

< Ass, Bss >r −8(n+ 1)∑

p<q

< Apq, Bpq >r,

where the following identities have been used:

∑p<q[< App, Bpp >r + < Aqq, Bqq >r= (n− 1)

∑ns=1 < Ass, Bss >r,∑n

s=1

∑nk 6=s,k=1 < Ask, Bsk >r= 2

∑p<q < Apq, Bpq >r,∑

p<q

∑k 6=p,q(< Apk, Bpk >r + < Aqk, Bqk >r) = 2(n− 1)

∑p<q < Apq, Bpq >r .

References

[1] Stephen C. Anco. Hamiltonian flows of curves in G/SO(N) and vector solitonequations of mKdV and sine-Gordon type. SIGMA Symmetry Integrability Geom.Methods Appl., 2:Paper 044, 17 pp. (electronic), 2006.

[2] Stephen C. Anco. Group-invariant soliton equations and bi-Hamiltonian geometriccurve flows in Riemannian symmetric spaces. J. Geom. Phys., to appear.

[3] V. I. Arnol′d. Mathematical methods of classical mechanics, volume 60 of GraduateTexts in Mathematics. Springer-Verlag, New York, 199? Translated from the 1974Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second(1989) edition.

[4] Andreas Arvanitoyeorgos. An introduction to Lie groups and the geometry of ho-mogeneous spaces, volume 22 of Student Mathematical Library. American Mathe-matical Society, Providence, RI, 2003. Translated from the 1999 Greek original andrevised by the author.

[5] Radha Balakrishnan, A. R. Bishop, and R. Dandoloff. Geometric phase in theclassical continuous antiferromagnetic Heisenberg spin chain. Phys. Rev. Lett.,64(18):2107–2110, 1990.

[6] Luigi Bianchi. Lezioni di geometria differenziale., volume VIII. Springer-Verlag,1894.

[7] Richard L. Bishop. There is more than one way to frame a curve. Amer. Math.Monthly, 82:246–251, 1975.

[8] F. E. Burstall. Isothermic surfaces: conformal geometry, Clifford algebras andintegrable systems. In Integrable systems, geometry, and topology, volume 36 ofAMS/IP Stud. Adv. Math., pages 1–82. Amer. Math. Soc., Providence, RI, 2006.

[9] F. E. Burstall, D. Ferus, K. Leschke, F. Pedit, and U. Pinkall. Conformal geometryof surfaces in S4 and quaternions, volume 1772 of Lecture Notes in Mathematics.Springer-Verlag, Berlin, 2002.

135

136 References

[10] Shiing Shen Chern and Keti Tenenblat. Foliations on a surface of constant curvatureand the modified Korteweg-de Vries equations. J. Differential Geom., 16(3):347–349(1982), 1981.

[11] Claude Chevalley. Theory of Lie groups. I, volume 8 of Princeton Mathematical Se-ries. Princeton University Press, Princeton, NJ, 1999. Fifteenth printing, PrincetonLandmarks in Mathematics.

[12] A. Doliwa and P. M. Santini. An elementary geometric characterization of theintegrable motions of a curve. Phys. Lett. A, 185(4):373–384, 1994.

[13] I. Ya. Dorfman and A. S. Fokas. Hamiltonian theory over noncommutative ringsand integrability in multidimensions. J. Math. Phys., 33(7):2504–2514, 1992.

[14] Irene Dorfman. Dirac structures and integrability of nonlinear evolution equations.Nonlinear Science: Theory and Applications. John Wiley & Sons Ltd., Chichester,1993.

[15] P. G. Drazin and R. S. Johnson. Solitons: an introduction. Cambridge Texts inApplied Mathematics. Cambridge University Press, Cambridge, 1989.

[16] V. G. Drinfel′d and V. V. Sokolov. Lie algebras and equations of Korteweg-de Vriestype. In Current problems in mathematics, Vol. 24, Itogi Nauki i Tekhniki, pages81–180. Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1984.

[17] Charles Ehresmann. Les connexions infinitesimales dans un espace fibredifferentiable. In Colloque de topologie (espaces fibres), Bruxelles, 1950, pages 29–55. Georges Thone, Liege, 1951.

[18] Luther Pfahler Eisenhart. A treatise on the differential geometry of curves andsurfaces. Dover Publications Inc., New York, 1960.

[19] L. D. Faddeev and L. A. Takhtajan. Hamiltonian methods in the theory of solitons.Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1987. Translatedfrom the Russian by A. G. Reyman [A. G. Reıman].

