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Multiplicative geometric structures Henrique Bursztyn, IMPA (joint with Thiago Drummond, UFRJ) Workshop on EDS and Lie theory Fields Institute, December 2013

Outline: 1. Motivation: geometry on Lie groupoids 2. Multiplicative structures 3. Infinitesimal/global correspondence 4. Examples and applications

1. Motivation Lie groupoids are often equipped with compatible geometry...

1. Motivation Lie groupoids are often equipped with compatible geometry... functions: f C (G), f (gh) = f (g) + f (h) (1-cocycles)

1. Motivation Lie groupoids are often equipped with compatible geometry... functions: f C (G), f (gh) = f (g) + f (h) (1-cocycles) Vector fields: infinitesimal automorphisms (X gh = l g (X h ) + r h (X g ))

1. Motivation Lie groupoids are often equipped with compatible geometry... functions: f C (G), f (gh) = f (g) + f (h) (1-cocycles) Vector fields: infinitesimal automorphisms (X gh = l g (X h ) + r h (X g )) Poisson-Lie groups: m : G G G Poisson map.

1. Motivation Lie groupoids are often equipped with compatible geometry... functions: f C (G), f (gh) = f (g) + f (h) (1-cocycles) Vector fields: infinitesimal automorphisms (X gh = l g (X h ) + r h (X g )) Poisson-Lie groups: m : G G G Poisson map. Symplectic groupoids: graph(m) G G G Lagrangian submf.

1. Motivation Lie groupoids are often equipped with compatible geometry... functions: f C (G), f (gh) = f (g) + f (h) (1-cocycles) Vector fields: infinitesimal automorphisms (X gh = l g (X h ) + r h (X g )) Poisson-Lie groups: m : G G G Poisson map. Symplectic groupoids: graph(m) G G G Lagrangian submf. Differential forms: ω = p1 ω m ω + p2 ω = 0.

1. Motivation Lie groupoids are often equipped with compatible geometry... functions: f C (G), f (gh) = f (g) + f (h) (1-cocycles) Vector fields: infinitesimal automorphisms (X gh = l g (X h ) + r h (X g )) Poisson-Lie groups: m : G G G Poisson map. Symplectic groupoids: graph(m) G G G Lagrangian submf. Differential forms: ω = p1 ω m ω + p2 ω = 0. Complex Lie groups: m : G G G holomorphic map.

1. Motivation Lie groupoids are often equipped with compatible geometry... functions: f C (G), f (gh) = f (g) + f (h) (1-cocycles) Vector fields: infinitesimal automorphisms (X gh = l g (X h ) + r h (X g )) Poisson-Lie groups: m : G G G Poisson map. Symplectic groupoids: graph(m) G G G Lagrangian submf. Differential forms: ω = p1 ω m ω + p2 ω = 0. Complex Lie groups: m : G G G holomorphic map. Contact structures, distributions, projections (e.g. connections)...

1. Motivation Lie groupoids are often equipped with compatible geometry... functions: f C (G), f (gh) = f (g) + f (h) (1-cocycles) Vector fields: infinitesimal automorphisms (X gh = l g (X h ) + r h (X g )) Poisson-Lie groups: m : G G G Poisson map. Symplectic groupoids: graph(m) G G G Lagrangian submf. Differential forms: ω = p1 ω m ω + p2 ω = 0. Complex Lie groups: m : G G G holomorphic map. Contact structures, distributions, projections (e.g. connections)... Recurrent problem: infinitesimal counterparts, integration...

Some cases have been considered, through different methods...

Some cases have been considered, through different methods... [1] Crainic: Differentiable and algebroid cohomology, van Est isomorphisms, and characteristic classes, Comment. Math. Helv. (2003) [2] Drinfel d: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations. Soviet Math. Dokl. (1983). [3] Lu, Weinstein: Poisson Lie groups, dressing transformations, and Bruhat decompositions. J. Differential Geom. (1990). [4] Weinstein: Symplectic groupoids and Poisson manifolds. Bull. Amer. Math. Soc. (1987). [5] Mackenzie, Xu: Lie bialgebroids and Poisson groupoids. Duke Math. J. (1994), [6] Mackenzie, Xu: Integration of Lie bialgebroids. Topology (2000). [7] B., Crainic, Weinstein, Zhu: Integration of twisted Dirac brackets, Duke Math. J. (2004). [8] B., Cabrera: Multiplicative forms at the infinitesimal level. Math. Ann. (2012). [9] Crainic, Abad: The Weil algebra and Van Est isomorphism. Ann. Inst. Fourier (2011). [10] Laurent, Stienon, Xu: Integration of holomorphic Lie algebroids. Math.Ann. (2009). [11] Iglesias, Laurent, Xu: Universal lifting and quasi-poisson groupoids. JEMS (2012). [12] Crainic, Salazar, Struchiner: Multiplicative forms and Spencer operators.

