Some examples of algebraic Poisson and symplectic groupoids

Jiang-Hua Lu

The University of Hong Kong

Global Poisson, September 24, 2020

(based on joint work with Victor Mouquin and Shizhuo Yu)

1 / 27 A class of polynomial Poisson algebras

Warm-up: systematical examples of polynomial Poisson brackets:

Constant Poisson brackets on C[x1, x2,..., xl ]:

{xi , xj } = cij , cij ∈ C;

Linear Poisson brackets:

X (k) (k) {xi , xj } = cij xk , cij ∈ C; k

(Quadratic) log-canonical Poisson brackets:

{xi , xj } = cij xi xj , cij ∈ C;

Higher order? Hard to make up in general.

2 / 27 A class of Poisson polynomial algebras

Theorem (Elek-L. 2012, 2019) Every sequence of simple reflections

u = (s1, s2,..., sl )

in a Weyl group gives rise to a polynomial Poisson structure on Cl .

1 Each C[xi ,..., xj ] for 1 ≤ i ≤ j ≤ l is a Poisson sub-algebra:

{xi , xj } = cij xi xj + fij (xi+1,..., xj−1). log-canonical terms

2 Has a natural torus T -action by Poisson automorphisms.

3 Has finitely many T -leaves and thus also an open T -leaf.

4 Can be computed by a computer program (by Balazs Elek).

3 / 27 A class of polynomial Poisson algebras

Example 1: Type G2 with simple generators s1, s2: the sequence u = (s1, s2, s1, s2, s1, s2) gives a Poisson bracket on C[x1,..., x6]:

{x1, x2} = −3x1x2, {x1, x3} = −x1x3 − 2x2, 2 {x1, x4} =0 − 6x3 , {x1, x5} = x1x5 − 4x3,

{x1, x6} =3 x1x6 − 6x5, {x2, x3} = −3x2x3, 3 2 {x2, x4} = −3x2x4 − 6x3 , {x2, x5} =0 − 6x3 ,

{x2, x6} =3 x2x6 − 18x3x5 + 6x4,

{x3, x4} = −3x3x4, {x3, x5} = −x3x5 − 2x4, 2 {x3, x6} =0 − 6x5 , {x4, x5} = −3x4x5, 3 {x4, x6} = −3x4x6 − 6x5 , {x5, x6} = −3x5x6.

Terms in red are the log-canonical terms.

4 / 27 A class of polynomial Poisson algebras

Example 2: G2 again, with u = (s1, s2, s1, s2, s1, s2, s2, s1, s2, s1, s2, s1), give a Poisson bracket on C[x1,..., x12]. Some examples:

3 2 {x1, x8} = −3x1x8 − 6x7 , {x1, x9} =0 − 6x7 ,

{x1, x10} =3 x1x10 − 18x7x9 + 6x8, {x1, x11} =3 x1x11 − 6x7, 3 2 {x1, x12} =6 x1x12 − 6, {x2, x8} =0 − 6x6x7 + 6x7 , 2 {x2, x9} = x2x9 − 6x6x7 + 4x7,

{x2, x10} =3 x2x10 − 18x6x7x9 + 6x6x8 + 6x9,

{x2, x11} =2 x2x11 − 6x6x7 + 2, {x2, x12} =3 x2x12 − 6x6, 3 3 2 {x3, x8} =3 x3x8 − 18x4x6x7 + 6x5x7 + 18x4x7 , 2 2 {x3, x9} =3 x3x9 − 18x4x6x7 + 6x5x7 + 12x4x7,

{x3, x10} =6 x3x10 − 54x4x6x7x9 + 18x4x6x8 + 18x5x7x9 + 18x4x9 − 6x5x8 − 6,

∃ open symplectic leaf given by φ1(x1,..., x12) 6= 0, φ2(x1,..., x12) 6= 0 for some φ1, φ2 ∈ C[x1,..., x12] (with Lie-theoretical meaning).