[20] Douglas R. Farenick and Barbara A. F. Pidkowich. The spectral theorem in quater-nions. Linear Algebra Appl., 371:75–102, 2003.

[21] A. S. Fokas and P. M. Santini. Recursion operators and bi-Hamiltonian structuresin multidimensions. II. Comm. Math. Phys., 116(3):449–474, 1988.

[22] Allan P. Fordy and Peter P. Kulish. Nonlinear Schrodinger equations and simpleLie algebras. Comm. Math. Phys., 89(3):427–443, 1983.

[23] B. Fuchssteiner and A. S. Fokas. Symplectic structures, their Backlund transforma-tions and hereditary symmetries. Phys. D, 4(1):47–66, 1981/82.

References 137

[24] Benno Fuchssteiner. The Lie algebra structure of nonlinear evolution equationsadmitting infinite-dimensional abelian symmetry groups. Progr. Theoret. Phys.,65(3):861–876, 1981.

[25] Clifford S. Gardner, John M. Greene, Martin D. Kruskal, and Robert M. Miura.A method for solving the korteweg-deVries equation. Phys. Rev. Letters, 19:1095–1097, 1967.

[26] I. M. Gel′fand and L. A. Dikiı. Asymptotic properties of the resolvent of Sturm-Liouville equations, and the algebra of Korteweg-de Vries equations. Uspehi Mat.Nauk, 30(5(185)):67–100, 1975.

[27] I. M. Gel′fand and L. A. Dikiı. A Lie algebra structure in the formal calculus ofvariations. Funkcional. Anal. i Prilozen., 10(1):18–25, 1976.

[28] I. M. Gel′fand and I. Ja. Dorfman. Hamiltonian operators and algebraic structuresassociated with them. Funktsional. Anal. i Prilozhen., 13(4):13–30, 96, 1979.

[29] I. Z. Golubchik and V. V. Sokolov. Integrable equations on Z-graded Lie algebras.Teoret. Mat. Fiz., 112(3):375–383, 1997.

[30] Metin Gurses, Atalay Karasu, and Vladimir V. Sokolov. On construction of recur-sion operators from Lax representation. J. Math. Phys., 40(12):6473–6490, 1999.

[31] R. Hasimoto. A soliton on vortex filament. J. Fluid Mechanics, 51:477–485, 1972.

[32] Sigurdur Helgason. Differential geometry, Lie groups, and symmetric spaces, vol-ume 80 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt BraceJovanovich Publishers], New York, 1978.

[33] V. G. Kac. Automorphisms of finite order of semisimple Lie algebras. Funkcional.Anal. i Prilozen., 3(3):94–96, 1969.

[34] David J. Kaup and Alan C. Newell. An exact solution for a derivative nonlinearSchrodinger equation. J. Mathematical Phys., 19(4):798–801, 1978.

[35] Shoshichi Kobayashi and Katsumi Nomizu. Foundations of differential geometry.Vol I. Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963.

[36] Norihito Koiso. The vortex filament equation and a semilinear Schrodinger equationin a Hermitian symmetric space. Osaka J. Math., 34(1):199–214, 1997.

[37] B. G. Konopel′chenko. Nonlinear integrable equations, volume 270 of Lecture Notesin Physics. Springer-Verlag, Berlin, 1987. Recursion operators, group-theoreticaland Hamiltonian structures of soliton equations.

[38] Y. Kosmann-Schwarzbach and F. Magri. Lax-Nijenhuis operators for integrablesystems. J. Math. Phys., 37(12):6173–6197, 1996.

138 References

[39] Boris A. Kupershmidt. KP or mKP, volume 78 of Mathematical Surveys and Mono-graphs. American Mathematical Society, Providence, RI, 2000. Noncommutativemathematics of Lagrangian, Hamiltonian, and integrable systems.

[40] G. L. Lamb, Jr. Solitons on moving space curves. J. Mathematical Phys.,18(8):1654–1661, 1977.

[41] Serge Lang. Fundamentals of differential geometry, volume 191 of Graduate Textsin Mathematics. Springer-Verlag, New York, 1999.

[42] Joel Langer and Ron Perline. Poisson geometry of the filament equation. J. Non-linear Sci., 1(1):71–93, 1991.

[43] Joel Langer and Ron Perline. Geometric realizations of Fordy-Kulish nonlinearSchrodinger systems. Pacific J. Math., 195(1):157–178, 2000.

[44] Peter D. Lax. Integrals of nonlinear equations of evolution and solitary waves.Comm. Pure Appl. Math., 21:467–490, 1968.

[45] Jean-Louis Loday. Cyclic homology, volume 301 of Grundlehren der Mathematis-chen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 1992. Appendix E by Marıa O. Ronco.

