Multiplicative geometric structures Henrique Bursztyn, IMPA (joint - - PowerPoint PPT Presentation

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 (Xgh = lg(Xh) + rh(Xg))

• 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 (Xgh = lg(Xh) + rh(Xg)) ⋄ 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 (Xgh = lg(Xh) + rh(Xg)) ⋄ 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 (Xgh = lg(Xh) + rh(Xg)) ⋄ Poisson-Lie groups: m : G × G → G Poisson map. ⋄ Symplectic groupoids: graph(m) ⊂ G × G × G Lagrangian submf. ⋄ Differential forms: ∂ω = p∗

1ω − m∗ω + p∗ 2ω = 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 (Xgh = lg(Xh) + rh(Xg)) ⋄ Poisson-Lie groups: m : G × G → G Poisson map. ⋄ Symplectic groupoids: graph(m) ⊂ G × G × G Lagrangian submf. ⋄ Differential forms: ∂ω = p∗

1ω − m∗ω + p∗ 2ω = 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 (Xgh = lg(Xh) + rh(Xg)) ⋄ Poisson-Lie groups: m : G × G → G Poisson map. ⋄ Symplectic groupoids: graph(m) ⊂ G × G × G Lagrangian submf. ⋄ Differential forms: ∂ω = p∗

1ω − m∗ω + p∗ 2ω = 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 (Xgh = lg(Xh) + rh(Xg)) ⋄ Poisson-Lie groups: m : G × G → G Poisson map. ⋄ Symplectic groupoids: graph(m) ⊂ G × G × G Lagrangian submf. ⋄ Differential forms: ∂ω = p∗

1ω − m∗ω + p∗ 2ω = 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(∧qTG ⊗ ∧pT ∗G).

• 2. Multiplicative tensors

Consider G ⇒ M, τ ∈ Γ(∧qTG ⊗ ∧pT ∗G). TG ⇒ TM, T ∗G ⇒ A∗ are Lie groupoids; also their direct sums.

• 2. Multiplicative tensors

Consider G ⇒ M, τ ∈ Γ(∧qTG ⊗ ∧pT ∗G). TG ⇒ TM, T ∗G ⇒ A∗ are Lie groupoids; also their direct sums. Consider the Lie groupoid G = (⊕qT ∗G) ⊕ (⊕pTG) ⇒ M = (⊕qA∗) ⊕ (⊕pTM).

• 2. Multiplicative tensors

Consider G ⇒ M, τ ∈ Γ(∧qTG ⊗ ∧pT ∗G). TG ⇒ TM, T ∗G ⇒ A∗ are Lie groupoids; also their direct sums. Consider the Lie groupoid G = (⊕qT ∗G) ⊕ (⊕pTG) ⇒ M = (⊕qA∗) ⊕ (⊕pTM). View τ ∈ Γ(∧qTG ⊗ ∧pT ∗G) as function ¯ τ ∈ C ∞(G): (α1, . . . , αq, X1, . . . , Xp)

¯ τ

→ τ(α1, . . . , αq, X1, . . . , Xp).

• 2. Multiplicative tensors

Consider G ⇒ M, τ ∈ Γ(∧qTG ⊗ ∧pT ∗G). TG ⇒ TM, T ∗G ⇒ A∗ are Lie groupoids; also their direct sums. Consider the Lie groupoid G = (⊕qT ∗G) ⊕ (⊕pTG) ⇒ M = (⊕qA∗) ⊕ (⊕pTM). View τ ∈ Γ(∧qTG ⊗ ∧pT ∗G) as function ¯ τ ∈ C ∞(G): (α1, . . . , αq, X1, . . . , Xp)

¯ τ

→ τ(α1, . . . , αq, X1, . . . , Xp). Definition: τ is multiplicative if ¯ τ ∈ C ∞(G) is multiplicative. (1-cocycle)

Infinitesimal description?

❛

t ❛ ❛ ❛

Infinitesimal description? For functions: multiplicative f ∈ C ∞(G) ⇌ µ ∈ Γ(A∗), dAµ = 0.

❛

t ❛ ❛ ❛

Infinitesimal description? For functions: multiplicative f ∈ C ∞(G) ⇌ µ ∈ Γ(A∗), dAµ = 0. Given a ∈ Γ(A), Lar f = t∗(µ(a)).

❛

t ❛ ❛ ❛

Infinitesimal description? For functions: multiplicative f ∈ C ∞(G) ⇌ µ ∈ Γ(A∗), dAµ = 0. Given a ∈ Γ(A), Lar f = t∗(µ(a)). For tensors τ ∈ Γ(∧qTG ⊗ ∧pT ∗G), want ¯ µ : Γ(A) → C ∞(M), L❛r ¯ τ = t∗(¯ µ(❛)), for ❛ ∈ Γ(A). ❛

Infinitesimal description? For functions: multiplicative f ∈ C ∞(G) ⇌ µ ∈ Γ(A∗), dAµ = 0. Given a ∈ Γ(A), Lar f = t∗(µ(a)). For tensors τ ∈ Γ(∧qTG ⊗ ∧pT ∗G), want ¯ µ : Γ(A) → C ∞(M), L❛r ¯ τ = t∗(¯ µ(❛)), for ❛ ∈ Γ(A). Enough to use particular types of sections ❛ ∈ Γ(A)...