5 / 27 A class of Poisson polynomial algebras

Today’s Talk:

Some background on these polynomial Poisson structures;

Construct algebraic symplectic groupoids for them.

Part I: Some basic concepts;

Part II: A general construction;

Part III: Specializing to one example;

Part IV: In the (grand) scheme of things ......

6 / 27 Part I: Some basic concepts

1. T -Leaves.

If T is a torus, a T -Poisson manifold is a Poisson manifold (X , πX )

with a T -action preserving πX .

A T -orbit of symplectic leaves, or a T -leaf for short, of a T -Poisson

manifold (X , πX ) is a sub-manifold L of X of the form [ L = tΣ, t∈T

where Σ is a symplectic leaf of (X , πX ), and the map

T × Σ −→ L, (t, x) 7−→ tx

is a submersion.

7 / 27 Part I: Some basic concepts

2. Mixed Product Poisson Structures.

Let Y = Y1 × · · · × Yn with projections

pi : Y −→ Yi , i ∈ [1, n]. A Poisson structure π on Y is called a mixed product if

pi (π) is a well-defined Poisson structure on Yi for each i;

∗ ∞ ∞ Equivalently, if each pi C (Yi ) ⊂ C (Y ) is a Poisson sub-algebra.

In coordinates for X × Y , X ∂ ∂ X ∂ ∂ π = αij (x) ∧ + βkl (y) ∧ ∂xi ∂xj ∂yk ∂yl i

8 / 27 Part I: Some basic concepts

3. Poisson groupoids and symplectic groupoids.

A Poisson groupoid is a Lie groupoid θ± :Γ ⇒ M with a Poisson structure π such that the graph of the groupoid multiplication

0 0 0 {(γ, γ , γγ ): θ−(γ) = θ+(γ )} ⊂ Γ × Γ × Γ

is a coisotropic sub-manifold of (Γ × Γ × Γ, π × π × (−π)). In such

a case, πM := θ+(π) = −θ−(π) is a well-defined Poisson structure on M. Denote the Poisson groupoid as (Γ, π) ⇒ (M, πM ).

A symplectic groupoid is a Poisson groupoid (Γ, π) ⇒ (M, πM ), where π is non-degenerate and dim Γ = 2 dim M;

Basic problem in Poisson geometry: To construct interesting Poisson groupoids and symplectic groupoids.

9 / 27 Part II: A general construction

Examples of Poisson groupoids:

1 Pair Poisson groupoid( X × X , πX × (−πX )) ⇒ (X , πX );

2 A Poisson Lie group is a Poisson groupoid over a one point space;

Can combine these two examples to construct gauge Poisson groupoids:

A Poisson action of a Poisson Lie group (G, πG ) is a Poisson map

(X × G, πX × πG ) → (X , πX ) that is also a right Lie group action. In such a case, have quotient Poisson manifolds

(X /Q, πX /Q ) (assuming that X /Q is smooth),

where Q is a coisotropic subgroup of (G, πG ), and πX /Q is the

projection of πX to X /Q.

10 / 27 Part II: A general construction

A General Construction, I: Gauge Poisson Groupoids

Given free proper Poisson Lie group action of (G,πG ) on (X ,πX ), have

(X /G, πX /G ), quotient of (X , πX ) by G;

((X × X )/G, π), quotient of πX × (−πX ) by diagonal G-action; (X × X )/G ⇒ X /G, gauge groupoid with

source : (X × X )/G → X /G :[x1, x2] 7→ [x1],

target : (X × X )/G → X /G :[x1, x2] 7→ [x2], unit : X /G → (X × X )/G, [x] 7→ [x, x],

inverse : (X × X )/G → (X × X )/G :[x1, x2] 7→ [x2, x1],

multiplication: [x1, x2] · [x3, x4] = [x1g, x4], where x2g = x3.

Lemma (Reference?)

As above, ((X × X )/G, π) ⇒ (X /G, πX /G ) is a Poisson groupoid.