[46] Fernando Lund and Tullio Regge. Unified approach to strings and vortices withsoliton solutions. Phys. Rev. D (3), 14(6):1524–1535, 1976.

[47] F. Magri and C. Morosi. On the reduction theory of the Nijenhuis operators andits applications to Gel′fand-Dikiı equations. In Proceedings of the IUTAM-ISIMMsymposium on modern developments in analytical mechanics, Vol. II (Torino, 1982),pages 599–626, 1983.

[48] F. Magri, C. Morosi, and O. Ragnisco. Reduction techniques for infinite-dimensionalHamiltonian systems: some ideas and applications. Comm. Math. Phys., 99(1):115–140, 1985.

[49] G. Marı Beffa, J. A. Sanders, and Jing Ping Wang. Integrable systems in three-dimensional Riemannian geometry. J. Nonlinear Sci., 12(2):143–167, 2002.

[50] Robert M. Miura, Clifford S. Gardner, and Martin D. Kruskal. Korteweg-de Vriesequation and generalizations. II. Existence of conservation laws and constants ofmotion. J. Mathematical Phys., 9:1204–1209, 1968.

[51] Albert Nijenhuis. Xn−1-forming sets of eigenvectors. Nederl. Akad. Wetensch. Proc.Ser. A. 54 = Indagationes Math., 13:200–212, 1951.

[52] Richard S. Palais. The cohomology of Lie rings. In Proc. Sympos. Pure Math., Vol.III, pages 130–137. American Mathematical Society, Providence, R.I., 1961.

References 139

[53] Peter Petersen. Riemannian geometry, volume 171 of Graduate Texts in Mathemat-ics. Springer, New York, second edition, 2006.

[54] K. Pohlmeyer. Integrable Hamiltonian systems and interactions through quadraticconstraints. Comm. Math. Phys., 46(3):207–221, 1976.

[55] J. Scott Russell. Report on waves. Inverse Problems, pages 311–390, 1844.

[56] Jan A. Sanders. Normal form in filtered Lie algebra representations. Acta Appl.Math., 87(1-3):165–189, 2005.

[57] Jan A. Sanders and Jing Ping Wang. Combining Maple and Form to decide onintegrability questions. Comput. Phys. Comm., 115(2-3):447–459, 1998.

[58] Jan A. Sanders and Jing Ping Wang. Integrable systems in n-dimensional Rieman-nian geometry. Mosc. Math. J., 3(4):1369–1393, 2003.

[59] Jan A. Sanders and Jing Ping Wang. Integrable systems in n-dimensional conformalgeometry. J. Difference Equ. Appl., 12(10):983–995, 2006.

[60] P. M. Santini and A. S. Fokas. Recursion operators and bi-Hamiltonian structuresin multidimensions. I. Comm. Math. Phys., 115(3):375–419, 1988.

[61] Norihito Sasaki. Differential geometry and integrability of the Hamiltonian systemof a closed vortex filament. Lett. Math. Phys., 39(3):229–241, 1997.

[62] R. Sasaki. Soliton equations and pseudospherical surfaces. Nuclear Phys. B,154(2):343–357, 1979.

[63] R. W. Sharpe. Differential geometry, volume 166 of Graduate Texts in Mathemat-ics. Springer-Verlag, New York, 1997. Cartan’s generalization of Klein’s Erlangenprogram, With a foreword by S. S. Chern.

[64] Michael Spivak. A comprehensive introduction to differential geometry. Vol. II.Publish or Perish Inc., Wilmington, Del., second edition, 1979.

[65] M. H. Stone. On one-parameter unitary groups in Hilbert space. Ann. of Math.(2), 33(3):643–648, 1932.

[66] Antoni Sym. Soliton surfaces and their applications (soliton geometry from spec-tral problems). In Geometric aspects of the Einstein equations and integrable sys-tems (Scheveningen, 1984), volume 239 of Lecture Notes in Phys., pages 154–231.Springer, Berlin, 1985.

[67] Keti Tenenblat and Chuu Lian Terng. Backlund’s theorem for n-dimensional sub-manifolds of R2n−1. Ann. of Math. (2), 111(3):477–490, 1980.

[68] Chuu-Lian Terng. Soliton equations and differential geometry. J. DifferentialGeom., 45(2):407–445, 1997.

[69] Chuu-Lian Terng and Gudlaugur Thorbergsson. Completely integrable curve flowson adjoint orbits. Results Math., 40(1-4):286–309, 2001. Dedicated to Shiing-ShenChern on his 90th birthday.

