Lie theory of multiplicative structures.

Transcription

1 Lie theory of multiplicative structures. T. Drummond (UFRJ) Workshop on Geometric Structures on Lie groupoids - Banff, 2017

2 Introduction Goal: Survey various results regarding the Lie theory of geometric structures on Lie groupoids presented from a unified perpective.

3 Introduction Goal: Survey various results regarding the Lie theory of geometric structures on Lie groupoids presented from a unified perpective. Outline Early examples of multiplicative structures on Lie groupoids focusing on three aspects:

4 Introduction Goal: Survey various results regarding the Lie theory of geometric structures on Lie groupoids presented from a unified perpective. Outline Early examples of multiplicative structures on Lie groupoids focusing on three aspects: 1. Compatibility of the structure with the multiplication;

5 Introduction Goal: Survey various results regarding the Lie theory of geometric structures on Lie groupoids presented from a unified perpective. Outline Early examples of multiplicative structures on Lie groupoids focusing on three aspects: 1. Compatibility of the structure with the multiplication; 2. Differentiation (description of the infinitesimal data);

6 Introduction Goal: Survey various results regarding the Lie theory of geometric structures on Lie groupoids presented from a unified perpective. Outline Early examples of multiplicative structures on Lie groupoids focusing on three aspects: 1. Compatibility of the structure with the multiplication; 2. Differentiation (description of the infinitesimal data); 3. Integration (infinitesimal-global correspondence).

7 Introduction Goal: Survey various results regarding the Lie theory of geometric structures on Lie groupoids presented from a unified perpective. Outline Early examples of multiplicative structures on Lie groupoids focusing on three aspects: 1. Compatibility of the structure with the multiplication; 2. Differentiation (description of the infinitesimal data); 3. Integration (infinitesimal-global correspondence). Unifying framework: general theory of multiplicative tensor fields (joint with H. Bursztyn).

8 Introduction Goal: Survey various results regarding the Lie theory of geometric structures on Lie groupoids presented from a unified perpective. Outline Early examples of multiplicative structures on Lie groupoids focusing on three aspects: 1. Compatibility of the structure with the multiplication; 2. Differentiation (description of the infinitesimal data); 3. Integration (infinitesimal-global correspondence). Unifying framework: general theory of multiplicative tensor fields (joint with H. Bursztyn). van Est for differential forms with coefficients (joint with A. Cabrera).

9 The beginning: Poisson-Lie groups Drinfel d (1983); Lu, Weinstein(1990) G Lie group, π X 2 (G) Poisson structure.

10 The beginning: Poisson-Lie groups Drinfel d (1983); Lu, Weinstein(1990) G Lie group, π X 2 (G) Poisson structure. Compatibility: m : G G G is a Poisson map π gh = R h (π g ) + L g (π h ) π : G 2 g cocycle

11 The beginning: Poisson-Lie groups Drinfel d (1983); Lu, Weinstein(1990) G Lie group, π X 2 (G) Poisson structure. Compatibility: m : G G G is a Poisson map π gh = R h (π g ) + L g (π h ) π : G 2 g cocycle Infinitesimal data: δ : Λ g Λ +1 g derivation δ([u, v]) = [δ(u), v] + [u, δ(v)], u, v g (cocycle equation) δ 2 = 0 Correspondence: L u π = δ(u)

12 The beginning: Poisson-Lie groups Drinfel d (1983); Lu, Weinstein(1990) G Lie group, π X 2 (G) Poisson structure. Compatibility: m : G G G is a Poisson map π gh = R h (π g ) + L g (π h ) π : G 2 g cocycle Infinitesimal data: δ : Λ g Λ +1 g derivation δ([u, v]) = [δ(u), v] + [u, δ(v)], u, v g (cocycle equation) δ 2 = 0 Correspondence: L u π = δ(u) (g, g ) is a Lie bialgebra: g, g are compatible Lie algebras.

13 The beginning: Poisson-Lie groups Drinfel d (1983); Lu, Weinstein(1990) G Lie group, π X 2 (G) Poisson structure. Compatibility: m : G G G is a Poisson map π gh = R h (π g ) + L g (π h ) π : G 2 g cocycle Infinitesimal data: δ : Λ g Λ +1 g derivation δ([u, v]) = [δ(u), v] + [u, δ(v)], u, v g (cocycle equation) δ 2 = 0 Correspondence: L u π = δ(u) (g, g ) is a Lie bialgebra: g, g are compatible Lie algebras. Integration: Cocycles.

