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Home , Lie algebroid, Lie groupoid

Lie algebroids and Lie

Kirill Mackenzie

Sheffield, UK

Indian Statistical Institute, Kolkata December 2012 1. Lie groups and Lie algebras

Lie groups are important because they can be linearized and because they encode symmetry.

I Given a Lie G there is a g = T1(G)

I Given a morphism of Lie groups F : G → H there is a morphism of Lie algebras F∗ : g → h

I This is a , the Lie functor.

Thus (F2 ◦ F1)∗ = (F2)∗ ◦ (F1)∗ and (idG)∗ = idg .

So Lie groups can be linearized, and this is one of the keys to the classification of Lie groups. Lie groups have a lot of symmetry. This is a major reason for their importance. But, after all, not everything in nature is symmetric. Groupoids allow the linearization process to be applied to geometric situations with little or no symmetry. As a first step, consider the symmetries of bundle structures. 2. Symmetries of vector bundles

In we typically look not at a single vector space, but at (say) all the tangent spaces to a surface, or all the fibres of a , and we want to pass from one tangent space to another, by parallel translation or other means. It is not enough to consider a single tangent space or fibre. Instead consider all the isomorphisms between the various tangent spaces or fibres. For a vector bundle E on base M , write Φ(E) for the set of all isomorphisms ξ : Ex → Ey for x, y ∈ M . These isomorphisms obey laws which are very similar to the axioms for a group:

I Associativity: ζ(ηξ) = (ζη)ξ whenever the products (=ordinary composition of maps) are defined

I There is an identity isomorphism idEx for each x ∈ M ; denote it 1x −1 I Each ξ : Ex → Ey has an inverse ξ : Ey → Ex

The new feature is that multiplication ηξ is only defined when the source of η equals the target of ξ .

For ξ : Ex → Ey call x the source and y the target and write α(ξ) = x , β(ξ) = y . These properties are the model for the axioms. 3. Definition of

A Lie groupoid G ⇒ M consists of

I a M of ‘points’ and a manifold G of ‘arrows’ between the points

I source (α) and target (β ) maps G → M , which are surjective and submersions

I a (smooth) multiplication hg defined when α(h) = β(g) which I is associative

I has an identity 1x for each x ∈ M

g 1αg = g, 1βg g = g

−1 I each g has an inverse g

−1 −1 g g = 1αg , gg = 1βg 4. More examples

I For a vector bundle E with a fibrewise metric g the set of all ξ ∈ Φ(E) which preserve g is a Lie groupoid Φ(E, g). Likewise with fibrewise orientations, complex structures, ... All these examples are transitive. 0 0 I Pair groupoid: M × M with (z, y )(y, x) = (z, x) when y = y .

I of a connected manifold

I Fundamental groupoid of a (regular or singular).

I Given any smooth action G × M → M of a G , the action groupoid is G × M with β(g, x) = gx , α(g, x) = x and multiplication

(h, gx)(g, x) = (hg, x)

Write G <7 M .

I (More later)

The general concept of Lie groupoid — not necessarily transitive — unifies many disparate concepts in differential geometry. 5. The of a Lie groupoid

I For a Lie group G take g := T1(G). I For a Lie groupoid G ⇒ M , restrict TG to the identity elements; get T1M G , a vector bundle on M . Right-translations Rg map α-fibres to α-fibres. So take the kernel of (α): → T T1M G TM . Call this AG .

I Each X ∈ g defines a right-invariant vector I Each X ∈ ΓAG defines a right-invariant −→ −→ −→ −→ field X by X (g) = Xg. vector field X on G by X (g) = Xg. −→ −→ −→ That is, X (hg) = X (h)g for all h, g . That is, X is α-vertical and −→ −→ X (hg) = X (h)g for all h, g . −→ −→ I Each right-invariant vector field is X for I Each right-invariant vector field is X for some X ∈ g. some X ∈ ΓAG .

I Bracket of right-invariant vector fields is I Bracket of right-invariant vector fields is right-invariant. right-invariant. −−−→ −→ −→ I Define bracket on ΓAG by I Define bracket on g by [X, Y ] = [ X , Y ]. −−−→ −→ −→ [X, Y ] = [ X , Y ].

I AG is the Lie algebroid of G . I g is the Lie algebra of G . 6. The anchor

One new feature: for f ∈ C∞(M) and X, Y ∈ ΓAG ,

[X, fY ] = f [X, Y ] + a(X)(f )Y , where a: AG → TM (the anchor) is the restriction of T (β): TG → TM to AG . −−−−→ −→ −→ Proof: By definition of the bracket, [X, fY ] = [ X , fY ]. −→ −→ −→ Now fY = (f ◦ β)Y by definition of . By a standard identity for vector fields, −→ −→ −→ −→ −→ −→ [ X , (f ◦ β)Y ] = (f ◦ β)[ X , Y ] + X (f ◦ β)Y .

