weyl branching lie algebrasweyl orbit-orbit branching rules for lie algebras martin thoma department...
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Weyl orbit-orbit branching rules for Lie algebras
Martin Thoma
Department of Physics
McGill University, Montreal
July 1997
A thesis submitted to the Faculty of Graduate Studies and Research
in partial fulfilment of the requirements of the degree of
Doctor of Philosophy.
@Martin Thoma 1997
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Contents
A bst ract v
Résumé vii
Acknowledgment s ix
Preface x
Introduction 1
1.1 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Simple Lie algebras (Structure Theory) . . . . . . . . . . . . . . . . . 3
1.3 Classical Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . - 3
1.4 Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Weyl group . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 9
1.7 Branching rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7.1 A description of the problern . . . . . . . . . . . . . . . . . . . 10
1.7.2 Known results . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Orbit-orbit branching mies . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Orbit-Orbit Branching Rules for Classicai Lie Algebras 18
Orbit-orbit branching rules for families of classical Lie algebra-
subalgebra pairs 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2 The algebra-subalgebra pair Cm+. > Cm $ C. . . . . . . . . . . . . 21
2.3 The algebra-subalgebra pair Dm+ . 3 Dm $ Dn . . . . . . . . . . . . 24
. . . . . . . . . . . . 2.4 The algebra-subaigebra pair Bm+. > Dm $ B. 27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Concluding remarks 30
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgrnents 31
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References 31
Orbit-orbit branching rules between classical simple Lie algebras and
maximal reduct ive subalge bras 32
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Introduction 33
2.7 The algebra-subalgebra pair A,+, +l > A. $ A. $ u(1) . . . . . . 34
. . . . . . . . . . . 2.8 The algebra-subalgebra pair Bm+l > B. $ u(1) 37
. . . . . . . . . . . 2.9 The algebra-subalgebra pair Cm+1 > A, $ u(1) 39
2.10 The algebra-subalgebra pair > Dm $ u(1) . . . . . . . . . . . 42
. . . . . . . . . . . 2.11 The aigebra-subalgebra pair > A, $ u(1) 44
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Conclusions 46
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References -47
2.13 Orbit-orbit generating function . . . . . . . . . . . . . . . . . . . . . 49
Complete branching rules for the family of algebra-subaigebra pairs
SO(n) 3 SO(n . 2) x U ( l ) 53
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction 55 . . 3.2 The algebra-subalgebra B, > B,. x U(1); n 2 1 . . . . . . . . aa
3.3 The algebra-subalgebra D, > Dnmi x U(1); n 2 3 . . . . . . . . . 59
3.4 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 Orbit-Orbit Branching Rules for Affine Kac-Moody Algebras 65
. . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Affine untwisted algebras 66
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 AffineCVeylgroup 68
. . . . . . . . . . . . . . . . . . . . . . 4.3 Highest weight representations 69
. . . . . . . . . . . . . . . . 4.4 Subalgebras of untwisted affine algebras 71
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Branching rules 74
. . . . . . . . . . . . . . . . . . . . 4.6 Affine orbit-orbit branching rules 74 -- . . . . . . . . . . . 4.7 The algebra-subalgebra pair cm!+ . 3 c:) $ cf) , a
4.8 The algebrn-subalgebra pair DEL, > D:) $ D:) . . . . . . . . . . . 79
. . . . . . . . . . . 4.9 The algebra-subalgebra pair B::, > D;) $ B:) 84
. . . . . . . . . . . 4.10 The algebra.subalgebra, pair B$!+, > D:) $ Ai1) 88
. . . . 4.11 The algebra-subalgebra pair Am:,,, > A$ $ ~ f ' $ ~ ( 1 ) " ) 89
. . . . . . . . . . 4.12 The algebra-subalgebra pair B:),, > BE $ u ( l ) ( l ) 92
4.13 The algebra-subalgebra pair c::, > A$) $ ~(1)'~) . . . . . . . . . . 95
4.14 The algebra-subalgebra pair l3:L1 > D$ $ u( l ) ( l ) . . . . . . . . . . 97
. . . . . . . . . . 4.15 The algebra-subalgebra pair lltll > A$) $ ~(1)'~' 101
5 Summary and Conclusion 105
A Summary of Properties of Classical Simple Lie Algebras 107
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1 A , . n > l 109
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 B,. n z 3 111
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A 3 C,. n z 2 114
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A1 D n . n 2 4 116
Bibliography 119
Abstract
This thesis is devoted to branching rules for Lie algebras. that is the description
of decompositions of algebra representations upon restriction to a subalgebra. and
cmsists of three major parts.
In the first part the Weyl orbit-orbit branching rules are calculated for al1 classical
simple Lie algebra - maximal regular reductive subalgebra pairs:
c m + , 3 Cm @ e n ,
Dm+, 3 D m @ Rn,
B m + n 3 D m @ Bn.
- L + n + i 3 -4m @ A n $ ~ ( l ) ,
B m + l 2 Bm 8 ~ ( l ) ,
c m + , 3 -4, e3 ~ ( 1 ) .
D m + l 3 -Am @ ~ ( 1 ) .
The branching rules are given in terms of integrity bases and compatibility rules.
In the second part we use results from the first part to derive the complete branch-
ing rules ( i. e. represent ation-representation branching rules) for the algebra subalge-
bra series
so (n ) 3 ro(n - 2) $ ~ ( 1 ) .
The branching rules are given in terms of generating functions.
The third part is in character similar to the first part - the Weyl orbit-orbit
branching rules are computed for affine algebra-subalgebra pairs obtained from the
pairs listed above by afnnization. The rules are presented in terms of integrity bases
USTRACT
and compatibility rules.
Résumé
Cette thèse est consacrée aux règles de branchement pour des algèbres de Lie. C'est à
dire à la description de décompositions de représentations d'algèbres en se restreignant
à une sous-algèbre. Elle est divisée en trois parties.
Dans la première partie, les règles de branchement orbite-orbite de Weyl sont
calculées pour toutes les paires formées d'une algèbre de Lie simple classique et d'une
sous-algèbre régulière réductive maximale:
cm,, 3 Gn t3 cn, Dm+n 3 D m @ D n .
Bm+* 3 D m fB B*,
- -L+n+ l 3 -4, @ -4, a3 4. B m + i 3 Brn@u(l).
C m + l 3 A 4 r n @ ~ ( l ) ,
~ m + l 3 Dm a3 u i l ) ,
D,+i 3 A, @ ~ (1 ) .
Les règles de branchement sont données en termes de bases d'intégrité et de règles de
compatibilité.
Dans la deuxième partie, nous utilisons les résultats de la première partie pour
obtenir les règles de branchement complètes (i.e. les règles de branchement représenta-
tion-représentation) pour les séries algèbre et sous-algèbre
so(n) > ?io(n - 2) $ u(1).
Les règles de branchement sont données en termes de fonction génératrices.
La troisième partie est similaire à la deuxième. Les règles de branchement orbite-
vii
R ESCMÉ ... Vl l l
orbite de Weyl sont calculées pour des paires algèbre et sous-algèbre affines obtenues
de la liste ci-haut par affinisation. Les règles sont présentées en termes de bases
d'intégrité et de règles de compatibilité.
Acknowledgment s
1 would like to thank my supervisor Professor Robert T. Sharp for proposing the
topics of this thesis as well as for his patience and guidance throughoiit the work. 1
woiild also like to thank him for the financial support.
Further. I would like to thank:
Bucky Balls (François, Yiri. Uikko, Erol. Colin. Morten. Oleh. Rainer. Dimit re.
Mikko. Yick. Pascal. . . . ) for the soccer games and the ~'disçussion" sessions in TH
aftenvards .
Many other people from the Physics Department who made my stay here much
more enjoyable - Graham, Kostas, Sean. Bao. Alex. Youreddine. Jason, Martin. May.
Judith. Denis. Harold. . . . (and their girlfriends. boyfriends. wives. . . . 1.
The CRM people (Winternitz's. Jureo's. Jacqueline, Yannis. . . . ).
Paula. Yancy, and Joanne for their help in the bureaucratic jungle: and Paul and
Juan for their help with the cornputers.
!vlartinii's. Hiib's and al1 other "falcons" for al1 the volleyball matches.
Al1 'VI A.R.C. people for the volleyball practices.
Pavla, Melody, Zbynék. and Zdenëk for al1 the long ''Irish'' evenings and nights.
Stéphane for transforming the Abtract into le Résumé.
A very special thank you to rny parents, my brother LuboS, and my wife Zorka
without whom this thesis would have never been written.
Preface
The problem of computing branching rules. i.e. decompositions of algebra representa-
tions upon restriction to a subalgebra, is one of the well recognized and often studied
problems in representation theory. Yethods for computing branching rules have been
known for a long time. but usually they involve large amounts of calculations for each
algebra representation. Analytic solution for al1 representations or rank independent
solutions are rare.
In 1989 Patera and Sharp [PS89] came with the idea of dividing the computation
of branching rules into three steps. The first step is a decomposition of the algebra
representation in question into a sum of Weyl orbits. The second step is a calculation
of Weyl orbit-orbit branching rules (ie. decomposing algebra Weyl orbits into subal-
gebra Weyl orbits). The third step is assembling the subalgebra Weyl orbits obtaineci
in the previous step into subalgebra representations. This idea was later extended to
affine algebras by Bégin [BéggO].
The main topic of this thesis is the computation of Weyl orbit-orbit branching
rules for most subalgebras of classical simple Lie algebras and their affinizations. The
thesis is divided into four chapters and one appendix.
In the first chapter we review some basic definition from the theory of Lie algebras
over the field of complex numbers and their representations. We also define more
precisely the problem to be studied and recall some known results.
In the second chapter we calculate the Weyl orbit-orbit branching rules for ail
classicd simple Lie algebra - maximal regular reductive subalgebra pairs:
cm,, 2 GI @ cm
In the third chapter we use the results from the first part to derive the cnmplete
branching rules (ie. representation-representation branching rules) for the algebra
subalgebra series
m ( n ) 3 S O ( ~ - 2) $ u(1 ) .
The branching rules are given in terms of generating functions.
In the fourth chapter we review basic definitions and results from the theory of
untwisted affine Kac-Ziloody algebras and compute the Weyl orbit-orbit branching
rules for affine algebra-affine subalgebra pairs obtained from the pairs listed a bow
by affinization. The rules are presented in terms of integrity bases and compatibility
rules.
The appendix contains a summary of properties of classical simple Lie algebras
and their affinizations.
This thesis includes three papers CO-authored by my supervisor Professor Robert
T. Sharp and published or submitted for publication in the Journul of Mathemati-
cal Physics. In such a case the "Guidelines for Thesis Preparation" of the Faculty
of Graduate Studies and Research of >lcGill University require the following five
paragraphs to be reproduced in full:
Candidates have the option of including, as a part of the thesis. the text
of one or more papers submitted or to be submitted for publications, or
the clearly-duplicated text of one or more published papers. These texts
must be bound as an integral part of the thesis.
If this option is chosen, connecting t ext s t hat provide logical bridges
between the different papers are mandatory. The thesis must be
written in such a way that it is more than a mere collection of manuscripts:
in other rvords. results of a series of papers must be integrated.
The thesis must still conform to al1 other requirements of the "Guidelines
for Thesis Preparation". The thesis must include: .A Table of Con-
tents. an abstract in English and French. an introduction which clearly
states the rationale and objectives of the study. a review of the literature.
a final conclusion and summary. and a thorough bibliography or reference
list .
Additional material must be provided where appropriate (e.g. in appen-
dices) and in sufficient detail to allow a clear and precise judgement to
be made of the importance and originality of the research reported in the
t hesis.
In the case of manuscripts CO-authored by the candidate and others. the
candidate is required to make an explicit statement in the thesis
as to who contributed to such work and to what extent. Su-
pervisors must attest to the accuracy of such statements at the doctoral
oral defense. Since the task of the examiners is made more difficult in
these cases, it is in the candidate's interest to make perfectly clear the
responsibilities of al1 the authors of the CO-authored papers.
The material contained in the first chapter is by no rneans original. The problem
of computing Weyl orbit-orbit branching mles was suggested to me by my supervisor
Professor R. T. Sharp. The results contained in the included papers (Chapters 2
and 3) were obtained independently by the author and Professor Sharp and Later
compared. The results about affine algebras, i.e. those included in the fourth chapter,
were obtained by the author quite independently.
Chapter 1
Introduction
In this chapter we will review some basic definitions and results from the theory of Lie
algebras and their representations. They can be found in almost any standard test-
book on Lie theory. e.g.. [Hum72], [FH91]. [Ser92]. [SerS7], [Jac79]. [Cor84]. [Cah8-l].
[Fuc92]. Further. we will state the problem studied in this thesis and review the
previously known results.
1.1 Lie algebras
In this section. we give some basic definitions from the theory of Lie algebras.
Definition 1.1 A Lie algebm g is a vector space over afield F endozued with a bilinear
map [ , ] : g x g + g (called "Lie bracket" or "comrnutator") such that
[x, x] = O for every x in g (1.1)
and
[x, [y, il] + [y, (2, 111 + [z . [x, y]] = O for al1 X . y, 2 in g. ( 1-21
Unless specified otherwise, we shall consider only F = @ (cornplex numben). The
property (1.1) is cailed antisymmetry because it implies that [x, y] = -[y. x] for al1
x , y in g. The property (1.2) is called the Jacobi identity.
Deflnit ion 1.2 A su bspace i j of g is called a su balgebra of g if [q , $1 C t) . A subspace
0' of g i s called an d e 0 1 of g if [b', g] C II'.
Definition 1.3 A Lie algebra g is called sobable if there exzsts a positive integer n
such that the n th element of the derived series O(") = g. g(') = [g. 01. . . . , p i - [g? g ( k ) ] , . . . mnishes. i.e. gr") = O . A Lie dgebra g is called nilpotent if there erists
a positive integer rn such that the mth elenaent of the lower central series go = 8.
g' = [g.$~] , .. . . = [ g . g k ] . . . . uanishes. i.e. gm = 0.
Definition 1.4 The maximal solzable ideal c of g is called the radical of g.
Definition 1.5 A Lie algebra g is called si-mple if it has no ideals except itself and O
and [g , g] # O (or, equivalently, dim g > 1).
Definition 1.6 A Lie algebra g is called sernisimple if g # O and the radical r of g
equals to zero.
Theorem 1.7 Euery sernisimple Lie algebra can be written as a direct sum of si*mple
ideals.
The following Levi Theorem shows the importance of the semisimple and solvable
algebras.
Theorem 1.8 (Levi decomposition) Let g be any finite-dimensional Lie algebra
and r its radical. Then there exists a semzsimple subalgebra s of g such that g = s $ r
(direct sum of uector spaces) and [S. t] C t, i.e., g is a semidirect sum of Lie algebras
5 and r.
Thus the classification of al1 finite-dimensional Lie algebras might be doae in three
steps
1. classification of al1 simple (and thus al1 semisimple) Lie algebras
2. classification of al1 solvable Lie algebras
3. classification of al1 ways these algebras can be "combined" into new algebras.
Only the first step has been accomplished completely - the finite-dimensional simple
Lie algebras over the field of complex numbers have been classified by W.Killing and
E.Cartan in the 19th century. These algebras fa11 into four infinite series A,. B,. C,,
D, (classical Lie algebras) and five exceptional Lie algebras G2, F4, Es, E7, E8.
CHAPTER 1. INTRODUCTION
Definition 1.9 A Lie algebra g i s called reductive il its adjoint representation ad
(ad: g + End g, ad: x ct [x, .]) i s completely reducible. (See below for the definition
of representation and reducibdity.)
It can be shown that a Lie algebra is reductive if and only iE
1. Every finite-dimensional representation of g is completely reducible. or
2. The algebra g is a direct sum of a semisimple and an abelian subalgebras. (Any
abelian algebra is a direct sum of one or more one-dimensional Lie algebras
denoted in this work by n(1) or Di .)
1.2 Simple Lie algebras (Structure Theory)
Definition 1.10 A Cartan subalgebm of a Lie algebra g i s a nilpotent subalgebra
which equals i t s nonnalizer ({x E g ( [x, t) ] C b)). I ts dimension is the rank of Q.
where 4' is the space dual to the Cartan subalgebra. If Ea # 0 and a # O then o! is
called a root and E, a root space corresponding to a. The set of al1 roots is denoted
A and contains k = rank g elements (so called simple roots) ai, al , . . . , <rk such that
this set spans 4* and every root can be uniquely written as a linear combination of
the simple roots with integer coefficients. Moreover, for any root these coefficients
are either al1 non-negative (positive root) or al1 non-positive (negative root) . The set
of al1 positive (negative) roots is denoted A+ (A-).
The algebra g can be written as
which are called the root space decomposition and the triangular decomposition,
respect ively.
A symmetric bilinear form on g defined by
(x, y) = Sr(adx o ady) , for al1 X, 9 E 8 (1 .3)
is called the Killing form. The restriction of this form to the Cartan subalgebra is
nondegenerate (in fact. up to a multiple. it is the unique form with such a property).
Thus it gives a natural pairing between t) and b* and also a bilinear form on b*.
This form plays the role of a scalar product in the geornetrical picture of the Cartan
subalgebra.
For every root a there exists a unique dernent t , E f) satisfying
(t,,h) =<r(h) . forevery h E t ) .
The Killing form on fi* is then defined by
(a. (3) = ( t , . t ~ ) .
For any two roots a and 3 the following nurnber can be defined:
(a . 3) < (2.3 >=?- ' ( A J) '
For simple roots a, , 1 5 i 5 k. these numbers are integers and form the Cartan
matrix -4:
(for the classical Lie algebras, the Cartan matrices are given in Appendix A).
Theorem 1.11 With the above notation, the clgebra g is generated by 3k elernents
ei e,, E E,,. f i e-,, E E-,,, hi h, E 9, 1 5 i 5 k (= rank g) which satisfy
the fol lowing commutation relations:
where the last relation is called the Serre relation.
In fact, these relations together with the root çystem (or Cartan matrix) define
uniqueiy the Lie algebra in question. The generators ei, fi and hi form. together
with other root vectors, the Chevalley-Weyl basis which will be used throughout this
work.
We recall the following two important relations:
and
The scalar product defined in 1.7 is also the basis for the representation of the root
system in an Euclidean space with orthonormal basis el, . . . , e,. rn 2 rank g. The
change-of-basis matrices for the simple root basis, the orthonormal basis and for the
fiindamental weight basis (introduced below) for the classical Lie algebras are given
in Xppendix .A.