Key facts: ⋄ Information about L❛r ¯ τ encoded in Lar τ ∈ Γ(∧qTG ⊗ ∧pT ∗G), iar τ ∈ Γ(∧qTG ⊗ ∧p−1T ∗G), it∗ατ ∈ Γ(∧q−1TG ⊗ ∧pT ∗G), for a ∈ Γ(A), α ∈ Ω1(M). t

Key facts: ⋄ Information about L❛r ¯ τ encoded in Lar τ ∈ Γ(∧qTG ⊗ ∧pT ∗G), iar τ ∈ Γ(∧qTG ⊗ ∧p−1T ∗G), it∗ατ ∈ Γ(∧q−1TG ⊗ ∧pT ∗G), for a ∈ Γ(A), α ∈ Ω1(M). ⋄ The map t∗ : C ∞(M) → C ∞(G) restricts to Γ(∧qA ⊗ ∧pT ∗M) → Γ(∧qTG ⊗ ∧pT ∗G), χ ⊗ α → χr ⊗ t∗α

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

As a result of L❛r ¯ τ = t∗(¯ µ(❛)), ¯ µ completely determined by D : Γ(A) → Γ(∧qA ⊗ ∧pT ∗M), l : A → ∧qA ⊗ ∧p−1T ∗M, r : T ∗M → ∧q−1A ⊗ ∧pT ∗M, t t t

As a result of L❛r ¯ τ = t∗(¯ µ(❛)), ¯ µ completely determined by D : Γ(A) → Γ(∧qA ⊗ ∧pT ∗M), l : A → ∧qA ⊗ ∧p−1T ∗M, r : T ∗M → ∧q−1A ⊗ ∧pT ∗M, such that Lar τ = t∗(D(a)), iar τ = t∗(l(a)), it∗ατ = t∗(r(α)). Leibniz-like condition: D(fa) = fD(a) + df ∧ l(a) − a ∧ r(α)

As a result of L❛r ¯ τ = t∗(¯ µ(❛)), ¯ µ completely determined by D : Γ(A) → Γ(∧qA ⊗ ∧pT ∗M), l : A → ∧qA ⊗ ∧p−1T ∗M, r : T ∗M → ∧q−1A ⊗ ∧pT ∗M, such that Lar τ = t∗(D(a)), iar τ = t∗(l(a)), it∗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❛r ¯ τ = t∗(¯ µ(❛)), ¯ µ completely determined by D : Γ(A) → Γ(∧qA ⊗ ∧pT ∗M), l : A → ∧qA ⊗ ∧p−1T ∗M, r : T ∗M → ∧q−1A ⊗ ∧pT ∗M, such that Lar τ = t∗(D(a)), iar τ = t∗(l(a)), it∗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) − iρ(b)D(a) (3) r(Lρ(a)α) = a.r(α) + iρ∗(α)D(a) (4) iρ(a)l(b) = −iρ(b)l(a) (5) iρ∗αr(β) = −iρ∗βr(α) (6) iρ(a)r(α) = −iρ∗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) − iρ(b)D(a) (3) r(Lρ(a)α) = a.r(α) + iρ∗(α)D(a) (4) iρ(a)l(b) = −iρ(b)l(a) (5) iρ∗αr(β) = −iρ∗βr(α) (6) iρ(a)r(α) = −iρ∗αl(a). Here Γ(A) acts on Γ(∧•A ⊗ ∧•T ∗M) via a.(b ⊗ α) = [a, b] ⊗ α + b ⊗ Lρ(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) − iρ(b)D(a) (3) r(Lρ(a)α) = a.r(α) + iρ∗(α)D(a) (4) iρ(a)l(b) = −iρ(b)l(a) (5) iρ∗αr(β) = −iρ∗βr(α) (6) iρ(a)r(α) = −iρ∗αl(a). Here Γ(A) acts on Γ(∧•A ⊗ ∧•T ∗M) via a.(b ⊗ α) = [a, b] ⊗ α + b ⊗ Lρ(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(∧qTG ⊗ ∧pT ∗G) multiplicative and (D, l, r), where D : Γ(A) → Γ(∧qA ⊗ ∧pT ∗M), Leibniz-like condition, l : A → ∧qA ⊗ ∧p−1T ∗M, r : T ∗M → ∧q−1A ⊗ ∧pT ∗M,