14 Poisson groupoids Weinstein(1988); Mackenzie, Xu(1994,2000) G Lie groupoid, π X 2 (G) Poisson structure.

15 Poisson groupoids Weinstein(1988); Mackenzie, Xu(1994,2000) G Lie groupoid, π X 2 (G) Poisson structure. Compatibility: Graph(m) G G G is coisotropic

16 Poisson groupoids Weinstein(1988); Mackenzie, Xu(1994,2000) G Lie groupoid, π X 2 (G) Poisson structure. Compatibility: Graph(m) G G G is coisotropic π : T G T G is a groupoid morphism

17 Poisson groupoids Weinstein(1988); Mackenzie, Xu(1994,2000) G Lie groupoid, π X 2 (G) Poisson structure. Compatibility: Graph(m) G G G is coisotropic π : T G T G is a groupoid morphism Infinitesimal data: δ : Γ( A) Γ( +1 A) derivation δ 2 = 0 δ([a, b]) = [δ(a), b] + [a, δ(b)], a, b Γ(A) (A, A ) is a Lie bialgebroid: A, A are compatible Lie algebroids.

18 Poisson groupoids Weinstein(1988); Mackenzie, Xu(1994,2000) G Lie groupoid, π X 2 (G) Poisson structure. Compatibility: Graph(m) G G G is coisotropic π : T G T G is a groupoid morphism Infinitesimal data: δ : Γ( A) Γ( +1 A) derivation δ 2 = 0 δ([a, b]) = [δ(a), b] + [a, δ(b)], a, b Γ(A) (A, A ) is a Lie bialgebroid: A, A are compatible Lie algebroids. Correspondence: π (t df ) = δ 0 (f ), L a π = δ 1 (a)

19 Poisson groupoids Weinstein(1988); Mackenzie, Xu(1994,2000) G Lie groupoid, π X 2 (G) Poisson structure. Compatibility: Graph(m) G G G is coisotropic π : T G T G is a groupoid morphism Infinitesimal data: δ : Γ( A) Γ( +1 A) derivation δ 2 = 0 δ([a, b]) = [δ(a), b] + [a, δ(b)], a, b Γ(A) (A, A ) is a Lie bialgebroid: A, A are compatible Lie algebroids. Correspondence: π (t df ) = δ 0 (f ), L a π = δ 1 (a) Integration: Lie bialgebroid π A : T A TA is a Lie algebroid morphism.

20 Examples Poisson manifolds. (M, π M ) Poisson manifold (T M, TM) is a Lie bialgebroid: δ : Γ( T M) Γ( +1 T M) is the de Rham differential. Integration yields the symplectic groupoid.

21 Examples Poisson manifolds. (M, π M ) Poisson manifold (T M, TM) is a Lie bialgebroid: δ : Γ( T M) Γ( +1 T M) is the de Rham differential. Integration yields the symplectic groupoid. Triangular Lie bialgebroids: r Γ( 2 A) defines a multiplicative bivector field by π = r r

22 Examples Poisson manifolds. (M, π M ) Poisson manifold (T M, TM) is a Lie bialgebroid: δ : Γ( T M) Γ( +1 T M) is the de Rham differential. Integration yields the symplectic groupoid. Triangular Lie bialgebroids: r Γ( 2 A) defines a multiplicative bivector field by π = r r Integrable [r, r] = 0 (Classical Yang-Baxter equation)

23 Examples Poisson manifolds. (M, π M ) Poisson manifold (T M, TM) is a Lie bialgebroid: δ : Γ( T M) Γ( +1 T M) is the de Rham differential. Integration yields the symplectic groupoid. Triangular Lie bialgebroids: r Γ( 2 A) defines a multiplicative bivector field by π = r r Integrable [r, r] = 0 (Classical Yang-Baxter equation) In this case, δ = [r, ].

24 Differential 2-forms Weinstein (1987); Karasev(1987); Bursztyn,Crainic,Weinstein,Zhu(2004). ω Ω 2 (G), φ Ω 3 (M), dω = s φ t φ.

