Poisson Lie 2-groups and Lie 2-bialgebras

Transcription

1 Poisson Lie 2-groups and Lie 2-bialgebras PING XU Kolkata, December 2012 JOINT WITH ZHUO CHEN AND MATHIEU STIÉNON

2 1 Motivation 2 Strict 2-groups & crossed modules of groups 3 Strict Poisson Lie 2-groups & Lie 2-bialgebras 4 Weak Lie 2-bialgebras 5 Universal lifting theorem for Lie 2-groups

3 1 Motivation 2 Strict 2-groups & crossed modules of groups 3 Strict Poisson Lie 2-groups & Lie 2-bialgebras 4 Weak Lie 2-bialgebras 5 Universal lifting theorem for Lie 2-groups

4 DEFINITION: A Poisson Lie group is a Lie group G endowed with a multiplicative Poisson structure, i.e. a multiplicative field of bivectors π which is Poisson. DEFINITION: A Lie bialgebra is a vector space g endowed with a compatible structures of Lie algebra ( 2 g g) and Lie coalgebra (g 2 g). THEOREM (DRINFELD): The category of all Lie bialgebras is isomorphic to the category of all connected, simply-connected Poisson Lie groups.

5 QUESTION: What about 2-groups?

6 1 Motivation 2 Strict 2-groups & crossed modules of groups 3 Strict Poisson Lie 2-groups & Lie 2-bialgebras 4 Weak Lie 2-bialgebras 5 Universal lifting theorem for Lie 2-groups

7 strict 2-groupoid: Γ 2 α Γ 1 f, g Γ 0 x, y Horizontal and vertical multiplications: x f g α y f 1 f 2 x g 1 α 1 y g 2 α 2 z h 1 β 1 h 2 β 2 + many natural conditions. strict 2-group: Γ 2 = D particular case where Γ 0 =, Γ 1 = G, and

8 Other characterization of a strict 2-group: D G is a double groupoid in the sense of Mackenzie

9 DEFINITION: A strict Lie 2-group is a Lie groupoid D G where D and G are Lie groups and D D m D is a morphism of Lie groupoids. G G m G

10 crossed module of groups: Θ Φ G homomorphism of groups together with action of G on Θ by automorphisms: α g α satisfying Examples: 1 G Ad Aut(G) 2 1 G 3 Z (G) 1 Φ( g β) = g Φ(β) g 1 Φ(α) β = α β α 1

11 equivalence between strict 2-groups and crossed modules of groups: THEOREM (R. BROWN): crossed modules Θ Φ G bijection strict 2-groups D G D = G Θ semi-direct product of groups (g, α) (h, β) = (gh, h 1 (α)β) D = G Θ G transformation groupoid for the Θ-action on G given by g α = gφ(α) s(g, α) = g (g, α) (h, β) = (g, αβ) t(g, α) = gφ(α) if h = gφ(α)

12 strict Lie 2-algebra: A crossed module of Lie algebras consists of a pair θ, g of Lie algebras, a homomorphism φ : θ g and a g-action on θ by derivations satisfying φ( x u) = [x, φ(u)] and Φ(u) v = [u, v] for all x g and u, v θ. strict Lie 2-algebra = crossed module of Lie algebras

13 1 Motivation 2 Strict 2-groups & crossed modules of groups 3 Strict Poisson Lie 2-groups & Lie 2-bialgebras 4 Weak Lie 2-bialgebras 5 Universal lifting theorem for Lie 2-groups

14 multiplicative vector fields: G = Lie group Π X k (G) is multiplicative if Π gh = L g Π h + R h Π g Γ M = Lie groupoid A M = its Lie algebroid Π X k (Γ) is multiplicative if is a 1-cocycle w.r.t. groupoid Π C (T Γ Γ (k) Γ T Γ) T Γ Γ (k) Γ T Γ A M (k) M A

15 DEFINITION: A strict Poisson Lie 2-group is a strict Lie 2-group D G where D is endowed with a Poisson structure π, which is multiplicative w.r.t. both the group and the groupoid structures. DEFINITION: A strict Lie 2-bialgebra (or crossed module of Lie bialgebras) is a pair of crossed modules of Lie algebras in duality θ φ g and g φ θ such that the pair (g θ, θ g ) is a Lie bialgebra. g θ = semi-direct product of Lie algebras

16 EXAMPLE: Given a crossed module θ φ g and r 2 θ such that x [r, r] = 0, x g, we have [[r, r], u] = 0, u θ and [[φ(r), φ(r)], x] = 0, x g. Hence (θ, θ ) is a Lie bialgebra with r-matrix r and (g, g ) is a Lie bialgebra with r-matrix φ(r).

17 THEOREM: The category of connected, simply-connected, strict Poisson Lie 2-groups is isomorphic to the category of strict Lie 2-bialgebras.

18 1 Motivation 2 Strict 2-groups & crossed modules of groups 3 Strict Poisson Lie 2-groups & Lie 2-bialgebras 4 Weak Lie 2-bialgebras 5 Universal lifting theorem for Lie 2-groups

19 DEFINITION (BAEZ,CRANS): A (weak) Lie 2-algebra is a 2-term L -algebra, i.e. a sequence (l k ) k Z of homomorphisms l k : k V V of degree k 2 (where V = V 0 V 1 is a 2-term graded vector space) satisfying a certain compatibility condition. Given a (weak) Lie 2-algebra V = V 0 V 1 with l 3 = 0, set g := V 0 and θ := V 1. l 1 : θ φ g l 2 : { g g [,] g θ g θ l 3 : g g g 0 θ g is a Lie algebra: [x, y] = l 2 (x, y) θ is a Lie algebra: [u, v] = l 2 (u, l 1 (v)) (weak) Lie 2-algebra with l 3 = 0 strict Lie 2-algebra