• 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(∧qTG ⊗ ∧pT ∗G) multiplicative and (D, l, r), where D : Γ(A) → Γ(∧qA ⊗ ∧pT ∗M), Leibniz-like condition, l : A → ∧qA ⊗ ∧p−1T ∗M, r : T ∗M → ∧q−1A ⊗ ∧pT ∗M, 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(∧qTG ⊗ ∧pT ∗G) multiplicative and (D, l, r), where D : Γ(A) → Γ(∧qA ⊗ ∧pT ∗M), Leibniz-like condition, l : A → ∧qA ⊗ ∧p−1T ∗M, r : T ∗M → ∧q−1A ⊗ ∧pT ∗M, 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−1A), 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−1A), 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] (δ0 = r, δ1 = D)

(q, 0): multiplicative multivector fields Infinitesimal components become: δ : Γ(∧•A) → Γ(∧•+q−1A), 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] (δ0 = r, δ1 = 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−1T ∗M, ν : A → ∧pT ∗M,

(0, p): multiplicative differential forms Infinitesimal components become: µ : A → ∧p−1T ∗M, ν : A → ∧pT ∗M, such that iρ(a)µ(b) = −iρ(b)µ(a), µ([a, b]) = Lρ(a)µ(b) − iρ(b)dµ(a) − iρ(b)ν(a), ν([a, b]) = Lρ(a)ν(b) − iρ(b)dν(a).

(0, p): multiplicative differential forms Infinitesimal components become: µ : A → ∧p−1T ∗M, ν : A → ∧pT ∗M, such that iρ(a)µ(b) = −iρ(b)µ(a), µ([a, b]) = Lρ(a)µ(b) − iρ(b)dµ(a) − iρ(b)ν(a), ν([a, b]) = Lρ(a)ν(b) − iρ(b)dν(a). (µ = l, ν = D − dµ)

(0, p): multiplicative differential forms Infinitesimal components become: µ : A → ∧p−1T ∗M, ν : A → ∧pT ∗M, such that iρ(a)µ(b) = −iρ(b)µ(a), µ([a, b]) = Lρ(a)µ(b) − iρ(b)dµ(a) − iρ(b)ν(a), ν([a, b]) = Lρ(a)ν(b) − iρ(b)dν(a). (µ = l, ν = D − dµ) E.g. symplectic groupoids and Poisson structures...

(1, p): vector-valued forms Ω(G, TG)

(1, p): vector-valued forms Ω(G, TG) Infinitesimal components become (D, τA, τM), D : Ω•(M, A) → Ω•+p(M, A), τA : ∧•T ∗M ⊗ A → ∧•+p−1T ∗M ⊗ A, τM ∈ Ωp(M, TM), plus compatibilities...

(1, p): vector-valued forms Ω(G, TG) Infinitesimal components become (D, τA, τM), D : Ω•(M, A) → Ω•+p(M, A), τA : ∧•T ∗M ⊗ A → ∧•+p−1T ∗M ⊗ A, τM ∈ Ωp(M, TM), plus compatibilities... (D0 = D, (τA)0 = l, τM = r∗)

(1, p): vector-valued forms Ω(G, TG) Infinitesimal components become (D, τA, τM), D : Ω•(M, A) → Ω•+p(M, A), τA : ∧•T ∗M ⊗ A → ∧•+p−1T ∗M ⊗ A, τM ∈ Ωp(M, TM), plus compatibilities... (D0 = D, (τA)0 = l, τM = r∗)

GLA relative to Frolicher-Nijenhuis bracket on multiplicative Ω•(G, TG)...

Particular case p = 1: J : TG → TG

Particular case p = 1: J : TG → TG Components are (D, JA, JM), JA : A → A, JM : TM → TM, D : Γ(A) → Γ(T ∗M ⊗ A).

Particular case p = 1: J : TG → TG Components are (D, JA, JM), JA : A → A, JM : TM → TM, D : Γ(A) → Γ(T ∗M ⊗ A). Can analyze 1

2[J, J] = NJ ∈ Ω2(G, TG) infinitesimally...

Particular case p = 1: J : TG → TG Components are (D, JA, JM), JA : A → A, JM : TM → TM, D : Γ(A) → Γ(T ∗M ⊗ A). Can analyze 1

2[J, J] = NJ ∈ Ω2(G, TG) infinitesimally...

E.g. holomorphic Lie groupoids ⇋ holomorphic Lie algebroids... complex Lie group: Lie bracket complex bilinear

Particular case p = 1: J : TG → TG Components are (D, JA, JM), JA : A → A, JM : TM → TM, D : Γ(A) → Γ(T ∗M ⊗ A). Can analyze 1

2[J, J] = NJ ∈ Ω2(G, TG) 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 : TG → TG Components are (D, JA, JM), JA : A → A, JM : TM → TM, D : Γ(A) → Γ(T ∗M ⊗ A). Can analyze 1

2[J, J] = NJ ∈ Ω2(G, TG) 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...

Thank you