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
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