[70] Chuu-Lian Terng and Karen Uhlenbeck. Schrodinger flows on Grassmannians. InIntegrable systems, geometry, and topology, volume 36 of AMS/IP Stud. Adv. Math.,pages 235–256. Amer. Math. Soc., Providence, RI, 2006.

[71] J.A.M. Vermaseren. Symbolic Manipumation with FORM, Tutorial and referencemanual. CAN, 1991.

[72] Jing Ping Wang. Generalized Hasimoto transformation and vector sine-Gordonequation. In SPT 2002: Symmetry and perturbation theory (Cala Gonone), pages276–283. World Sci. Publ., River Edge, NJ, 2002.

[73] Joseph A. Wolf. Spaces of constant curvature. Publish or Perish Inc., Houston, TX,fifth edition, 1984.

[74] Yukinori Yasui and Waichi Ogura. Vortex filament in a three-manifold and theDuistermaat-Heckman formula. Phys. Lett. A, 210(4-5):258–266, 1996.

[75] Yukinori Yasui and Norihito Sasaki. Differential geometry of the vortex filamentequation. J. Geom. Phys., 28(1-2):195–207, 1998.

[76] V. E. Zakharov and A. V. Mikhaılov. Relativistically invariant two-dimensionalmodels of field theory which are integrable by means of the inverse scattering prob-lem method. Soviet Phys. JETP, 74(6):1953–1973, 1978.

[77] V.E. Zakharov and L.D. Faddeev. Korteweg-de Vries equation: a completely inte-grable hamiltonian system. Funct. Anal. Appl., 5:280–287, 1971.

140

Summary

Many of the equations and systems which now are called integrable have been knownin differential geometry. One of them is the famous sine-Gordon equation which wasderived to describe the pseudospherical surfaces. The Backlund transformation generatesan infinite-dimensional ‘symmetry group’ acting on the set of pseudospherical surfacesand the permutability theorem of Bianchi, shows the possibility of writing down explicitsolutions starting with a simple surface. Another one is the Liouville equation describingminimal surfaces in Euclidean space. For physicists, the prototype examples of integrablesystems are the Korteweg-De Vries equation and the nonlinear Schrodinger equation.

A starting point from which all this rich structure can be derived is a zero-curvatureformulation of the underlying problem or the Lax representation of nonlinear equations.The zero curvature representation has a transparent geometrical origin. In differentialgeometry, the embedded surface is the Gauss-Codazzi equation represented as a com-patibility condition of linear equations for the moving frame.

The connection between geometry and integrable systems is clarified by Hasimotoin 1972. He found the transformation between the equations governing the curvatureand torsion of a thin vortex filament moving in a fluid and the NLS equation. In factHasimoto constructed the complex function of the curvature and torsion of the curveand showed that if the curve evolves according to the vortex filament equation, thenthis function solves the cubic nonlinear Schrodinger equation. One can find, throughthe Hasimoto transformation, the recursion operator for NLS hierarchy as well. Lateris showed that the Hasimoto transformation is induced by a gauge transformation fromthe Frenet frame to the parallel or natural frame.

Generalizing these result to the motion of a curve in the Riemannian manifold withconstant curvature following an arc-length preserving geometric evolution, gives rise tothe evolution of its curvature and torsion which proved to be Hamiltonian flow. By usingthe parallel frame, one can find the recursion operator as well as the Hamiltonian andSymplectic operators. This can be done equivaletly by Cartan structure equation havinga Cartan connection which is specified according to the frame we choose. Similarly thismethod can be used in conformal geometry as well.

In this thesis, we consider the symplectic geometry defined by the homogeneous spacewhich indeed is identified with projective quaternionic space. We study the Cartan

141

structure equation, and see that choosing natural or parallel frame, one can find thetime evolution of invariants of a family of curves embedded in the homogeneous space,the recursion operator, Nijenhuis operator as well as the Hamiltonian and symplecticoperators. Replacing the unknown variables with trivial symmetries, one can get annoncommutative integrable system. We also express all those operators in terms ofLie bracket, Killing form and projections on the underlying subspaces. The method isemployed is enough general to say that choosing a “right frame” would lead the Cartanstructure equation to integrable equations together with all geometric operators.

Generalizing the Drinfel’d-Sokolov method to the symplectic geometry, we find theLax representation of the equation we found.

142

Download - Integrable Systems in Symplectic Geometryjansa/thesis.pdffriends in Eindhoven, among them are, Amir Hossein Gharamarian, Kambiz Pournazeri and his wife Sima Nasr, Parsa Beigi, Neda

Top Related