1.3 Classical Lie algebras
The four infinite families of finite-dimensional simple Lie algebras mentioned above
are called classical Lie algebras:
0 -4, = e l ( n + l ) . n 3 1
0 B, = so(2n + 1). n 2 2
0 C, =sp(2n), n 2 3
0 D, = 40(2n), n > 4.
The algebras Bi, Cl, C2, Dl, D2, and LI3 have been omitted to avoid repetitions
(Ai = Bi = Cl, B2 = C2, A3 z D3) and because Dl and Dz are not simple (Dl is
one dimensional abelian and D2 ci AI $ ;IL ).
These algebras have, of course, the structure described in Theorem 1.11 in the
previous section with the Cartan matrices given in Appendix A.
1.4 Representat ions
In this section we overview some basic definitions and theorems about representations
of Lie algebras.
Definition 1.12 Let V be a vector space and End V the ring of its endomorphisms.
The ring End Y with a cornmutator defined by [x. y] = xy - yx is a Lie algebra. A
hornomorphism g : g + End V is called a representation of 0. (Hence. .we can talk
a bout g -modules.)
Definition 1.13 A representation g : g + End V is called irreducible (or simple) if
V # O and the only subspaces invariant under the action of g are O and I.'. Other-
wise. i t is called reducible. The representation Q is said to be completely reducible
(or fully reducible. or semisimple) if euery invariant subspace of I' has an invariant
complement (or. equiualently, g is a direct sum of irreducible representations).
Theorem 1.14 ( H . Weyl) Euery finite-dimensional 6 . e . . dim L' < 30) representa-
tion of a semisimple Lie algebra is completely reducible.
We Say that a representation g is a representation with a highest weight .\ (or a
cyclic module) if there exists a vector v,, E V (called the highest weight vector) and
a functional A E k' (called the highest weight) such that
and
V = LL(g),u* .
The algebraLL(0) in the last equation is the universal enveloping algebra of g. Le. the
tensor algebra of g modulo the ideal generated by the elements of the form x @ y -
y @ x - [x, y], x, y E g . By equat ion (1.18), the whole g-module V is generated by
applying al1 possible sequences of elements from g to VA.
It can be shown that any such highest weight module V can be written as a
direct sum of weight subspaces VA (for al1 v E VA, ljv = X($)u) with of the form
X = A - ~ f = ~ liai. l i E Z':
v = $ VI + (1.19) k .l=ii-xs=, lia,
A weight X E h* is called an integral weight if X[hi) E L. 1 5 i 5 k . and a
dominant weight if ,\(hi) 2 0. 1 5 i 5 k.
Theorem 1.15 A representation g of a simple Lie algebra g is irreducible finite di-
mensional if and only if it is G hhighest weight representation .with a dnmhinant integral
highest weight.
Given two representations gl on VI and go? on one can form a new representation
g on I' = C; 8 b, called the tensor product of gl and QI, by
The new representation is in general reducible even if the two original representations
are irreduci ble. But if and 02 are irreducible finit e dimensional representations wit h
highest weights .Il and A?. respectively. then g contains an irreducible component
with the highest weight .il + .12. We cal1 this irreducible component the stretched
product of representations and 02.
Let di be a weight such that di (h j ) = J i j , 1 5 i, j 5 k. Then di is called the ith
fundamental weight and the corresponding highest weight module is the ith funda-
mental representation. Any finite dimensional representation can be obtained from
the fundamental representations by taking appropriate tensor products and separat-
ing the irreducible component with the highest weight (ie. the stretched product of
the representations) .
'lote: The problem of decomposing a tensor product of two representations into
irreduci ble components is one of the important pro blems of represent ation t heory and
is also closely related to Our problem of finding the branching rules:
0 Decomposing of a tensor product of two representations of an algebra g can be
viewed as calculating the branching rule for the diagonal subalgebra g of g $ g.
It has been shown ([Whi65] and references therein) for the A,+, > -4, $ A,
algebra-subalgebra pair that the coefficients m,,,, in the decomposition of the
tensor product of two irreducible representations Q,, and LI, of A,+.
and the coefficients rn;,,,,, in the branching
are very closely related (they are, in fact, the same if one introduces some
straightforward mapping between the representations of -A,+, and -4, (and
1.5 Subalgebras
The study of subalgebras of Lie algebras is a vast subject with relevancy to both
mathematical and physical parts of Lie theory. We are interested only in a srna11
fraction of al1 subalgebras: maximal reductive (or simisimple) regular subalgebras of
classical simple Lie algebras.
Definition 1.16 We say that a subalgebra fi of g is maximal if there is no subalgebm
4' of g, b' # g , b' # h satisfyingg > h' > h.
We can restrict our attention to the maximal subalgebras as the branching rules for
any non-maximal subalgebra can be obtained by combining the results for a chain of
algebra - maximal subalgebra pairs connecting it to the algebra.
Definition 1.17 A semisimple svbalgebra g' of a semisimple Lie algebra g is called
reguhr if the embedding maps the Cartan subalgebm of g' into the Cartan subalgebra
of g and the root spaces of g' into root spaces of g.
CHAPTER 1. INTRODUCTIOX 9
This definition as well as a method of finding these subalgebras and theis classification
has been given by Dynkin [DynXb, DynXa] (but origindy already in [BDSM]) and
reviewed, e.g., by [Cah84]. From this classification, one can easily obtain the following
list of maximal reductive regular subalgebras (we cal1 a reductive subalgebra regular
if its semisimple part is a regular subalgebra and the abelian part is mapped to the
Cartan subalgebra) :
1 Algebra 1 Su balge bras
Dk + Dn-Ev k = 2 . 3 . . ...-, 2 n odd.(:,n even)
D,-1 + ~ ( 1 )
+ ~ ( 1 ) . if n > 4
We are interested in the list only up to linear equivalency, Le. , we consider two
embeddings of a subalgebra equivalent if the branching rules are the same.
1.6 Weyl group
In this section we give a definition and recall some properties of Weyl group. For
each root a of g we can define a mapping
sa : fi* -t fi* x r - t x - 2 ~ a .
The Weyl group W of g is the group generated by Sa, cr simple:
W = (Sa 1 a a simple root of 8)
Note:
1. W = (Sa 1 a rootofg).
2. This is an "algebraic" definition of the Weyl group. There is also an *analytic"
definition (on the level of Lie groups).
The structure of Weyl groups is best described in the orthogonal b a i s ei of b'
and is given for each classical family in Xppendix A. For the classical Lie algebras
the structure is that of permutation groups Sn and its "variations".
The Weyl groups play a significant role in representation theory and also in math-
ematical physics due to the following properties:
1. Every finite-dimensional representation (more precisely, the set of weights of such
a representation) is invariant under the action of W. i.e. for evpry element s E W
and for every weight X of a representation Q the functional SA is a weight of the same
representation.
2. The Weyl group determines the structure of Verma modules (generalization of
representations with highest weight) (see [BGG88] for details).
3. The Weyl groups of simple finite-dimensional Lie algebras constitute a large part
of finite groups generated by reflections (Coxeter groups) . More precisely. aside from
the Weyl groups there are only the following Coxeter groups: dihedral groups 1,.
(n = 5 . n 2 7). Ha, and HA.
1.7 Branching rules
1.7.1 A description of the problem
One of the problems frequently studied in representation theory is a computation of
branching rules. i. e. reduction of represent ations of an algebra (group) wit h respect
to a subalgebra (subgroup). When dealing with fully reducible representations of the
algebra, it is enough to study only the restriction of the irreducible representations.
This is the case for finite dimensional representations of reductive algebras and also
for the highest weight representations of afine algebras studied below. The problern
is best defined when the representations in question are among the fully reducible
representations of the subalgebra (again, this will be the case for al1 subalgebras
CHA PTER 1. INTRO D UCTION 11
considered in this work).
More specifically, let Q be an irreducible representation of an algebra g and let
g' be a subalgebra of g. Thus 2 furnishes a representation of g' which is, in general,
reducible. It can be written as a direct sum of irreducible representations of g' and the
coeilicients (multiplicities) in this decomposition give the solution to the branching
problem. In the literature, there are many different notations used to write these
decompositions; among the standard ones are the following two:
and
1
where $s are the irreducible representations of the subalgebra and mi's are the
mult i plicit ies . Branching rules are interesting from both mathematical and physical points of
view due to connection to the following topics:
a Symmetriea of perturbed systems in quantum mechanics (the symmetry group
of the perturbed Hamiltonian is a subgroup of the symmetry group of the un-
perturbed Hamiltonian) .
Dynamical symmetry of quantum mechanical systems
r Symmetry breaking
Interna1 labeiing problem
r Gelfand-Zeitlin patterns
r Tensor product decomposition (Clebsch-Gordan coefficients, external labeling
pro blem) : The decomposition of a t ensor (Kronecker) product of two represen-
tations can be viewed as a computation of the branching rules for the algebra
- diagonal subalgebra pair g $ g > g.
CHAPTER 1. INTRODUCTION 12
O Weight multiplicity. Solving the branching rules for the Cartan subalgebra of
a semisimple Lie algebra solves the weight multiplicity problem for the given
represent at ion.
1.7.2 Known results
In general. one can Say that the usefulness of branching rules is directly proportional
to their simplicity. An example of the simplest possible rules is the following theorem
by Boerner [Boe'iO] (Theorem 1.1) :
Theorem 1.18 The irreducible integral representations of the / d l linear groTup
GL (n . C) remain irreducible on restriction to one of thc foollotuing subgroups:
- the real linear group GL ( n , W)
- the unimodular group SL(n. C)
- the real unimodular group SL(n. W)
- the unitary group C,
- the unimodular unitary group SC@).
There are two important surveys of branching rules which consider also algebra-
subalgebra pairs studied in this thesis. The first one is by Whippman [Whi65], where
the following cases are studied: > -4, $ A,, B, > Dm. Cm+I > Cm ( r n =
1, 2 ) , D,+l > .-Lm, and Cm+l > dm (and some other. which are not relevant to our
work). The results are given in terms of some inequalities (usually simple) which
must be satisfied by the highest weight labels of the subalgebra representations in the
decomposition. Let us quote at least the following three well known theorems [BoeiO]
(modified) (al1 weights are given in the appropriat e orthogonal bases) :
Theorem 1.19 If a representation +,, of il, îs considered as a representa-
tion of the subulgebra A,-i (embedded regularly) then it decomposes and its decompo- An-i sition contains every svch representation Q ( , ~ exactly once whose indices satisfy
the conditions
CHAPTER 1. INTROD CTCTIOX 13
(For more detaiis see also [H4165] and [MicÏO], or a- literature on Gelfand-Zeitlin
patterns.)
Theorem 1.20 If tue restrict the rep~esentations of B,, to the subalgebra D,, . then
the irreducible representations decompose in the fol&oving way:
luhere the sum is over al& systems mi . . . . . rn; which sutisfy the
imn
inequalities
The m> are integral or half-integral, according to what the mj are.
Theorern 1.21 If we restrict the representations of Dn to the subalgebra B,- . then
the irreducible representations decompose in the following way:
where the sum is over al1 systems mi, . . . . rn; which satisfy the inequalities
The mi are integral or half-integral, according to what the mj are. (Note: The em-
bedding considered is not regular and as such is not considered in our work.)
T heorem 1.22 ([Heg67]) On restrictzng a representation @fil ,..., m n ) of CR to the
subalgebra Cn-i one has the followzng splitting into irreducible representations Cm-1
@(mi ..... rn:-, of en-,:
with mi and rnf integers.
CHA PTER 1. IXTROD UCTION
'iote that these theorems are fundamental for the Gelfand-Zeitlin patterns for the
corresponding groups.
Al1 the algebra-subalgebra pairs considered in our work (and some others as well)
have been studied in the second survey by King in [Kin751 (without the u(1 ) factors)
and in [Kin821 (with the u(1 ) factors). The formulae given in these two papers involve
operations on S-functions (Schur functions. special symrnetric functions) and are not
as compact as the formulas mentioned above (but have some other advantages. e.y..
rank independency and validity for more algebra-subalgebra pairs).
For some branching rules there exist solutions in terms of generating functions (or
in terms of an integrity basis and compatibility rules. which is more or less equivalent) .
These are reviewed in [MP81] and given mainly in [PS8O]. [GPSiS], [CS80], [ShaiO].
and [PSSP] . The aglebra-subalgebra pairs for which this solution is known are: -4, > d n - l , -4, 3 --Ln-? $ -41, Br, > Dn. Dn > Bn-l, Ca > Cn-i $ -41. & > - A l q B2 > -A2. .A3 > C2, -L1 > C2, -As > C3, C3 > A2, -4 > .A2 $ Al $ ~ ( 1 ) . and some others
involving the exceptional Lie algebras.
'iloreover. in Chapter 3 the branching rules for the algebra-subalgebra pairs B, 3
Bn-l 8 u(1 ) and D, > D,-l$ u ( 1 ) are given in generating function form. They are
computed using the orbit-orbit branching rules given below in this chapter.
The branching rules can also be calculated using one of several existing coniputer
programs, e.g.:
1. LIE was developed by Arjeh M. Cohen and his colleagues from the the corn-
puter algebra group at the Centre for Mathematics and Computer Science in
Amsterdam. The program, its description and manual can be obtained from
http:/ /www .can.nl/SystemaOverview/Special/ G r o u p T h e o L i E /index.html.
It was written in C and so should be available for most Unix and Unix-like
platforms.
2. Schur was wntten by Brian G. Wybourne. It was also written in C and is
available for PC's and several Unix platforms. More information about this
program can be found on http://smc.vnet.net/Schur.html.
3. simpLie was developed by W. G. ZilcKay. J. Patera and D. W. Rand and is
available for Macintosh only. Unlike the two programs described above this
one can also calculate the orbit-orbit branching rules. A short description is
available at ht tp://~~~tv.crrn.umontreal.ca/~rand/simpLie. ht ml.
Unfortunately, the author has not have the opportunity to use any of these programs.
There are also quite extensive tables with branching rules. e.g.. [41P81].
1.8 Orbit-orbit branching rules
In this section. we define Weyl orbits and add few comments on the calculation of
the orbit-orbit branching rules presented in the next chapter.
Definition 1.23 Given a aweight X let
The set WX is called the W e y l orbit of A.
Every finite dimensional representation of a reductive Lie algebra can be written as a
direct sum of Weyl orbits. Each orbit contains precisely one dominant weight - called
the highest weight of the orbit: the components of the highest weight serve as orbit
labels. They (including rnultiplicities) are the summing indices in the decomposition
of a representation into Weyl orbits.
In 1989, J. Patera and R.T. Sharp [PS89] studied a new way of computation
of branching rules for representations of Lie algebras - via Weyl orbits. Thep also
proved that upon restriction to a subalgebra each Weyl orbit decomposes into a
direct sum of subalgebra Weyl orbits, and, together with F. Gingras, calculated some
of the necessary (Weyl) orbit-orbit branching rules [GPS92, GPS91, GPS931. In this
work, the orbit-orbit branching rules for al1 maximal reductive regular subalgebras of
classical simple Lie algebras ([TS96d, TS96bl) are obtained.
Some of the orbit-orbit branching rules calculated below were known. In [GPS92]
and [GPS91], the orbit-orbit branching rules between each simple Lie algebra of rank
at most 5 and its equal rank aubalgebras are given. Also treated are the generic case
-1, > x G'(l), the subjoining F4 > B3 x LT(l), ail equal-rank subalgebras of
Eg. and the pair Ea > Es x -+. We have used these results to check the validity of
our formulas for low rank cases. The results generally agree ercept for the following
misprints (for notation see the cited papers):
Case 3.34 Cs > C3 x C2
Q is not compatible with h3 and is compatible with j4.
a Case 3.17 BLI > A4 should read Bq > D4
Case 3.29 Bs > -q3 x C1
g3 in the tables should read h3.
a Case 3.26 .A5 2 4 x L7(1)
e; = [Oi00, %] should read e$ = [OiOO. $1.
a Case 3.28 .-la > d2 x d2 x C(1)
a,) = . . . shodd read a,)* = . . ., 6 = [Ol, IO,O] shodd read , = [01101,0],
and in the 1 s t table b 2 ) should read bf" and b!) should read by)'.
a Case 3.12 & 3 A2 x U(1)
In the last table bl should read b?, and the compatibility rules should be as
follows: ai is compatible with b2, c2> d3, e3 and 4, a; is compatible rvith b2. c:.
da, e; and 4, also, b2 is compatible with d3 and 4, c? is compatible with d3
and e ~ . and c; is compatible with e j and 4; al1 other pairs are incompatible.
a Case 3.39 D5 > 4 x U(1)
d3 is compatible with g&
All results from this work can be used also for the compact real form of the algebras
involved ( su(n) for d ( n , C), 40(2n+ 1) for 50(2n+l, C), ep(n) for sp(n, C) , and 54272)
for so(Zn,@)) as the algebra-subalgebra relations remain the same and there is a
one-to-one correspondence between the finite-dimensional complex representations of
a complex simple Lie algebra and its compact real form.
Our work cannot be generalized to highest weight modules with non-integral high-
est weights (e.g. Verma modules) as these are. in general, not invariant under the
action of the Weyl group (or. more precisely. their weight system is not invariant).
The decompositions of orbits calculated in the next chapter can be checked by
computing the dimension of both the algebra and subalgebra orbits. This cornparison
confirms our results but does not give any new insight as it involves only the well
known Vandermonde convolution formula
Chapter 2
Orbit-Orbit Branching Rules for Classical Lie Algebras
In this chapter. two papers coauthored by my supervisor Professor Robert S. Sharp
are presented. In the first paper the orbit-orbit branching rules for the classical sim-
ple algebra-maximal semisimple regular subalgebra pairs are derived and presented in
terrns of integrity bases and compatibility rules. In the second paper the same prob-
lem is solved in a similar manner for maximal regular reductive but not semisimple
subalgebras of classical Lie algebras. The presentation of branching rules in terrns
of an integrity basis and compatibility rules is equivalent to a presentation in terms
of generating functions but the transformation is not always simple. In the closing
section of this chapter we discuss this transformation.
