Examples of Poisson brackets

Weinstein SymposiumIHP, July 18 - 20, 2013

Jiang-Hua Lu

Department of MathematicsThe University of Hong Kong

July 18, 2013

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Definition of Poisson algebras

Definition. A Poisson algebra over a field k is acommutative k-algebra A together with a k-bilinearskew-symmetric map {, } : A× A→ A, called thePoisson bracket, such that for all a, b, c ∈ A,

Leibniz rule :

{a, bc} = b{a, c}+ c{a, b},Jacobi identity :

{a, {b, c}}+ {b, {c , a}}+ {c , {a, b}} = 0.

A Poisson space is a space X with a Poisson algebrastructure on its algebra of functions.

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Question

Why are Poisson structures interesting?

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Answer

“They become interesting if you study them longenough”

—Alen Weinstein, 1985

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Examples?

In local coordinates (x1, x2, . . . , xn), { , } is

determined by the

(n2

)functions

{xi , xj}, 1 ≤ i < j ≤ n,

which must satisfy

(n3

)non-linear PDEs.

Poisson structures are hard to make up.

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Standard Examples

0) {xi , xj} is cnstant for all 1 ≤ i < j ≤ n;

1) {xi , xj} is linear for all 1 ≤ i < j ≤ n;

2) {xi , xj} = cijxixj for constants cij .

3) X or X/G , where X is symplectic, G acting onX by symplectomorphisms

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Examples in Nature?

Quantum spaces give rise to Poisson structures assemi-classical limits;

Quantum groups give rise to Poisson structures on(Lie) groups and related spaces.

Goal of talk: To explain some of these Poissonspaces.

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Basic constructions

A Lie bialgebra is a Lie algebra g together with acompatible Lie algebra structure on g∗.

Constructions:

I Lie bialgebras =⇒ Poisson Lie groups: pairs(G , π), where G is a Lie group and π amultiplicative Poisson structure on G ;

I Quotiens.

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Three classes

I Real semi-simple Lie groups =⇒ real analyticPoisson structures;

I Complex semi-simple Lie groups =⇒holomorphic Poisson structures;

I Reductive algebraic groups =⇒ algebraicPoisson structures, even over positivecharacteristics.

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Common features

I “Natural Darboux type” coordinates;

I Finitely many T -leaves for some torus T .

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Real semi-simple Lie groups

G0, real semisimple Lie group;

G = (G0)C, complexification of G0;

U ⊂ G , maximal compact subgroup of G ;

K0 = U ∩ G0, maximal compact subgroup of G0;

B, flag variety of G .

Example: G0 = SL(n,R), G = SL(n,C), U =SU(n), K0 = SO(n).

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Spaces with natural real analytic Poisson structures:

I G0: one standard multiplicative structure foreach open G0-orbit in B;

I non-compact symmetric space G0/K0;

I compact symmetric space U/K0;

I flag manifold B.

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Poisson structures on B coming from G0

I A choice of a Borel subgroup B in an openG0-orbit in B defines a Poisson structure on B;

I T0-orbits of symplectic leaves are connectedcomopnents of (G0−orbits) ∩ (B−orbits);

I There are finitely many G0-orbits and B-orbitsin B; B-orbits parametrized by the Weyl group.

I Matsuki correspondence between G0-orbits andK = (K0)C-orbits in B;

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I The geometry and combinatorics of K -orbitclosures are essential in representations of G0,eg. Vogan-Kazdan-Lusztig polynomials;

I Computations of T0-leaves can be done usingAtlas of Lie groups.

I When G0 = U , the Poisson structure on B isclosely related to Kostant’s harmonic formsand Schubert calculus on B.

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Reductive algebraic groups

G , connected reductive algebraic group over k = k;

Spaces carrying natural algebraic Poisson structures:

I G and the wonderful compactification G of G ;

I (Twisted) conjugacy classes of G ;

I Grothendieck simultaneous resolution of G ;Unipotent varieties; Steinberg fibers;

I Flag varietties G/B , G/P , and products

G/B × G/B × · · · × G/B ;

I Bott-Samelson varieties.15 / 32

Examples coming from Bott-Samelson varieties

Let S be the set of simple reflections in Weyl groupW , e ∈ W the identity element.

I A sequence u = (s1, s2, . . . , sn) in S defines aBott-Samelson variety of dim = n:

Zu = Ps1 ×B Ps2 ×B · · · ×B Psn/B .

