LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES

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1 LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES BENJAMIN HOFFMAN 1. Outline Lie algebroids are the infinitesimal counterpart of Lie groupoids, which generalize how we can talk about symmetries of smooth manifolds. Unlike Lie algebras, not every Lie algebroid can be integrated to a Lie groupoid, though Rui Loja Fernandes and Marius Crainic have recently given the conditions under which a Lie algebroid may be integrated. Poisson geometry investigates smooth manifolds M with a certain Lie algebra structure on the ring of smooth functions C (M). In this talk, I will describe the connection between these two subjects, and how knowing when a Lie algebroid is integrable tells us something about Poisson (and symplectic) geometry. 2. Poisson Geometry 2.1. Motivating example. Consider M = R 3 with coordinates x, y, z. We want to define a new way {, } of multiplying smooth functions on M that satisfies the conditions: anti-symmetric R-linear Jacobi identity Leibniz identity (First three conditions give Lie algebra structure...recall definition of a Lie algebra). To do so it s enough to specify the bracket on the coordinate functions (Taylor). Let {x, y} = z, {y, z} = x, {z, x} = y. Can check that this satisfies Jacobi. Notice that this gives us a way of associating a vector field X f to a smooth function f: X f g = {g, f}. So we can do Hamiltonian dynamics. {f, g} = 0 means f constant on flow of X g. The Jacobi identity tells us that {, } is preserved by the flows of Hamiltonian vector fields, and that the map f X f is a Lie algebra anti-homomorphism: [X f, X g ] = X f,g. Date: November 10,

2 2 BENJAMIN HOFFMAN 2.2. Poisson manifolds. A manifolds satisfying the conditions listed above on its ring of smooth functions is a Poisson manifold. Symplectic manifolds. (1) Definition (2) Strong restrictions: Even dimensional, orientable, closed implies nontrivial second cohomology. (3) Hamiltonian mechanics: Gives a bundle isomorphism T M T M: for f C (M), X f X(M) defined to be the unique vector field such that ω(x f, ) = df( ). Closed iff Jacobi. (4) Example: Cotangent bundles. Darboux theorem. Trivial poisson structure {, } = 0 Lie-Poisson manifolds: Given a (finite dimensional) Lie algebra g, g can be identified with the linear functions on g, and induces a Poisson structure on g : Let v 1,..., v n be a basis for g, and µ 1,..., µ n be the dual basis for g. There are structure constants c k ij for g such that [v i, v j ] = c k ij v k. Use these to define on g : {f, g} = c k ijµ k f µ i g µ j Note: Independent of choice of coordinates: g is canonically a Poisson manifold. Gives embedding of Lie algebras g C (g ). Jacobi holds because it holds for g. Coordinate-free version, for µ g : {f, g}(µ) = µ, [df µ, dg µ ] g. Last example gives 1-1 correspondence between linear Poisson structures on R n and n- dimensional Lie algebras. In example: M = su(2)( ) (traceless matrices A with A +A = 0). Identified via Killing form. [ ] ix y + iz A = y + iz ix 2.3. Basic Properties. Singular foliation by symplectic leaves (Partition into manifolds of different dimensions) which determines the Poisson structure. In example: leaves are spheres centered at the origin (plus the origin). These are the (co)adjoint orbits. Holds true for general Lie-Poisson manifolds. In particular, symplectic leaves in su(n) are flag manifolds. Gives a bundle map T M T M. Compare to symplectic manifolds (invert bundle isomorphism) Conclude: Poisson manifolds are natural generalizations of symplectic manifolds. They also arise in nature (Lotka-Volterra equations for population dynamics) Theorem (Karasev, Weinstein): Every Poisson manifold M has a surjective submersive map from a symplectic manifold S that preserves Poisson brackets (ie,