Orbit-orbit branching rules for families of classical Lie algebra-subalgebra pairs
M. Thoma and R.T. Sharp Physics Department, McGill University,
Montreal, Quebec, H3A 2T8, Canada
April 1, 1996
A bst ract
Complete orbit-orbit branching rules are derived for each classical algebra-
maximal subalgebra pair Cm+, > Cm $ C,, B,+, > Dm $ Bn+ Dm+, > Dm $ D,. Since each pair is equal-rank. and algebra and subalgebra Weyl
sectors line up. the integrity basis in each case consists of the subalgebra
orbits contained in the fundamental or bits of the aigebra.
Journal of Mathematzcal Physics 37 (l996), no. 9, -1750-4757.
Introduction
In physics it is generally useful. when possible, to reduce an object of interest to
smaller "building blocks". We are thinking of the reduction of irreducible represen-
tations (IR's) of a simple, or semisimple. Lie algebra to Weyl orbits (W-orbits. or
simply orbits). a device which has seen little exploitation so far in applications of
group theory.
.A particular use of W-orbits is as an intermediate stage in finding branching rules
between IR's of algebra and subalgebra. The procedure consists of three steps:
1. Reduction of the algebra IR into algebra W-orbits; this can be done by a re-
cursive routine [1.2] or better by a procedure described Later in this paragraph
under step 3; extensive tables exist [3].
2. Reduction of algebra W-orbits to subalgebra W-orbits. the subject of this paper.
which treats algebra-subalgebra pairs listed in the abstract .
3. .-\ssembling subalgebra orbits into subalgebra IR's: the W-orbits of the sub-
algebra IR'S inay be lifted, one IR at a time, starting with the highest. from
the collection of subalgebra orbits. hlternatively, each subalgebra orbit can be
written directly as a superposition of subalgebra IR's [SI. The relevant orbit-IR
triangular matrix (for the algebra) can be inverted for a solution of step 1 above.
These steps have been applied to a few low-rank Kac-Moody algebras to obtain IR-IR
branching rules [0].
Usually, in representation theory, it is simplest to use a fundamental weights
basis in weight space. In the present context we find it more convenient to use an
orthonormal weights basis for the most part; it is easier then to recognize to which
algebra W-orbit a given subalgebra W-orbit belongs.
We complete this section with some information from recent papers which give
orbit-orbit branching rules for some low-rank algebras [4,6]: the t hree families to be
considered here have the properties that algebra and subalgebra have the same rank
and that Weyl sectors of algebra and subalgebra line up, that is. each subalgebra
Weyl sector contains only complete algebra sectors. Shen the elementary subalgebra
orbits consist of the subalgebra orbits lying in the fundamental orbits of the algebra:
they form the integrity basis from which al1 subalgebra orbits are formed by taking
stretched products (orbit labels al1 additive). Vie then have only to find which pairs
of elementary orbits are compatible and which incompatible to complete the solution
of the problem.
2.2 The algebra-subalgebra pair Cm+, > C , $ C ,
We treat this family first because it is simplest.
As basis vectors in weight space we use the 1 = m + n orthonormal vectors e;(i =
1. . . .. 1). Weyl reflections of Cl consist of sign reversals ei + -ei and interchanges
The simple roots are ûi = ei - ei, l . ( i = 1. . . . . 1 - 1) and cq = 2ei: the extended
simple root is û.0 = -2ei (for simplicity we have multiplied the simple roots. and the
fundamental weights below. by 2;). The fundamental weights di are given in terms
of the simple roots by the inverse Cartan matrix.
and thus in terms of the orthonormal basis we find
We label a W-orbit by the components Xj of its highest weight in a fundamental
weights basis. The fundamental orbit [il (A j = 6, (the Kronecker delta)) has high-
est weight wi. According to (2.2) the weights of the orbit [il consist of al1 linear
combinations of i distinct eh? each with coefficient f 1.
The simple roots of Ci = Cm+,, Cm and C, are shown in Fig. 2.1. The simple
roots cri of Cm and a; of Cn are given in terms of those of of Ci by
m rn-1 m-2 j 1 I k n-2 n-1 n -...+....-...*...- O 1 2 m-j m-1 m m+l m+k 1-2 1-1 i
Figure 2.1: The Dynkin-Coxeter diagram for Cm+, > Cm $ C,. The numbers belon- the diagram label the simple roots of Cl ( 1 = rn + n) . Those to the Ieft above label those of Cm, and those to the right above label those of C,.
Hence the fundamental weights of Cm and C, are respectively
The elementary Cm $ C, W-orbits are thosc contained in the fundamental orbits
[il of Cl. We denote one of them by b: k). the direct product of the fundamental
orbits b] of Cm and [k] of C,; the ranges of j and of k = i - j are specified below (in
(2 .5)) . CVe may write
min ( lm)
j is the number of eh in a weight of [il for which 1 5 h 5 m and k is the number in
the range m + 1 5 h $ 1 : for j = 0. b] is the zero or point orbit of Cm and for k = 0.
[k] is the zero orbit of C,. The elementary orbits b; k] are shown in tabular form in
Fig. 2.2.
The elementary orbits constitute the integrity b a i s for al1 subalgebra orbits:
higher orbits are stretched (orbit labels additive) products of powers of the elementary
It remains to find the compatibility rules, Le., the answer to the question "Which
pairs of elementary orbits can appear together in a product yielding a higher orbit?".
Consider the stretched product of the elernentary orbits Ij; k] and b'; k'] which belong
respectively to the CI orbits [il and [il] with i = j + k and i' = j' + k'. We may
suppose i' > i (two elementary orbits in the same fundamental orbit are known to
Figure 2.2: The elementary Cm+, > Cm $ Cn W-orbits. The elementary orbit b: k] belongs to the ( j + k)th fundamental W-orbit of Cm+,. The eiernentary orbits b: k] and kt: kt ] are compatible if j' 2 j . k' 2 k (we suppose j' + kt > - j + k ) .
be incompatible). Our product must belong to the Cl orbit [i. i'] which has labels
Ah = &hi + bhil ( i t h and i'th labels unity, other labels zero): each weight of [i. if] has i
eh with coefficient f 2 and it - i with coefficient f 1; the other l - i' eh have coefficient
O. The stretched product [j; k] - b'; kt] belongs to the Cm $ C, orbit [j. j': k. kt ] with
Cm labels X i = 6hj + bhjl and Cn labels = bhk + bhr. kVe may suppose j' 2 j
(otherwise interchange the roles of Cm and C,); among the first rn eh's there are then
j with coefficient f 2 and j' - j with coefficient f 1. In the last n eh's there are thus
i - j = k with coefficient f 2 and i f - i - j' + j = k' - k with coefficient f 1. It follows
that k' 2 k; two elementary orbits are incompatible if one lies above and to the right
of the other in Fig. 2.2.
Our solution for the Cm+, > Cm $ C, orbit-orbit branching rules is now complete.
For the subalgebra orbit content of the general Ci orbit [A1,. . . , XI] select a sequence
of elementary orbits, one [ j i t : kit] frorn each diagonal jit + kir = i' = constant for which
Ait # O. Each one chosen must be compatible with the preceding one Li: ki] in the
list. i.e., j i t 2 ji, kit 2 ki. Each such sequence corresponds to one subalgebra orbit
[A: . . . . , Am: A i . . . . . A:] with A', = Ei Aishjii X i = Ci Xibhki (the stretched product of
the chosen mutually compatible elementary orbits, the ith one used X i times).
2.3 The algebra-subalgebra pair Dm+, > Dm @ D,
In this section and the next (B,,, > Dm @ B.), to Save space, we sçtate our resiilts
without detailed proofs; the proofs are very similar to those given in $2.2 for Cm+, > Cm $ C,. and only slightly more complicated.
We assume here that rn and n are both greater than 1. Otherwise ive would have
to take Dm and/or D, to be u ( 1 ) and the details would be rather different: we hope
to treat > Dm $ u (1 ) in a later paper.
.As basis vectors in weight space we use I = rn + n orthonormal vectors ei ( i =
1. . . . . I l . Weyl reflections for Di consist of interclianges ei +t e j and two sign reversals
at a time ei -t - e i , e j + -e j .
The simple roots are ai = ei - ei+l, ( i = 1.. . . . l - 1) and cul = el-1 + el. The
extended simple root is a0 = -el - e ~ . The fundamental weights di are given in terms
of the simple roots by the inverse Cartan matrix (see Eq. (2.1)) and thus in terms of
the orthonormal basis we find
The W-orbit labels are defined as in $2.2 following Eq. (2.1). According to (2.6) the
weights of the fundamental orbit [il, i = 1, . . . , I - 2 consist of al1 linear combinations
of i distinct eh with coefficients 4~1; the weights of [l - 11 and [ I l consist of linear
combinations of al1 e h , an odd number with coefficient - and the rest with coefficient
+$ - for [Z - 11, an even number with coefficient -+ - and the rest with coefficient ++ - for [ I l .
The simple roots of Di. Dm and Dn are shown in Fig. 2.3. The simple roots a; of
i> m-j
I...
Figure 2.3: The Dynkin-Coxeter diagram for Dm+, > Dm $ LI,. The numbers below the nodes label the simple roots of Di (1 = m + n). Those t o the left above label those of Dm, and those to the right above label those of D,.
Dm and a! of 3, are given in terrns of Di simple roots by
Then the fundamental weights ui: of D m and of D, are respectively
d: = - eh, j = l ..... m - 2 .
The elementary Dm $ D, orbits are those contained in the fundamental orbits [il
of Di. We denote one of them by [j: k], the direct product of the orbit [j] of Dm and
[k] of D,, to be specified below. We may write
[m: nl + [ml; n]
CH,-IPTER 2. CLASSICAL ALGEBRAS 26
In the first of Eqns. (2.9). j is the number of eh in a weight of [il for which 1 5 h 5 rn
and k = i - j is the number in the range m + 1 5 h 5 1: for j = O b] is the zero orbit
of Dm and for j = 1.. . . . m - 2 it is j t h fundamental orbit; for j = rn - 1 it is the
Dm orbit for which the last two labels. the ( m - 1) t h and mth, are unity and the rest
zero: for j = m there are two Dm orbits [ml. [m']. The first. [ml. has the rnth label
2 and the rest O. while the second. [rn'], has the (m - l ) t h label 2 and the rest O.
The statements in the preceding sentence are al1 valid with the replacements j + k.
m u n. For i = 1 - 1 (the second of Eqns. (2.9)) there are two elementary orbits.
[m; n'] and [ml: n]: for i = 2 (the third of Eqns. (2.9)) there are also two. [m: n] and
[ml: dl: here [ml means the mth fundamental orbit of Dm and [m'] is
similarly with the replacement m tt n.
Fig. 2 .A shows the elementary subalgebra orbits descri bed above.
the ( m - 1) th :
Except for the
Figure 2.4: The elementary Dm+, > Dm $ D, W-orbits. Which ones belong to a Dm+, orbit [il and which pain are compatible are stated in the text.
four in the lower right hand corner those for which j + k = i belong to the orbit [il of
Di. The two a t the bottom of the nth column belong to [l - 11 while the two at the
right of the mth row belong to [ I l . To complete the description of our solution we rnust now give the compatibility
rules.
Consider a pair of elernentary orbits in Fig. 2.4. of which neither is in the lower
right corner, Le., belongs to the (1 - 1) th or Ith fundamental orbit of Dl and not
both of which lie in the bottom row with j = m or m' nor both in the right coliimn
with k = n or nt. Then they are incompatible if and only if one lies above and to
the right of the other. If both are in the right column and at most one in the lower
right corner they are compatible if and only if both are labekd n or both labeled n'.
Similarly if both are in the bottom row and at rnost one in the lower right corner.
they are compatible if and only if both are labeled m or both labeled rn'. Those in
the lower corner are compatible with al1 those for which j 5 m - 1 and k 5 n - 1.
Finally both in the ( 1 - 1) t h fundamentai orbit of Di are compatible with both in the
Ith fundamental orbit.
For the complete orbit-orbit branching rules consider a Di orbit. For each non-
zero label Xi select one elementary subalgebra orbit from the i th fundamental orbit
of DI so that al1 the elernentary orbits chosen are mutually compatible. Then form
the stretched product (orbit labels additive) of the elementary orbits chosen. the ith
one used Xi times. Each such choice of elementary orbits gives one Dm $ D, orbit in
the Di orbit [ X I , . . . , X i ] .
2.4 The algebra-subalgebra pair B,+, > D m $ B n
We suppose that m > 1 since Di would be u ( l ) and the details different. We hope
to deal with Bm+L > B,,, $ u(1) in a future paper.
As basis vectors in weight space we use 1 = rn + n orthonormal vectors eh ( h =
1, . . . , 1 ) . Weyl refiections for Bl consist of sign reversals ei -t -ei and interchanges
ei # e j .
CHAPTER 2. CLASSICAL ALGEBRAS 28
The simple roots are ai = ei - ei+l, (i = 1, . . . , 2 - 1) and = el: the extended
simple root is a0 = -el - e2. The fundamental weights are given in terms of the
simple roots by means of the inverse Cartan matrix and in terms of the orthonormal
basis we have
The W-orbit labels are the components X i of the highest iveight of the orbit in a
fundamental weights basis. Accordhg to (2.10) the weights of the fundamental orbit
[il, i = 1.. . . . l - 1 consist of al1 linear combinations of i distinct eh with coefficients
f 1: the weights of the last fundamental orbit [Z] consist of linear combinations of al1
eh with coefficients fi. - The simple roots of 4. Dm and B, are shown in Fig. 2.5. The simple roots û) of
Figure 2.5: The Dynkin-Coxeter diagram for B,,, > Dm $ B,. The numbers below the nodes label the simple roots of Bi (1 = rn + n) . Those to the left above label those of Dm, and those to the right above label those of B,,.
Dm and a; of B, are given in terms of the Bl simple roots by
a: = a,+, j = l . .... m,
a;I = ~ , + b , k = l , . . .,n .
CHAPTER 2. CLASSICA L AL GEBRAS
Shen the fundamental weights (c:. of Dm and i ~ y of B, are respectively m
J. = - I 1 et,. j = 1 ,..., r n - 2 ,
The eiementary Dm $ Bn orbits are those contained in the fundamental orbits [il
of Bi. They may be written b: k], the direct product of a Dm orbit [j] and a B, orbit
Explicitly we have min ( i , r n )
[ I l > [m: n] + [ml:n]. the number of eh in a weight of [il for which 1 5 h < rn and k = i - j is the
number in the range m + 1 5 k 5 1. For j = O. b] is the zero orbit of Dm and for
k = O, [k] is the zero orbit of B,. For j = 1.. ... m - 2, b] is the j t h fundamental
orbit of Dm and for k = 1.. ... n - 1. [k] is the kth fundamental orbit of B,. For
j = m - 1 b] is the Dm orbit with the (m - l ) t h and mth labels unity and the rest
zero; for j = m there are two Dm orbits. [ml, which has the rnth label 2 and the rest
0, and [ml], which ha3 the (m - 1)th label 2 and the rest O. For k = n, the B, orbit
[n] has the nth label 2 and the rest O. For i = 1 , there are two Dm $ Bn orbits [m: n]
and [m'; n] where [ml and [ml] are respectively the rnth and (m - 1)th fundamental
orbits of Dm and [n] is the nth fundamental orbit of B,.
It remains to give the compatibility rules for BI > Dm $ Bn; they are very similar
to those for Ci > Cm $ C, and Dl > Dm $ D,.
If not more than one of a pair lies in the bottom row of Fig. 2.6, i.e.. if j # m
for one of the pair, then they are incompatible if either lies above and to the right
of the other, otherwise they are compatible. If both lie in the bottom row they are
compatible if both are labelled m or both labelled m'; otherwise they are incompatible.
Figure 2.6: The elementary B,+, > Dm $ B, lV-orbits. The elementary orbit Ij: k] belongs to the ( j + k) th fundamental IV-orbit of &+, (two of them when k = m ) .
The instructions for finding a complete set of compatible orbits and hence the
complete orbit-orbit branching rules are the same as those for Ci > Cm $ C, in $2.2
and for Di 2 Dm $ D, in $2.3.
ing remarks
We hope in the near future to publish orbit-orbit branching rules for compact algebra-
subalgebra families in which the subalgebra is reductive but not semisimple, i .e..
contains a u ( 1 ) factor. We are thinking of > A, $ A, $ u ( l ) , Cm+i > A, $ u(l) and > A, $ u(1). Shen we hope to deal with families of Kac-
Moody algebra-subalgebras; we had started on this problem when we noticed that
the classical problem had never been solved and decided it should be dealt with first.
Acknowledgment s
The work was supported in part by the Y'ational Science and Engineering Research
Couiicil of Canada and by the Fonds FCAR du Québec.
References
[l] R.V. !doody and J. Patera, "Fast recursion fomula for weight multiplicities".
Bull. Am. Math. soc.7, 237-242 (1982)
[2] M.R. Bremner. "Fast compzltation of weight multiplicities" J . Symbol. Comput. 2.
337-362 (1986).
[3] M.R. Bremner. R.V. Moody and J. Patera, "Tables of dominant cweight multiplie-
dies for representatzons of simple Lie algebras", ( M . Dekker, New York, 1985).
[5 ] J. Patera and R.T. Sharp. "Branching rules for representations of si-mple Lie
algebras through Weyl group orbit reduction". J. Phys. A: MathGen. 22. 2329-
2340 (1989).
[6] F. Bégin and R.T. Sharp, "Weyl orbits and their expansions in irreducible repre-
sentations for a8ne Kac-Moody alge bras ", J. Math. Phys. 33,2343-2356 (1992).
[A] F. Gingras, J. Patera and R.T. Sharp, "Orbit-orbit b~unching rules between simple
low-rank algebras and equal rank subalgebras", J. Math. Phys. 33. 16184626
(1992).
Orbit-orbit branching rules between classical
simple Lie algebras and maximal reductive
subalgebras
M. Thoma and R.T. Sharp Physics Department, McC-ill University.
Montreal. Quebec, H3A 2T8, Canada
Abst ract
Complete orbit-orbit branching rules are found for the classical algebra-
maximal reductive subalgebra pairs .4,+,+1 > -4, $ -4, $ ~ ( 1 ) . B,,.,+i >
Bm@u(l)- C m + i 3 -L@u(~). D m + l 3 D m e ~ ( 1 ) . and Dmii > .-L,$u(l).
Journal of Mathematical Physics 37 (1996), no. 12, 6370-6581.