I A subexpression γ of u, i.e.,

γ = (γ1, γ2, . . . , γn),

where γj ∈ {sj , e}, defines an open subsetOγ ∼= Cn with coordinates (x1, x2, . . . , xn);

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Facts:

I The n-fold product of the standardmultiplicative Poisson structure on G descendsto a well-defined Poisosn structure Πu on Zu.

I In the coordinates (x1, x2, . . . , xn) on Oγ,

{xi , xj} ∈ Z[x1, x2, . . . , xn], ∀ 1 ≤ i < j ≤ n,

so have a Poisson bracket on k[x1, x2, . . . , xn]for any field k;

I Finitely many T -leaves!

I Computer program by Balazs Elek.17 / 32

Example 3. G = G2, s = (s1, s2, s1, s1, s2) andγ = (s1, s2, e, s1, e),

{x1, x2} = −3x1x2

{x1, x3} = 2x2x23 + x1x3

{x1, x4} = −4x2x3x4 − x1x4 − 2x2

{x1, x5} = 6x3x34x2

5 + 6x24x2

5 + 6x2x3x5

{x2, x3} = 3x2x3

{x2, x4} = −3x2x4

{x2, x5} = 6x34x2

5 + 3x2x5

{x3, x4} = −2x3x4

{x3, x5} = 3x3x5

{x4, x5} = 3x4x5.

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Let’s check the Jacobi identity:

{{x1, x2}, x5} = −18x2x3x34x2

5 − 18x1x34x2

5 − 18x2x24x2

5

− 18x22x3x5 − 9x1x2x5,

{{x2, x5}, x1} = −72x3x64x3

5 − 72x54x3

5 − 18x2x3x34x2

5

+ 18x1x34x2

5 + 18x2x24x2

5 − 18x22x3x5

+ 9x1x2x5,

{{x5, x1}, x2} = 72x3x64x3

5 + 72x54x3

5 + 36x2x3x34x2

5

+ 36x22x3x5

{{x1, x2}, x5}+ {{x2, x5}, x1}+ {{x5, x1}, x2} = 0.

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Another example. G = G2,w = (s1, s2, s1, s1, s2, s1, s1, s1, s2),γ = (s1, s2, e, s1, s2, e, s1, s1, e),

{x1, x2} = −3x1x2

{x1, x3} = 2x2x23 + x1x3

{x1, x4} = −4x2x3x4 − x1x4 − 2x2

{x1, x5} = −6x3x34 − 6x2x3x5 − 6x2

4

{x1, x6} = 6x3x24x2

6 + 2x2x3x6 + 4x4x26 − x1x6

{x1, x7} = −12x3x24x6x7 − 2x2x3x7 − 6x3x

24

− 8x4x6x7 + x1x7 − 4x4

{x1, x8} = 12x3x24x6x8 + 2x2x3x8 + 8x4x6x8 − x1x8

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{x1, x9} = −6x3x5x36x3

7x38x2

9 + 18x3x4x26x3

7x38x2

9

+ 18x3x5x36x2

7x28x2

9 − 18x3x5x26x2

7x38x2

9

− 54x3x4x26x2

7x28x2

9 + 36x3x4x6x27x3

8x29

+ 6x26x3

7x38x2

9 − 18x3x5x36x7x8x

29

+ 36x3x5x26x7x

28x2

9 − 18x3x5x6x7x38x2

9

+ 54x3x4x26x7x8x

29 − 72x3x4x6x7x

28x2

9

+ 18x3x4x7x38x2

9 − 18x26x2

7x28x2

9 + 12x6x27x3

8x29

+ 6x3x5x36x2

9 − 18x3x5x26x8x

29 + 18x3x5x6x

28x2

9

− 6x3x5x38x2

9 − 18x3x4x26x2

9 + 36x3x4x6x8x29

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− 18x3x4x28x2

9 + 18x26x7x8x

29 − 24x6x7x

28x2

9

+ 6x7x38x2

9 − 18x3x24x6x9 − 6x2

6x29

+ 12x6x8x29 − 6x2

8x29 − 6x2x3x9 − 12x4x6x9

{x2, x3} = 3x2x3

{x2, x4} = −3x2x4

{x2, x5} = −6x34 − 3x2x5

{x2, x6} = 6x24x2

6

{x2, x7} = −12x24x6x7 − 6x2

4

{x2, x8} = 12x24x6x8.