−→ −→ Next, X (f ◦ β) = a(X)(f ) ◦ β because X projects to a(X) under β . Now reassemble the steps. This construction, the Lie functor, extends the basic construction of Lie theory to situations which do not have the symmetry of a group. 7. Examples

G = M × M the pair groupoid. 1M = ∆M the diagonal. T (α) sends Y ⊕ X ∈ T(x,x)(M × M) to X . So AG = TM with the standard structure.

For G = Π(M) the fundamental The Lie algebroid is TM . The endpoint map groupoid of a connected M . Π(M) → M × M is a morphism with discrete kernel.

For G = Φ(E) the full frame Sections D of AΦ(E) are first-order diff operators groupoid of a vector bundle E D :ΓE → ΓE for which there exists a vector field x on on M , M such that D(fs) = fD(s) + x(f )s for all f ∈ C∞(M), s ∈ ΓE .

For G = Φ(E, g) the frame Sections D of AΦ(E, g) are first-order diff operators groupoid of a vector bundle E D :ΓE → ΓE as above such that on M with a fibre metric g . g(D(s1), s2) + g(s1, D(s2)) = x g(s1, s2). 8. Examples, continued

For G the fundamental The Lie algebroid is the tangential distribution D ⊆ TM . groupoid of a foliation on manifold M .

For an action groupoid AG = M × h with anchor (m, X) 7→ X †(m) the G = H <7 M ⇒ M . fundamental vector field map (infinitesimal action). Sections are V : M → h. • [V1, V2] = L † (V2) − L † (V1) + [V1, V2] V1 V2

TP For the groupoid corresponding Lift the action of G on P to TP and form G , a vector to a P bundle on G = M . P(M, G, p). TP Sections of G correspond to G -invariant vector fields on P . TP Anchor is T (p): TP → TM quotiented down to G . Anchor is surjective. Atiyah sequence:

P×g TP G / / G / / TM 9. Abstract Lie algebroids

Defn: A Lie algebroid is a vector bundle A on base M together with a bracket of sections ΓA × ΓA → ΓA and a map a: A → TM such that

I the bracket of sections makes ΓA an R-Lie algebra, ∞ I [X, fY ] = f [X, Y ] + a(X)(f )Y for X, Y ∈ ΓA, f ∈ C (M)

I a[X, Y ] = [aX, aY ] for X, Y ∈ ΓA.

There are Lie algebroids which do not arise from any Lie groupoid (the Integrability Problem). Most of the examples described above can be given in an abstract form, without assuming the existence of an underlying Lie groupoid. The theory now divides. A Lie algebroid is transitive if the anchor is surjective. The kernel is then a vector bundle and there is a short exact sequence

L / / A / / TM .

The transitive theory is very closely related to connection theory. I and others will talk about this later. The general case is bound up with Poisson geometry. 10. Poisson structures

All the preceding theory and examples existed before the middle of the 1980s. Lie groupoid and Lie algebroid theory was then completely transformed by the introduction of the methods of Poisson geometry and the associated notion of symplectic groupoid. This was due, independently, to Karasëv, Weinstein and Zakrzewski. Very quick summary (others will say more):

1 The dual A∗ of a Lie algebroid A has a Poisson structure. Recall: the dual of a Lie algebra has a linear Poisson structure; a cotangent bundle has a symplectic structure. Poisson structures dual to Lie algebroid structures send linear functions to linear vector fields and pullback functions to vertical vector fields. A Poisson structure on a vector bundle E with these properties is the dual of a Lie algebroid structure on E∗ .

2 A Poisson structure on a manifold P induces a Lie algebroid structure on T ∗P . If this Lie algebroid is integrable, then it integrates to a symplectic groupoid, that is, a Lie groupoid with a symplectic structure such that the graph of multiplication is Lagrangian in a suitable sense.

For any symplectic groupoid G ⇒ M , the base manifold M has a canonical Poisson structure and T ∗M =∼ AG as Lie algebroids. 11. Poisson structures

A Poisson structure on a manifold P is a bracket of functions { , } which makes ∞ C (P) an R-Lie algebra and which is such that {u, vw} = {u, v}w + {u, w}v for all u, v, w ∈ C∞(P). ∞ So each u ∈ C (P) generates a vector field Xu := {u, −}. This Xu depends only on ∗ du and du 7→ Xu extends to a map ΓT P → ΓTP defined by v du 7→ vXu . Denote this map by #. Define a bracket on ΓT ∗P by [du, dv] = d{u, v} and extending so that the Leibniz rule

[ϕ, wψ] = w[ϕ, ψ] + ϕ#(w)ψ holds. This bracket makes T ∗P a Lie algebroid on base P with anchor #. So a Poisson structure on P induces a Lie algebroid structure on T ∗P . Are such Lie algebroids integrable ? Not always, but if T ∗P is integrable, then there is ∼ ∗ a symplectic groupoid G ⇒ P with AG = T P . 12. Symplectic groupoids

Defn: A symplectic groupoid is a Lie groupoid G ⇒ P together with a symplectic structure ω on G such that the graph of the multiplication

G = {(hg, h, g) | α(h) = β(g)} is a Lagrangian submanifold of G × G × G . I’ll explain these terms in a moment. First, Thm: P has a (unique) Poisson structure such that β : G → P is a Poisson map. Further, T ∗P =∼ AG as Lie algebroids.