CHAPTER 2. CLASSICAL ALGEBRAS
2.6 Introduction
In a recent paper [l], to be referred t o as 1. we found orbit-orbit branching rules
for the classical algebra-subalgebra pairs Cm+, > Cm $ C,, Bm+, > Dm $ B,!
Dm+, > Dm $ D,. Our motivation here is similar, and we refer to 1 for an account
of it.
CVe now consider the algebra-subalgebra pairs Am+,+ > A, $A, $ ~ ( 1 ) . Bmci > B, $ ~ ( l ) , Cm+[ > -4, $ u (1 ) . Dm+[ > Dm $ u ( 1 ) . > Am $ u ( 1 ) . in which the
subalgebra contains the factor u(1 ) : this completes the cases in which the algebra is
classical and the subalgebra is maximal and equal-rank (we regard u (1) as contri but ing
1 to the subalgebra rank). As for the cases considered in 1, explicit representation-
representation branching rules for the present algebra-subalgebra pairs are not known
for general values of rn, n. Weyl sectors of algebra and subalgebra line up. Le.. each
subalgebra sector contains only complete algebra sectors.
In recent publications [2.3] it is shown that for the cases iinder consideration
(equal rank algebra-subalgebra pairs. with Weyl sectors lining up) the integrity basis
(elementary Weyl orbits), in terms of which the subalgebra Weyl orbits (or. simply.
orbits) contained in al1 algebra orbits may be erpressed as stretched products (al1
orbit labels additive, ie.. each algebra and subalgebra label of the product orbit is
equal to the sum of the corresponding labels of the factor orbits) consists entirely of
the subalgebra orbits contained in the fundamental orbits of the algebra. LVe label
an orbit by the components of its highest weight in a fundamental weights basis: a
fundamental orbit is one which has a fundamental weight as its highest weight (one
label is unity, the rest zero).
We use orthonormal vectors ei for the most part as our basis in weight space rather
than the more commonly used fundamental weights - it is easier then to recognize
to which algebra orbit a subalgebra orbit belongs.
The u(1) label, u, is centered so that its sum over al1 states of an algebra orbit
vanishes, and its scale is such that the spacing between adjacent values in the same
algebra orbit is unity. Unlike other orbit labels u can take negative and fractional
values.
For each algebra-subalgebra pair we define a u ( 1 ) fundamental weight i.,. It is
orthogonal to the other subalgebra fundamental weights (and roots) and points in the
u ( 1 ) direction. The presence of ud, in the weight of a state indicates that its u ( 1 )
label is u .
The branching rules given in 1 for B,+, > Dm $ B, and Dm,, > Dm $ D, are
inapplicable for rn = 1. when Dm becomes u ( 1 ) ; that is why B,+I > B, $ u ( 1 ) and
Dm+1 > Dm $ u(1) are treated here.
2.7 The algebra-subalgebra pair A,+,+1 > A, $
A n @ 4 1 )
-4s basis vectors in weight space we use the 1 + 1 ( I = rn + n + 1) orthonormal vectors
e ; , i = 1,. . . , l + 1. The Weyl group is generated by the interchanges ei +t e j .
The simple roots of .Ai are ûi = ei - ei+i. i = 1. . . . . I . The fundamental rveiglits
are expressed in terms of the simple roots by the reciprocal of the Cartan matris
in terms of the orthonormal basis we find
ai and wi are al1 orthogonal to &ll e h and hence lie in an I = rn + n + l dimensional
weight space.
For any weight of the fundamental orbit [il according to (2.15) the number of en
with coefficient (1 - i + 1) / (1+ 1) is i and the number with coefficient -il([ + 1) is
1 - i + 1 .
The simple roots of Ai, A, and A, are shown in Fig. 2.7. In terms of the simple
roots of Al the simple mots ai of A, and ai of A, are cri = aj, j = 1, . . . , m. and
Figure 2.7: The Dynkin-Coxeter diagram for > -4, $ -4, @ u ( 1 ) . The numbers below the diagram label the simple roots of .-Li = The first m above label those of -4, and the last n above label those of A,. The direction unlabeled above corresponds to that of ~ ( 1 ) . orthogonal to the other I - 1 = m + n simple roots of Ai.
= am+k+l. k = 1, . . . . n. The fundamental weights ~ i : of -4, and J[ of -4, are
The fundamental u(1) weight is
m+ 1 1+ 1
J, = (m + 1)-l 1 eh - (n + 1)-' C e h . h= 1 h=m+2
The algebra weight [XI. A?. . . .. X I ] becomes, in the subalgebra basis. [XI. A?. . . . . A,: \ / m+2. . . . , X I : u], where
The elementary -4, $ il, $ u(1) orbits are those contained in the fundamental
-4, orbits [il. A representative one, Ij: k], in [il has as the weight of its highest state
d! , + ,, + uw,, where
min (m+l , i )
[il> 1 [ j ; i - j ] . j=max(O,i-n- 1)
When j = O or rn + 1 the point orbit of A, is understood; similady for the ;In orbit
when k = O or n + 1. It may be checked that the weights wi + w{ + uw, have i e's
with coefficient (Z - i + l ) / ( l + 1) and 1 - i + 1 with coefficient - i / ( l + 1) and therefore
belong to the -4, orbit [il; moreover the weights of [j; k] exhaust those of [il.
It remains to find the compatibility rules between pain of elementary orbits.
Consider the stretched product [j; k] kt ; kt]; it implies the orbit Ij, j': k. kt] of -4, $
-4, $ u ( 1 ) where the -4, orbit b, j'] is that with labels Xh = dhj + d ; h j t and sirnilarly
for the orbit [k. k'] of -4,. The pair U; k], b': k'] is compatible if and only if their
stretched product belongs to the orbit [i. i'] of -4, where i = j + k and i' = j' + k f .
We may suppose i' > i (two elementary orbits in the same .4/ orbit are known to bc
incompatible). We may d s o suppose j' > j (otherwise interchange the roles of A, and
.-Ln). Xow a weight of the -4, orbit [i. il] has i e's with coefficient (21 - i - i' + 2)& + 1 ) .
il-i with coefficient ( 1 - i - i t+ 1)/(1+1) and 1 - i f + 1 with coefficient - ( i + i f ) / ( l + 1).
The A, orbit b. j'] has weight u: + +;, + [ ( j + j t)(1 - m ) - ( k + k') (m + l ) ] [ ( l +
l )(n + l)]-1 CF!: eh; we have included the part of the u(1) weight component for
which h 2 rn + 1. There are j e's with coefficient (21 - i - i' + 2) /(Z + 1 ) . j' - j with
coefficient ( 1 -i-i1+l)/(2+l) and m+l-j' with coefficient - ( i+i t ) / ( l+l ) . Cornparing
this with the number of e's in the .-Il orbit with each coefficient we conclude that in
the -4, part of the weight there are k e's with coefficient (21 - i - i' + 2)/ (1 + 1 ). k' - k
with coefficient ( 2 - i - i f + 1 ) / ( 1 + 1 ) and n - k' + 1 with coefficient -(i + i1) /(2 + 1 ) .
This is possible only if k' 2 k. So [j; k] and [j': k'] are compatible if and only if j' 2 j
and k f 2 k (recall i l > i ) . The compatibility condition is shown diagrammatically in
Fig. 2.8. Two elementary orbits are incompatible if one lies above and to the right of
the other.
So the orbit-orbit branching rules are compiete. To get the -1, $ -4, $ u(1) orbit
content of the Ai orbit [XI, A Z , . . . , XI], choose a series of boxes Li; ki], one from each
diagonal row i = ji + ki for which X i # O, such that each is compatible with the
last preceding i (and therefore with ail preceding i). Each such series contributes
one subalgebra orbit [ A i , . . . , A&; A'[, . . . , A t ] with X i = Ci .Aibhji, = Ci Xibhki (the
stretched product of the chosen rnutually compatible orbits); the u ( l ) labels are also
additive. The same strategy can be used to get the subalgebra orbit content of an
Figure 2.8: The elementary > -4, $ .-In $ u(1) W-orbits. The elementary orbit b; k] belongs to the ( j + k)th fundamental orbit of Two elementary orbits are compatible only if one lies below and/or t o the right of the other.
arbitrary algebra orbit for the algebra-subalgebra pairs considered in the next four
sections: we will not repeat it for them.
2.8 The algebra-subalgebra pair B,+l > B,$u(l)
In this and the next three sections (2.8-2.11), to Save space. we state our results
without detailed proofs: the proofs are very similar to those given in s2.7 for > -4, $ -4, $ ~ ( 1 ) . The roots of B, (1 = m + 1) are f e i , i = 1. .. . . I and &ei k' e j .
i = 1.. . . , l - 1 and j = i + 1'. . . , [ where ei, i = 1,. . . , l are orthonormal vectors:
the prime above is to indicate that the I signs are independent. The Weyl group is
generated by sign reversals ei -t -ei and interchanges e i +t ej .
The simple roots of Bi are ai = ei - ei+l, i = 1,. . . , 1 - 1 and al = el. The
fundamental weights are given by
In any weight of the fundamental orbit [il, i = 1. . . . . 1 - 1. the number of e h with
coefficient A l is i. the nurnber with coefficient O is I - i: in the fundamental orbit [ I l . al1 L eh have coefficient kt.
The simple roots of Bi and B, are shown in Fig. 2.9. The simple roots of B, are
Figure 2.9: The Dynkin-Coxeter diagram for B,+I > B, $ u(L). The numbers below the diagram label the simple roots of Br = B,+I. Those above label those of B,. The direction unlabeled above corresponds to that of ~ ( 1 ) . orthogonal to the simple roots and fundamental weights of B,.
The fundamental weights of B, are
The fundament al u (1) weight is
The algebra weight [XI, A*, . . . , X I ] becomes, in the subalgebra basis, [A2, X3! . . . , X I :
IL] ' wit h
CHAPTER 2. CLASSICAL .-IL GEBRAS 39
The elementary B, $ u(1) orbits are those contained in the fundamental Bi orbits
[il. The first rn orbits contain 3 elementary orbits each and the lth contains 2:
where [i:1] = [i:O], i = l ..... m - 1 .
[m : i] = [2rn:0].
[ i : 2 ] = [i-1:1]. i = 1 . .... m. [ i : 31 = [i - 1:-11. i = 1. . . . .m.
[1 : 21 = [m; 31, 1 [l: 31 = [m;-z].
The numbers i. i - 1. rn mean the corresponding B, fundamental orbits; 2rn means
the B, orbit whose mth label is 2. the others O: i - 1 means the point orbit when
i = 1. The symbol on the right of the semicolon is the u(1) label of the orbit.
Two elementary orbits : k] and b' : kt], with j' > j , are compatible if k = 1.
if k and k' are equal: if k f 1 and k' # k they are incompatible: see Fig. 2.10.
Figure 2.10: The elementary subalgebra orbits for Bm+i > B, $ u(1) . Those in t ith column belong to the ith fundamental orbit of BI = Bmci One in the first row is compatible with al1 those to its right, in any row. One in the second or third row is compatible with one to its right only if they are in the same row.
2.9 The algebra-subalgebra pair Cmcl > A,$ u(1)
The roots of Cl (1 = m + l ) are f ei, 1 5 i < 1 and i e i Y e j , 1 5 i < j 2 1, where ei , ej
are orthonormal weight space vectors. Weyl reflections are generated by sign reversais
ei + -ei and interchanges ei t+ e j . The simple roots are ai = e;-e;+l. i = 1.. . . . l-1.
and cri = 2ei (for simplicity we have multiplied them and the fundamental weights
below by fi). In terms of the orthonormal basis the fundamental weights are
Any weight of the fundamental orbit [il, with highest weight dir i , has i e's with coeffi-
cient A l , 1 - i with coefficient O.
The simple roots of Cl and A, are shown in Fig. 2.11. Those of -4, are aj = a;.
Figure 2.11: The Dynkin-Coxeter diagram for Cm,l > A, $ u ( 1 ) . The numbers below the diagram label the simple roots of Ci = Cm+i. Those above label those of A,. The direction unlabeled above corresponds to t hat of u( 1). orthogonal to the simple roots and fundamental weights of -4,.
i = 1.. . . . m. The fundamental weigbts 4 of -4, are
The fundamental u ( 1 ) weight is
The algebra weight [Xi.. . . , A,] becomes, in the subalgebra basis. [XI . . . . . u]
with
The elementary A, $ u(1) orbits are the ones contained in the fundamental Cl
orbits [il. A representative one, b; k]: in [il has as the weight of its highest state
d j + ~ l - c + UW,, where
CK4PTER 2. CLASSIC-4L ALGEBRAS
Thus
The elernentary orbit [j, k]. where k = i - j . has -4, orbit labels
where dhl or bh,l-k is tdwn as zero when j or k respectively is O or i. The u label of
the orbit b. k], suppressed in (2.34). is given by (2.33). When i = 1. j and 1 - k are
equal and non zero A, orbit labels are al1 2.
It can be shown that two elementary orbits [ j . k] and kt, kt] (FR niay suppose
i' = j' + k' > i = j + k) are compatible if and only if j' > j and k' 2 k (only one of
the 2 signs can be =). The elementary orbits and their compatibility rules are shown
graphically in Fig. 2.12.
Figure 2.12: The elementary Cm+i > A, $ u(1) orbits [ j , k]. The orbit [ j , k] belongs to the ith= ( j + k)th fundamental orbit of Ci = Cmci Two elementary orbits are incompatible if and only if one lies to the right and above the other.
2.10 The algebra-subalgebra pair > Dm $
The roots of Dl ( 1 = rn + 1) are i e i f' e j , 1 5 i < j < 2 where ei, e j are orthonormal
weight space vectors. The Weyl group is generated by two sign reversals at a tirne
ei + -ei . ej + - e j . and interchanges ei +t e j .
The simple roots are aii = ei-ei+l, i = 1,. . . , L I . cri = el-l+e,. The fundamental
weights are 1
Thus a weight of the orbit [il,
with coefficient 0; one of [l - 1
with coefficient i: one of [l] has
coefficient 4. -
i < 1 - 2, has i eh with coefficient f 1. the rest
] has an odd number with coefficient - i , the rest - an even number with coefficient -il the rest with
The simple roots of Di and Dm are shown in Fig. 2.13. The simple roots of Dm
Figure 2.13: The Dynkin-Coxeter diagram for > Dm $ u(1) . The numbers below the diagram label the simple roots of Dl (1 = rn + 1). Those above label those of Dm. The direction unlabeled above corresponds to that of u(l), orthogonal to the simple roots and fundamental weights of Dm.
are LY: = a i + ~ . i = 1.. . . , m. The fundamental weights of Dm are
The fundamental u ( 1 ) weight is
.-\ DI weight [XI. . . . . XI], in a Dm $ u (1) basis, becomes [A?, A3, . . . , X I : U] with
The elementary Dm $ il(1) orbits are those contained in the fundamental Di orbits
[il. The first 1 - 2 orbits contain 3 elementary orbits each: the ( 1 - 1) th and 1 t h each
contains 2:
where
[ i : 11 = [i;O], i = 1 $ . . . , 1 - 3 . [l - 2 : 11 = [m - 1, rn; O] (the last two Dm labels are 1, the rest O)
[i : 2 ] = [ i -1:1] , i = 1 ,.... l - 2 , [i:3] = [i-1:-11, i = 1 , ..., 1 - 2 ,
[ i - 1 : 2 ] = [ m - l ; ) ] , 1 [ l - 1 : 3 ] = [ n a : - 5 ] ,
[i : 21 = [m: $ 1 , [1:3] = [m-1;-61.
The numbers i, i - 1, m, m - 1 mean the corresponding Dm fundamental orbits. i.e..
the corresponding Dm orbit label is 1. The number to the right of the semicolon is
the u(1) label.
Two elementary orbits : k] and [j' : k t ] . with j' > j , are compatible if k = 1 or
if k and kt are equal or if j = m and jt = m + 1; otherwise they are incompatible: see
Fig. 2.14.
Figure 2.14: The elementary subalgebra orbits for > Dm $ u (1). Those in the i th column belong to the ith fundamental Dl orbit. 1 = m + 1. One in the first row is compatible with al1 those to its right. in any row. Each in rows 2, 3 is compatible with those to its right in the same row. and both in the mth column are compatible with both in the (m + 1)th column.
2.11 The algebra-subalgebra pair > A, $
4 1 )
The roots and fundamental weights of Dl (1 = m + 1) are of course the same as those
given in the first paragraph of 52.10.
The simple roots of 4 and A, are shown in Fig. 2.15. The simple roots of A,
Figure 2.15: The Dynkin-Coxeter diagram for > A, $ u(1). The numbers below the nodes label the simple roots of Di (1 = m + 1). Those above label those of A,. The direction unlabeled above corresponds to that of z l ( l ) , orthogonal to the roots and fundamental weights of A,.
are cri = ai, i = 1. . . . , m. The fundamental weights of A, are
The fundamental u ( 1 ) weight is
The DI weight [XI. . . . . X I ] becomes. in the -4, @ u ( 1 ) basis. [AL. . . . . u ] . with
The elementary A, $ u ( 1 ) orbits are those contained in the fundamental Dl orbits
[il. We find
In (2.45) the elementary orbit [j. k]. with k = i - j . is that with labels
ie., the non-zero (unit) labels are the j t h and (1 - k)th: if j (or k ) is O or 1 the
corresponding Kronecker 6 in (2.47) is O. The u label of the orbit [ j , k]. suppressed
In (2.46) [;il means the integer part of ii, i.e., l i or i ( i - 1) according to whether i
is even or odd. The elementary orbit [i - 2 j , 2 j] is t hat with labels
i. e.,
d in
is
there is one non-zero (unit) label, the (i - 2j)th: if i - 2 j is O or 1 the Kronecker
(2.49) is O and we have the point or zero orbit of A,. The u-label of [i - 2 j , 2 j ]
CHAPTER 2. CLASSICAL ALGEBRLSS 46
The elernentary orbits and their compatibility rules are shown graphically in
Fig. 2.16 and 2.17. for 1 even and odd respectively.
Figure 2.16: The elementary > A, $ u(1) orbits for 1 = m + 1 even. Two orbits are compatible only if one is to the right of and/or below the other. or, in case one of them is in the diagonal row 1 - 1. if either of its squares is to right of and/or below the other or to the left of and/or above the other.