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{x2, x9} = −6x5x36x3

7x38x2

9 + 18x4x26x3

7x38x2

9

+ 18x5x36x2

7x28x2

9 − 18x5x26x2

7x38x2

9

− 54x4x26x2

7x28x2

9 + 36x4x6x27x3

8x29

− 18x5x36x7x8x

29 + 36x5x

26x7x

28x2

9

− 18x5x6x7x38x2

9 + 54x4x26x7x8x

29

− 72x4x6x7x28x2

9 + 18x4x7x38x2

9 + 6x5x36x2

9

− 18x5x26x8x

29 + 18x5x6x

28x2

9 − 6x5x38x2

9

− 18x4x26x2

9 + 36x4x6x8x29 − 18x4x

28x2

9

− 18x24x6x9 − 3x2x9,

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{x3, x4} = −2x3x4

{x3, x5} = −3x3x5

{x3, x6} = x3x6

{x3, x7} = −x3x7

{x3, x8} = x3x8

{x3, x9} = −3x3x9

{x4, x5} = −3x4x5

{x4, x6} = 2x5x26 + x4x6

{x4, x7} = −4x5x6x7 − x4x7 − 2x5

{x4, x8} = 4x5x6x8 + x4x8

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{x4, x9} = 6x6x37x3

8x29 − 18x6x

27x2

8x29

+ 6x27x3

8x29 + 18x6x7x8x

29 − 12x7x

28x2

9

− 6x5x6x9 − 6x6x29 + 6x8x

29 − 3x4x9

{x5, x6} = 3x5x6

{x5, x7} = −3x5x7

{x5, x8} = 3x5x8

{x5, x9} = 6x37x3

8x29 − 18x2

7x28x2

9 + 18x7x8x29

− 6x5x9 − 6x29

{x6, x7} = −2x6x7, {x6, x8} = 2x6x8

{x6, x9} = −3x6x9, {x7, x8} = 2x7x8 − 2

{x7, x9} = −3x7x9, {x8, x9} = 3x8x9

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{x1, {x5, x9}}= −72x3x5x

36x6

7x68x3

9 + 216x3x4x26x6

7x68x3

9

+ 432x3x5x36x5

7x58x3

9 − 216x3x5x26x5

7x68x3

9

− 1296x3x4x26x5

7x58x3

9 + 432x3x4x6x57x6

8x39

+ 72x26x6

7x68x3

9 − 1080x3x5x36x4

7x48x3

9

+ 1080x3x5x26x4

7x58x3

9 − 216x3x5x6x47x6

8x39

+ 3240x3x4x26x4

7x48x3

9 − 2160x3x4x6x47x5

8x39

+ 216x3x4x47x6

8x39 − 432x2

6x57x5

8x39

+ 144x6x57x6

8x39 + 36x3x

25x3

6x37x3

8x29

+ 1440x3x5x36x3

7x38x3

9 − 2160x3x5x26x3

7x48x3

9

26 / 32

+ 864x3x5x6x37x5

8x39 − 72x3x5x

37x6

8x39

− 108x3x4x5x26x3

7x38x2

9 − 4320x3x4x26x3

7x38x3

9

+ 4320x3x4x6x37x4

8x39 − 864x3x4x

37x5

8x39

+ 1080x26x4

7x48x3

9 − 720x6x47x5

8x39 + 72x4

7x68x3

9

− 216x3x24x6x

37x3

8x29 − 108x3x

25x3

6x27x2

8x29

+ 108x3x25x2

6x27x3

8x29 − 1080x3x5x

36x2

7x28x3

9

+ 2160x3x5x26x2

7x38x3

9 − 1296x3x5x6x27x4

8x39

+ 216x3x5x27x5

8x39 + 324x3x4x5x

26x2

7x28x2

9

− 216x3x4x5x6x27x3

8x29 + 3240x3x4x

26x2

7x28x3

9

27 / 32

− 4320x3x4x6x27x3

8x39 + 1296x3x4x

27x4

8x39

− 36x5x26x3

7x38x2

9 − 1440x26x3

7x38x3

9

+ 1440x6x37x4

8x39 − 288x3

