Lie groupoids and Lie algebroids

Songhao Li

University of Toronto, Canada

University of Waterloo July 30, 2013

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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

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Lie groupoid

Lie groupoid A Lie groupoid G over the base manifold M is a category such that

I the set of objects is M, and the set of arrows G is a manifold;

I the arrows are invertible;

I the source s and target t are submersions;

I 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

I the set of objects is M, and the set of arrows G is a manifold;

I the arrows are invertible;

I the source s and target t are submersions;

I the multiplication m and the identity id are smooth. The structure maps are summarized by the following commutative diagram: i s  G × G m / G o id M s t A t

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, −1 I for x ∈ M, the source fiber of x is s (x), and the target fiber is t−1(x); −1 I G ⇒ M is source-connected, if s (x) is connected for each x ∈ M; −1 I G ⇒ M is source-simply-connected, if s (x) is connected and simply-connected for each x ∈ M.

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Examples: Lie groupoids

1. Lie group If the base M is a point, then G is a Lie group.

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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.

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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: I Pair(M) = M × M;

I s : Pair(M) → M, (x, y) 7→ x;

I t : Pair(M) → M, (x, y) 7→ y;

I id : M → Pair(M), x 7→ (x, x);

I m : Pair(M)s×t Pair(M) → Pair(M), ((x, y), (y, z)) → (x, z).

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Examples: Lie groupoids

4. Fundamental groupoid For a connected manifold M, we define the fundamental groupoid Π1(M) → M:

I Π1(M) is the homotopy classes of paths;

I s :Π1(M) → M, γ 7→ γ(0);

I t :Π1(M) → M, γ 7→ γ(1);

I id : M → Π1(M), x 7→ id(x) where id(x) is the constant path at x;

I m :Π1(M)s×t Π1(M) → Π1(M) is the concatenation of paths.

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Examples: Lie groupoids

4. Fundamental groupoid For a connected manifold M, we define the fundamental groupoid Π1(M) → M:

I Π1(M) is the homotopy classes of paths;

I s :Π1(M) → M, γ 7→ γ(0);

I t :Π1(M) → M, γ 7→ γ(1);

I id : M → Π1(M), x 7→ id(x) where id(x) is the constant path at x;

I m :Π1(M)s×t Π1(M) → Π1(M) is the concatenation of paths.

Remark Note that (s, t):Π1(M) → Pair(M) is a Lie groupoid morphism.

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Examples: Lie groupoids

5. General linear groupoid For a vector bundle E → M, we define the general linear groupoid GL(E) ⇒ M: I GL(E) '{(x, y, φxy ) | x ∈ M, y ∈ M} ∼ where φxy : Ex → Ey is an invertible linear transformation;

I s : GL(E) → M, (x, y, φxy ) 7→ x;

I t : GL(E) → M, (x, y, φxy ) 7→ y;

I id : M → GL(E), x 7→ (x, x, idxx );

I m : GL(E)s×t GL(E) → GL(E),
(x, y, φxy ), (y, z, φyz ) 7→ (x, z, φyz ◦ φxy ).

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Examples: Lie groupoids

5. General linear groupoid For a vector bundle E → M, we define the general linear groupoid GL(E) ⇒ M: I GL(E) '{(x, y, φxy ) | x ∈ M, y ∈ M} ∼ where φxy : Ex → Ey is an invertible linear transformation;

I s : GL(E) → M, (x, y, φxy ) 7→ x;

I t : GL(E) → M, (x, y, φxy ) 7→ y;

I id : M → GL(E), x 7→ (x, x, idxx );

I m : GL(E)s×t GL(E) → GL(E),
(x, y, φxy ), (y, z, φyz ) 7→ (x, z, φyz ◦ φxy ).

Remark A Lie groupoid action of G ⇒ M on a vector bundle E → M is a Lie groupoid morphism ρ : G → GL(E).

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Examples: Lie groupoids

6. Holonomy groupoid For a vector bundle E → M, we define the holonomy groupoid Hol(E) ⇒ M: I Hol(E) = {(γ, φst ) | γ ∈ Π1D. ∼ where φst : Eγ(0) → Eγ(1)} is an invertible linear transformation;

I s : Hol(E) → M, (γ, φst ) 7→ γ(0);

I t : Hol(E) → M, (γ, φst ) 7→ γ(1);

I id : M → Hol(E), x 7→ (id(x), idxx );

I m : Hol(E)s×t Hol(E) → Hol(E), ((γ, φst ), (σ, ψst )) 7→ (γ ◦ σ, φst ◦ ψst ).

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Examples: Lie groupoids

6. Holonomy groupoid For a vector bundle E → M, we define the holonomy groupoid Hol(E) ⇒ M: I Hol(E) = {(γ, φst ) | γ ∈ Π1D. ∼ where φst : Eγ(0) → Eγ(1)} is an invertible linear transformation;

I s : Hol(E) → M, (γ, φst ) 7→ γ(0);

I t : Hol(E) → M, (γ, φst ) 7→ γ(1);

I id : M → Hol(E), x 7→ (id(x), idxx );

I m : Hol(E)s×t Hol(E) → Hol(E), ((γ, φst ), (σ, ψst )) 7→ (γ ◦ σ, φst ◦ ψst ).

Remark There is an obvious surjective groupoid morphism Hol(E) → GL(E) which covers Π1(M) → Pair(M).

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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: I Gauge(P) = (P × P)/G. For u, v, v 0, w ∈ P such that p(v) = p(v 0) = x,

I s([u, v]) = p(u);

I t([u, v]) = p(v);

I id(x) = ([v, v]); 0 I 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)

Li (Toronto) Groupoids and algebroids Waterloo 2013 10 / 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.

