Lecture

Symplectic leaves of a Poisson lie

group

Symplectic leaves of a simple lie

group with standardPoisson lie

structure double Bruhat cells

Characters as Poisson commuting functionson symplectic leaves characteristic systems

Coxeter Toda systems

Factorization method solving equations

of motion

Degenerate integrability of characteristic

systems

Sy ewesiP ps

Let G be a Poisson lie group G be its

Poisson lie dual G be its double

Consider

G D a Dca G

Gtc DCG it acts on DCG hey

left multiplications it acts on DIG a

we have a lie algebra homorphism

of Vectfola a

and DCG G is locally isomorphic to G

This is why we have

The There exists a tie algebra homomorphism

y VectfG local action of Gt on G set

Vet G Vet DCG G

Tg

This action is called the dressingaction ofatton G

Thin Symplectic leaves of G are orbits

of G action i e they are leaves of

the tangent distribution on G given bythe dressing g action

ExampII G simple with the standardPoisson lie structure As we know

D G Gx G standard PL structure

as a manifold and a groupis factorizable

Gcs Gx G g g g

G G x G Cbt b ft b

are Poisson lie subgroups

Because

G let b 31 Edo IT

we have

DCG G GIN x GIN H

where he H acts on GIN x GIN as

h LgaNt ga NH g Nth gzN ht

Bruhat decompositions

G U Ba Nt G U B r NCW

U EW

choose representatives in c NIHCG of

elements of the Weyl group then

G1Nt tfw H Nta ie c G

GIN E U H NE ir a G

VE W

Here NI In c N I ii n ie E N

NI In e N I n is e N t

Proposition g Nt g N HH E LHNI ie x HNINH

iff g e G Bu B h Bu B

Proof Iw

Let

6g fG orbit through g g G't

theorem If g c a then

0g HIM a ux NI x N I

ProofLet g C G

vi e

g h n ie n't ti n i n

then

0g H orbit through xN orbit through

hntie Nt Int is N J H

L H orbit through h NI i Nt.HN irN H

We have

h hNIiN hNIirN JH

h Nti Nt Ee J voi Ck Nini N H

i e H action is not free

The stabilizer Huw then I nicht h

a r

EBB

The symplectic leaf 5g C G through

G E G k G is the preimage of 0g

The symplectic leaves of G

are level surfaces of

Cnut g II Dewi wilgjttbwi.uiw.ggt ti Wi E Ker article c f Eiwi Z

Corollary dim Sgc G

elm relist codimlicercuitidD

dustsheetson GUN

G Sten Cl

D B si B I ftp I

L int

B B n B si B Iexp Lt Fi Fi Eitho

similarly

BE B n BoB Ethe profile is determined by r

Let N Si Six reduced decomposition

13 BY i3 LHexplt Fia expltic Fie

Coordinate charts reduced decompositions

of N

Coordinate tranformations on intersection

cluster mutations consequence ofSerre relations

2 G BUB n B r B

Bub I KAYdetermined

by a

determined by V

is 14 73,41determinedby v

BUBNB vB.tl hEEEEfFEfp

determined by u

G 3 GUY zgjzu Sir sikN Sjn r Sjc

cluster coordinate charts

G Ii33j3 Hexp snEj explseEje

expltafin exp their

Coordinate transformations clustermutations

Thm FZ Ghifizy ReutteW1 tr

theorem R.KZ 4 Coordinates si ti Hare Poisson flat

xa xp Pap xx Xp12 symplectic leaves are cluster

subvarities in GUN

Integrablesysteursgerreratedbycharacters

Proposition In a quasitriangular Poisson

lie group central functions Poisson

commute

Proofi fj hgh fi g i l z

fi f23 g e p g dfncg nd feelg 7

52 Adf22 dfa g A dfdg

O

because Adg Idf g Ad fdg

dfidg A dfolg

There are re rank G independent central

functions on a simple lie group G

Independent functions remain independent

when restricted to a symplectic leaf

Completeintegrabilityukendimlfl

2rolinal

S C G Elul t l ly t codimlverlurtidl

Coxeter elements of W e Sii Sir

in.ir 3 a permutation of 21 r3

di G r and we have

rintependentPoissoncommutingfunctions

Cluster coordinates on S o

g D II pidgin firlgir T

di SLIM oC G o i th da triple PE H

gt t YT Yt H KKR 2000

I

Poisson brackets between XIxtiixtj3 dx i.li 3 0

4x4 Xj 3 2dicijxtiljHamiltonians

Hk Chak g Coxeter Toda systems

For g Sen k 1 this gives relativistic

Toda

Solutionstoequationotion

thx is central we want to

describe flow lines of NH at DH

algebraically

a For x c G consider

glt explt OHHc G

Here OH X H RxH

G is a factorizable Poisson lie group

glH gilt g HI Cgt o g IT

g I E G B C G

theorem the flow line of the Hamiltonianvector field generated by H passing

through x at t o is

x It gilt Ix gilt

Rematke It works for all symplecticleaves not necessary Coxeter

Superintegrability onother

symphedicl eaves

On each symplectic leafwe have r independent Poisson

commuting functions

theorem R 2002

when dem S zr central functions

generate a superintegrable system

s Sx 5 TtdgSItdgdimlst

dimllsxskadfftdin.IS FdaHere

Ajit the set of H orbitsintersecting A

assuming AC X and Haets on X

summaryG simple Lie group

Standard Poisson lie structure

Symplectic leaves cluster varitieswith flat Poisson structure

Functions on G Ada a 616 6 6

Poisson commute and define

integrable systems with r independent

Poisson commuting integralsLiouville integrable on Coxter

symplectic leaves

superintegrableon others

factorization of exp tongsolves equations of motion

ReinasWorks for a dimensional lie

algebras and lie groups

cluster coordinates forKac Moody groups H Williams

Factorization dynamicsX G G

xig gtg.ms gigDescends to symplectic leaves

On Coxeter symplectic leaves in

cluster XF coordinates

Hxti XI

xxi Hist H x x

Cij