To see what the definition means, use the tangent groupoid TG ⇒ TP . This is the groupoid obtained by applying the tangent functor to the structure of G ⇒ P . For a map f : M → N the tangent map TM → TN is denoted T (f ). (Not df .) So the source of the tangent groupoid is T (α), the object inclusion is T (1) and the multiplication Y • X of two tangent vectors with T (α)(Y ) = T (β)(X) is T (κ)(Y , X) where κ: G ×M G → G is the multiplication in G . Because the tangent functor preserves diagrams, the groupoid axioms follow immediately.

If G is a group, then TG is a group and Y • X = T (Rg )(Y ) + T (Lh)(X) for Y ∈ Th(G) and X ∈ Tg (G). However no Lie group has a symplectic groupoid structure. 13. Symplectic groupoids, p2

1 The Lagrangian condition on the graph G means that dim G = 2 dim(G × G × G) and that ω(X , Y ) = 0 for all X , Y ∈ T G . So −ω(Y1 • X1, Y2 • X2) + ω(Y1, Y2) + ω(X1, X2) = 0 (∗) for all Xi , Yi ∈ TG for which the multiplications are defined.

Set Xi = Yi = T (1)(xi ) where xi ∈ TP . Then (∗) becomes

−ω(T (1)(x1), T (1)(x2)) + ω(T (1)(x1), T (1)(x2)) + ω(T (1)(x1), T (1)(x2)) = 0 so ω(T (1)(x1), T (1)(x2)) = 0 for all x1, x2 ∈ TP . Next,

dim(G ) = dim(G ×P G) = 2 dim G − dim P

1 so the dimension condition on the graph gives dim P = 2 dim G . This proves that 1P is Lagrangian in G .

−1 Next consider any X1, X2 . Write xi = T (α)(xi ). Then with Yi = Xi we have

−1 −1 −ω(T (1)(x1), T (1)(x2)) + ω(X1 , X2 ) + ω(X1, X2) = 0

−1 −1 −1 So ω(X1 , X2 ) = −ω(X1, X2), which shows that inversion g 7→ g is antisymplectic. 14. Symplectic groupoids, p3

Now consider ω at an identity 1m of G . Every tangent vector at 1m has the form X + T (1)(x) where X ∈ AG and x ∈ TP . So

ω(X + T (1)(x), Y + T (1)(y)) = ω(X, Y ) + ω(X, T (1)(y)) − ω(Y , T (1)(x)).

Write ω : AG ×P TP → R for the restriction of ω . Note that the rank of AG and TP are equal. If ω is not non-degenerate then there is a y ∈ TP , y 6= 0, such that ω(X, T (1)(y)) = 0 for all X ∈ AG . It follows that ω(X + T (1)(x), T (1)(y)) = 0 for all X ∈ AG, x ∈ TP and so ω would be degenerate. Hence ω is a non-degenerate pairing. So we get an isomorphism of vector bundles AG → T ∗M , denote it W . Using the anchor of AG we have a ◦ W −1 : T ∗P → TP . This is a Poisson structure on P and β is a Poisson map. Further, W is an isomorphism of Lie algebroids. 15. Symplectic groupoids, Examples

There are two ‘easy’ examples of symplectic groupoids:

I the pair groupoid M × M for M a symplectic manifold

I the fundamental groupoid of a symplectic manifold

The most instructive example is the cotangent groupoid of a Lie group. For G a Lie group, T ∗G is a symplectic groupoid on g∗ . The source and target are the translations to the identity:

hβ(ϕg ), Y i = hϕ, Ygi,

hα(ϕg ), Xi = hϕ, gXi,

∗ for ϕg ∈ Tg G and X, Y ∈ g. The multiplication is ψh ϕg = ψ ◦ T (Lg−1 ) = ϕ ◦ T (Rh−1 )

∗ ∗ The identity element for θ ∈ g is θ ∈ T1 G . This extends to any Lie groupoid G ⇒ M . There is a Lie groupoid structure ∗ ∗ ∗ T G ⇒ A G and with the standard structure on T G this is a symplectic groupoid. 16. Further developments

There is a concept of Poisson Lie group; these were introduced as a semi-classicl limit of quantum groups and are relevant to work in integrable systems. The basic ideas for Poisson Lie groups resemble that of symplectic groupoids and the two concepts were unified into the notion of Poisson groupoid (Weinstein, 1988).

For a Poisson groupoid G ⇒ P it is still true that P has a unique Poisson structure making β : G → P a Poisson map, but it is not true in general that T ∗P =∼ AG . Instead there is a Lie algebroid structure on A∗G and the dual of the anchor ∗ ∗ ∗ a∗ : A G → TP gives a morphism of Lie algebroids a∗ : T P → AG . The two Lie algebroid structures, on AG and A∗G , are related in rather difficult ways and the study of the relationship has led to the concepts of

I (and Courant algebroid)

I Dirac structure

I double Lie algebroid.

These are the subject of a great deal of ongoing work.