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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.

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

I For a connected manifold M, the tangent bundle TM itself is a Lie algebroid.

I Both the pair groupoid Pair(M) and the fundamental groupoid Π1(M) integrates the tangent algebroid TM.

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Examples: Lie algebroids

4. General linear algebroid For a vector bundle E → M of rank r, we define the general linear algebroid gl(E):

I Let E be the Euler vector field, we have

Γ(gl(E)) = {X ∈ Γ(TE) | LEX = 0}. (2.3)

I The anchor a : gl(E) → TM is the pushforward of the projection E → M.

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Examples: Lie algebroids

Remark

I The general linear algebroid gl(E) fits into the following exact sequence 0 −→ V −→ gl(E) −→a TM −→ 0 (2.4) where V is the vector bundle whose sections are the vertical vector fields.

I Both the holonomy groupoid Hol(E) and the general linear groupoid GL(E) integrates the general linear algebroid gl(E).

I A Lie algebroid action A → M on a vector bundle E → M is a Lie algebroid morphism from A to gl(E).

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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.

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Category of integrations

I 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].

I For an integrable Lie algebroid A, the source-connected groupoids integrating A form a category Gpd(A).

I The initial object in Gpd(A) is the source-simply-connected groupoid integrating A; the terminal object, if exists, is the adjoint groupoid.

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Category of integrations

I 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].

I For an integrable Lie algebroid A, the source-connected groupoids integrating A form a category Gpd(A).

I 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 sc simply-connected Lie group, Λ(π1(G )).

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Category of tangent integrations

I 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(π1(M, x)). I The equivalence is given by

−1 G ⇒ M 7→ t∗(π1(s (x), id(x))). (2.6)

I The fundamental groupoid Π1(M) is the source-simply-connected integration, and corresponds to the trivial group {id} inside π1(M, x). I The pair groupoid Pair(M) is the adjoint integration, and corresponds to π1(M, x).

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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

[X1 ··· Xm, Y1 ··· Yn] X i+j ˆ ˆ (3.1) = (−1) [Xi , Yj ]X1 ··· Xi ··· XmY1 ··· Yj ··· Yn. i,j

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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

[X1 ··· Xm, Y1 ··· Yn] X i+j ˆ ˆ (3.1) = (−1) [Xi , Yj ]X1 ··· Xi ··· XmY1 ··· Yj ··· Yn. i,j

Poisson manifold A Poisson manifold is a manifold M with a bivector π ∈ X2(M) such that

[π, π] = 0. (3.2)

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Poisson algebroid

Poisson algebroids ∗ For a Poisson manifold (M, π), we define the Poisson algebroid Tπ M: ∗ ∗ I Tπ M = T M; ] ∗ I the anchor is π : T M → TM, α 7→ ιαπ; I the bracket is the Koszul bracket

[α, β] = Lπ](α)β − Lπ](β)α − dπ(α, β). (3.3)

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Poisson algebroid

Poisson algebroids ∗ For a Poisson manifold (M, π), we define the Poisson algebroid Tπ M: ∗ ∗ I Tπ M = T M; ] ∗ I the anchor is π : T M → TM, α 7→ ιαπ; I the bracket is the Koszul bracket

[α, β] = Lπ](α)β − Lπ](β)α − dπ(α, β). (3.3)

Coisotropic submanifold

I A coisotropic submanifold of (M, π) is a submanifold C ⊂ M such ] ∗ that π (T M|C) ⊂ TC. ∗ I The conormal bundle N C is a Lie subalgebroid of the Poisson ∗ algebroid Tπ M.

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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) ∈ Gs×t G} (3.4)

is coisotropic with respect to σ ⊕ σ ⊕ −σ.

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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) ∈ Gs×t G} (3.4)

is coisotropic with respect to σ ⊕ σ ⊕ −σ.

I The pushforward π = s∗(σ) = −t∗(σ) is Poisson on M.

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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) ∈ Gs×t G} (3.4)

is coisotropic with respect to σ ⊕ σ ⊕ −σ.

I 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.

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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) ∈ Gs×t G} (3.4)

is coisotropic with respect to σ ⊕ σ ⊕ −σ.

I 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.

I For a symplectic groupoid (G, σ), the graph Graph(m) is Lagrangian.

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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 ] = LX d∗Y − LY 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.

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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 ] = LX d∗Y − LY 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)

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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.

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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 Π1(M) with the Poisson structrure σ = (s, t)∗(π ⊕ −π) is a Poisson groupoid.

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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 Π1(M) with the Poisson structrure σ = (s, t)∗(π ⊕ −π) is a Poisson groupoid.

I The Lie bialgebroid of (Pair(M), π ⊕ −π) and (Π1(M), σ) is ∗ (TM, Tπ M). I If (M, π) is symplectic, then both (Pair(M), π ⊕ −π) and (Π1(M), σ) are symplectic groupoids.

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Examples: Poisson groupoids

∗ 3. (T M, ω0) ⇒ (M, 0) For a manifold M with the zero Poisson structure, the cotangent bundle ∗ T M with the canonical symplectic structrue ω0 is a symplectic groupoid.

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Examples: Poisson groupoids

∗ 3. (T M, ω0) ⇒ (M, 0) For a manifold M with the zero Poisson structure, the cotangent bundle ∗ T M with the canonical symplectic structrue ω0 is a symplectic groupoid.

∗ ∗ 4. (T G, ω0) ⇒ (g , πkks) Let G be a Lie group, and let g be its Lie algebra. ∗ I The dual g is equipped with the KKS Poisson structure πkks. ∗ I The cotangent bundle (T G, ω0) is a symplectic groupoid of ∗ (g , πkks). I The source and the target are left and right translations.

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