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 ]: fxi ; xj g = cij ; cij 2 C; Linear Poisson brackets: X (k) (k) fxi ; xj g = cij xk ; cij 2 C; k (Quadratic) log-canonical Poisson brackets: fxi ; xj g = cij xi xj ; cij 2 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: fxi ; xj g = 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]: fx1; x2g = −3x1x2; fx1; x3g = −x1x3 − 2x2; 2 fx1; x4g =0 − 6x3 ; fx1; x5g = x1x5 − 4x3; fx1; x6g =3 x1x6 − 6x5; fx2; x3g = −3x2x3; 3 2 fx2; x4g = −3x2x4 − 6x3 ; fx2; x5g =0 − 6x3 ; fx2; x6g =3 x2x6 − 18x3x5 + 6x4; fx3; x4g = −3x3x4; fx3; x5g = −x3x5 − 2x4; 2 fx3; x6g =0 − 6x5 ; fx4; x5g = −3x4x5; 3 fx4; x6g = −3x4x6 − 6x5 ; fx5; x6g = −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 fx1; x8g = −3x1x8 − 6x7 ; fx1; x9g =0 − 6x7 ; fx1; x10g =3 x1x10 − 18x7x9 + 6x8; fx1; x11g =3 x1x11 − 6x7; 3 2 fx1; x12g =6 x1x12 − 6; fx2; x8g =0 − 6x6x7 + 6x7 ; 2 fx2; x9g = x2x9 − 6x6x7 + 4x7; fx2; x10g =3 x2x10 − 18x6x7x9 + 6x6x8 + 6x9; fx2; x11g =2 x2x11 − 6x6x7 + 2; fx2; x12g =3 x2x12 − 6x6; 3 3 2 fx3; x8g =3 x3x8 − 18x4x6x7 + 6x5x7 + 18x4x7 ; 2 2 fx3; x9g =3 x3x9 − 18x4x6x7 + 6x5x7 + 12x4x7; fx3; x10g =6 x3x10 − 54x4x6x7x9 + 18x4x6x8 + 18x5x7x9 + 18x4x9 − 6x5x8 − 6; 9 open symplectic leaf given by φ1(x1;:::; x12) 6= 0; φ2(x1;:::; x12) 6= 0 for some φ1; φ2 2 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; t2T 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 2 [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; ∗ 1 1 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<j k<l X @ @ + γik (x; y) ^ : @xi @yk i;k 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 f(γ; γ ; γγ ): θ−(γ) = θ+(γ )g ⊂ 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 2 [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 2 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 = f[g1; g2;:::; g2n]Fe2n : g1g2 ··· g2n 2 B−g ⊂ 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.