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Lecture 7: Lie Group Actions on Symplectic Manifolds

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Lecture 7: Lie Group Actions on Symplectic Manifolds LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS Contents 1. A crash course on Lie groups and Lie algebras 1 2. A crash course on Lie group actions 4 3. Symplectic actions 7 4. Lie algebra cohomology 9 NOTE: The proofs of the results stated in first two sections below can be found in my Lie group course notes at http://staff.ustc.edu.cn/˜ wangzuoq/Lie13/Lie.html 1. A crash course on Lie groups and Lie algebras { Lie groups. Roughly speaking, • A group is a set with simple algebraic structure (multiplication and inverse) • A manifold is a set with nice geometry (locally looks like Rn) • A smooth manifold is a manifold on which one can do analysis and a Lie group is an organic integration of all these structures: Definition 1.1. A Lie group G is a smooth manifold with a group structure, so that the group multiplication map (g1; g2) 7! g1g2 is smooth. Remark. Suppose G is a Lie group. • One can prove that the inversion map g 7! g−1 is automatically smooth. • Any element a 2 G gives rise to three natural diffeomorphisms on G, namely the left multiplication La : G ! G; g 7! ag; the right multiplication Ra : G ! G; g 7! ga; and the conjugation −1 ca : G ! G; g 7! aga : 1 2 LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS Example. Any vector space is a Lie group under vector addition. Example. The circle S1 ⊂ C and the punctured complex plane C∗ ⊂ C are Lie groups under complex multiplication. Example. The real torus Tn = S1 × · · · × S1 ⊂ Cn and the complex torus (C∗)n are Lie groups. Example. The general linear group GL(n; R) of all nonsingular matrices is a Lie group under matrix multiplication. Definition 1.2. A Lie group homomorphism ' : G ! H between two Lie groups is a smooth map that preserves the group multiplication, i.e. '(g1g2) = '(g1)'(g2) for all g1; g2 2 G. Definition 1.3. A subgroup ι : H,! G of a Lie group G is called a Lie subgroup if it is a Lie group with respect to the induced group operation, and the inclusion map ι is a smooth immersion. Note that we don't require H to be embedded submanifold of G. For example, one can easily construct a dense curve in T2 which is a Lie subgroup. Definition 1.4. A closed Lie subgroup of G is a Lie subgroups H of G that is also an embedded submanifold. One can show that any closed Lie subgroup must be closed subset of G. Con- versely, Theorem 1.5 (Cartan's closed subgroup theorem). Any closed subgroup H of a Lie group G is a closed Lie subgroup. This is a very powerful tool in determine whether a group is a Lie group. { Lie algebras associated to Lie groups. By definition a Lie algebra is a vector space g together with an anti-symmetric bilinear bracket [·; ·]: g × g ! g which satisfies the Jacobi identity [[X; Y ];Z] + [[Y; Z];X] + [[Z; X];Y ] = 0: Lie algebras arises naturally as the linearization of a Lie group. There are several equivalent ways to describe the Lie algebra g associated to a given Lie group G. • First one can regard g as a Lie subalgebra of (Vect(G); [·; ·]). Definition 1.6. A vector field X on G is called left invariant if 8a 2 G, dLa(Xg) = Xag: Fact: If X; X0 are left invariant vector fields on G, so is [X; X0]. In other words, the space of left invariant vector fields on G form a Lie subalgebra (whose dimension equals the dimension of G) of the (infinitely dimensional) Lie algebra of all vector fields on G. LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS 3 Definition 1.7. The Lie algebra g = Lie(G) of a Lie group G is g = fall left invariant vector fields on Gg; where the Lie bracket is the usual Lie bracket between vector fields. • Note that left invariant vector fields on G are in one-to-one correspondence with vectors in TeG: any vector Xe 2 TeG determines uniquely a left invari- ant vector field X on G via Xa = dLa(Xe). So as a vector space g = TeG: The Lie bracket of two vectors X; Y 2 TeG is defined to be [X; Y ] := ad(X)Y; where ad : TeG ! End(TeG) is defined as follows: For each g 2 G, the conjugation map cg : G ! G maps e to e. So its