2. Multiplicative tensors Consider G M, τ Γ( q T G p T G).

2. Multiplicative tensors Consider G M, τ Γ( q T G p T G). T G TM, T G A are Lie groupoids; also their direct sums.

2. Multiplicative tensors Consider G M, τ Γ( q T G p T G). T G TM, T G A are Lie groupoids; also their direct sums. Consider the Lie groupoid G = ( q T G) ( p T G) M = ( q A ) ( p TM).

2. Multiplicative tensors Consider G M, τ Γ( q T G p T G). T G TM, T G A are Lie groupoids; also their direct sums. Consider the Lie groupoid G = ( q T G) ( p T G) M = ( q A ) ( p TM). View τ Γ( q T G p T G) as function τ C (G): (α 1,..., α q, X 1,..., X p ) τ τ(α 1,..., α q, X 1,..., X p ).

2. Multiplicative tensors Consider G M, τ Γ( q T G p T G). T G TM, T G A are Lie groupoids; also their direct sums. Consider the Lie groupoid G = ( q T G) ( p T G) M = ( q A ) ( p TM). View τ Γ( q T G p T G) as function τ C (G): Definition: (α 1,..., α q, X 1,..., X p ) τ τ(α 1,..., α q, X 1,..., X p ). τ is multiplicative if τ C (G) is multiplicative. (1-cocycle)

Infinitesimal description?

Infinitesimal description? For functions: multiplicative f C (G) µ Γ(A ), d A µ = 0.

Infinitesimal description? For functions: multiplicative f C (G) µ Γ(A ), d A µ = 0. Given a Γ(A), L a r f = t (µ(a)).

Infinitesimal description? For functions: multiplicative f C (G) µ Γ(A ), d A µ = 0. Given a Γ(A), L a r f = t (µ(a)). For tensors τ Γ( q T G p T G), want µ : Γ(A) C (M), for a Γ(A). L a r τ = t ( µ(a)),

Infinitesimal description? For functions: multiplicative f C (G) µ Γ(A ), d A µ = 0. Given a Γ(A), L a r f = t (µ(a)). For tensors τ Γ( q T G p T G), want µ : Γ(A) C (M), for a Γ(A). L a r τ = t ( µ(a)), Enough to use particular types of sections a Γ(A)...

Key facts: Information about L a r τ encoded in L a r τ Γ( q T G p T G), i a r τ Γ( q T G p 1 T G), i t ατ Γ( q 1 T G p T G), for a Γ(A), α Ω 1 (M).

Key facts: Information about L a r τ encoded in L a r τ Γ( q T G p T G), i a r τ Γ( q T G p 1 T G), i t ατ Γ( q 1 T G p T G), for a Γ(A), α Ω 1 (M). The map t : C (M) C (G) restricts to Γ( q A p T M) Γ( q T G p T G), χ α χ r t α

As a result of L a r τ = t ( µ(a)), µ completely determined by

As a result of L a r τ = t ( µ(a)), µ completely determined by D : Γ(A) Γ( q A p T M), l : A q A p 1 T M, r : T M q 1 A p T M,

As a result of L a r τ = t ( µ(a)), µ completely determined by D : Γ(A) Γ( q A p T M), l : A q A p 1 T M, r : T M q 1 A p T M, such that L a r τ = t (D(a)), i a r τ = t (l(a)), i t ατ = t (r(α)). Leibniz-like condition: D(fa) = fd(a) + df l(a) a r(α)

As a result of L a r τ = t ( µ(a)), µ completely determined by D : Γ(A) Γ( q A p T M), l : A q A p 1 T M, r : T M q 1 A p T M, such that L a r τ = t (D(a)), i a r τ = t (l(a)), i t ατ = t (r(α)). Leibniz-like condition: D(fa) = fd(a) + df l(a) a r(α) We call (D, l, r) the infinitesimal components of τ.