For the (1 - 1)th fundamental orbit we associate with the elementary orbit [rn - 2 j , 2 j ] also the labels [m - 2 j - 1 , 2 j + 11 (except when m - 2 j = O ) , so it is represented
by two boxes in Figs. 2.16, 2.17. Two elernentary orbits b, k], b', k'] are incompatible
i f j ' > j and k'< k o r i f j f < j and k l > k. Incase j + k = na or j'+kl= m these
2.12 Conclusions
This work concludes the computation of branching rules for Weyl orbits of classical
Lie algebras and their maximal reductive regular subalgebras. Work on a similar
Figure 2.17: The elementary > -Am $ u(l) orbits for 1 = m + 1 odd. The compatibility condition for two orbits is the same as that in the caption of Fig. 2.16.
problem for Kac-Moody algebras is in progress.
Acknowledgment s
The work was supported in part by the Xational Science and Engineering Research
Council of Canada and by the Fonds FCAR du Québec.
References
[1] M. Thomz and R.T. Sharp, "Orbit-orbit branching rules for families of classical
Lie algebra-suba2gebra pairs", .Y. Math. Phys. 37, 1996, t o appear. We refer to
this paper as 1.
[2] J. Patera and R.T. Sharp, "Branching rules for representations of simple Lie
algebras throagh Weyl group orbit reduction", J . Phys. A: Math.Gen. 22. 2329-
2340 (1989).
[3] F. Gingras. J. Patera and R.T. Sharp, "Orbit-orbit branchzng rules between simple
bw-rank algebras and equal rank subalgeb~as". J . !dath. Phys. 33. 1618-1626
i 1992).
CHAPTER 2. CL ASSIC AL AL GEBRAS
2.13 Orbit-orbit generat ing function
In this section, we shall explain how to obtain an orbit-orbit gener
the data presented above.
ating fi
Generating functions are a very efficient tool of combinatorial mathematics. It is
not only a very good "storage" for an infinite number of numbers but it also gives
us the opportunity to compute with al1 of them at the same time. The original and
simplest definition is as follows:
Definition 2.1 Let a = al , a2, . . . . a,. . . . be a sequence of numbers. The I f o n a l )
power series X
is called the generating function for the sequence a.
This definition has an obvious generalization for a's with more indices via power series
in more variables.
The orbit-orbit generating function rvas introduced in [PS89]. it is a rational
function F(d. B) whose power expansion satisfies the following definition:
Definition 2.2 Let g' (rank g' = n ) be a subalgebra of g (rank g = m). Let
-4 = (-4, A?, . . . ,A,) and B = (Bi, B 2 , . . . , B,) be dummy uariables (which c a r y as
exponents the algebra orbit labels ai and subalgebm orbit labels bi, respectively), and let
a = (ai, a?, . . . , a,) and b = (b l , b 2 , . . . . b,) be the Dynkin labels of the highest weights
of Weyl orbits of g and g'. respectiuely (ie., the highest weights of the orbits are
[Cgi aiwi] and [C:=, biwi], respectively). Also, let A' = nE"=,;' and B~ = ny==L BQ;. The (formal) power sen'es
such that the nurnber cab is the rnultiplicity of the subalgebra orbzt [Cy.'=, biuj] in the de-
composition of the algebra orbit [Cg, aiwi] M called the orbit-orbit generating fvnction
for the algebm-subulgebra pair g > g'.
In the following, we will describe a construction of the orbit-orbit generating
functions for the algebra-subalgebra pairs considered above in this chapter. The sets
of mutually compatible elementary orbits given above are the starting point. The
algorithm is suitable for computer. it should be easy to implement it in computer
algebra systems. e.g. in REDUCE or Mathematica.
Let v i . j = 1,. . . , m. i = 1.. . . . mj. be a variable representing the ith elementary
orbit obtained from the decornposition of the j th fundamental algebra orbit. For
the construction. it is useful to introduce also the following notation: let O; be a
variable representing the j t h elementary orbit of the kth set of mutually compatible
orbits (i.e., it is equal to one of the variables u i , i = 1, . . . . mi). Let us define a
pre-generating function - it h a (after an appropriate substitution described below)
the required properties as long as the orbit to be decomposed is not degenerate ( L e . .
none of the Dynkin labels of its highest weight is zero):
where the sum is over al1 sets of mutually compatible orbits. Each rnernber of the
sum corresponds to one subalgebra orbit in the decomposition.
We have to remove the overcounting which occurs for degenerate orbits. We will
do it in several steps (more exactly. in rn steps). First. we will consider orbits with
one zero in the Dynkin label, then with two. three. etc. The situation is simplified by
the fact that when one is decomposing a degenerate orbit, each subalgebra orbit is
overcounted by the same amount. From the projection we know that each subalgebra
orbit can occur no more than once in decomposition of any algebra orbit (we consider
only equal-rank subalgebras and the projection matrix has rank m).
This can be seen as follows. The stabilizer of any weight il in the Weyl group W
(ie., the subgroup H of W for which hl\ = A, for al1 h E H) is generated by Sa such
that S,ii = A and a is a simple root of 0.
We can decompose W into left classes WH, ut E W with respect to H. The
Lagrange theorem tells us that the number of such classes is 1 W : HI = 1 Wl/l HI.
Clearly. the number of classes is equal to the number of weights in
Weyl orbit.
Each subalgebra orbit in a decomposition is counted IH( times
31
. the corresponding
(degeneracy of any
weight. i.e.. the number each weight of the orbit is obtained when the full LVeyl group
W is applied to the highest weight.) (HI can be easily counted. because it is again a
Weyl group for some semi-simple Lie aigebra. A possible way is to take the Dynkin
diagram of g and delete al1 the vertices corresponding to a's such that Sa $ H and
count the size of the Weyl group of the remaining diagram ( L e . , multiply the sizes
of PVeyl groups of al1 connected subdiagrams). Moreover, the size of the Weyl group
corresponding to the diagram made h m the previously deleted points is equal to the
size of the corresponding Weyl orbit.
We will start with removing the overcounting for orbits with one Dynkin label
vanishing. Let
I
u{=O if 3 p # i i such that LJ; =op,
1 otherwisc
(2.34)
where k runs over al1 sets of mutually compatible orbits. and "" indicates an omitted
variable.
We will continue with removing the overcounting for orbits with two zero Dynkin
labels.
ug=O if 3p#i i , i2 such that 1-1 (2.55) v:=op and 1 otherwise
where k runs over al1 sets of mutually compatible orbits, and "checked" variables are
omitted.
Similarly. we construct functions 3,, for i = 3 . 4 . . . . rn - 1 (corrections for orbits
with t Dynkin labels equal to zero)
and finally also Fm:
This function should be expressed in the variables 4.
Theorem 2.3 A b e r the substitution u: = 1 - vhere b are the labels of the
highest weight of the r th elementary orbit obtained from the ith fundamental algebra
orbit. the function 3 = 3, constructed above is the orbit-orbit generating function
for the alge bra-su balgebra pair in question.
Proof: This can be verified by expanding ail terms in power series (using (1 -
x)-'= ~ + X + X ' + Z ~ + . . . = C ' & X ~ ~ o T x = A ~ B ' ) .
Chapter 3
Complete branching rules for the family of algebra-subalgebra pairs
In this chapter. a paper coauthored by my supervisor Professor Robert T. Sharp is
presented. In this paper the (representation-representation) branching rules for the
algebra-subalgebra pairs so (n) > 50(n - 2) $ u (1 ) are derived and presented in terms
of generating functions.
Complete branching rules for the family of algebra-subalgebra pairs
S O ( n ) 3 S O ( n - 2 ) x U(1)
M. Thoma and R.T. Sharp Physics Department, McGill University,
Montreal, Quebec, H3A 2T8, Canada
November 4, 1996
Abst ract
Complete branching rules for the algebra-subalgebra pairs B, > Bn-i x
L(1) and D, > D,-I x U(1) are given in generating function form.
Submitted t o J. Math. Phys.
3.1 Introduction
The algebra chains ivhose single steps are Bn > B,-l x C'(1) and D, > Dn-I x
U(l) respectively are important because. unlike the more conventional chain B, > D, > Bn-l.. .. there is no loss of rank at any stage (the U ( l ) factor is regarded as
contributing 1 to the subgroup rank). Thus the chains. continued down to L'(1)".
give al1 the weight multiplicities in a representation of the original algebra Bn or
LIn. Formally one could substitute the B,-, (or D,J character generator into the
B, > B,-l x L'(1) (or Dn > D,-l x L W ( l ) ) branching rules generating function to
obtain the Bn (or D,) character generator. In Section 3.2 we deal with the algebra-
subalgebra pair B,, > Bn-i x U(1). The pair D,, > Dn-i x C(1) is treated by ver-
similar methods in Section 3.3. Section 3.4 contains some concluding remarks.
We might remark that Our results for some low rank cases agree with those given
earlier (Ref. [l], Sec. V, B3 > B2 x U(1) as SO(7) > SO(5) x L'(1); Ref. [2]. Sec. 3.
Case 4. B2 > BI x U(1) as O ( 5 ) > SC'(2): Ref. [3], D3 > D2 x L'(1) as SL'(4) > SC(2) x SL'(2) x CT( 1).
3.2 The algebra-subalgebra B, > Bndl x U(1); n z l
We begin by deriving the branching rules generating function for B, > Bn-1 by
substituting that for D, > B,-l into that for B, > 0,. The branching rules for Bn > Dn and D, > B,-l are well known; see for example Boerner [.II. Here we use Cartan
labels for group representations, rather than Boerner's notation. Thus the irreducible
representation (IR) ( A l . . . . , An) is that whose highest weight is Cy='=, A i d i where the
fundamental weight wi is the highest weight of the ith fundamental representation.
The branching rules for B, > D, are given by the generating function
Bo in (3.1) is taken to be unity. The coefficient of A" Bb = ny-L_, - A;' n,",, B ~ J in the
expansion of (3.1) is the multiplicity of the D, IR ( b ) = ( b l , . . . . b,) in the Bn IR
(a ) = ( a . . . , a ) Similarly the multiplicity of the Bn-i IR (c) = (cl. . . . . cn - 1 ) in
the D, IR (b) = (bl.. . . . b,) is given by the generating function
Co in (3.2) is taken to be unity. Substituting (3.2) into (3.1). Le. . evaluating
gives the branching rules generating function for Bn > B,+ The symbol in (3.3) 8"
is an instruction to retain the coefficient of n)=I B), the product of the zero degree
powers of Bj. The evaluation of (3.3) leads to
where 3, = 3: = a4iCi-l 9 i = 1, .. . q n 9
aj = AiCj. j = l , . . . , n - 2 , a n - 1 = .4n-LC:-l (3.5)
j = A'ljCj-2, j = 2 , . . . , n - 1,
= .4ncn-2.
The coefficient of AaCc in the expansion of (3.4) is the multiplicity of the BnVi IR ( c )
in the B, IR (a). Eq. (3.4) gives the desired branching rules for Bn > Bn-1 x G(1)
except that the U(1) labels are missing. Each elementary multiplet ,di, QI, Q j ? '/j
must be provided with a factor UU giving its U(1) label u, carried by the dummy
variable U. The required factors can be found by determining the branching rules for
the B, fundamental IR'S (i), i = 1,. . . , n, and for the IR whose nth label is 2, the
rest O.
We do this by expanding the IR's in question in Weyl orbits. and then. using the
known ([j]) orbit-orbit branching rules for B, > B,-I x L'(1) we have the B, IR
erpanded in Bn-i x U(1) orbits. Finally we assemble the BnmI x C(1) orbits into
B,-[ x C(1) IR's.
We first treat the fundamentai IR's (i) . i = 1. . . . . n - 2. ljsing the methods
erplained in Ref. [6], we find the following expansion of the B, fundamental IR (ij .
i = 1.. . . . n - 2, in B, orbits:
( i ) = [il + [i- 11 + ( n - i + 2)[i - 21 + . . * . (3.6)
What Brernner. Patera. and Moody [6] cal1 the S.P. (scalar product. i.e.. magnitude
squared of its weights) of an orbit is J i for the fundamental orbit [il, i = 1. . . . . n - 1.
Esamination of the elementary multiplets contained in ( i) , namely. the Bn-i IR'S ( i ) .
( i - 1). ( i - 2) shows we need only the three terms retained on the right hand side of
(3.6). (Including the L:(l) contribution to the S.P., -LU< can ooly increase. or leave
unchanged. the S.P.'S of the subalgebra orbits.) According to Ref. [5] the Bn- x C( 1)
content of the fundamental B, orbit [il is given by
[il > [i: O] + [i - 1: 11 + [i - 1: -11, (3.7)
where. on the right hand side. i. or i - 1. labels the ith or ( i - 1) th fundamental orbit
of Bn-i; the number following the semicolon is the LT( l ) label. From (3.6) we get
(i) > [ i ;O]+[ i -1 ; l ]+[ i -1 : -1]+
+[i - 1:0] + [i - 2: 11 + [i - 2: -1]+
+(n - i + 2) [[i - 2;0] + [i - 3: 11 + [i - 3; - I I ] + (3-8)
Vie are not retaining on the right hand side of (3.8) orbits whose S.P. is less then
-In - 8; recall the U(1) contribution to the S.P. is 4u2. The presence of [i; O], [i - 1: 11,
[i- 1; -11 on the right of (3.8) implies the Bn-I x U(1) IR's (i; O), ( i - 1; l ) , ( i - 1; -1).
Their expansions in Bn-i x C T ( l ) orbits yield
[i; O ] + [i - 1;0] + (n - i + l)[i - 2;0] + . . . + [ i - l ; I ] + [ i - 2 ; 1 ] + ( n - i + 2 ) [ i - 3 ; 1 ] + ... (3.9)
+ [i - 1;-1]+ [i -2;-11 + (n - i + 2 ) [ i - 3;-1]+ .. . .
Subtracting (3.9) from (3.8) leaves only [i - 2: O] which implies the presence of the
B,-l x C(1) IR ( i - 2: 0).
Thus we get the B, > Ba-1 x L T ( l ) decomposition of the fundamental B, IR (i).
i = l . . . , n - 3 ,
( i ) 2 (i: O) + ( 1 - 1: 1) + ( i - 1: -1) + ( i - 2: O) (3.10)
which gives the C(1) labels of the elementary multiplets ai, 3,. 3; and yi as O. 1. - 1.
O respectively. For i = 1, (i - 2: 0) is to be omitted and ( i - 1. k1) means the scalar
IR of B,-: with u = Il. At the end of this section are given the ~ i e w versions of al1
the ai, 3i. 3:, 7, including their C(1) factors.
A similar analysis of the B, fundamental IR ( n - 1) yields the branching rule
The B,-i x LT(l) IR ( 2 ( n - 1)) is that with the (n - 1)th label 2. the rest O. This
shows that an-[ and Yn-i have U(1) label O. while 3,4 and 3;-, have labels f 1.
The branching rule for the IR (n) is found to be
1 1 (n) 3 (n - 1: ,) + ( n - 1: -2) ,
so Jn, 3; have the U(1) labels A$. - For the IR (2n) (nth label 2, the rest 0) the branching rule turns out to be
which determines the U(1) label of 7, to be O.
Thus the B, > B,.,-l x U(1) branching rules generating function H(-li; Cj: Cr) is
again given by Equation (3.4) but with ai, now given by
pi = AiCi-IU, i = l , ..., n - 1 ,
@: = AiCi-lU-19 i = l , . . . , n - 1 , p,, = A,C, , -~U~,
1 = A,CnblU-~,
ai and yi are as given in Equations (3.5).
3.3 The algebra-subalgebra D, > D,-l x U(1); 1123
This section follows closely the treatment given in Section 3.2 to Bn > Bn-l x L'(1).
To obtain the branching rules generating function for D,, > D,-i we substitute into
the D, > B,-l branching rules generating function G(B, , Ck) given by (3.2) the
B,-l 3 D,-l branching rules generating function given by (3.1 j with n reduced by
1. The result is. with Dn labels bi carried by Bi. D,-l labels di, carried by Dc
with
The coefficient of l3'Lld in (3.15) is the multiplicity of the Dn-1 IR (d) in the D, IR
(b) , so we have the branching rules for D, II D,-[ x ü(1), except that the U ( l ) labels
are missing. The eiementary multiplets gi7,, Sf, ai, Yi must be provided with factors
Uu with the U(1) label u carried by U. The needed factors can be found by finding
the LI, > DndL x U(1) branching rules for the fundamental Dn IR's (i) . i = 1. . . . . n.
and also the IR (n - 1, n) whose iast two labels are 1, the rest O.
We expand the IR's in question in Weyl orbits, then, using the known, cf [j],
D, > D,-i x U(1) orbit-orbit branching rules we have the D, IR's expanded in
x U ( l ) orbits. Then we assemble the x U(1) orbits into Dn-i x U(1) IR'S.
We treab first the fundamental IR's (i). i = 1. . . . . n - 3. Using the methods
explained in [6], we expand the D, IR's in D, orbits:
(i) = [il + ( n - i + 2)[ i - 21 + . . . . (3.17)
Esamination of the elementary multiplets contained in ( i ) . namely the D,- IR'S (i) .
( i - 1). ( i - 2). shows that we need only the two terms retained on the right hand side
of (3.17). According to [SI . the D,-I x C(1) content of the fundamental D, orbit [il.
i = l , . . . , n - 3 , is
[il 3 [i: O ] + [i - 1: 11 + [i - 1: -11 . (3.18)
Using (3.18) to expand the orbits in (3.17) into D,-i x u'(1) orbits we find
( 1 ) > [ i : ~ ] + [ i - 1 : l ] + [ i - 1 ; -1]+(n-i+2) {[i - 2:0] + [i - 3; 11 + [i - 3: -I]}+- .
(3.19)
Yow according to (3.17) the orbit content of certain D,-l x CT(l) IR's is
( i ;O) > [i:O]+ ( n - i + l ) [ i - 2 : 0 ] + - . ( i - 1 : > [ i - 1:1] + ( n - i + Z ) [ i - 3 : 1 ] + - .
(3.20) ( - 1 ) 3 [ i - 1 : - l ] + ( n - i + 2 ) [ i - 3 : - 1 ] + - .
( i -2:O) > [ i - 2 ; 0 ] + - a .
Comparing with (3.19) we have the decomposition of the D. IR (i) into D,-l x U(1)
IR's, where i = 1 , . ..,n - 3:
( 1 ) 3 (i; O) + ( i - 1: 1) + ( i - 1; -1) + ( i - 2: O) . (3.21)
We see that for i = 1, . . . , n - 3, ai and have U(1) labels O and Ji, fi: have C'(1)
labels f 1, respectively.