11 / 27 Part II: A general construction

A General Construction, II: Series of Quotient Poisson Manifolds:

Given closed Poisson Lie sub-group Q ⊂ (G, πG ) and Poisson action

(Q, πG ) × (Y , πY ) −→ (Y , πY ), (q, y) 7→ qy,

projection of πG × πY is a well-defined Poisson structure on

def −1 G ×Q Y = (G × Y )/Q, (g, y)q := (xq, q y).

Given closed Poisson Lie sub-groups Qj ⊂ (G, πG ), j ∈ [1, n], have

successive quotients G/Qn, G ×Qn−1 (G/Qn), ··· , ··· , ··· ,

Zn = G ×Q1 × · · · × G ×Qn−1 G/Qn,

n the quotient of G by Q1 × · · · × Qn−1 × Qn by action

−1 −1 (g1, g2,..., gn) · (q1, q2,..., qn) = (g1q1, q1 g2q2,..., qn−1gnqn),

n n with quotient Poisson structure πZn , projection of (πG ) on G .

12 / 27 Part II: A general construction

Combining the two constructions: (Summary of Part II)

A closed Poisson Lie sub-group Q ⊂ (G, πG ) gives a series: for n ≥ 1, n n z }| { z }| { Xn = G ×Q · · · ×Q G, Yn = G ×Q · · · ×Q G /Q = Xn/Q, and 2n n n z }| { z }| { z }| { X2n = G ×Q · · · ×Q G = G ×Q · · · ×Q G ×Q G ×Q · · · ×Q G = Xn ×Q Xn.

Theorem (L.-Mouquin-Yu 2020)

For any n ≥ 1, (X2n, πX 2n ) ⇒ (Yn, πY n ) is a Poisson groupoid with

source θ+ : X2n → Yn, [g1,..., g2n]X 2n 7→ [g1,..., gn]Y n ; −1 −1 target θ− : X2n → Yn, [g1,..., g2n]X 2n 7→ [g2n ,..., gn+1]Y n ; −1 −1 unit : Yn → X2n, [g1,..., gn]Y n 7→ [g1,..., gn, gn ,..., g1 ]X 2n ; −1 −1 inverse : X2n → X2n, [g1,..., g2n]X 2n 7→ [g2n ,..., g1 ]X 2n ; 0 0 0 product: for γ = [g1,..., g2n]X 2n and γ = [g1,..., g2n]X 2n composable, 0 0 0 0 0 0 γγ = [g1,..., gn, gn+1 ··· g2ng1 ··· gngn+1, gn+2,..., g2n]X 2n .

13 / 27 Part III: Specializing to an example

Fix G: connected complex semi-simple Lie group with g = Lie(G);

(B, B−): pair of opposition Borel sub-groups of G;

T = B ∩ B−: maximal torus of G;

W = NG (T ): Weyl group of (G, T );

h , ig: invariant symmetric non-degenerate blinear form on g.

Well-known fact: Above data give rise to standard classical r-matrix rst ∈ g ⊗ g and Poisson Lie group (G, πst), where

L R πst = rst − rst.

Both B and B− are Poisson Lie sub-groups of (G, πst).

Now apply the general construction in Part II to the pair B ⊂ (G, πst).

14 / 27 Part III: Specializing to an example

Apply Part II to B ⊂ (G, πst): For any integer n ≥ 1, have

n 2n z }| { z }| { Fn = G ×B · · · ×B G /B and Fe2n = G ×B · · · ×B G

with respective quotient Poisson structures πn and πe2n. Introduce

Γ2n = {[g1, g2,..., g2n]Fe2n : g1g2 ··· g2n ∈ B−} ⊂ Fe2n.

Theorem (L. -Mouquin-Yu, 2020)

For each n ≥ 1, (Fe2n, πe2n) is a Poisson groupoid over (Fn, πn) with

source θ+ : Fe2n → Fn, [g1,..., g2n]Fe2n 7→ [g1,..., gn]F n ; −1 −1 target θ− : Fe2n → Fn, [g1,..., g2n]Fe2n 7→ [g2n ,..., gn+1]F n ; unit, inverse, multiplication, ··· , ··· , ··· .