25 Differential 2-forms Weinstein (1987); Karasev(1987); Bursztyn,Crainic,Weinstein,Zhu(2004). ω Ω 2 (G), φ Ω 3 (M), dω = s φ t φ. Compatibility: Graph(m) G G G is isotropic: m ω = pr1 ω + pr 2 ω

26 Differential 2-forms Weinstein (1987); Karasev(1987); Bursztyn,Crainic,Weinstein,Zhu(2004). ω Ω 2 (G), φ Ω 3 (M), dω = s φ t φ. Compatibility: Graph(m) G G G is isotropic: m ω = pr1 ω + pr 2 ω Infinitesimal data: µ : A T M, µ(a), ρ(b) = µ(b), ρ(a) µ([a, b]) = L ρ(a) µ(b) i ρ(b) dµ(a) i ρ(b) i ρ(a) φ

27 Differential 2-forms Weinstein (1987); Karasev(1987); Bursztyn,Crainic,Weinstein,Zhu(2004). ω Ω 2 (G), φ Ω 3 (M), dω = s φ t φ. Compatibility: Graph(m) G G G is isotropic: m ω = pr1 ω + pr 2 ω Infinitesimal data: µ : A T M, µ(a), ρ(b) = µ(b), ρ(a) µ([a, b]) = L ρ(a) µ(b) i ρ(b) dµ(a) i ρ(b) i ρ(a) φ Equivalent to the image of (ρ, µ) : A TM T M being isotropic; closed under the φ-twisted Courant bracket.

28 Differential 2-forms Weinstein (1987); Karasev(1987); Bursztyn,Crainic,Weinstein,Zhu(2004). ω Ω 2 (G), φ Ω 3 (M), dω = s φ t φ. Compatibility: Graph(m) G G G is isotropic: m ω = pr1 ω + pr 2 ω Infinitesimal data: µ : A T M, µ(a), ρ(b) = µ(b), ρ(a) µ([a, b]) = L ρ(a) µ(b) i ρ(b) dµ(a) i ρ(b) i ρ(a) φ Equivalent to the image of (ρ, µ) : A TM T M being isotropic; closed under the φ-twisted Courant bracket. Correspondence: i a ω = t µ(a)

29 (twisted) Symplectic: (Cattaneo-Xu, 2004) ω non-degenerate Pre-symplectic: (Bursztyn-Crainic-Weinstein-Zhu, 2004) π = ρ µ 1 : T M TM (twisted) Poisson structure { (ρ, µ) : A TM T M dim(g) = 2 dim(m) ( ) ker(ω) ker(ds) ker(dt) = 0 (twisted) Dirac structure

30 (twisted) Symplectic: (Cattaneo-Xu, 2004) ω non-degenerate Pre-symplectic: (Bursztyn-Crainic-Weinstein-Zhu, 2004) π = ρ µ 1 : T M TM (twisted) Poisson structure { (ρ, µ) : A TM T M dim(g) = 2 dim(m) ( ) ker(ω) ker(ds) ker(dt) = 0 (twisted) Dirac structure Recently, ( ) interpreted as ω : T G T G being a Morita map (Del Hoyo, Ortiz(2017)).

31 (twisted) Symplectic: (Cattaneo-Xu, 2004) ω non-degenerate Pre-symplectic: (Bursztyn-Crainic-Weinstein-Zhu, 2004) π = ρ µ 1 : T M TM (twisted) Poisson structure { (ρ, µ) : A TM T M dim(g) = 2 dim(m) ( ) ker(ω) ker(ds) ker(dt) = 0 (twisted) Dirac structure Recently, ( ) interpreted as ω : T G T G being a Morita map (Del Hoyo, Ortiz(2017)). Integration: Explicit construction of 2-form on the Weinstein groupoid G(A). (Cattaneo-Felder (2000) ; Crainic, Fernandes (2003))