Similarly for the D, IR (n - 2) we find the decomposition into D,-I x U(1) IR's:
So the elementary multiplets C Y , ~ , */n-2 have U(1) labels O, 1, -1, 0.
respect ively.
The D, IR ( n - 1) decomposes as
from which we read the LT( l ) labels of 3,-1 and a,-1 as fi, respectively.
The last fundamental D, IR ( n ) decomposes as
according to which the L r ( l ) labels of 3;-, and -,, are respectively &$.
Finally. we need the decomposition of the Dn IR ( n - 1. n). in which the last
two labels are 1 and the rest O. Proceeding through x C(1) orbits as for the
preceding Dn IR's we find
(n-1,n) > (72-2. n-l;l)+(n-2,n-l:-l)+(S(n-2):0)+(2(n-1):0)+(~-~:0)
(3.23)
where (n - 2. n - 1) is the D,-l IR with the 1 s t two labels 1. the rest O. (2(n - 2 ) )
and ( 2 ( n - 1)) are the Dn-I IR's with the second last or last label respectively 3. the
rest O. According to the D, > Dn-i branching rules generating function (3.13) the
five terms on the right hand side of (3.25) are 3n-l$~-,. Jn, 3 n - l m ( n > < I ~ - ~ J ~ - ~ , ?,- i .
respectively, where we have used the U ( l ) labels given by Equationa (3.23). (3.24) to
make the identifications. So we conclude that ;7, and 7,-1 have C(1) labels -1 and
O respectively.
Thus we conclude that the D, > DnT1 x U( l ) branching rules are still given by
(3.13) but with !3; = BiDi-lCI i = 1, ..., n - 2 , 3f = BiDi-iU-l, i = 1 , . . . . 12 - 2 ,
3 = B,D,-~u~, A = Bn-LBnDn-2Dn-lCI-1,
an-1 = B,,-&&~-~,
= B,D,-~c'-~. The other elementary multiplets al1 have ü(1) label O and are as given by Equa-
tion (3.16).
3.4 Concluding remarks
The branching rules ive have found are in large measure independent of n and the
same for B, and D,. Specifically if we consider only B, representations for which the
last two labels are zero and Dn representations for which the last three labels vanish.
the branching rules for Bn to Bn-I x C(1) and for Dn to D,-l x C(1) are the same
for al1 cases with the same values of the representation labels.
Substitution of the branching rules for Bn-i > Bn-2 x L T ( l ) (or Dn-1 > Rn-? x
C(1)) into those for Bn > B,-l x C(1) (or Dn > D,- x U(1)) gives branching rules for
Bn > BR-* x ~ ~ ( 1 ) ~ (or Dn > x ~ ' ( 1 )~ ) . When the C ' ( I ) ~ compnents of weight
are rotated by 4Z degrees in the ~ ( 1 ) ~ plane they become weights of -4:; this affords
a relatively simple approach to the branching rules problem for B, > B,,-2 x -4: (or
D, > Dn-2 x -4:). This problem has been considered in the B, context in [10.11].
We are using the results of this paper to compute character generators for B, and
D, with low values of n. as explained in Section 3.1. Because the branching rules
generating functions for B,, > B,- x U(1) and Dn > Rn- x U(1) have no numerator
terms as such (each numerator consists of just one term which is a product of factors
appearing in the denominator of that term). the character generators will have this
same form. This differs. for exampie. from Gaskell's [7] version of the character
generator in which interior states (belonging to lower orbits) of the fundamental IR'S
appear in numerators only; in our opinion our version leads to simplifications when
the character generator is used to determine basis states (character states) for higher
representations.
The results of the present paper could have been derived using methods analogous
to those used in Ref. [8] for Cn > x Ai branching rules (induction on the
representation labels using Weyl's character formulas). but the method used here is
considerably less complicated.
CHAPTER 3. COMPLETE BRd4,VCHliVG R ULES
Acknowledgments
The work was supported in part by the National Science and Engineering Research
Council of Canada and by the Fonds FCAR du Québec.
References
[Il J. Patera. R.T. Sharp and R. Slansky, "On a new relation between semisimple Lie
algebras". J. Math. Phys. 21. 2335-2341 (1980).
[2] R.S. Sharp and C S . Lam. "Interna1 labeling problem", J. Math. Phys. 10. 3033-
2038 (1969).
[3] R.T. Sharp, "SCT(n - 2) x SCT(2) x U ( l ) bases for SLy(n) ". J. Math. Phys. 13.
183-186 (1972).
[4] H. Boerner, b'Representations of groups ". Chap.VI1, Yorth-Holland Publishing CO.
(Amsterdam, 1963).
[5]M. Thoma and R.T. Sharp, "Orbit-orbit branching rules between classical simple
Lie alge bras and maximal reductiue subalge bras ". J. Math. Phys. 37. 6570-6381
(1996).
[6] M.R. Bremner, R.V. Moody and J. Patera. 'Tables of dominant ~weight rnulti-
plicities for representations of simple Lie algebms!'. Marcel1 Dekker. Inc. ('lew
York, 1985).
[7] R.W. Gaskell, "Character generator for compact semisimple Lie groups ". J. Math.
P h p 24, 2379-2386 (1983).
[8] R.T. Sharp, "Interna2 Zabelling: the classical gmups", Proc. Camb. Phil. Soc. 68.
[9] H. Weyl, "The classical groups ", Princeton University Press (Princeton, 1946)
[IO] H. De Meyer, P. De Wilde. G. Vanden Berghe. " SO(2n + 1) i n an S0(2n - 3) @
SU(2) @ SU(2) basis: I. Reduction of the symmetric representations ". J . Phys.
A 15 . no. 9. 2663-2676 (1982).
[Il] G. Vanden Berghe, H. De Meyer. P. De Wilde. ' S 0 (2n + 1) in an S 0 (2n - 3 ) 8
SU(2) @ SU(2) basis: II. Detuiled study of the symmetric representations of the
Ço(7) group". J. Phys. A 13 , no. 9, 2677-2686 (1982).
Chapter 4
Orbit-Orbit Branching Rules for Affine Kac-Moody Algebras
In this chapter. we will study orbit-orbit branching rules between untwisted affine
Kac-Moody algebras and their untwisted affine subalgebras.
Kac-Moody algebras were discovered independently by Victor G. Kac and Robert
V. Yoody around 1961. Since then, these algebras have enjoyed significant attention
by both mathematicians and physicists. Let us recall, for example. their appearance
in the theory of two dimensional spin systems on lattices or in current algebras.
The main references for the mathematical point of view are [KacSq. [CorS-l].
[MacBl], and [K'IIPSSO], we found also very useful the paper [J'iI85]. The applications
to physics are reviewed in [DolBq, [Do193], and [G086]; more details can be found in
[Fuc92j.
The finite dimensional simple Lie algebras are characterized by Cartan matrices
together with relations (1 -10)-(1 .l3). These Cartan matrices A are characterized by
the following conditions (1 5 i, j < n) :
and the determinant of A as well as the determinants of al1 the principal minors of A
must be positive. Kac-Moody algebras are obtained by relaxing the condition on the
CHAPTER 4. AFFINE AL GEBRAS 66
matrix -4 (namely the condition on det-4) while keeping the relations (1. IO)-( 1.13).
(One can generalize even further and assign a Lie algebra (contragredient Lie algebra)
to every cornplex n x n matrix.)
This chapter deals only with so called &ne algebras, ie., those algebras for which
detA=O, but al1 the principal minors have positive determinants. Among the affine
algebras. we are interested only in untwisted affine dgebras which have Cartan ma-
trices closely related to the Cartan matrices of finite dimensional simple Lie algebras
- their Cartan matrices are obtained by adding a zeroth column and a zeroth row to
the classical matrices. These additional colurnn and row correspond to the negat ive
of the highest root of the classical algebra.
4.1 Affine untwisted algebras
From now 011 by g'affine algebra" we always mean "affine untwisted Kac-Moody alge-
bra." Fortunately. the theory of these algebras is very similar to the theory of finite
dimensional simple Lie algebras. and many definitions can be immediately used also
in the affine case. The affine Kat-Moody algebras have a simple and for our purposes
very useful realization via central extensions of loop algebras. For any simple finite
dimensional Lie algebra g we define the following algebra:
with the following commutation relations:
for all x, y E g and j, k E Z. This is the affine untwisted Kac-Moody algebra
corresponding to the algebra g .
CHAPTER 1. AFFINE AL GEBR4S
.As in the case of the finite algebras, we have a triangular decomposition
O=ii+$i&i-.
where
The Cartan subalgebra 6 of 8 can be taken as
Its dual 6' hm several comrnon bases:
5* = span{ao = 6 - ae. ai. a*, . . . . a,, .Io} (4.13)
= span{Ao, A l , ilq, . . . . !in, 6 ) (4.14)
= span{Ao, el , el , . . . . e,, 6) . (4.15)
where ai( l @ h ) = a i ( h ) . h E t). (we use the same notation for the finite dimensional
roots and their (trivial) affine extensions). <r i ( c ) = O. ai(d) = O ( 1 5 i 5 n in al1
cases). Also 6 ( l O h ) = 0. 6(c) = 0. and 6(d) = 1. The functionals ili. O 5 i 5 n.
(fundamental weights) are defined as follows: . l i(hj) = 6,, O 5 i, j 5 n. -ii ( d ) = 0.
O 5 i 5 n (for 1 5 i < n ili is a trivial extension of the finite dimensional fundamental
weight wi). The ci's (1 < i 5 m, rn 2 n) are trivial extensions of the ci's used for
finit e dimensional algebras.
Any reductive Lie algebra g can be written as a sum of simple (or abelian) ideals
where go is the only abelian ideal. For such an algebra we define the corresponding
untwisted afnne algebra in the following way. Let a, k 2 1, be
The algebra
is the untwisted affine Kac-Moody algebra corresponding to g.
A similar construction for go yields the oscillator (Heisenberg) algebra ([ICR~T].
[Kacg4 especially the Remark 12.8). In this thesis ne need only the affinization
of the one dimensional Lie algebra 1(1) mhich we will denote by u(l)('). sce below.
The algebra u(l)(') is a Lie algebra (over C) spanned by {a,. n E Z, c, ci) with the
following coinmutation relations:
[d.a,] = na,.
This agrees with the above definition (via loop algebras) for g = ~ ( 1 ) . Also the
theory of irreducible representations of u ( l ) ( l ) fits that of untwisted affine algebras.
An irreducible representation is generated by a cyclic vector u and labelled by the
eigenvalues of u with respect to a0 and c. Both of these eigenvalues can be any
complex numbers. the one belonging to c should be nonvanishing.
4.2 Affine Weyl group
Similarly to the finite case we define t.he affine Weyl group as
It is a remarkable fact that the structure of this group is known for al1 affine Lie
algebras. In fact, it is this structure that gave affine algebras their name (for the
same reason are they sometimes called the euclidean algebras):
(semidirect product with T being the normal subgroup), where W is the classical Weyl
group which acts classicdy on span(A1, As, . . . , A,) and trivially on span{Ao, 61, and
T is an abelian group composed from elements t a , cr E M, where -CI is the scaled root
lattice of g
(these lattices are given explicitly in Appendix A). The action of ta on 6' is given by
the following formula:
Given this description of the Weyl group. it is clear that it is advantageous to use the
orthonormal basis for the classical part of 6'. We know the action of W in this baçis and T consists of simple translations corn-
plicated only by the quadratic expression for the depth (the negative of the coefficient
of 6). For a weight X E 6' the depth is given by the following formula [BS92]:
where do is the zero depth (an arbitrary constant which can be set as needed. usually
in such a way that the depth of the highest weight of the orbit or representation is
equal to O ) . L = X(c) is the level (a linear combination of the weight labels: is constant
on a Weyl orbit) of the highest weight of the Weyl orbit, and Q(X) is the length of
the finite part of X squared (easily computed in the orthogonal basis or using the
quadratic form matrix Q [BMP85] given in hppendix A).
that for the algebra u ( l ) ( l ) the Weyl group is trivial (of order one).
Highest weight representations
The representations we are interested in are a direct generalization of the irreducible
highest weight representations of classical Lie algebras (but are infinite dimensional).
We Say that a representation g is a highest weight representation with the highest
weight A E ij* if the underlying space V is generated by a vector U A (highest weight
vector) satisQing the following two conditions:
We are interested only in so called integrable representations. It can be shown that
every integrable irrediicible highest weight repre~ent~ation has the highcst wigh t of
the form n
where Ah E Z>. AL > 0. and p is a complex number. Such a representation is
isomorphic to a so called standard representation with highest weight
The structure of these representations is quite well understood. They are invariant
under the action of the Weyl
group orbits. Each such orbit
group and so can be written as a direct sum of Weyl
contains exactly one weight of the form
(highest weight), where Ak E Z' and p' E Es. and is isomorphic to a Weyl orbit with
the highest weight
It was shown by Bégin in [Bég90] that. upon restriction to a subalgebra. each algebra
Weyl orbit decomposes into a sum of subalgebra Weyl orbits (this is equivalent of the
theorem given in [Pst391 for finite algebras and holds under sorne fairly general condi-
tions, e.g., when the Weyl orbits of the finite algebras %ne up" which is always the
case in this thesis). This suggests the idea of computing representation-representation
branching rules in a way similar to that suggested in the first part of this thesis - via
Weyl orbits. For this we need to know the d e s for the decomposition of an algebra
Weyl orbit upon restriction to a subalgebra - the (Weyl) orbit-orbit branching rules.
CHAPTER 4. AFFINE A L G E B U S 71
4.4 Subalgebras of untwisted affine algebras
In studying subalgebras of untwisted &ne Lie algebras it seems quite natural !re-
calling the realization in terrns of centrally extended loop algebras) to consider the
following types of subalgebras:
1. @[t, t-LI BQ: g' $ Cc $ @d c g, where g' is a subalgebra of g. This case is very
closely related to the one studied in this thesis.
2. C[t. t-LI 0 (c g $ Cc c 3. In this case nothing important in the representation
theory changes hecause al1 the irreducible representations remain irreducible
upon restriction to these subalgebras. But a lot of information about these
represent ations is lost due to infinite weight multiplicities.
3. C[tm, t-m] @ê g $ Cc $ Cd c i. where m E M. prime. These are the so called
winding subalgebras studied , e.g., in [KW90]. [HKLPSl] . They are interest-
ing because the subalgebras are isomorphic to the algebras yet the branching
rules are nontrivial thanks to the nontrivial embeddings. Also. one c m get
representations of higher levels than the level of the original representation.
4. g i l @cc g C 5. This is the underlying finite dimensional Lie algebra.
Let g' and g be simple Lie algebras (finite dimensional) and g' c g. Let f : 8' + g
be the embedding of g' into g. Shen the mapping f , + 5 defined by
( t @ ) = t j 8 f (x') 9 X I E g' ,
f ( d t ) = d . dl E g', d E g ,
f(d) = j / c , cf € gr , c E g
is an embedding of # into 5. Here ji is the Dynkin's index of the embedding f defined
where ( . ) stands for the invariant bilinear form on the appropriate algebra. This
definition naturally extends to the case where the subalgebra (or even the algebra)
is reductive (for the one dimensional subalgebra we define the index of embedding to
be 1).
FVe will usually specify such an embedding by giving explicitly a mapping f speci-
Qing the embedding of the root system of g' into the root system of g (e.9.. by giving
!(a:) for al1 simple roots a: of 0'). Then the corresponding embedding f is
where ai: are the simple subalgebra roots. Usiiig the commutation relations. one can
extend this definition of f to g'. The affine embedding f is obtained via the extension
descri bed in (4.3 1)-(4.33).
The embedding of the algebra ~ ( 1 ) is specified by giving the direction .* f(crl)" orthogonal to al1 other subalgebra roots (there is. of course. no root of the algebra
u ( 1 ) but we found this notation useful).
We will study al1 affine untwisted algebra - affine subalgebra pairs obtained by
affinization from the simple algebra - maximal reductive subalgebra pairs considered
in the first part of this thesis. The indices of embedding (given in the second column)
can be easily computed using the concrete form of the embeddings f
or can be found in, e.g., [BB87]): 'l' 3 cp @ CA1), (Il 1) Cm+* 'l' 3 D$) @ Di1) , (1,l) Dm+n
(1) B,,, > DG1 $ B:')? (1,l) (except for n = 1 when they are (1,2). recall that
Bl - --hl 3 A;) @ Ah1) $ u ( l ) ( ? (1,l.-) -Im+n+i
(1 ) B,,, 1 B$ $ u(!)(~). (L-) (1 1 Cm,! 3 -4%) $ u( l ) ( l ) , (2.9
D!+~ 3 D:) e .(y, (1,-1
D E L , 3 -4;) @ u ( l ) ( l ) , (1.-)
Xote that al1 of these embeddings ercept the first one are so called conforma1
embeddings which are important for some physical applications. In these embeddings
the central charge of the Virasoro algebra associated to the affine algebras by the
Sugawara construction does not change. The conforma1 embeddings were classified
by several authon. e.g.. [BBW].
The projection of a weight A.
of a representation of the algebra is
It can be seen from the explicit form of f given above ((4.31)-(4.33)) that p' = p
and that A;, . . . .A',, are given by the finite dimensional projection matrix (as used in
the first part of the thesis). Also, if ha, = & ajh j (aH is the highest root of the
algebra), hLh = x;:, a>h), f (h i ) = C;,, /3jihr , O 5 j n', and if Yi is defined by
When calculating the finite dimensional projection matrix, we need the following
expression of ha, a E A, in terms of h,,'s, ai simple roots (a = C:=, a i a i ) :
2 2 2 " n
C aitai = C (ai, ai) a.h h, = - (ta, t y a = - (a, a) ta = - t Clii 7 (4.40)
(a, 4 i=l i=i ( 0 7 0 )
where the equality a(h ) = ( t a , h) , h 4 implies ta = Cy!, aitai.
4.5 Branching rules
The branching rules for some of the subalgebras considered here (D:;, > Di) $LI(') n q
(1 B,+, > D:) p. A:!+,+, 2 -4;) $ n $ u(I)(~), CLL, 1 -4:) U ( I ~ , mi^ 3 -4:' @ u ( i ) ( ' ) ) have been studied by other methods before [KW88], [KW88]. [LuS-L].