7x58x3

9

− 72x2x3x37x3

8x29 + 648x3x

24x6x

27x2

8x29

− 108x3x24x2

7x38x2

9 + 108x3x25x3

6x7x8x29

− 216x3x25x2

6x7x28x2

9 + 108x3x25x6x7x

38x2

9

+ 432x3x5x36x7x8x

39 − 1080x3x5x

26x7x

28x3

9

+ 864x3x5x6x7x38x3

9 − 216x3x5x7x48x3

9

− 144x4x6x37x3

8x29 − 324x3x4x5x

26x7x8x

29

28 / 32

+ 432x3x4x5x6x7x28x2

9 − 108x3x4x5x7x38x2

9

− 1296x3x4x26x7x8x

39 + 2160x3x4x6x7x

28x3

9

− 864x3x4x7x38x3

9 + 108x5x26x2

7x28x2

9 − 72x5x6x27x3

8x29

+ 1080x26x2

7x28x3

9 − 1440x6x27x3

8x39

+ 432x27x4

8x39 + 216x2x3x

27x2

8x29 − 648x3x

24x6x7x8x

29

+ 216x3x24x7x

28x2

9 − 36x3x25x3

6x29 + 108x3x

25x2

6x8x29

− 108x3x25x6x

28x2

9 + 36x3x25x3

8x29 − 72x3x5x

36x3

9

+ 216x3x5x26x8x

39 − 216x3x5x6x

28x3

9 + 72x3x5x38x3

9

+ 432x4x6x27x2

8x29 − 72x4x

27x3

8x29 + 108x3x4x5x

26x2

9

29 / 32

− 216x3x4x5x6x8x29 + 108x3x4x5x

28x2

9 + 216x3x4x26x3

9

− 432x3x4x6x8x39 + 216x3x4x

28x3

9 − 108x5x26x7x8x

29

+ 144x5x6x7x28x2

9 − 36x5x7x38x2

9 − 432x26x7x8x

39

+ 720x6x7x28x3

9 − 288x7x38x3

9 − 216x2x3x7x8x29

+ 108x3x24x5x6x9 + 216x3x

24x6x

29 − 108x3x

24x8x

29

− 432x4x6x7x8x29 + 144x4x7x

28x2

9 + 36x3x34x9

+ 36x5x26x2

9 − 72x5x6x8x29 + 36x5x

28x2

9 + 72x26x3

9

− 144x6x8x39 + 72x2

8x39 + 72x2x3x5x9 + 72x2x3x

29

+ 72x4x5x6x9 + 144x4x6x29 − 72x4x8x

29 + 36x2

4x9

30 / 32

− 324x3x24x6x

27x2

8x29 − 432x3x5x

36x7x8x

39

+ 1080x3x5x26x7x

28x3

9 − 864x3x5x6x7x38x3

9

+ 216x3x5x7x48x3

9 + 72x4x6x37x3

8x29

+ 1296x3x4x26x7x8x

39 − 2160x3x4x6x7x

28x3

9

+ 864x3x4x7x38x3

9 − 108x5x26x2

7x28x2

9

+ 72x5x6x27x3

8x29 − 1080x2

6x27x2

8x39

+ 1440x6x27x3

8x39 − 432x2

7x48x3

9

− 108x2x3x27x2

8x29 + 324x3x

24x6x7x8x

29

+ 72x3x5x36x3

9 − 216x3x5x26x8x

39 + 216x3x5x6x

28x3

9

− 72x3x5x38x3

9 − 216x4x6x27x2

8x29 − 216x3x4x

26x3

9

31 / 32

+ 432x3x4x6x8x39 − 216x3x4x

28x3

9 + 108x5x26x7x8x

29

− 144x5x6x7x28x2

9 + 36x5x7x38x2

9 + 432x26x7x8x

39

− 720x6x7x28x3

9 + 288x7x38x3

9 + 108x2x3x7x8x29

+ 108x3x24x5x6x9 − 108x3x

24x6x

29 + 216x4x6x7x8x

29

+ 36x3x34x9 − 36x5x

26x2

9 + 72x5x6x8x29 − 36x5x

28x2

9

− 72x26x3

9 + 144x6x8x39 − 72x2

8x39

− 36x2x3x29 − 72x4x6x

29 .

{x1, {x5, x9}}+ {x9, {x1, x5}}+ {x5, {x9, x1}} = 0.

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