20 Lie bialgebras and the big bracket: [Lecomte, Roger; Kosmann-Schwarzbach] Take a vector space g. Endow the graded vector space k (g g ) with the graded Poisson bracket of degree -2 characterized by {x, ξ} = ξ x {x 1, x 2 } = 0 {ξ 1, ξ 2 } = 0 for all x, x 1, x 2 g and ξ, ξ 1, ξ 2 g. k (g, g ) is a Lie bialgebra with bracket b 2 2 g g and cobracket c 2 2 g g s := b 2+c 2 3 (g g ) satisfies {s, s} = 0

21 Odd big bracket: Take a graded vector space V = k Z V k. Endow the graded symmetric algebra E (V ) := S (V [1] V [2]) with the graded Poisson bracket of degree -3 characterized by {x i, ξ j } = δ ij ξ j x i x i V i, ξ j (V j ) ; {x, y} = 0 x, y V ; {ξ, η} = 0 ξ, η V. S (V [1] V [2]) = Γ( T (V )) as vector spaces {, } is the Schouten bracket of polyvector fields

22 Let V = V 0 V 1 with V 0 = g and V 1 = θ. DEFINITION: A (weak) Lie 2-bialgebra on V = V 0 V 1 consists in an element ε E(V ) 4 such that {ε, ε} = 0. with ε = φ + l l l 3 + c c c 3 E(V ) 4 φ θ g l 1 2 g g g l 2 2 g θ θ l 3 g g g θ c 1 2 θ θ θ c 2 2 g g θ c 3 g θ θ θ θ φ g is a Lie 2-algebra θ φ g is a Lie 2-coalgebra (i.e. g φ θ is a Lie 2-algebra) together with compatibility relations

23 PROBLEM: DEFINITION: consisting of Integrate (weak) Lie 2-bialgebras. A quasi-poisson Lie 2-group is a triple a Lie 2-group G Θ G; a multiplicative bivector field π X 2 (G Θ); and a 1-cocycle η : G 3 θ Γ(Lie(G θ G)) satisfying 1 2 [π, π] = η η and [π, η ] = 0. THEOREM: connected simply connected quasi-poisson 2-groups bijection (weak) Lie 2-bialgebras with l 3 = 0 REMARK: The Lie 2-groups integrating weak Lie 2-algebras are simplicial manifolds satisfying Kan conditions [Getzler, Henriques]

24 1 Motivation 2 Strict 2-groups & crossed modules of groups 3 Strict Poisson Lie 2-groups & Lie 2-bialgebras 4 Weak Lie 2-bialgebras 5 Universal lifting theorem for Lie 2-groups

25 G = Lie group Π X k (G) is multiplicative if Π gh = L g Π h + R h Π g When endowed with the Schouten bracket, the multiplicative polyvector fields on G form a graded Lie algebra X (G). D k (g) := {deg k-1 derivations of Gerstenhaber algebra g} δ D k (g) δ g : g k g is a 1-cocycle When endowed with the graded commutator, D (g) becomes a graded Lie algebra. THEOREM (DRINFELD): X (G) = D (g) (as graded Lie algebras)

26 Γ M = Lie groupoid A M = its Lie algebroid Π X k (Γ) is multiplicative if Π C (T (k) Γ Γ Γ T Γ) is a 1-cocycle w.r.t. groupoid T Γ Γ Γ T Γ A M M A (k) (k) X k (Γ) = space of multiplicative k-vector fields on Γ D k (A) = space of degree k-1 derivations of the Gerstenhaber algebra A THEOREM (IGLESIAS PONTE, LAURENT-GENGOUX, XU): X (Γ) = D (A) (as graded Lie algebras)

27 G Θ G = 2-group associated to crossed module Θ G X (G Θ) = graded Lie algebra of polyvector fields on G Θ which are multiplicative w.r.t. both the group and the groupoid structures D (g θ) = some graded Lie algebra whose elements of degree k are couples of Chevalley-Eilenberg 1-cocycles ω : θ k θ δ : g g k 1 θ compatible in a certain sense THEOREM (CHEN, STIÉNON, X): X (G Θ) = D (g θ) (as graded Lie algebras)

28 G Θ G = 2-group associated to crossed module Θ G A = G θ G = its Lie algebroid 1 Π X k (G Θ) multiplicative w.r.t. groupoid structure 2 Iglesias, Laurent, Xu: ˆδ : Γ( 0 A) Γ( k 1 A) s.t. ˆδ(fg) = f ˆδ(g) + ˆδ(f )g 3 ˆδ Γ(T G k 1 A) = C (G, g k 1 θ) 4 ˆδ : G g k 1 θ is in fact a 1-cocycle for Lie group G 5 δ : g g k 1 θ is a 1-cocycle for Lie algebra g 1 Θ = Θ {1} is a Lie subgroup of Θ G 2 Π X k (G Θ) multiplicative w.r.t. group structure 3 [1] + [2] = Π Θ X k (Θ) 4 Drinfeld: ω : θ k θ is a 1-cocycle

29 How the Universal Lifting Theorem is used to integrate Lie 2-bialgebras to quasi-poisson 2-groups: X 2 (G Θ) π X 3 (G Θ) η η coming from η : G 3 θ 1 2 [π, π] = η η and [π, η ] = 0 D 2 (g θ) { ω : θ 2 θ c 1 2 δ : g g θ c2 2 { D 3 ωη : θ (g θ) 3 θ δ η : g g 2 θ obtained from η : g 3 θ c 3 using φ : θ g ε = φ + l l2 2 + l 3 satisfies {ε, ε} = 0 0= + c c2 2 + c 3

30 Thank You