[LL95], and are. in general, fully determined by the modular and asymptotic proper-
ties of the branching functions jsee iKW88] for definitions and details j . Unfort unately.
this does not mean that it is easy to calculate the branching rules from these con-
ditions (especially for higher level representations). There are also extensive tables
[KMPSSO]. The orbit to orbit branching rules were caiculated in [Bég90] but only for
algebras of rank a t most 3 (the rank is the order of the corresponding Cartan matrix).
4.6 Affine orbit-orbit branching rules
The procedure of computing the orbit-orbit branching rules for the affine algebras is
very similar to the finite dimensional case. First, we select the algebra-subalgebra pair
and the appropriate embedding. Second, from the embedding we find the projection
matrix (prescription for the reduction of an algebra weight to a subalgebra weight).
Third. we find al1 the weights in the fundamental algebra orbits ( i . e . with highest
weight Ai, O 5 i n ) . select those which are projected on dominant subalgebra
weights (and become so the highest weights of subalgebra CVeyl orbits) and find the
projections. Finally, we repeat the third step for algebra orbits with highest weight
1 + ilj O , j n. i # j. This gives us the compatibility rules (as in the finite
dimensional case).
The description of the Weyl orbits is best done in the orthonormal basis for the
weight space of the underlying finite dimensional algebra because in this basis the
action of the Weyl group is easily described. It is also very useful to use the constancy
of the level on any Weyl orbit. For a weight X = XiAi + p6 the level is an integer
linear combination of Xo, . . . , A, with nonzero coefficient by Xo (the coefficients are
given in Appendix A). If we compute the level for the highest weight of the Weyl
orbit, we can use it to compute Xo for any weight of the orbit froin the %nite" part
of the weight (Le . , A l , . . . ,An) . The coefficient p of 6 (the negative of the depth of the
weight) can be computed from the depth of the highest weight of the orbit and from
the finite part of the weight using formula (4.24). Hence. keeping these two facts in
mind, we can conclude that a weight in a Weyl orbit is fully specified by its finite
part A l . . . . .An. We will often write (Ao: A L . . . . , A,: p ) . or (AO; AL,. . . . or even
( A L , . . . . A,), for z:=o A i i i i + pb (the level and thus also Xo is understood from the
context).
4.7 The algebra-subalgebra pair ciin 1 c:)@c(~) 11
k.Ve will treat the pair ciin > Cm) $ CA1) in detail and then briefly sumrnarize the
results for the other pairs.
It is easy t o find the embedding of the root system of Cm $ Cn (ûi and &y) into
that of Cm+, (specified are the embeddings of the simple roots) :
This embedding extends into the embedding of the Cartan subalgebras and then. A A
together with the general formulas f^(dt) = d, f (c',) = j,c, and f (8,) = jnc. gives
the projection mat r ix for this algebra-subalgebra pair:
In this case both of the indices of embedding j, and are equal to 1.
CHAPTER 4. AFFINE ALGEBR4S 76
Thus an algebra weight X = (Ao: Al, . . . . An; p ) is projected on a dominant subal-
gebra weight if and only if
If we use the orthogonal basis el. . . . . e,+, in the weight space of Cm+, ( this change of
basis is given in .Appendix A) these conditions become (recall X = Xo.io -i XI=;" n,ei +
L a + a . . . 2 a,,, 2 O .
where L = Xo + X i + . . . + A,+, is the level of the weight A.
For the algebra cm;, the lattice M. which appesrs in the definition of the affine
Weyl group, equals m f n
M = { 5 uiei 1 ui E 2~ . } (4.46)
Using this. we obtain the following description of the fundamental orbits O 5
( m + n - i) ai's are in 2 2 . L = 1. . (4.47) i al 's are in 2Z + 1
Yotice that we have omitted the precise description of Xa 's and p's as they can be
easily obtained from the finite dimensional (spatial) part of each weight and from the
level L:
where the aY's are the comarks given in Appendix A. Hence? one gets for the algebra
where ai is the first coordinate in the orthogonal basis. The depth d A = -p of a
weight X in an orbit with highest weight A of depth zero is given by
(recall that the level L is constant on any Weyl orbit). The easiest way to calculate
Q(X) and Q(A) is in the orthogonal b a i s as (ei, ej) is well known and simple for al1
i and j . For example, for the fundamental weights iii we get
Similarly the orbits [ili + :lj], O 5 i < j 5 rn + n. (with the highest rveight + .ij) contain the following weights:
/ i al's are in 4Z + 2. 1 ( j - i) al's are in 4 L f 1,
Using the description, the selection rules, and the projection given above. we find
the branching rules for the fundamental orbits (these give us the elementary orbits)
and also for the orbits [Ai + hj]. The computation and results are very similar to those
for the corresponding finite dimensional algebra-subalgebra pair Cm+, > Cm $ C,,:
With the notation introduced in the last equation, we can see that these branching
rules are identical to those of the finite dimensional algebras! (Except. of course.
for the existence of the zeroth fundamental orbit.) Moreover, after decomposing the
orbits [Ai + +Aj] (or checking which pairs of elementary orbits form products belonging
to these orbits), we have found that even the compatibility rules are very similar. In
fact, we can use a picture and its description similar to Figure 2.2 given above for
the Cm+, > Cm $ C, pair. Two elementary orbits [p; q] and [r; s], O 5 p, r m,
O I r, s 5 n are compatible if and only if p 2 r and q 5 s (or vice versa). In Figure
4.1 this means that two orbits are compatible if and only if one of thern lies below
and to the right of the other (this includes the position in the sarne column or in the
The rule for decomposing higher algebra orbits is similar as for the finite dimen-
sional cases. Let k be an index labelling the sets of mutually compatible orbits and let
Figure 4.1: The elementary ci:, > C:) $ Ci1) Weyl orbits.
be the highest weight of the elementary orbit which cornes from the ith fundamen-
ta1 algebra orbit and belongs to the kth set of mutually compatible elernentary orbits.
Then the decornposition of the algebra orbit with highest weight A = AiAi is
[A] 3 A i A , 1
where the sum over k runs over al1 sets of mutually compatible orbits.
Since the treatment of the other algebra-subalgebra pairs is very similar, in the
following sections we d l only summarize the results.
CHAPTER 4. -4FFINE -4LGEBR;IS 19
(1) W $ ~ ( l i 4.8 The algebra-subalgebra pair Dm+, > Dm 11
Embedding:
Projection matrix:
The content of the fundamental Weyl orbits [Ai ] . O < i n. for algebras D,
(instead of Dm+.) is given by:
i n n
[Al] = X = X o A o + 1 aiel + pb L = 1 , al1 al's are in Z, al E 2 2 + 1 l=l l= 1
2 < i < n - 2 :
[ili] = { A = AoAo + 2 aiel + p6 l=l
i al's are in 2Z+ 1, L = 2, (4.59) other ai's are in 2 2
[i\,-l~ = { A = ~ ~ i \ ~ + 2 aiel + ps l=l
Similarly, the orbits [Ai + A j ] , O 5 i < j 5 n, (with highest weight ili + A,)
al1 ai's are in Z + $, L = 1, a[ E 2 2 + + } (-4.60)
n
[AnI= { h = h o i \ o + ~ a r e i + p 6 I = 1
contain the following weights:
al1 al's are in Z + i, L = l . &aai E 2 2 + $ } (4.61)
n one al is in 2 2 + 1, } (4.62) l= 1
i al's are in 32 k 1. L = 3, other ai's are in 3 Z
I xî='=, al E 2~ + ( i mod 2) J
f i k - n h e r of ai9s is other al's are in 2 2 + f,
nm6d.l x;=,ai E 2S+ 1 + (4.64)
L = 2, even number of ai's is in '2Z - other al's are in ZZ + 1. Cî=L al E 22 + zrn;d3
(4.65)
odd number of a, 's are in 2 2 - 7. other ai's are in 22 + h.
n m0d4 x î = 1 a i ~ 2 z + 1 + T i (4.68)
i al 's are in 32 f 1, L = 3. other ai's are in 3 2 ,
CLl al E 2 2 + ( ( i + 1) mod 2) (4.66)
i al's are in 4 2 + 2. L = 4, ( j - i) al's are in 4 Z f 1.
other ai's are in 4 2 (4.69)
[ + ] = X = X o i i o + C aiel + / ~ b l= 1
i ai's are in 3 2 + 4, other al's are in 32 i, = 3, C;=l al E 2L+
(n-2) rnod 2 +(zmod2)+- ] (4.70)
L = 2, even number of ai's are in 2Z - other al's are in 2 2 + 1,
nmod4 - al E 2 2 + -7
I i al's are in 3Z + 3. I other ai's are in 3 2 f 4. L = 3, (4.71) CF=, al E 2Z+
+jimod2) + 1 al is in 22 .
[A,-, + A,] = (A = + oie, + ps L = 2. /= 1
other ai's are in 2 2 + 1
The conditions for projection on a dominant weight (in orthogonal basis. Le . .
.\ = XoAo + xzn aie; + p6) are:
and
The decornpositions of the fundamental orbits then are:
Finally. we get the ompatibility rules:
With the notation introduced above and with [l. O] z [O1. 11, [O. 11 [l. 0'1. two
elementary orbits are compatible if they fit one of the follo~ving situations:
[p : q] and [r: s] for al1 O p 5 r 5 m and O 5 q 5 s 5 n
[P: QI and [ml: s] for al1 O 9 p 5 rn - 1 and O 5 q 5 s 5 n
a [ml: q] and [m'; s] for al1 O 5 q, s 2 n
[O': 0'1 and [r; s] for al1 1 5 r 5 rn and 1 5 s 5 n
0 [Wq] and [ r : s ] for al1 1s r m and 15 q 5 s 5 n
0 [O1: q] and [O1: s] for dl O I 9, s < n
0 [p: 01] and [ r ; 0'1 for al1 O 5 p. r I m
[O1; q] and [ml; s] for al1 O 5 q 5 s 5 n
[p ; 01] and [m': s] for al1 O 5 p < rn - 1 and 1 5 s 5 n
[ml: 01] and [m'; s] for al1 1 5 s 5 n
[p, nt] and [T, n'] for al1 O 5 p. r 5 m
[p, 0' and [T, nl] for al1 O 5 p 5 r 5 rn
a [O1,q] and [r,nl] for al1 14 r < m and0 I q I n - 1 .
CHAPTER 4. AFFINE ALGEBRAS
(1) Figure 4.2: The elementary Dm+, > D:) $ D ( I ) n Weyl orbits.
CVe can draw Figure 4.2 similar to the one from the finite dimensional case. Two
orbits not both of which lie in the first column, first row, last column, or in the last
row are compatible if either lies below and to the right of the other (this includes
orbits in the same column or row). Two orbits in the first row are compatible if and
only if both of them have the first index O' or both of them have the first index O.
Similarly for the other exceptional row and columns: two orbits in the last row are
compatible if and only if they have the same first index (m or m'); two orbits in the
first column are compatible if and only if they have the same second index (O or 0'):
two orbits in the last column are compatible if and only if they have the same second
index (n or nt).
Embedding: f(û:) = c r i , l< iS rn -1 ,
m+n
!(a',) = a,-i + 2 1 aj . j=m
Projection matrix:
The content of the fundamental Weyl orbits [Ai] . O 5 i 5 n. for algebras B,
(instead of B,,,) is:
[:Il] = ( A = AoAo + aiel + p6 l= 1
Similarly, the orbits [hi + ilj], O i < j 5 n (with highest weight :Ii+iIj) contain
L = 1, al1 ai's are in Z, al E ?Z i 1 (4.84) 1=1
[?\il = { A = ~o + 5 aie, + ~6 1=1
the following weights:
i al's are in 2Z + 1, L = 2. other are in 2 2 (4.83)
one al is in 2 2 + 1, } (4.87) l= L
n i ai's are in 3Z f 1. .\ = AoA0 f alel + = 3. other aiSs are in 32.
I = I Cî=, al E 2 2 + ((i + l)rnod2) (4.89)
n
,\ = Xoi\o + aiel + pS 1=1
i al's are in 3 2 31 1 , L = 3, other ai's are in 3 2 . (4.88)
Cî., ai E 2 2 + (imod2)
n
X = Aoi\o + alel + p6 1 L = 2. al1 ai's are in Z + f l= 1
} (4.92)
{ n
[.Ii +.ij] = X = XoAo + x a i e l + p 6 I= 1
i al's are in 4Z + 2. L = 4. ( j - i) ai's are in 4 Z i 1.
other ai's are in 4Z
The conditions for projection on a dominant weight (in orthogonal basis. L e . .
(4.90) n
= + C a l e l +p6 1 L = 2. allai's are in Z + 4 l= 1
} (4.91)
n
= '01'0 + C alel f p6 1 = i
The decompositions of the fundamental orbits then are:
i al's are in 3 2 + i, , } ,493 , = 3 q other aiYs are in 3 2 *
[Ao] 2 [Ah + AO)] $ [A; + A: - 6'1 [O; O] $ [O', 0'1 (4.97)
Compati bility rules:
With the notation introduced above (notice that the sum of indices in each elementary
orbits gives the number of the fundamental algebra orbit from which it originates).
two elementary orbits are compatible if they fit one of the following situations (for this
occasion only we use also t his notation: [l . O] e [O1. 11, [O. 11 n [l .Ot] , and [O1: O: ] = [Ot: O] = [O; 01]):
a [ p : q ] and rml;s] f o r a i l O s p < m - 1 a n d O < q < s I n
a [m'; q] and [ml; s] for al1 O 5 q, s 5 n
0 [O1; 0'1 and [r: S] for al1 1 5 r 5 m and 1 5 s 5 n
[0';q] and [ C S ] for al1 15 r 5 rn and 1s q 5 s 5 n
[Ot; q] and [O1; s] for d l O 5 q, s 5 n
[p; Ot] and [r ; 0'1 for al1 O 5 p, r 5 m
CHAPTER 4. AFFIXE ALGEBRAS
0 [O1: q] and [ml: s] for a11 0 5 q 5 s 5 n
[p: 01] and [.nt'; s] for al1 O 5 p 5 rn - 1 and I 5 s < n
[m': 01] and [ml: s] for al1 1 5 s 5 n.
We can draw Figure 4.3 similar to the one from the finite diinensional case. Two
orbits. not both of them in the first row. last row. or first column. are compatible (as
in the finite-dimensional case) if either lies below and to the right of the other ( t his
includes the orbits in the same row or in the same column). Two orbits in the first
row are compatible if and only if both of them have the first index O' or both of theni
have the first index O. Similarly for the other exceptional row and colurnn: two orbits
in the last row are compatible if and only if they have the same first index ( m or m'):
two orbits in the first column are compatible if and only if they have the same second
index (O or 0').
m-1,O m-1,l m-l f m-l,3 m-1,4
m-1,O '
m,O1 m J m J m J m,d t m g in9 m:4
Figure 4.3: The elementary ~ g f > Di) $ BA') Weyl orbits.
4.10 The algebra-subalgebra pair B:!, > DE) $
The algebrs-subalgebra pair B::, > Di) $ Ai1) is obviously a special case of the
pairs BE!, > Dm) $ B L ~ ) for n = 1 but it must be treated separately as the index
of embedding of the second subalgebra changes from 1 to 2 (this happens due to the
convention that the highest root is normalized to 2, i.e.. (aH. c r H ) = 2). Most of
what follows can be obtained frorn the previous section (&in > D:) $ ~ ( " 1 n~ just by
taking n = 1 but the projection of algebra weights on subalgebra weights is different
and so are the highest weights of the elementary orbits.
Embedding:
j=m
f(aI') = û m + l .
Projection matrix:
The conditions for projection on a dominant weight (in orthogonal basis, a'. e . .
and
CHAPTER 4. AFFINE -4LGEBRAS
The decompositions of the fundamental orbits then are:
[ilo] > [Ah + 21\01 $ [Ai + 21\: - 61 [O; O ] $ [Ot . 0'1 (4.106)
[:Il] > [A; + 2113 $ [.Io + 2dy] G [l: O ] $ [O: 11 (4.107)
I
1 [k ; i - k ] $ [mt, O] $ [i: O f ] $ [m': Ot] k=i-1
Compati bility rules:
With the notation introduced above the compatibility rules are exactly the same as (1) those for the B,,, > Di) $ BiL) pair.
4.11 The algebra-subalgebra pair A:!,,~ > A:) $
Embedding: f (ai) = ai, 15 !(a:) = Qm+l+i
The direction conesponding to the u (1) label, i. e., orthogonal to al1 roots of -4, and
Projection matrix:
(The constant norrnalizing the u ( 1 ) label is chosen as in the first part of the thesis.)
The content of the fundamental Weyl orbits [A,], O 5 i 5 n for algebras -4, is:
Sirnilariy the orbits [hi + .ij]. O 5 i < j 5 n (with highest weight i l i + A i j ) contain
the following weights:
( j 4) ai's are in 2 2 + 1 - z. 1 a, other al's are in 2 2 - L = 2 , n+i
The conditions for projection on a dominant subalgebra weight (in orthogonal
basis, i.e.. A = Xoho + CE^^^^ aie; + pd) are:
and
CHAPTER 4. AFFINE ALGEBRAS 91
The decompositions of the fundamental orbits then are ( O 5 i 5 m + n + 1):
where
min (i,m+ 1)
C [ a , b : k ; i - k ] ,
is the depth of the elementary orbits.
Compatibility rules:
With the notation introduced above, two elementaryorbits [ a, b : k; i - k] j - 11, i < j are compatible if they fit one of the following situations:
and [c. d : l :
a a and cl and b and d have the same parity (pair-wise), and k 5 1 and i- k 5 j - 1
(this resembles the finite dimensional case, which corresponds to a = b = c =
d =O), or
a a and c have the same parity, b and d have different parities and 11 - kl + (n + 1) - l j - 1 - ( i - k ) l = j - i l or
a a and c have different parities, b and d have the same parity and (m + 1) - 1 2 - kl + l j -1- ( z - k)l = j -i, or
a a and c have different parities, b and d have different parities and (m + 1) - 12-kt- ( n + l ) + l j - 2 - ( i - k ) [ = j - i .