Furthermore, Γ2n is a wide Poisson sub-groupoid of (Fe2n, πe2n) ⇒ (Fn, πn).

Today’s main objects of interests:(Γ 2n, πe2n)⇒(Fn, πn). 15 / 27 Part III: Specializing to an example

A name for (Γ2n, πe2n) ⇒ (Fn, πn): Let N ⊂ B be the unipotent radical.

B = F1 = G/B: flag variety of G, with Poisson structure π1; A := G/N: decorated flag variety of G, with Poisson structure 0 o π1 := πG/N . Let A = B−N/N ⊂ A (open). ∼ 2n−1 o As varieties, have isomorphism Θb 2n :Γ2n = C2n := B × A :

Θb 2n([g1, g2,..., g2n]Fe2n ) = (g1·B,..., g1 ··· g2n−1·B, g1g2 ··· g2n·N).

∼ n Similarly, have isomorphismΘ n : Fn = B . 2 n 2 Define πn = Θn(πn) ∈ X (B ) and πb2n = Θb 2n(πe2n) ∈ X (C2n). n Call (Γ2n, πe2n) ⇒ (Fn, πn) or (C2n, πb2n) ⇒ (B , πn) the n’th configuration Poisson groupoid of flags of G.

Theorem (L.-Mouquin, 2016)

1 πn is a mixed product of n-copies of π1. 2 0 πb2n is a mixed product of (2n − 1)-copies of π1 and one π1. 16 / 27 Part III: Specializing to an example

n A decomposition of (Fn, πn): For u = (u1,..., un) ∈ W , let u O = (Bu1B) ×B · · · ×B (BunB)/B ⊂ Fn. F Bruhat decomposition G = u∈W BuB gives the disjoint union G u Fn = O (of T-Poisson sub-manifolds). u∈W n

Lemma

Using root sub-groups and reduced words of each ui , one has

u ∼ l(u) O = C , l(u) = l(u1) + ··· + l(un),

l(u) resulting in a polynomial Poisson structure πu on C .

u O ⊂ Fn is called a generalized Schubert (or Bruhat) cell; u πn := πn|Ou is called the standard Poisson structure on O ; Ou =∼ Cl(u) gives Bott-Samelson coordinates on Ou; −1 Earlier G2 examples: u = (s1, s2, s1, s2, s1, s2) or (u, u ). 17 / 27 Part III: Specializing to an example

u Goal of talk: Construct algebraic symplectic groupoids of (O , πn).

Some history:

For n = 1, Ou = BuB/B ⊂ G/B. L.-Mouquin (2018) proved that

u,u G = BuB ∩ B−uB− ⊂ (G, πst) (double Bruhat cell)

u u,u is a Poisson groupoid over (O , π1) and symplectic leaves of G u are symplectic groupoids over (O , π1) (using Kogan-Zelevinsky);

For arbitrary u ∈ W n, Mouquin (2019) showed that the generalized double Bruhat cell G u,u is a Poisson groupoid (in many ways) over u (O , πn). No statement on symplectic leaves.

General phenomenon: For Poisson structures related to (G, πst), T -leaves are easier to determine (and often finitely many), but symplectic leaves are trickier.

18 / 27 Part III: Specializing to an example

Main result for today’s talk (Summary of Part III):

Recall the configuration Poisson groupoid θ± : (Γ2n, πe2n) ⇒ (Fn, πn). F u n u Recall Fn = u∈W n O . For u ∈ W , have full sub-groupoid over O

u −1 u −1 u u Γ = θ+ (O ) ∩ θ− (O ) ⇒ O .

u Fact: Γ is a single T -leaf of (Γ2n, πe2n). Theorem (L.-Mouquin-Yu, 2020)

n u u For each u ∈ W , (Γ , πe2n) ⇒ (O , πn) is a Poisson groupoid; u u u the symplectic leaf Σ of πe2n through the identity section O ,→ Γ u u is a symplectic groupoid (Σ , πe2n) ⇒ (O , πn); u,u ∼ u u,u u isomorphisms G =Γ give Mouquin’s Poisson groupoids G ⇒O .