32 Further works Kosmann-Schwarzbach: Multiplicativity, from Lie groups to generalized geometry. preprint arxiv: (2015). Mackenzie, Xu: Classical lifting processes and multiplicative vector fields. Quarterly J. Math. (1998). Iglesias, Marrero: Jacobi groupoids and generalized Lie bialgebroids, J. Geom. Phys (2003). Crainic, Zhu: Integrability of Jacobi and Poisson structures, Ann. Inst. Fourier (Grenoble) (2007). Grabowski, Rotkiewicz: Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys. (2009). Laurent, Stienon, Xu: Integration of holomorphic Lie algebroids. Math.Ann. (2009). Arias Abad, Crainic: The Weil algebra and the Van Est isomorphism. Ann. Inst. Fourier (Grenoble) (2011). Bursztyn, Cabrera: Multiplicative forms at the infinitesimal level. Math. Ann. (2012). Iglesias, Laurent, Xu: Universal lifting and quasi-poisson groupoids. JEMS (2012). Jotz Lean, Ortiz: Foliated groupoids and infinitesimal ideal systems, Indag. Math. (2014) Crainic, Salazar, Struchiner: Multiplicative forms and Spencer operators. Math Z (2015). Cabrera, Marcut, Salazar: A construction of local Lie groupoids using Lie algebroid sprays. preprint arxiv: (2017).

33 General tensor fields Idea: Treat tensor fields as functions on Whitney sum of tangent and cotangent bundles.

34 General tensor fields Idea: Treat tensor fields as functions on Whitney sum of tangent and cotangent bundles. Advantages: Uniform description of multiplicativity, simple linear data and integration as morphisms:

35 General tensor fields Idea: Treat tensor fields as functions on Whitney sum of tangent and cotangent bundles. Advantages: Uniform description of multiplicativity, simple linear data and integration as morphisms: (Compatibility) f : G R multiplicative f (gh) = f (g) + f (h). (Linear data) µ : A R, µ([a, b]) = L ρ(a) µ(b) L ρ(b) µ(a). (Correspondence) L a f = t µ(a).

36 General tensor fields Idea: Treat tensor fields as functions on Whitney sum of tangent and cotangent bundles. Advantages: Uniform description of multiplicativity, simple linear data and integration as morphisms: (Compatibility) f : G R multiplicative f (gh) = f (g) + f (h). (Linear data) µ : A R, µ([a, b]) = L ρ(a) µ(b) L ρ(b) µ(a). (Correspondence) L a f = t µ(a). Adaptable method to other contexts (e.g. van Est for differential forms with coefficients)

37 General tensor fields Idea: Treat tensor fields as functions on Whitney sum of tangent and cotangent bundles. Advantages: Uniform description of multiplicativity, simple linear data and integration as morphisms: (Compatibility) f : G R multiplicative f (gh) = f (g) + f (h). (Linear data) µ : A R, µ([a, b]) = L ρ(a) µ(b) L ρ(b) µ(a). (Correspondence) L a f = t µ(a). Adaptable method to other contexts (e.g. van Est for differential forms with coefficients) Cost: Work on different groupoid.

38 Compatibility G M Lie groupoid and τ Γ( p T G q T G) a (q, p) tensor field. View τ as a function on G = ( p T G) ( q T G), (U 1,..., U p, ξ 1,..., ξ q ) cτ τ(u 1,..., U p, ξ 1,..., ξ q ). G is a Lie groupoid over M = ( p TM) ( q A ).

39 Compatibility G M Lie groupoid and τ Γ( p T G q T G) a (q, p) tensor field. View τ as a function on G = ( p T G) ( q T G), (U 1,..., U p, ξ 1,..., ξ q ) cτ τ(u 1,..., U p, ξ 1,..., ξ q ). G is a Lie groupoid over M = ( p TM) ( q A ). Definition: τ is multiplicative if c τ C (G) is a multiplicative function.

40 Previous appearences in the literature: (q, 0) case: multivector fields - (Iglesias, Laurent, Xu, 2012). c π multiplicative ( π π ( 1) q+1 π ) (ξ 1,..., ξ q ) = 0, for ξ i N (graph(m)). (0, p) case: differential forms - (Bursztyn, Cabrera, 2012) c ω multiplicative m ω = pr1 ω + pr2 ω.