As a result of L a r τ = t ( µ(a)), µ completely determined by D : Γ(A) Γ( q A p T M), l : A q A p 1 T M, r : T M q 1 A p T M, such that L a r τ = t (D(a)), i a r τ = t (l(a)), i t ατ = t (r(α)). Leibniz-like condition: D(fa) = fd(a) + df l(a) a r(α) We call (D, l, r) the infinitesimal components of τ. How about cocycle equations?

Cocycle equations for (D, r, l): (1) D([a, b]) = a.d(b) b.d(a) (2) l([a, b]) = a.l(b) i ρ(b) D(a) (3) r(l ρ(a) α) = a.r(α) + i ρ (α)d(a) (4) i ρ(a) l(b) = i ρ(b) l(a) (5) i ρ αr(β) = i ρ βr(α) (6) i ρ(a) r(α) = i ρ αl(a).

Cocycle equations for (D, r, l): (1) D([a, b]) = a.d(b) b.d(a) (2) l([a, b]) = a.l(b) i ρ(b) D(a) (3) r(l ρ(a) α) = a.r(α) + i ρ (α)d(a) (4) i ρ(a) l(b) = i ρ(b) l(a) (5) i ρ αr(β) = i ρ βr(α) (6) i ρ(a) r(α) = i ρ αl(a). Here Γ(A) acts on Γ( A T M) via a.(b α) = [a, b] α + b L ρ(a) α

Cocycle equations for (D, r, l): (1) D([a, b]) = a.d(b) b.d(a) (2) l([a, b]) = a.l(b) i ρ(b) D(a) (3) r(l ρ(a) α) = a.r(α) + i ρ (α)d(a) (4) i ρ(a) l(b) = i ρ(b) l(a) (5) i ρ αr(β) = i ρ βr(α) (6) i ρ(a) r(α) = i ρ αl(a). Here Γ(A) acts on Γ( A T M) via a.(b α) = [a, b] α + b L ρ(a) α (Redundancies...)

3. The infinitesimal/global correspondence Let G M be s.s.c., A M its Lie algebroid.

3. The infinitesimal/global correspondence Let G M be s.s.c., A M its Lie algebroid. Theorem: (B., Drummond) 1-1 correspondence between τ Γ( q T G p T G) multiplicative and (D, l, r), where D : Γ(A) Γ( q A p T M), l : A q A p 1 T M, r : T M q 1 A p T M, Leibniz-like condition,

3. The infinitesimal/global correspondence Let G M be s.s.c., A M its Lie algebroid. Theorem: (B., Drummond) 1-1 correspondence between τ Γ( q T G p T G) multiplicative and (D, l, r), where D : Γ(A) Γ( q A p T M), l : A q A p 1 T M, r : T M q 1 A p T M, Leibniz-like condition, satisfying (1) (6).

3. The infinitesimal/global correspondence Let G M be s.s.c., A M its Lie algebroid. Theorem: (B., Drummond) 1-1 correspondence between τ Γ( q T G p T G) multiplicative and (D, l, r), where D : Γ(A) Γ( q A p T M), l : A q A p 1 T M, r : T M q 1 A p T M, Leibniz-like condition, satisfying (1) (6). (more general tensors, coefficients in reps...)

(q, 0): multiplicative multivector fields

(q, 0): multiplicative multivector fields Infinitesimal components become: δ : Γ( A) Γ( +q 1 A), such that δ(ab) = δ(a)b + ( 1) a (q 1) aδ(b) δ([a, b]) = [δa, b] + ( 1) ( a 1)(q 1) [a, δb]

(q, 0): multiplicative multivector fields Infinitesimal components become: δ : Γ( A) Γ( +q 1 A), such that δ(ab) = δ(a)b + ( 1) a (q 1) aδ(b) δ([a, b]) = [δa, b] + ( 1) ( a 1)(q 1) [a, δb] (δ 0 = r, δ 1 = D)

(q, 0): multiplicative multivector fields Infinitesimal components become: δ : Γ( A) Γ( +q 1 A), such that δ(ab) = δ(a)b + ( 1) a (q 1) aδ(b) δ([a, b]) = [δa, b] + ( 1) ( a 1)(q 1) [a, δb] (δ 0 = r, δ 1 = D) E.g. (quasi-)poisson groupoids and (quasi-)lie bialgebroids...