19 / 27 Part IV: In the (grand) scheme of things

u Why are the (O , πn)’s interesting? Why their symplectic groupoids?

1. Generalized Schubert cells as basic building blocks

The decomposition

G u (Fn, πn) = (O , πn) u∈W n

gives a paving of (Fn, πn) by generalized Schubert cells.

Theorem (L.-Yu, 2019)

Many Poisson homogeneous spaces of (G, πst), including

(G, πst), (G/B, πG/B ), (G/T , πG/T ),

have open covers by (T -mixed products of) generalized Schubert cells.

20 / 27 Part IV: In the (grand) scheme of things

u u u 2. A prominent feature of (O , πn), (Σ , πe2n) ⊂ (Γ , πe2n):

A symmetric T -Poisson CGL, defined by Goodearl-Yakimov, is a polynomial Poisson algebra (A = C[x1, x2,..., xl ], { , }) with a T -action by Poisson algebra automorphisms, such that

1 × each xi is a weight vector with weight λi ∈ HomC(T , C ); 2 ∃ h1,..., hl ∈ Lie(T ) s.t. λi (hi ) 6= 0 for all i ∈ [1, l], and

{xi , xj } ∈ λi (hj )xi xj + C[xi+1,..., xj−1], 1 ≤ i < j ≤ l.

Can extend the Poisson structure π from Cl to π ./ 0 on Cl × T via

{xi , ξj } = cij xi ξj , {ξj , ξk } = 0, i ∈ [1, l], j, k ∈ [1, dim T ].

Theorem (Elek-L. 2012, 2019) u (O , πn) is a symmetric T -Poisson CGL in Bott-Samelson coordinates.

21 / 27 Part IV: In the (grand) scheme of things

n −1 −1 −1 For u = (u1,..., un) ∈ W , define u = (un ,..., u1 ), and (u,u−1) (u,u−1) Oe = {[g1, g2,..., g2n]F 2n ∈ O : g1g2 ··· g2n ∈ B−B}.

Lemma

−1 n (u,u ) (u,u−1) For any u ∈ W , Oe is an open symplectic leaf of (O , π2n).

Theorem (L.-Mouquin-Yu, 2020) n For any u = (u1,..., un) ∈ W , have Poisson isomorphism

u ∼ (u,u−1) I : (Γ , πe2n) = (Oe × T , π2n ./ 0),

−1 −1 u (u,u ) (u,u ) and I (Σ ) ⊂ Oe × T is a finite cover of Oe by projection.

Conclusion:

n u u For each u ∈ W , (Γ , πe2n) and (Σ , πe2n) are respectively localizations and specializations of symmetric Poisson CGLs. 22 / 27 Part IV: In the (grand) scheme of things

−1 Example: For G2 with (u, u ) = (s1, s2, s1, s2, s1, s2, s2, s1, s2, s1, s2, s1):

−1 1 recall polynomial Poisson structure π on O(u,u ) =∼ C12;

−1 (u,u ) 12 2 open symplectic leaf Oe ⊂ C defined by

φ1(x1,..., x12) 6= 0 and φ1(x1,..., x12) 6= 0;

3 extend π to π ./ 0 on C12 × C2 by

{ξ1, ξ2} = 0, {xi , ξj } = λij xi ξj , i ∈ [1, 12], j = 1, 2.

−1 u ∼ (u,u ) 14 4 Γ = Oe × T ⊂ C is given by

φ1(x1,..., x12) 6= 0, φ1(x1,..., x12) 6= 0, ξ1 6= 0, ξ2 6= 0;

5 symplectic leaf Σu ⊂ Γu ⊂ C12 × (C×)2 is given by 2 2 ξ1 = φ1(x1,..., x12), ξ2 = φ2(x1,..., x12), −1 a 4-to-1 cover of the symplectic leaf O(u,u ) ∼ 12 . e = Cφ1φ26=0

23 / 27 Part IV: In the (grand) scheme of things

3. In connection with cluster algebras

Goodearl-Yakimov theory (2018): Every symmetric T -Poisson CGL extension is naturally a cluster algebra.