41 Previous appearences in the literature: (q, 0) case: multivector fields - (Iglesias, Laurent, Xu, 2012). c π multiplicative ( π π ( 1) q+1 π ) (ξ 1,..., ξ q ) = 0, for ξ i N (graph(m)). (0, p) case: differential forms - (Bursztyn, Cabrera, 2012) c ω multiplicative m ω = pr1 ω + pr2 ω. For vector-valued forms τ Γ( q T G T G), we have that τ is multiplicative τ : } T G {{ T G } T G is a groupoid morphism q times

42 Infinitesimal components Let A be the Lie algebroid of G Cocycle: Multiplicative τ Γ( p T G q T G) µ Γ(A ), dµ = 0 L a c τ = t µ, a, for a Γ(A). }{{} C (M)

43 Infinitesimal components Let A be the Lie algebroid of G Cocycle: Multiplicative τ Γ( p T G q T G) µ Γ(A ), dµ = 0 L a c τ = t µ, a, for a Γ(A). }{{} C (M) Γ(A) is generated by 3 types of sections ua, va, vα, parametrized by a Γ(A), α Ω 1 (M) (DVB theory).

44 Infinitesimal components Let A be the Lie algebroid of G Cocycle: Multiplicative τ Γ( p T G q T G) µ Γ(A ), dµ = 0 L a c τ = t µ, a, for a Γ(A). }{{} C (M) Γ(A) is generated by 3 types of sections ua, va, vα, parametrized by a Γ(A), α Ω 1 (M) (DVB theory). Evaluation maps: D : Γ(A) C (M), D(a) = µ, ua l : A C (M), l(a) = µ, va r : T M C (M), r(α) = µ, vα.

45 (D, l, r) }{{} take values in Γ( p i T M q j A) C (M). (0,0),(1,0),(0,1)

46 (D, l, r) }{{} take values in Γ( p i T M q j A) C (M). (0,0),(1,0),(0,1) They are the infinitesimal components of τ.

47 (D, l, r) }{{} take values in Γ( p i T M q j A) C (M). (0,0),(1,0),(0,1) They are the infinitesimal components of τ. D(fa) = fd(a) + df l(a) r(df ) a (Leibniz equation)

48 (D, l, r) }{{} take values in Γ( p i T M q j A) C (M). (0,0),(1,0),(0,1) They are the infinitesimal components of τ. D(fa) = fd(a) + df l(a) r(df ) a (Leibniz equation) Pull-back by target map t : C (M) C (G) restricts to T : Γ( p T M q A) Γ( p T G q T G) β X t β X.

49 (D, l, r) }{{} take values in Γ( p i T M q j A) C (M). (0,0),(1,0),(0,1) They are the infinitesimal components of τ. D(fa) = fd(a) + df l(a) r(df ) a (Leibniz equation) Pull-back by target map t : C (M) C (G) restricts to T : Γ( p T M q A) Γ( p T G q T G) β X t β X. What about L a c τ?

50 (D, l, r) }{{} take values in Γ( p i T M q j A) C (M). (0,0),(1,0),(0,1) They are the infinitesimal components of τ. D(fa) = fd(a) + df l(a) r(df ) a (Leibniz equation) Pull-back by target map t : C (M) C (G) restricts to T : Γ( p T M q A) Γ( p T G q T G) β X t β X. What about L a c τ? L ( ua) c τ = c (L a τ), L ( va) c τ = c (i a τ), L vα c τ = c (it α τ).

51 (D, l, r) }{{} take values in Γ( p i T M q j A) C (M). (0,0),(1,0),(0,1) They are the infinitesimal components of τ. D(fa) = fd(a) + df l(a) r(df ) a (Leibniz equation) Pull-back by target map t : C (M) C (G) restricts to T : Γ( p T M q A) Γ( p T G q T G) β X t β X. What about L a c τ? L ( ua) c τ = c (L a τ), L ( va) c τ = c (i a τ), L vα c τ = c (it α τ). So, L a τ = T (D(a)), i a τ = T (l(a)), i t ατ = T (r(α)).

52 IM equations dµ = 0 (D, r, l) satisfy (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).

53 IM equations dµ = 0 (D, r, l) satisfy (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( T M A) via a (α b) = L ρ(a) α b + α [a, b]

54 Let G M be s.s.c., A M its Lie algebroid.

55 Let G M be s.s.c., A M its Lie algebroid. Theorem: (Bursztyn, D.) There is 1-1 correspondence between τ Γ( q T G p T G) multiplicative and (D, l, r), where D : Γ(A) Γ( p T M q A), l : A p 1 T M q A, r : T M p T M q 1 A, Leibniz-like condition, satisfying (1) (6).