differential at e gives us a linear map Adg = (dcg)e : TeG ! TeG: In other words, we get a map (the adjoint representation of G) Ad : G ! Aut(TeG); g 7! Adg: Note that Ad(e) is the identity map in Aut(TeG). Moreover, since Aut(TeG) is an open subset in the linear space End(TeG), its tangent space at Id can be identified with End(TeG) in a natural way. Taking derivative again at e, we get (the adjoint representation of the Lie algebra g) ad : TeG ! End(TeG): • One can prove that any left invariant vector field X on G is complete. So the flow φt = exp(tX) exists for all t 2 R and is a one-parameter subgroup of G: exp(tX) exp(sX) = exp((t + s)X): Conversely, from any one-parameter subgroup φ of G one can construct a d left-invariant vector field on G through the vector Xe = dt jt=0φt. So g = fall one-parameter subgroups of Gg: The Lie bracket between exp(tX) and exp(tY ) can be defined to be the one-parameter subgroup generated by the vector @ @ exp(tX) exp(sY ) exp(−tX): @t t=0 @s s=0 So we associate to each Lie group G a Lie algebra g. Using the one-parameter subgroup exp(tX) associated to X one gets a natural map exp : g ! G from the Lie algebra g to G. This is called the exponential map. It is a very important tool in Lie theory. 4 LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS Example. The Lie algebra of GL(n; R) is gl(n; R), the space of all n × n matrices, with Lie bracket [A; B] = AB − BA. the exponential map is given explicitly by 1 1 exp : gl(n; ) ! GL(n; );A 7! eA = I + A + A2 + A3 + ··· : R R 2! 3! { Standard facts. Here we list some standard facts from Lie theory: • The differential d'e : g = TeG ! TeH = h of any Lie group homomorphism ' : G ! H at e is a Lie algebra homomorphism. • The exponential map exp : g ! G is a local diffeomorphism near 0, and is natural in the sense that for any Lie group homomorphism ' : G ! H, ' ◦ expg = exph ◦d'e. [Draw a commutative diagram!] • The Lie algebra h of a subgroup H of G is automatically a Lie subalgebra of g. Explicitly, h = fX 2 g j expg(tX) 2 H for all t 2 Rg: • This is a one-to-one correspondence between connected Lie subgroups of G and Lie subalgebras of g. • Any continuous homomorphism between Lie groups is smooth. • Suppose G is connected and simply connected. Then any Lie algebra homo- morphism ρ : g ! h lifts to a Lie group homomorphism ' : G ! H so that ρ = d'. • Any finitely dimensional Lie algebra is the Lie algebra of a unique connected and simply connected Lie algebra G. • Any connected abelian Lie group is of the form Rk × Tl. 2. A crash course on Lie group actions { Lie group actions. Definition 2.1. A smooth action of a Lie group G on a smooth manifold M is a group homeomorphism τ : G ! Diff(M) so that the evaluation map ev : G × M ! M; (g; m) = τ(g)(m) is smooth. For simplicity we will denote τ(g)(m) by g · m. Example. S1 acts on R2(= C) by rotations (scalar multiplications). Example. GL(n; R) acts on Rn by linear transformations. Example. The flow of any complete vector field on M is a smooth R-action on M. Conversely, any smooth R-action is the flow of a complete vector field. Example. G acts on G by left multiplication, right multiplication and by conjugation. LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS 5 Example. The adjoint action of G on g is Ad : G ! Aut(g); g 7! Adg; where Adg : g ! g is the differential of the conjugation c(g): G ! G; a 7! gag−1 at a = e. For the case G = GL(n; R), the adjoint action is given explicitly by −1 AdC (X) = CXC for C 2 GL(n; R) and X 2 gl(n; R). Example. The coadjoint action of G on g∗ is defined so that ∗ hAdgξ; Xi = hξ; Adg−1 Xi holds for all ξ 2 g∗ and X 2 g. [Check this is an action.] Definition 2.2. Let M and N be smooth manifolds with smooth G-action. A smooth map f : M ! N is called equivariant if it commutes with the given G- actions, i.e. f(g · m) = g · f(m) for all g 2 G and all m 2 M. { Orbits and stabilizers. Let τ : G ! Diff(M) be a smooth action. Definition 2.3. The orbit of G through m 2 M is G · m = fg · m j g 2 Gg ⊂ M; and the stabilizer (or isotropic subgroup) of m 2 M is Gm = fg 2 G j g · m = mg: Obviously if m; m0 lie in the same G-orbit, i.e.
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