L. Shen and D. Weng defined certain configuration spaces of flags v n Confe (A) for any v = (v1, v2,..., vn) ∈ W and showed that they are cluster varieties.

n u ∼ (u,u−1) Fact. For each u ∈ W ,Γ = Confe (A).

4. In connection with integrable systems

Theorem (L.-Mi, 2018)

(u,u−1) There is a completely integrable system on (Oe , π2n) all of whose Hamiltonian flows are defined on C.

24 / 27 Summary

Summary of today’s talk

1 Poisson Lie sub-group Q ⊂(G, πG ) gives series of Poisson gorupoids;

2 Applying to B ⊂ (G, πst), get the configuration Poisson groupoids

θ± : (Γ2n, πe2n) ⇒ (Fn, πn), n ≥ 1;

3 F u Fn = u∈W n O , decomposition into generalized Schubert cells

4 Each Ou =∼ Cl(u) gives rise to a polynomial Poisson algebra;

5 u −1 u −1 u Each Γ := θ+ (O ) ∩ θ− (O ) is a single T -leaf of (Γ2n, πe2n).

Main results: For each u ∈ W n,

u u (Γ , πe2n) ⇒ (O , πn) is a Poisson groupoid; the symplectic leaf Σu through the identity section Ou ,→ Γu is a u u symplectic groupoid(Σ , πe2n) ⇒ (O , πn).

25 / 27 Part IV: In the (grand) scheme of things

u u u Summary cont’d: (Σ ⊂ Γ , πe2n) ⇒ (O , πn) as the meeting ground

1 (This talk) Algebraic Poisson/symplectic groupoids;

2 (This talk) Localizations/specializations of symmetric Poisson CGLs

3 (Goodearl-Yakimov theory, Shen-Weng) Poisson cluster varieties;

4 (L.-Mi) Integrable systems with complete Hamiltonian flows;

5 (Mi, Mouquin, work in progress) Systematic quantization;

u u 6 (Remark:)Γ ⇒ O is isomorphic to a sub-groupoid of product of

u u u O × O ⇒ O (pair groupoid) and B × B− ⇒ {e} (group).

Question: How do these structures interact?

26 / 27 References

1 B. Elek and J.-H. Lu, Bott-Samelson varieties and Poisson-Ore extensions, IMRN, 2019. 2 K. Goodearl and M. Yakimov, Cluster algebras on nilpotent Poisson algebras, arXiv:1801.01963v2. 3 J.-H. Lu and Y. Mi, Generalized Bruhat cells and completeness of Hamiltonian flows of Kogan-Zelevinsky integrable systems, Kostant memorial volume, Progress in Mathematics, 326 (2018). 4 J.-H. Lu and V. Mouquin, Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras, IMRN, 2016. 5 J.-H. Lu and V. Mouquin, On the T-leaves of some Poisson structures related to products of flag varieties, Adv. Math., 2017). 6 J.-H. Lu and V. Mouquin, Double Bruhat cells and symplectic groupoids, Trans. Groups, 2018. 7 J.-H. Lu, V. Mouquin, and S. Yu, Configuration spaces, generalized double Bruhat cells, and symplectic groupoids, to appear in archive. 8 J.-H. Lu and S. Yu, Bott-Samelson atlases, total positivity, and Poisson structures on some homogeneous spaces, to appear in Selecta. 9 V. Mouquin, Local Poisson groupoids over mixed product Poisson structures and generalized double Bruhat cells, arXiv:1908.04044. 10 L. Shen, D. Weng, Cluster structures on double Bott-Samelson cells, arXiv: arXiv:1904.07992.

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