56 Multivector fields: Π Γ( q T G) { D : Γ(A) Γ( q A) r : T M q 1 A

57 { D : Γ(A) Γ( Multivector fields: Π Γ( q T G) q A) r : T M q 1 A Define δ : Γ( A) Γ( +q 1 A) via δ 0 (f ) = ( 1) q r(df ), δ 1 (a) = D(a) (Leibniz equation) δ(a b) = δ(a) b + ( 1) a (q 1) a δ(b).

58 { D : Γ(A) Γ( Multivector fields: Π Γ( q T G) q A) r : T M q 1 A Define δ : Γ( A) Γ( +q 1 A) via δ 0 (f ) = ( 1) q r(df ), δ 1 (a) = D(a) (Leibniz equation) δ(a b) = δ(a) b + ( 1) a (q 1) a δ(b). IM-equations equivalent to δ([a, b]) = [δ(a), b] + ( 1) ( a 1)(q 1) [a, δ(b)]. So, infinitesimal components are q-derivations of the Gerstenhaber algebra (Λ A, [, ], ). Iglesias,Laurent-Gengoux, Xu(2012)

59 Differential forms: ω Γ( p T G) { D : Γ(A) Γ( p T M) l : A p 1 T M

60 { D : Γ(A) Γ( Differential forms: ω Γ( p T G) p T M) l : A p 1 T M Define µ : A p 1 T M, ν : A p T M by µ = l, ν = D dµ

61 { D : Γ(A) Γ( Differential forms: ω Γ( p T G) p T M) l : A p 1 T M Define µ : A p 1 T M, ν : A p T M by µ = l, ν = D dµ IM-equations equivalent to ν([a, b]) = L ρ(a) ν(b) i ρ(b) dν(a) µ([a, b]) = L ρ(a) µ(b) i ρ(b) dµ(a) i ρ(b) ν(a) µ(b), ρ(a) = µ(a), ρ(b) So, infinitesimal components are IM p-forms. Arias Abad-Crainic(2011) ; Bursztyn-Cabrera(2012).

62 { D : Γ(A) Γ( Differential forms: ω Γ( p T G) p T M) l : A p 1 T M Define µ : A p 1 T M, ν : A p T M by µ = l, ν = D dµ IM-equations equivalent to ν([a, b]) = L ρ(a) ν(b) i ρ(b) dν(a) µ([a, b]) = L ρ(a) µ(b) i ρ(b) dµ(a) i ρ(b) ν(a) µ(b), ρ(a) = µ(a), ρ(b) So, infinitesimal components are IM p-forms. Arias Abad-Crainic(2011) ; Bursztyn-Cabrera(2012). Also, dω = s φ t φ ν(a) = i ρ(a) φ

63 Vector valued forms K Γ( p T G T G) D : Γ(A) Γ( p T M A) l : A p 1 T M A r : T M p T M

64 Vector valued forms K Γ( p T G T G) Closed under FN bracket D : Γ(A) Γ( p T M A) l : A p 1 T M A r : T M p T M graded Lie algebra structure on IM (1, p) forms (commutators of extensions)

65 Vector valued forms K Γ( p T G T G) Closed under FN bracket D : Γ(A) Γ( p T M A) l : A p 1 T M A r : T M p T M graded Lie algebra structure on IM (1, p) forms (commutators of extensions) D : Ω (M, A) Ω +p (M, A), l : Ω (M, A) Ω +p 1 (M, A).

66 Vector valued forms K Γ( p T G T G) Closed under FN bracket D : Γ(A) Γ( p T M A) l : A p 1 T M A r : T M p T M graded Lie algebra structure on IM (1, p) forms (commutators of extensions) D : Ω (M, A) Ω +p (M, A), l : Ω (M, A) Ω +p 1 (M, A). Nijenhuis torsion: In this case p = 1 and N K = 1 2 [K, K ] has infinitesimal components D 2 : Γ(A) Ω 2 (M, A) [D, l] : A T M A, [D, l](a) X = D X (l(a)) l X (D(a)) N r Ω 2 (M, TM).

67 p = 1 D : Γ(A) Γ(T M A), l : A A, r : TM TM.

68 p = 1 D : Γ(A) Γ(T M A), l : A A, r : TM TM. Complex structures: K 2 = Id l 2 = Id, r 2 = Id and T 01 connection : X+ir(X) a = l(d X (a)).

