Lie groupoids and Lie algebroids

Lie groupoids and Lie algebroids Songhao Li University of Toronto, Canada University of Waterloo July 30, 2013 Li (Toronto) Groupoids and algebroids Waterloo 2013 1 / 22

Outline Lie groupoid Lie groupoid Examples of Lie groupoids Lie algebroid Lie algebroid Examples of Lie algebroids Category of integrations Poisson groupoid Poisson manifold Poisson groupoid Lie bialgebroids Examples of Poisson groupoids and symplectic groupoids Li (Toronto) Groupoids and algebroids Waterloo 2013 2 / 22

Lie groupoid Lie groupoid Lie groupoid Lie groupoid A Lie groupoid G over the base manifold M is a category such that the set of objects is M, and the set of arrows G is a manifold; the arrows are invertible; the source s and target t are submersions; the multiplication m and the identity id are smooth. Li (Toronto) Groupoids and algebroids Waterloo 2013 3 / 22

Lie groupoid Lie groupoid Lie groupoid Lie groupoid A Lie groupoid G over the base manifold M is a category such that the set of objects is M, and the set of arrows G is a manifold; the arrows are invertible; the source s and target t are submersions; the multiplication m and the identity id are smooth. The structure maps are summarized by the following commutative diagram: G s t G m i G s id t M Li (Toronto) Groupoids and algebroids Waterloo 2013 3 / 22

Lie groupoid Lie groupoid Source-simply-connected Lie groupoid Notation For a Lie groupoid G over M, we denote it by G M. Source-simply-connected Lie groupoid For a Lie groupoid G M, for x M, the source fiber of x is s 1 (x), and the target fiber is t 1 (x); G M is source-connected, if s 1 (x) is connected for each x M; G M is source-simply-connected, if s 1 (x) is connected and simply-connected for each x M. Li (Toronto) Groupoids and algebroids Waterloo 2013 4 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 1. Lie group If the base M is a point, then G is a Lie group. Li (Toronto) Groupoids and algebroids Waterloo 2013 5 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 1. Lie group If the base M is a point, then G is a Lie group. 2. Bundle of Lie groups If the source s : G M and the target t : G M coincide, then G M is a bundle of Lie groups. 3. Pair groupoid For a connected manifold M, we define the pair groupoid Pair(M) M: Pair(M) = M M; s : Pair(M) M, (x, y) x; t : Pair(M) M, (x, y) y; id : M Pair(M), x (x, x); m : Pair(M) s t Pair(M) Pair(M), ((x, y), (y, z)) (x, z). Li (Toronto) Groupoids and algebroids Waterloo 2013 5 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 4. Fundamental groupoid For a connected manifold M, we define the fundamental groupoid Π 1 (M) M: Π 1 (M) is the homotopy classes of paths; s : Π 1 (M) M, t : Π 1 (M) M, γ γ(0); γ γ(1); id : M Π 1 (M), x id(x) where id(x) is the constant path at x; m : Π 1 (M) s t Π 1 (M) Π 1 (M) is the concatenation of paths. Li (Toronto) Groupoids and algebroids Waterloo 2013 6 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 4. Fundamental groupoid For a connected manifold M, we define the fundamental groupoid Π 1 (M) M: Π 1 (M) is the homotopy classes of paths; s : Π 1 (M) M, t : Π 1 (M) M, γ γ(0); γ γ(1); id : M Π 1 (M), x id(x) where id(x) is the constant path at x; m : Π 1 (M) s t Π 1 (M) Π 1 (M) is the concatenation of paths. Remark Note that (s, t) : Π 1 (M) Pair(M) is a Lie groupoid morphism. Li (Toronto) Groupoids and algebroids Waterloo 2013 6 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 5. General linear groupoid For a vector bundle E M, we define the general linear groupoid GL(E) M: GL(E) {(x, y, φ xy ) x M, y M} where φ xy : E x Ey is an invertible linear transformation; s : GL(E) M, (x, y, φ xy ) x; t : GL(E) M, (x, y, φ xy ) y; id : M GL(E), x (x, x, id xx ); m : GL(E) s t GL(E) GL(E), ( (x, y, φxy ), (y, z, φ yz ) ) (x, z, φ yz φ xy ). Li (Toronto) Groupoids and algebroids Waterloo 2013 7 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 5. General linear groupoid For a vector bundle E M, we define the general linear groupoid GL(E) M: GL(E) {(x, y, φ xy ) x M, y M} where φ xy : E x Ey is an invertible linear transformation; s : GL(E) M, (x, y, φ xy ) x; t : GL(E) M, (x, y, φ xy ) y; id : M GL(E), x (x, x, id xx ); m : GL(E) s t GL(E) GL(E), ( (x, y, φxy ), (y, z, φ yz ) ) (x, z, φ yz φ xy ). Remark A Lie groupoid action of G M on a vector bundle E M is a Lie groupoid morphism ρ : G GL(E). Li (Toronto) Groupoids and algebroids Waterloo 2013 7 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 6. Holonomy groupoid For a vector bundle E M, we define the holonomy groupoid Hol(E) M: Hol(E) = {(γ, φ st ) γ Π 1 D. where φ st : E γ(0) Eγ(1) } is an invertible linear transformation; s : Hol(E) M, t : Hol(E) M, (γ, φ st ) γ(0); (γ, φ st ) γ(1); id : M Hol(E), x (id(x), id xx ); m : Hol(E) s t Hol(E) Hol(E), ((γ, φ st ), (σ, ψ st )) (γ σ, φ st ψ st ). Li (Toronto) Groupoids and algebroids Waterloo 2013 8 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 6. Holonomy groupoid For a vector bundle E M, we define the holonomy groupoid Hol(E) M: Hol(E) = {(γ, φ st ) γ Π 1 D. where φ st : E γ(0) Eγ(1) } is an invertible linear transformation; s : Hol(E) M, t : Hol(E) M, (γ, φ st ) γ(0); (γ, φ st ) γ(1); id : M Hol(E), x (id(x), id xx ); m : Hol(E) s t Hol(E) Hol(E), ((γ, φ st ), (σ, ψ st )) (γ σ, φ st ψ st ). Remark There is an obvious surjective groupoid morphism Hol(E) GL(E) which covers Π 1 (M) Pair(M). Li (Toronto) Groupoids and algebroids Waterloo 2013 8 / 22