3 LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES 3 a symplectic realization). Think of as resolution of dynamics to a nondegenerate structure: T M T M is generally not injective T M T S T S is injective Can use existence of (local) symplectic realizations to prove PBW (see book by Weinstein and Cannas Da Silva). Ask: when can we preserve complete vector fields? 3. Lie Theory 3.1. Lie Groups and Groupoids. Recall a Lie Group is a manifold G that is a group, such that the group operations are smooth. Examples include matrix groups. Think of SU(n), matrices over C A with det(a) = 1 and A = A 1. We can find the Lie algebra of a Lie group, which we identify with the tangent space at the identity via right translations (doing right invariant vector fields here...see why later). The Lie algebra of SU(n) is su(n), which we looked at earlier. If we have a (finite dimensional) Lie algebra, we can ask if we can integrate it. Theorem (Lie III) says we can. We can think of a group as a category with one object where every arrow has an inverse (draw picture). This motivates us to define a groupoid, which is a small category where every arrow has an inverse (draw picture). Think equivalence relations, parametrizing equivalences. Can multiply things, some times. Definition: A Lie groupoid is a groupoid G over M (manifolds) with s and t arrows smooth submersions, inversion smooth, multiplication smooth (when it makes sense!). Note: M embeds in G. (Draw picture with s and t fibers over M.) Groups Pair groupoid M M. GL(E) of a vector bundle. Guage groupoid T G = G g (via right translation) s(g, ξ) = ξ, t(g, ξ) = Ad g(ξ), (g, ξ)(h, η) = (gh, η) (when it makes sense!) 3.2. Lie algebroids. Want to construct an infinitesimal counterpart to groupoids. Doing right translation only makes sense for vectors parallel to source fibers (draw picture of velocity vectors of arrows). But once we define a s-parallel vector field over the identity bisection in G (embedding of M into G as identity arrows), we can do right translation to get a vector field on G. Right invariant vector fields are closed under Lie bracket. (draw arrows on G-with-fibers picture). Define A(G) as the subbundle of T G M with fibers the vectors parallel to s-fibers. There is then a bracket on sections of A(G) (induced by

4 4 BENJAMIN HOFFMAN bracket on vector fields on G). Note that dt gives a map from A(G) to T M which preserves brackets. Lie Algebroids of examples: Lie algebras (good!) T M T (F (E))/Gl(n) T P/G T g Definition: A Lie algebroid E over M is a vector bundle with anchor map ρ : E T M covering the identity. We also want a Lie bracket on Γ(E) so that ρ is a Lie algebra homomorphism, and the Leibniz identity holds: [v, fw] E = f[v, w] E + (ρ(v) f)w TM lie algebra bundles with 0 anchor Involutive subbundles (foliations. anchor is inclusion) Question: Does Lie III hold here? Answer (Fernandes, Crainic): Not always. Obstruction is discreteness of monodromy groups, which can be defined for any Lie algebroids. 4. Applications to Poisson geometry Last example of Lie algebroid: Recall for a Poisson manifold we had a bundle map ρ : T M T M such that ρ(df) = X f This can be upgraded to a Lie algebroid with Koszul bracket [, ] on 1-forms: [α, β] = L ρ(α) (β) L ρ(β) α d(π(α, β)) (Here π is the bivector field induced by {, }. It has π(df, dg) = {f, g}.) Note: [df, dg] = d{f, g}. In general, image of anchor gives a singular integrable distribution in T M. Here, the image of the anchor gives the symplectic foliation. Question: When can we integrate the cotangent algebroid of a Poisson manifold? What would it look like? Theorem (Mackenzie and Xu, Crainic and Fernandes): If a Poisson manifold M is integrable, there is a unique s-simply connected Lie groupoid S integrating M. It has a symplectic structure compatible with multiplication, and s is Poisson and t is anti-poisson. Also, s is a complete symplectic realization. The converse is also true: if M admits a complete symplectic realization, it is integrable. In example: T su(2) integrates to T SU(2), with standard symplectic structure of a cotangent bundle and multiplication as above.

5 LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES 5 Interesting though: can perturb the bracket on su(2) by a function a(r); foliation remains the same. Scales symplectic area of leaves; new structure is integrable iff A(r) = 4πr a(r) has no critical points. 5. Notes I gave this talk twice, once for Olivetti and once for BUGCAT The first was an hour, and the second was a half hour. In the second I changed some things for the better. Also, I never really gave the examples of Lie algebroids or groupoids I have written down. I began by giving the following diagram as a heuristic for what the talk was about. Symplectic manifolds Poisson manifolds Lie algebroids???? Lie Groupoids Lie algebras Lie groups I also started out by talking about symplectic manifolds and hamiltonian mechanics, which seemed to work well. It motivated the Poisson bracket better than just dropping it.