69 p = 1 D : Γ(A) Γ(T M A), l : A A, r : TM TM. Complex structures: K 2 = Id l 2 = Id, r 2 = Id and T 01 connection : X+ir(X) a = l(d X (a)). N r = 0 M is complex N K = 0 D 2 = 0 flat [D, l] = 0 l parallel

70 p = 1 D : Γ(A) Γ(T M A), l : A A, r : TM TM. Complex structures: K 2 = Id l 2 = Id, r 2 = Id and T 01 connection : X+ir(X) a = l(d X (a)). N r = 0 M is complex N K = 0 D 2 = 0 flat [D, l] = 0 l parallel N K = 0 + (IM-equations) holomorphic Lie algebroid [, ] restricts to [, ] hol on Γ hol (A)

71 p = 1 D : Γ(A) Γ(T M A), l : A A, r : TM TM. Complex structures: K 2 = Id l 2 = Id, r 2 = Id and T 01 connection : X+ir(X) a = l(d X (a)). N r = 0 M is complex N K = 0 D 2 = 0 flat [D, l] = 0 l parallel N K = 0 + (IM-equations) holomorphic Lie algebroid [, ] restricts to [, ] hol on Γ hol (A) [, ] is C-linear.

72 p = 1 D : Γ(A) Γ(T M A), l : A A, r : TM TM. Complex structures: K 2 = Id l 2 = Id, r 2 = Id and T 01 connection : X+ir(X) a = l(d X (a)). N r = 0 M is complex N K = 0 D 2 = 0 flat [D, l] = 0 l parallel N K = 0 + (IM-equations) holomorphic Lie algebroid [, ] restricts to [, ] hol on Γ hol (A) [, ] is C-linear. So, holomorphic Lie groupoids holomorphic Lie algebroids Laurent-Gengoux, Stienon, Xu (2009)

73 p = 1 D : Γ(A) Γ(T M A), l : A A, r : TM TM. Complex structures: K 2 = Id l 2 = Id, r 2 = Id and T 01 connection : X+ir(X) a = l(d X (a)). N r = 0 M is complex N K = 0 D 2 = 0 flat [D, l] = 0 l parallel N K = 0 + (IM-equations) holomorphic Lie algebroid [, ] restricts to [, ] hol on Γ hol (A) [, ] is C-linear. So, holomorphic Lie groupoids holomorphic Lie algebroids Laurent-Gengoux, Stienon, Xu (2009) This framework can be applied to: Poisson quasi-nijenhuis structures, multiplicative projections, almost product structures...

74 Coefficients Tensors with values in a representation E are functions on G (E G) }{{}} M {{ E } G E M E Diff. forms: Crainic, Salazar, Struchiner (2015)

75 Coefficients Tensors with values in a representation E are functions on G (E G) }{{}} M {{ E } G E M E Diff. forms: Crainic, Salazar, Struchiner (2015) Infinitesimal-Global results Linear data: A cocycle µ on the Lie algebroid A E. Infinitesimal components: Extract more refined information using evaluation for a special set of generators.

76 Coefficients Tensors with values in a representation E are functions on G (E G) }{{}} M {{ E } G E M E Diff. forms: Crainic, Salazar, Struchiner (2015) Infinitesimal-Global results Linear data: A cocycle µ on the Lie algebroid A E. Infinitesimal components: Extract more refined information using evaluation for a special set of generators.

77 Coefficients Tensors with values in a representation E are functions on G (E G) }{{}} M {{ E } G E M E Diff. forms: Crainic, Salazar, Struchiner (2015) Infinitesimal-Global results Linear data: A cocycle µ on the Lie algebroid A E. Infinitesimal components: Extract more refined information using evaluation for a special set of generators. More general coefficients (Joint w/ L.Egea): Representation up to homotopy. Change E G by VB-groupoids. (Ehresmann connection, multiplicative distributions, forms with values in the adjoint representation.)

78 van Est Multiplicative functions are cocycles on a diff. complex (C (G), δ), where C k (G) = C (G (k) ), space of k composable arrows. δ 0 (f ) = s f t f, δ 1 (f ) = pr 2 f m f + pr 1 f,...