(0, p): multiplicative differential forms

(0, p): multiplicative differential forms Infinitesimal components become: µ : A p 1 T M, ν : A p T M,

(0, p): multiplicative differential forms Infinitesimal components become: µ : A p 1 T M, ν : A p T M, such that i ρ(a) µ(b) = i ρ(b) µ(a), µ([a, b]) = L ρ(a) µ(b) i ρ(b) dµ(a) i ρ(b) ν(a), ν([a, b]) = L ρ(a) ν(b) i ρ(b) dν(a).

(0, p): multiplicative differential forms Infinitesimal components become: µ : A p 1 T M, ν : A p T M, such that i ρ(a) µ(b) = i ρ(b) µ(a), µ([a, b]) = L ρ(a) µ(b) i ρ(b) dµ(a) i ρ(b) ν(a), ν([a, b]) = L ρ(a) ν(b) i ρ(b) dν(a). (µ = l, ν = D dµ)

(0, p): multiplicative differential forms Infinitesimal components become: µ : A p 1 T M, ν : A p T M, such that i ρ(a) µ(b) = i ρ(b) µ(a), µ([a, b]) = L ρ(a) µ(b) i ρ(b) dµ(a) i ρ(b) ν(a), ν([a, b]) = L ρ(a) ν(b) i ρ(b) dν(a). (µ = l, ν = D dµ) E.g. symplectic groupoids and Poisson structures...

(1, p): vector-valued forms Ω(G, T G)

(1, p): vector-valued forms Ω(G, T G) Infinitesimal components become (D, τ A, τ M ), D : Ω (M, A) Ω +p (M, A), τ A : T M A +p 1 T M A, τ M Ω p (M, TM), plus compatibilities...

(1, p): vector-valued forms Ω(G, T G) Infinitesimal components become (D, τ A, τ M ), D : Ω (M, A) Ω +p (M, A), τ A : T M A +p 1 T M A, τ M Ω p (M, TM), plus compatibilities... (D 0 = D, (τ A ) 0 = l, τ M = r )

(1, p): vector-valued forms Ω(G, T G) Infinitesimal components become (D, τ A, τ M ), D : Ω (M, A) Ω +p (M, A), τ A : T M A +p 1 T M A, τ M Ω p (M, TM), plus compatibilities... (D 0 = D, (τ A ) 0 = l, τ M = r ) GLA relative to Frolicher-Nijenhuis bracket on multiplicative Ω (G, T G)...

Particular case p = 1: J : T G T G

Particular case p = 1: J : T G T G Components are (D, J A, J M ), J A : A A, J M : TM TM, D : Γ(A) Γ(T M A).

Particular case p = 1: J : T G T G Components are (D, J A, J M ), J A : A A, J M : TM TM, D : Γ(A) Γ(T M A). Can analyze 1 2 [J, J] = N J Ω 2 (G, T G) infinitesimally...

Particular case p = 1: J : T G T G Components are (D, J A, J M ), J A : A A, J M : TM TM, D : Γ(A) Γ(T M A). Can analyze 1 2 [J, J] = N J Ω 2 (G, T G) infinitesimally... E.g. holomorphic Lie groupoids holomorphic Lie algebroids... complex Lie group: Lie bracket complex bilinear

Particular case p = 1: J : T G T G Components are (D, J A, J M ), J A : A A, J M : TM TM, D : Γ(A) Γ(T M A). Can analyze 1 2 [J, J] = N J Ω 2 (G, T G) infinitesimally... E.g. holomorphic Lie groupoids holomorphic Lie algebroids... complex Lie group: Lie bracket complex bilinear holomorphic vector bundle: flat, partial T 10 -connection

Particular case p = 1: J : T G T G Components are (D, J A, J M ), J A : A A, J M : TM TM, D : Γ(A) Γ(T M A). Can analyze 1 2 [J, J] = N J Ω 2 (G, T G) infinitesimally... E.g. holomorphic Lie groupoids holomorphic Lie algebroids... complex Lie group: Lie bracket complex bilinear holomorphic vector bundle: flat, partial T 10 -connection more: holomorphic symplectic/poisson groupoids, almost product...

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