Lie groupoid Examples of Lie groupoids Examples: Lie groupoids 7. Gauge groupoid For a principal G-bundle p : P M, we consider the diagonal action of G on P P: g(u, v) = (gu, gv). We define the gauge groupoid Gauge(P) M: Gauge(P) = (P P)/G. For u, v, v, w P such that p(v) = p(v ) = x, s([u, v]) = p(u); t([u, v]) = p(v); id(x) = ([v, v]); m ([u, v], [v, w]) = ([u, w]). Li (Toronto) Groupoids and algebroids Waterloo 2013 9 / 22

Lie algebroid Lie algebroid Lie algebroid Lie algebroid A Lie algebroid A over the base manifold M is a vector bundle A M with a Lie bracket [, ] on Γ(A) and an anchor map a : A TM that preserves the bracket and satisfies the Leibniz rule [X, fy ] = f [X, Y ] + a(x)(f )Y. (2.1) Lie functor For a Lie groupoid G M, the vector bundle Lie(G). = id ker (Ts : T G TM) (2.2) with the bracket on left invariant vector fields, and the anchor Tt : Lie(G) TM, is Lie algebroid. Li (Toronto) Groupoids and algebroids Waterloo 2013 10 / 22

Lie algebroid Examples of Lie algebroids Examples: Lie algebroids 1.Lie algebra If the base M is a point, then A is a Lie algebra. Li (Toronto) Groupoids and algebroids Waterloo 2013 11 / 22

Lie algebroid Examples of Lie algebroids Examples: Lie algebroids 1.Lie algebra If the base M is a point, then A is a Lie algebra. 2. Bundle of Lie algebras If the anchor a : A TM is trivial, then A is a bundle of Lie algebras. 3. Tangent algebroid For a connected manifold M, the tangent bundle TM itself is a Lie algebroid. Both the pair groupoid Pair(M) and the fundamental groupoid Π 1 (M) integrates the tangent algebroid TM. Li (Toronto) Groupoids and algebroids Waterloo 2013 11 / 22

Lie algebroid Examples of Lie algebroids Examples: Lie algebroids 4. General linear algebroid For a vector bundle E M of rank r, we define the general linear algebroid gl(e): Let E be the Euler vector field, we have Γ(gl(E)) = {X Γ(TE) L E X = 0}. (2.3) The anchor a : gl(e) TM is the pushforward of the projection E M. Li (Toronto) Groupoids and algebroids Waterloo 2013 12 / 22

