LECTURE7:LIEGROUPACTIONSONSYMPLECTICMANIFOLDSCONTENTS11.A crash course on Lie groups and Lie algebras42.AcrashcourseonLiegroupactions73.Symplecticactions94.Lie algebra cohomologyNOTE:Theproofs of theresults stated in first two sections below can be found inmyLiegroupcoursenotesathttp://staff.ustc.edu.cn/~wangzuoq/Liel3/Lie.html1.ACRASHCOURSEONLIEGROUPSANDLIEALGEBRAS 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 analysisand 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, sothat the group multiplication map (gi,g2) -→ gig2 is smooth.Remark.SupposeGisaLiegroup.. One can prove that the inversion map g g-1 is automatically smooth.. Any element a E G gives rise to three natural diffeomorphisms on G, namelythe left multiplicationLa: G →G,g ag,the right multiplicationRa: G→G,g -→ ga,and the conjugationCa:G→G,g → aga-11
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 R n ) • 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 ∈ 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 ca : G → G, g 7→ aga−1 . 1
2LECTURE7:LIEGROUPACTIONSONSYMPLECTICMANIFOLDSErample. Any vector space is a Lie group under vector addition.Erample. The circle Si C C and the punctured complex plane C* C C are Liegroupsundercomplexmultiplication.Erample. The real torus Tn = Si × ..- × si C Cn and the complex torus (C*)n areLie groups.Erample. The general linear group GL(n, R) of all nonsingular matrices is a Liegroup under matrixmultiplication.Definition 1.2. A Lie group homomorphism p : G -→ H between two Lie groupsis a smooth map that preserves the groupmultiplication,i.e. (g1g2)=(gi)(g2)for all gi, 92 E G.Definition 1.3.A subgroup t:H G of a Lie group G is called a Lie subgroupif it is a Lie group with respect to the induced group operation, and the inclusionmap t 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 T? which is a Lie subgroup.Definition 1.4. A closed Lie subgroup of G is a Lie subgroups H of G that is alsoan 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 Liegroup G is a closed Lie subgroup.This is a very powerful tool in determine whether a group is a Lie group.T Lie algebras associated to Lie groups.By definition a Lie algebra is a vector space g together with an anti-symmetricbilinear bracket [, ] : g x 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 severalequivalent 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), [, J).Definition 1.6. A vector field X on G is called left invariant if Va E G,dLa(Xg) = Xag:Fact:If X,X'are left invariant vector fields on G, so is [X,X'.In other words, the spaceof left invariantvector fields on G form a Liesubalgebra (whose dimension equals the dimension of G) of the (infinitelydimensional)Liealgebra ofallvectorfields onG
2 LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS Example. Any vector space is a Lie group under vector addition. Example. The circle S 1 ⊂ C and the punctured complex plane C ∗ ⊂ C are Lie groups under complex multiplication. Example. The real torus T n = S 1 × · · · × S 1 ⊂ C n 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 ϕ : G → H between two Lie groups is a smooth map that preserves the group multiplication, i.e. ϕ(g1g2) = ϕ(g1)ϕ(g2) for all g1, g2 ∈ 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 T 2 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. Conversely, 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 ∀a ∈ 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
3LECTURE7:LIEGROUPACTIONSONSYMPLECTICMANIFOLDSDefinition 1.7. The Lie algebra g = Lie(G) of a Lie group G isg = {all left invariant vector fields on G],where the Lie bracket is the usual Lie bracket between vector fields..Notethatleft invariantvectorfields on G arein one-to-onecorrespondencewith vectorsinT,G:anyvectorXeET.Gdeterminesuniquelyaleft invari-ant vector field X on G via Xa = dLa(Xe). So as a vector spaceg= T.G.The Lie bracket of two vectors X,Y eT.G is defined to be[X, Y] := ad(X)Y,where ad : T.G → End(T.G) is defined as follows: For each g E G, theconjugation map Cg : G -→ G maps e to e. So its differential at e gives us alinearmapAdg = (dcg)e : T,G -→ T,G.In other words, we get a map (the adjoint representation of G)Ad: G→Aut(T.G), g →AdgNote that Ad(e) is the identity map in Aut(T.G). Moreover, since Aut(T,G)is an open subset in the linear space End(TG), its tangent space at Id canbe identified with End(T.G) in a natural way. Taking derivative again at e,weget (the adjoint representation of the Liealgebrag)ad : T.G → End(TG).One can prove that any left invariant vector field X on G is complete.Sothe flow pt =exp(tx) exists for all t e R and is a one-parameter subgroupof G:exp(tX) exp(sX) = exp(t + s)X).Conversely, from any one-parameter subgroup of G one can construct aleft-invariant vector field onG through thevector Xe=lt=opt.Sog = [all one-parameter subgroups of G].The Lie bracket between exp(tX) and exp(tY) can be defined to be theone-parameter subgroup generated by the vectoro18exp(tX)exp(sY)exp(-tX).OtdsSo we associate to each Lie group G a Lie algebra g.Using the one-parametersubgroup exp(tX) associated to X one gets a natural map exp : g → G from theLie algebra g to G. This is called the erponential map. It is a very important toolin Lie theory