79 van Est Multiplicative functions are cocycles on a diff. complex (C (G), δ), where C k (G) = C (G (k) ), space of k composable arrows. δ 0 (f ) = s f t f, δ 1 (f ) = pr 2 f m f + pr 1 f,... The infinitesimal-global extends to VE : C (G) Γ( A ). VE(f )(u 1,..., u k ) = formula envolving Lie derivatives of f along the r.i. vect. fields u i.

80 van Est Multiplicative functions are cocycles on a diff. complex (C (G), δ), where C k (G) = C (G (k) ), space of k composable arrows. δ 0 (f ) = s f t f, δ 1 (f ) = pr 2 f m f + pr 1 f,... The infinitesimal-global extends to VE : C (G) Γ( A ). VE(f )(u 1,..., u k ) = formula envolving Lie derivatives of f along the r.i. vect. fields u i. (van Est; Weinstein-Xu; Crainic): G source k-connected, VE defines isomorphisms on cohomology up to degree k.

81 van Est for forms with coefficients Diff. complex: (Ω p (G ( ), t E), δ)

82 van Est for forms with coefficients Diff. complex: (Ω p (G ( ), t E), δ) ω Ω p (G, t E) multiplicative δω = 0. (m ω=pr 1 ω+g pr 2 ω.)

83 van Est for forms with coefficients Diff. complex: (Ω p (G ( ), t E), δ) ω Ω p (G, t E) multiplicative Embedding: (G E = T G T G (E G)). δω = 0. (m ω=pr 1 ω+g pr 2 ω.) Ω p (G (k), t E) C (G (k) ) subcomplex E

84 van Est for forms with coefficients Diff. complex: (Ω p (G ( ), t E), δ) ω Ω p (G, t E) multiplicative Embedding: (G E = T G T G (E G)). δω = 0. (m ω=pr 1 ω+g pr 2 ω.) Ω p (G (k), t E) C (G (k) ) subcomplex E Weil algebra: W k,p (A, E) is the space of sequences (c 0, c 1,... ) c i : Γ(A) Γ(A) Ω }{{} p i (M, S i A E) k i (skew + Leibniz cond.)

85 van Est for forms with coefficients Diff. complex: (Ω p (G ( ), t E), δ) ω Ω p (G, t E) multiplicative Embedding: (G E = T G T G (E G)). δω = 0. (m ω=pr 1 ω+g pr 2 ω.) Ω p (G (k), t E) C (G (k) ) subcomplex E Weil algebra: W k,p (A, E) is the space of sequences (c 0, c 1,... ) c i : Γ(A) Γ(A) Ω }{{} p i (M, S i A E) k i (skew + Leibniz cond.) k = 1: Elements of W 1,p (A, E) are (c 0, c 1 ), c 0 : Γ(A) Ω p (M, E), c 1 : A p 1 T M E

86 Embedding: W k,p (A, E) Γ( k A E ) through evaluation maps c i (a 1,..., a k i b 1,..., b i ) = µ(ua 1,..., ua k i, vb 1,..., vb i )

87 Embedding: W k,p (A, E) Γ( k A E ) through evaluation maps c i (a 1,..., a k i b 1,..., b i ) = µ(ua 1,..., ua k i, vb 1,..., vb i ) W k,p (A, E) is a subcomplex of the Chevalley-Eilenberg complex.

88 Embedding: W k,p (A, E) Γ( k A E ) through evaluation maps c i (a 1,..., a k i b 1,..., b i ) = µ(ua 1,..., ua k i, vb 1,..., vb i ) W k,p (A, E) is a subcomplex of the Chevalley-Eilenberg complex. Theorem: (Cabrera, D.) If G is source p-connected, the van Est map VE for G E induces isomorphism in cohomology for k p. VE Ω : H k (Ω p (G ( ), E)) H k (W,p (A, E))

89 Embedding: W k,p (A, E) Γ( k A E ) through evaluation maps c i (a 1,..., a k i b 1,..., b i ) = µ(ua 1,..., ua k i, vb 1,..., vb i ) W k,p (A, E) is a subcomplex of the Chevalley-Eilenberg complex. Theorem: (Cabrera, D.) If G is source p-connected, the van Est map VE for G E induces isomorphism in cohomology for k p. VE Ω : H k (Ω p (G ( ), E)) H k (W,p (A, E)) Explicit formulas: L ua L a, L va i a

90 Thank you.