Lie algebroid Examples of Lie algebroids Examples: Lie algebroids Remark The general linear algebroid gl(e) fits into the following exact sequence a 0 V gl(e) TM 0 (2.4) where V is the vector bundle whose sections are the vertical vector fields. Both the holonomy groupoid Hol(E) and the general linear groupoid GL(E) integrates the general linear algebroid gl(e). A Lie algebroid action A M on a vector bundle E M is a Lie algebroid morphism from A to gl(e). Li (Toronto) Groupoids and algebroids Waterloo 2013 13 / 22

Lie algebroid Examples of Lie algebroids Examples: Lie algebroids 5. Atiyah algebroid For a principal G-bundle P M, the Atiyah algebroid At(P) of P is the Lie algebroid of the gauge groupoid Gauge(P) P, which fits in the following short exact sequence: 0 P G g At(P) TM 0 (2.5) where P G g is the associated bundle. Li (Toronto) Groupoids and algebroids Waterloo 2013 14 / 22

Lie algebroid Category of integrations Category of integrations In general, it is not always true that a Lie algebroid integrates to a Lie groupoid. The integrability condition was given in [Crainic-Fernandes, 03]. For an integrable Lie algebroid A, the source-connected groupoids integrating A form a category Gpd(A). The initial object in Gpd(A) is the source-simply-connected groupoid integrating A; the terminal object, if exists, is the adjoint groupoid. Li (Toronto) Groupoids and algebroids Waterloo 2013 15 / 22

Lie algebroid Category of integrations Category of integrations In general, it is not always true that a Lie algebroid integrates to a Lie groupoid. The integrability condition was given in [Crainic-Fernandes, 03]. For an integrable Lie algebroid A, the source-connected groupoids integrating A form a category Gpd(A). The initial object in Gpd(A) is the source-simply-connected groupoid integrating A; the terminal object, if exists, is the adjoint groupoid. Lie groups For a Lie algebra g, the category of integrations Gpd(g) is equivalent to the lattice of normal subgroups of the fundamental group of simply-connected Lie group, Λ(π 1 (G sc )). Li (Toronto) Groupoids and algebroids Waterloo 2013 15 / 22

Lie algebroid Category of integrations Category of tangent integrations For a connected manifold M, the category of tangent integrations Gpd(TM) is equivalent to the lattice of normal subgroups of the fundamental group, Λ(π 1 (M, x)). The equivalence is given by G M t (π 1 (s 1 (x), id(x))). (2.6) The fundamental groupoid Π 1 (M) is the source-simply-connected integration, and corresponds to the trivial group {id} inside π 1 (M, x). The pair groupoid Pair(M) is the adjoint integration, and corresponds to π 1 (M, x). Li (Toronto) Groupoids and algebroids Waterloo 2013 16 / 22

Poisson groupoid Poisson manifold Poisson manifold Schouten Nijenhuis breacket For a manifold M, the bracket [, ] on vector fields X(M) may be extended to make the alternating multi-vectors X (M) a Gerstenhaber algebra [X 1 X m, Y 1 Y n ] = i,j ( 1) i+j [X i, Y j ]X 1 ˆX i X m Y 1 Ŷj Y n. (3.1) Li (Toronto) Groupoids and algebroids Waterloo 2013 17 / 22

Poisson groupoid Poisson manifold Poisson manifold Schouten Nijenhuis breacket For a manifold M, the bracket [, ] on vector fields X(M) may be extended to make the alternating multi-vectors X (M) a Gerstenhaber algebra [X 1 X m, Y 1 Y n ] = i,j ( 1) i+j [X i, Y j ]X 1 ˆX i X m Y 1 Ŷj Y n. (3.1) Poisson manifold A Poisson manifold is a manifold M with a bivector π X 2 (M) such that [π, π] = 0. (3.2) Li (Toronto) Groupoids and algebroids Waterloo 2013 17 / 22