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 = {all left invariant vector fields on G}, 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 ∈ TeG determines uniquely a left invariant vector field X on G via Xa = dLa(Xe). So as a vector space g = TeG. The Lie bracket of two vectors X, Y ∈ TeG is defined to be [X, Y ] := ad(X)Y, where ad : TeG → End(TeG) is defined as follows: For each g ∈ 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 ∈ 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 left-invariant vector field on G through the vector Xe = d dt|t=0φt . So g = {all one-parameter subgroups of G}. The Lie bracket between exp(tX) and exp(tY ) can be defined to be the one-parameter subgroup generated by the vector ∂ ∂t t=0 ∂ ∂s s=0 exp(tX) exp(sY ) exp(−tX). 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
4LECTURE7:LIEGROUPACTIONSONSYMPLECTICMANIFOLDSErample. 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 by1exp: gl(n,IR) →GL(n,IR), A-→e4 = I +A+A2A32!3! Standard facts.Here we list some standard facts from Lie theory:.The differential dpe:g =TG→ TH =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 O, andis natural in the sense thatfor any Lie group homomorphism :GH o exPg= exP, odpe. [Draw a commutative diagram!]: The Lie algebra h of a subgroup H of G is automatically a Lie subalgebra ofg.Explicitly,h=(XEgI expe(tX)EHforall teR). This is a one-to-one correspondence between connected Lie subgroups of Gand 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 p : g → h lifts to a Lie group homomorphism p : G -→ H so thatp=dp.Any finitely dimensional Lie algebra is the Lie algebra of a unique connectedand simply connected Lie algebra G.. Any connected abelian Lie group is of the form Rk × T'.2.A CRASHCOURSEONLIE GROUPACTIONS Lie group actions.Definition 2.1.A smooth action of a Lie group G on a smooth manifold M is agroup homeomorphism T : G→Diff(M) so that the evaluation mapev:G× M →M, (g,m)= T(g)(m)is smooth.Forsimplicitywewill denote(g)(m)byg·m.Erample. S1 acts on R2(= C) by rotations (scalar multiplications).Ecample. GL(n,R) acts on Rn by linear transformations.Erample.Theflow of any completevector field on M isa smoothR-action on M.Conversely, any smooth R-action is the flow of a complete vector field.Erample.Gacts on G by left multiplication,rightmultiplication and by conjugation
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 exp : gl(n, R) → GL(n, R), A 7→ e A = I + A + 1 2!A 2 + 1 3!A 3 + · · · . ¶ Standard facts. Here we list some standard facts from Lie theory: • The differential dϕe : g = TeG → TeH = h of any Lie group homomorphism ϕ : 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 ϕ : G → H, ϕ ◦ expg = exph ◦dϕe. [Draw a commutative diagram!] • The Lie algebra h of a subgroup H of G is automatically a Lie subalgebra of g. Explicitly, h = {X ∈ g | expg (tX) ∈ H for all t ∈ R}. • 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 homomorphism ρ : g → h lifts to a Lie group homomorphism ϕ : G → H so that ρ = dϕ. • 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 R k × T l . 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. S 1 acts on R 2 (= C) by rotations (scalar multiplications). Example. GL(n, R) acts on R n 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
LECTURE7:LIEGROUPACTIONSONSYMPLECTICMANIFOLDS5Erample. The adjoint action of G on g isAd :G→ Aut(g), g- Adg)where Adg : g -→ g is the differential of the conjugationc(g):G→G, agag-1at a = e. For the case G = GL(n, R), the adjoint action is given explicitly byAdc(X) = CXC-1for C e GL(n, R) and X e gl(n, R)Erample.The coadjoint action of Gong*is defined sothat(Ad's,X) = (s,Adg-1X)holds for all e g* and X e g. [Check this is an action.Definition 2.2.Let M and N be smooth manifolds with smooth G-action.Asmooth 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 e G and all m e M.TOrbits and stabilizers.Let T : G → Diff(M) be a smooth actionDefinition 2.3. The orbit of G through m E M isG.m= (g·m I gEG)c M,and the stabilizer (or isotropic subgroup) of m E M isGm=(gEGI g·m=m),Obviously ifm,m' lie in thesame G-orbit, i.e.m'=g·mfor someg EG, thenG·m =G·m'.So M can be decomposed into disjoint union of G-orbits.We defineM/G=the space of G-orbits,equipped with the quotient topology. In general, this quotient topology might bevery bad. For example, the quotient space of the R>o action on R (by multiplica-tions)has non-Hausdorff topology.However,if one put suitable assumptions on theaction, this quotient can have a nice geometric structure.Definition 2.4.A smooth G-action on M isfree if Gm=fe] forall mEM..locally free if Gm is discrete forall m EM.. transitively if Gm = M.The following definition is useful
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 AdC(X) = CXC−1 for C ∈ GL(n, R) and X ∈ gl(n, R). Example. The coadjoint action of G on g ∗ is defined so that hAd∗ g ξ, Xi = hξ, Adg−1Xi holds for all ξ ∈ g ∗ and X ∈ 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 Gactions, i.e. f(g · m) = g · f(m) for all g ∈ G and all m ∈ M. ¶ Orbits and stabilizers. Let τ : G → Diff(M) be a smooth action. Definition 2.3. The orbit of G through m ∈ M is G · m = {g · m | g ∈ G} ⊂ M, and the stabilizer (or isotropic subgroup) of m ∈ M is Gm = {g ∈ G | g · m = m}. Obviously if m, m0 lie in the same G-orbit, i.e. m0 = g · m for some g ∈ G, then G · m = G · m0 . So M can be decomposed into disjoint union of G-orbits. We define M/G = the space of G-orbits, equipped with the quotient topology. In general, this quotient topology might be very bad. For example, the quotient space of the R>0 action on R (by multiplications) has non-Hausdorff topology. However, if one put suitable assumptions on the action, this quotient can have a nice geometric structure. Definition 2.4. A smooth G-action on M is • free if Gm = {e} for all m ∈ M. • locally free if Gm is discrete for all m ∈ M. • transitively if G · m = M. The following definition is useful