We can draw Figure 4.4 similar to the one from the finite dimensional case. Two
elementary orbits [a, b : k; i - k] and [c, d : 1; j - 11 with a = c and b = d are compati-
ble if either lies below and to the right of the other (this includes orbits in the same
a , : 0,o
a,b: IBO
ab: 2 8
a,b: 3,O
L
ab: 4,O
qb: I,n+l
qb: 1J
qb: &3
a,b: 3J
qb: 4J
Figure 4.4: The elernentary .4~!+,+, 3 .-lm) $ .-ln) 8 u ( l ) ( l ) Weyl orbits for a. b E Z. (rn + l ) a + (n + l ) h = 0.
ab: OJ
a?b: 1J
a,b: 2J
a,b: 3,l
46: 4J
4b: 1,4
u,b: 2,4
a,b: 394
qb: 494
column or row). The same rule can be used when a # c and/or b # d but a and c.
and b and d have the same parities (ie. in the first case on the list above).
qb: OJ
a,b: 12
qb: 26
qb: 3 J
qb: 4.2
4.12 The algebra-subalgebra pair B:!, > BE) @ u(l)(l)
Embedding :
CHAPTER 4. AFFINE ALGEBMS
Projection matrix:
The conditions for projection on a dominant weight (in the orthogonal basis. i. e..
and
The decompositions of the fundamental orbits then are:
Compatibility rules:
With the notation introduced above two elementary orbits [a : pl and [b : q]. p < q
are compatible if they fit one of the following situations:
O p and q are either two of O. 1 and m + 1. or
We can visualize the elementary orbits as an infinite pile of levels such as the one
in Figure 4.5. An orbit in the ith column from the left (O 5 i < m +
Figure 4.5: The elementary B::~ > Bg) $ u(l ) ( l ) Weyl orbits,
1) cornes from
a E Z.
the ith fundamental algebra orbit. An orbit in any of the zeroth. first and ( m + 1) t h
column is compatible with any orbit in the other two columns. Two orbits from which
at least one is in the ith.. . . . ( m - 1)st column are compatible if and only if their
levels satisfy the conditions given above.
4.13 The algebra-subalgebra pair CE!, 2 A g $
u(l)il)
Embedding:
J(û.7) = 4 1 iai + 2amci . i= 1
Projection matrix:
The conditions for projection on a dominant weight (in the orthogonal basis. ie..
and
2L 2 al - a,+l . (4.131)
The decompositions of the fundamental orbits are (O 5 i 5 m + 1):
m-i
I m-i
for n odd, and
for a even.
Compati bility rules:
Let
for a E 2% and k = O. rn + 1 - i (these orbits are qua1 to some orbits with a - 1
and a + 1, respectively). Using the notation introduced above. two elementary orbits
[a : p. q] and [b : r, s] (belonging to the ith ( i = p + q) and j t h ( j = r + S. i < j ) orbit.
respcctively) are compatible if they fit one of the following situations:
We can visualize these elementary orbits as building with floors numbered by
integers and rooms labeled by [a : p, q] (the plan for floors with a E 2 2 + 1 is given in
Figure 4.6, and for a E 2 2 it is given in Figure 4.7). Two orbits on the same 0001- are
compatible if one of them lies below and to the right of the other on our plan (this
includes the orbits in the same row and in the same column). Two orbits on different
fioon are compatible if they satisfy the same condition (or, more precisely, their
projections to the ground floor satisQ this condition) and the distance between the
two floors is a multiple of four. Then there are compatibilities which cannot be easily
CHAPTER 4. AFFINE ALGEBMS
Figure 4.6: The elernentary C'!,.!il > -4;) $ i r ( l ) ( l 1 Weyl orbits for a E 2 2 + 1.
visulaized ( "secret tunnels" ) . These are descri bed by the new notation int roduced in
eqn. (4.136) and by the last two conditions on the list above. Yote that the finite
dimensional case corresponds precisely to the first floor.
4.14 The algebra-subalgebra pair D:!~ 3 DE) u(l)(l)
Embedding:
Figure 4.7: The elementary > A!.,? $ u(l)(') Weyl orbits for a E 2Z.
Projection matrix:
ai- -(m+l),
m+l a: -m m+1
O
a: -(?Tl-1),
m-1
The conditions for projection on a dominant weight (in the orthogonal basis. i.e..
t 4:
a: -(na-1),
m
and
a: *(nt-2),
na-2 s
. . . s
a: -(nt-2),
m-l
CHAPTER 4. AFFINE ALGEBR4S
The decompositions of the fundamental orbits then are:
Compatibility rules:
With the notation introduced above two elementary orbits [a : pl and [b : q] , p < q
are compatible if they fit one of the following situations:
CHAPTER 4. AFFINE -4LGEBMS
r p and q are either two of O. 1, rn and rn + 1, or
a b ~ 2 ~ + 1 . 2 ~ q ~ r n - l . p = O o r 1 . a n d a + b € 3 ~ f 1 .o r
O b ~ 2 ~ , 2 ~ q ~ r n - l , p = O o r l , a n d a + b ~ 3 Z , o r
a a ~ 2 Z + 1 . 2 ~ p ~ m - 1 . q = m o r m + 1 . a n d a + b ~ 3 % + 1 . o r
a a ~ 2 2 . 2 ~ p < m - l , q = r n o r m + l . a n d a + b ~ 3 Z o r 3 2 - 1 . o r
O 2 < p < q l r n - l a n d
- a E 2 Z + l , b ~ 2 ~ + + . a + b ~ 4 Z + S , o r
- a E 2 2 , b ~ 2 Z , a + b ~ - L Z , o r
- a ~ 2 Z , b ~ 2 Z + l .
We can visualize the elementary orbits as an infinite pile of levels such as the
one in Figure 4.8. An urbit in the ith column from the left (O 5 i 5
Figure 4.8: The elementary D:!+~ > Dm) $ u(l)(') Weyl orbits.
m + 1) cornes
a E Z .
from the ith fundamental algebra orbit. An orbit in any of the zeroth. first. mth
and ( m + i)st column is compatible with any orbit in the other three columns. Two
orbits in the 2nd.. . . , (rn - 1)st columns are compatible only if their levels satisfy the
conditions stated at the end of the list above. ,411 orbits on an even level a in these
m - 2 columns are compatible with al1 orbits on levels b satisfying b E 327, - a in the
zeroth and first columns, and al1 orbits on levels b satisf'ying b E 3 2 - a or 3L - a - 1
in the last and next to last columns. Similarly, al1 orbits on an odd level a in these
middle columns are compatible with al1 orbits on levels b satisfying b E 32 - a ir 1
in the zeroth and fint colurnns, and al1 orbits on levels b satisfying b E 3 2 - a + 1 in
the laat and next to l u t columns.
4.15 The algebra-subalgebra pair D:!~ > A ~ I 63 u(l)(l)
Embedding:
Projection matrix:
Conditions for projection on a dominant weight (in the orthogonal basis. ie..
.\ = Xod\o + CL,:' a;ei + p6):
and
The decompositions of the fundamental orbits then are: m t l r
a ~ 2 L k=l L
k ~ 2 Z t ( ( r n + 1) mod 2)
a€2Z k=l a ~ 2 2 2 - 1 k=L k~2%[(m+l) mod 2) kf 2Z
kE2Z-t-((m+ 1 ) rnod 1) k ~ 2 Z
3 1 C [a. i : k. i + k ] @ a €2Z k= 1
k~'LZ+((rn+ 1) mod 2 )
a ~ 2 Z k=l L
k€3%(rn mod 2)
k~2Z+(m mod 2) k ~ 2 % 1
k€2%((m+l) mod 2)
Compati bility rules:
With the notation introduced above two elementary orbits [a. i : p . q] and [b. j : r . s].
i < j . (g and s are not defined for i. j = 0, 1. m. m + 1) are compatible if they fit
one of the following situations:
a i isOor 1 and; iseither m o r m + l . o r
a i. j are 0 .1 or m. m + 1 and lp - rl = 1 or m. or
- a,& 2 2 , and
* a + b € - % o r
* a + b € 4 Z + 2 , i + j = r n + l , o r
- a.b E 2Z+ 1, and
* a + b ~ 4 z + 2 , o r
* a + b € - I ~ . i + j = m + l . o r
- a ~ S Z , b € 2 Z + l , a n d
r q _< r . and
a + b f 4Z-1 ,o r
a + b € - l Z + l a n d i + j = m + l . o r
* p 2 s, and
a + b E - % t 1 , o r
a + b ~ 4 Z - 1 a n d i + j = m + l .
Chapter 5
Summary and Conclusion
In this thesis we have obtained three kinds of results.
First. we computed the Weyl orbit-orbit branching rules for al1 maximal regular
reductive subalgebras of classical Lie algebras (without any restrictions on the rank
of the algebra). Previously, the results were known only for algebras of rank at most
five .
Second. we found the complete (representation to representation) branching rules
for the chains so(n) > m ( n - 2) $ ~ ( 1 ) . Previously. only the branching rules for
D, > Bn-i and Bn > LI, were known.
Third. we calculated the orbit to orbit branching rules for al1 untwisted affine Kac-
Moody algebra - untwisted affine Kac-Uoody subalgebra pairs obtained by affiniza-
tion from the fini te dimensional algebra-subalge bra pairs ment ioned above (aga in
without any restriction on the rank of the subalgebra) . Previously, t hese results were
known only for algebras of rank a t most three. I t is well known that the Dynkin
diagrams of affine untwisted algebras are more symmetric then those of finite dimen-
sional Lie algebras. It seems that such an increase of symmetry happens also for the
orbit-orbit branching rules (compare especially the pairs with semisimple subalgebra.
cg.. Dm+, > Dm $ D, and Di!, 3 Dg) $ DL1)).
There are several directions in which this research can continue:
1. Organizing the orbit-orbit branching rules according to the depth of the subalgebra
orbits.
2. Computing the orbit-orbit branching rules also for pairs which involve twisted
affine Kac-Moody algebras.
3. Expanding Weyl orbits in terms of representations. ie.. writing the orbits as lin-
ear combinations of highest weight representations (with both positive and negative
coefficients). This problem is equivalent to decomposing of the highest weight repre-
sentations into Weyl orbits. or to finding al1 dominant weights of these representations
including mult iplicit ies.
4. Finding the orbit-orbit branching rules in the explicit format of a generating fiinc-
tion.
5 . Studying the connection between our work and the theory of lattices (especially
building new lattices from old ones via gluing and shifting).
Appendix A
Summary of Properties of Classical Simple Lie Algebras
In this appendix we surnmarize properties of classical simple Lie algebras and their
affiniaations used throughout the main text. Most of these properties can be found
in any standard textbook on Lie algebras.
Extended Dynkin diagram: each node corresponds to one simple root CI, of
the finite dimensional algebra and the zeroth node corresponds to the negative of the
highest root. For the affine algebras the zeroth node belongs to the zeroth root. The
numbering of roots is that of Dynkin.
Extended Cartan matrix: Cartan matrix of the affine algebras. Without the
first column and Srst row it is the Cartan matrix of the finite dimensional Lie algebra.
Quadratic form rnatrix: a matrix formed by scalar products of fundamental
weights of the finite dimensional Lie algebras:
Roots, positive roots, simple roots, the highest root, and fundamental
weights: al1 of these objects are expressed in an orthogonal buis.
APPEIVDIX A. PROPERSIES OF SIMPLE LIE .4LGEBMS 108
Weyl group: the structure is described and the action on the dual of the Cartan
subalgebra is given. It is generated by reflections Sai. 1 5 i 5 n.
Congruence number: a constant for weights in one representation. Divides
representations into congruence classes and. also. describes the weight lattice modiilo
the root lattice.
Nul1 root 6: (for the affine algebras) a special linear combination of roots ao. a 1.
. . . , a,. The coefficients are the marks ai.
Coxeter number:
Central element c:(for the affine algebras) a linear combination of h,, . O 5 i 5 n:
the coefficients are the comarks a:. The level L ( A ) = h(c) = Cr., where
= 2 -- (a, *ai '
Xote that for al1 algebras a l = 1.
Dual Coxeter number: n
hV = Ca:. i=O
Affine Weyl group E: it is generated by Sa,. O 5 i $ n. The structure is
given. (In general it is );t, - W K M. where W is the classicd Weyl group and M iis
a lattice.) The action on the Cartan subalgebra of the affine algebras is described in
the text.
.ilPPENDl,iC -4. PROPERTIES OF SI&fPLE LIE -4LGEBR4S
A . l A,, n 2 1
Extended Dynkin diagram:
Extended Cartan matrix: =
Cartan matrix determinant (index of connection): det A = n + 1.
1 Inverse Cartan matrix: A-' = 3 x
.4PPE!VDI,'C A. PROPERTES OF SIMPLE LIE ALGEBRAS 110
Quadratic form matrk: same as the inverse Cartan matrix.
Roots: ei - e j . 1 5 i, j 5 n + 1. i # j : normalization of the orthogonal basis:
(ei, ej) = d i j , i, j = 1,. . . . n + 1.
Simple roots: ai = ei - e;+i. 1 5 i 5 n.
Highest root: c 2 ~ = ai + a? + . . . + an = el - e,+l.
Fundamental weights:
Change of basis in weight space (fundamental weights basis +t orthogonal basis)
(Classical) Weyl group: W Sncl (generated by interchanges ei tt ej): 1 WI =
( n + l)! .
Congruence number: C?J(A) = Al + 2X2 + 3X3 + . . . + nX, (mod ( n + 1)). where n
Coxeter number: h = n + 1. Dual Coxeter number: hV = + 1.
Nuil root (marks): 6 = a* + a l +a? +. . . +a,. Central element (comarks, level) : c = ho + hl + . . . + h,.
PROPERTIES OF SIMPLE LIE ALGEBRAS
1123
Extended Dynkin diagram:
Cartan matrix determinant (index of connection): det .A = 2.
Inverse Cartan matrix: A-1 =
APPEArDLY A. PROPERTES OF SIMPLE LIE ALGEBRAS
Quadratic form rnatrix: Q = $
ROO~S: *ei, 1 5 i 5 n : &ei f e j , 1 5 i < j 5 n: normalization of the orthogonal
basis: ( e i , e j ) = J i j , i, j = 1,. . . , n.
Highest root: aa = lal + 2a2 + . . . + 2an4 + 2cr, = el + e?.
Eùndament al weights:
Change of bais in weight space (fundamental weights basis H orthogonal basis)
(Classical) Weyl group: W "= Sn H (Z2)n (generated by interchanges ei +, ej and
sign reversals ei + -ei); 1 W 1 = n! 2".
n
Congruence number: C N ( h ) = A, (mod 2) , where :\ = 1 hiwi . i= 1
Coxeter number: h = 2n.
Duai Coxeter number: hV = 2n - 1.
Nul1 root (marks): 6 = a0 + ai + 2a2 + 2a3 + . . . + 2an-1 + 2a,.
APPENDIX A. PRO PERTIES OF SIhIPL E LIE -4 L GEBUS 113
Centrai element (comarks, level): c = ho + hl + 2h2 + 2h3 + . . . + 2hn-l + hn.
A PPEiVDIX -4. PROPERTIES OF SIMPLE LIE AL GEBRAS
A.3 C,, n 2 2
C, si sp(2n, C);
Cl 2 .Al, C2 2 B2;
dirnC, = 2n2 + n.
Extended Dynkin diagram:
wmm* O I
Extended Cartan matrix: .x =
Cartan matrix determinant (index of connection): det .A = 2.
Inverse Cartan matrix: = $
Quadratic form matrix: Q =
' 1 I 1 I * * * 1 1 1 2 2 2 - * - 2 2 1 2 3 3 * * * 3 3 1 2 3 4 * * * 4 4 . S . . , . . - . . . . . 1 2 3 4 * - - 1 n - 1 , 1 2 3 4 - . a n - l n
A P P E N D K A. PROPERTIES OF SIMPLE LIE AL GEBRAS
Roots: f 2ei, 1 5 i 5 n : &ei f e j , 1 5 i < j 5 n; normalization of the orthogonal
basis: (ei? e j ) = +Jij, i? j = 1.. . . , n.
Positive roots: 2e i , 15 i < n : e i f e j , 15 i < j 5 n.
Simple roots: ai = ei - e;+l. 1 5 i 5 n - 1: a, = 3e,.
Highest root: a~ = 2al f 2a2 i-. . . + 2a,-i +an = 2%
Fundamental weights: di = el + el +. . . + ei, 1 5 i 5 n.
Change of basis in weight space (fundamatal weights basis tt orthogonal basis)
(Classical) Weyl group: W n Sn K (&)* (generated by interchanges ei tt e j and
sign reversals ei -t -ei): 1 W ( = n! 2".
n
Congruence number: C?i(A) = L X i (mod 2) = XI + An -t X5 + . . . (mod 2 ) .
Coxeter number: h = 272,
Dual Coxeter number: hV = n + 1.
Nul1 root (marks): 6 = a0 + 2ai + 2a2 + . . . + + a,.
Central element (comarks, level): c = ho + hl + . . . + h,.
A.4 D,, n 2 4
Extended Dynkin diagram:
Extended Cartan matrix: d =
A
for n = 4, =
APPEIVDE A. PROPERTIES OF SIMPLE LIE ALGEBRAS
Inverse Cartan matrix:
Cartan matrix determinant (index of connection): det .A = 4.
Quadratic form matrix: same as the inverse Cartan rnatrix.
Roots: &ei k e j ! 1 < i < j 5 n: normalization of the orthogonal basis: (ci. e j ) = dij.
Positive roots: ei f e j , 1 5 i < j 5 n.
Highest root: = ai + 2a2 + . . . + 2an-? + andl + on = el + e?.
Fundamental weights:
Change of bais in weight space (fundamental weights basis +t orthogonal basis)
C aiei = C (a j - aj+i) wj + (a,-l + a,) un.
(Classical) Weyl group: W - Sn K (Z2)n-L (generated by two sign reversais at a
time ei -+ -ei , ej -+ -e j . and interchanges ei +t e,); IWI = n!2"-'.
,4PPEiVDl,iI' -4. PROPERTIES OF SIMPLE LIE -4 L GEBRAS
Congruence number (vector):
C'i(A) = + ( n - 2)X,4 + nX, (mod 4, + A, (mod 2) i= 1
Coxeter nurnber: h = 2n - 2.
Dual Coxeter number: h'' = 2n - 2.
Nul1 root (marks): 6 = a0 + al ; 2a2 + . . . + 2a,4 + Û , - I + a,..
Central element (comarks, level): c = ho + h l + 2h2 + . . . + 2h,-? + h,- 1 + h,.
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