Poisson groupoid Poisson manifold Poisson algebroid Poisson algebroids For a Poisson manifold (M, π), we define the Poisson algebroid T π M: T π M = T M; the anchor is π : T M TM, α ι α π; the bracket is the Koszul bracket [α, β] = L π (α)β L π (β)α dπ(α, β). (3.3) Coisotropic submanifold A coisotropic submanifold of (M, π) is a submanifold C M such that π (T M C ) TC. The conormal bundle N C is a Lie subalgebroid of the Poisson algebroid T π M. Li (Toronto) Groupoids and algebroids Waterloo 2013 18 / 22

Poisson groupoid Poisson groupoid Poisson groupoid Poisson groupoid A Poisson groupoid is a Lie groupoid G M with a Poisson structure σ on G such that the graph of multiplication Graph(m) = {(g, h, m(g, h)) (g, h) G s t G} (3.4) is coisotropic with respect to σ σ σ. The pushforward π = s (σ) = t (σ) is Poisson on M. Symplectic groupoid A symplectic groupoid is a Poisson groupoid (G, σ) (M, π) such that σ is non-degenerate. Li (Toronto) Groupoids and algebroids Waterloo 2013 19 / 22

Poisson groupoid Poisson groupoid Poisson groupoid Poisson groupoid A Poisson groupoid is a Lie groupoid G M with a Poisson structure σ on G such that the graph of multiplication Graph(m) = {(g, h, m(g, h)) (g, h) G s t G} (3.4) is coisotropic with respect to σ σ σ. The pushforward π = s (σ) = t (σ) is Poisson on M. Symplectic groupoid A symplectic groupoid is a Poisson groupoid (G, σ) (M, π) such that σ is non-degenerate. For a symplectic groupoid (G, σ), the graph Graph(m) is Lagrangian. Li (Toronto) Groupoids and algebroids Waterloo 2013 19 / 22

Poisson groupoid Lie bialgebroids Lie bialgebroid Lie bialgebroid Let (A, M, a, [, ]) be a Lie algebroid. If the dual bundle A carries a Lie algebroid structure (A, M, a, [, ] ) such that d [X, Y ] = L X d Y L Y d X, (3.5) where d is the A -differential and L is the Lie direvative with respect to A, then (A, A ) is a Lie bialgebroid. Lie functor for Poisson groupoid For a Poisson groupoid (G, σ) (M, π), the Lie algebroids A =. Lie(G) and A =. N (id(m)) form a Lie bialgebroid, and we write Lie(G, σ) = (A, A ). (3.6) Li (Toronto) Groupoids and algebroids Waterloo 2013 20 / 22

Poisson groupoid Examples of Poisson groupoids and symplectic groupoids Examples: Poisson groupoids 1. Pair groupoid For a Poisson manifold (M, π), the pair groupoid Pair(M) = M M with the Poisson structrure π π is a Poisson groupoid. 2. Fundamental groupoid For a Poisson manifold (M, π), the fundamental groupoid Π 1 (M) with the Poisson structrure σ = (s, t) (π π) is a Poisson groupoid. Li (Toronto) Groupoids and algebroids Waterloo 2013 21 / 22

Poisson groupoid Examples of Poisson groupoids and symplectic groupoids Examples: Poisson groupoids 1. Pair groupoid For a Poisson manifold (M, π), the pair groupoid Pair(M) = M M with the Poisson structrure π π is a Poisson groupoid. 2. Fundamental groupoid For a Poisson manifold (M, π), the fundamental groupoid Π 1 (M) with the Poisson structrure σ = (s, t) (π π) is a Poisson groupoid. The Lie bialgebroid of (Pair(M), π π) and (Π 1 (M), σ) is (TM, T π M). If (M, π) is symplectic, then both (Pair(M), π π) and (Π 1 (M), σ) are symplectic groupoids. Li (Toronto) Groupoids and algebroids Waterloo 2013 21 / 22

Poisson groupoid Examples of Poisson groupoids and symplectic groupoids Examples: Poisson groupoids 3. (T M, ω 0 ) (M, 0) For a manifold M with the zero Poisson structure, the cotangent bundle T M with the canonical symplectic structrue ω 0 is a symplectic groupoid. 4. (T G, ω 0 ) (g, π kks ) Let G be a Lie group, and let g be its Lie algebra. The dual g is equipped with the KKS Poisson structure π kks. The cotangent bundle (T G, ω 0 ) is a symplectic groupoid of (g, π kks ). The source and the target are left and right translations. Li (Toronto) Groupoids and algebroids Waterloo 2013 22 / 22