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LECTURE 7: LIE ACTIONS ON SYMPLECTIC

Contents 1. A crash course on Lie groups and Lie algebras 1 2. A crash course on actions 4 3. Symplectic actions 7 4. 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 ( and inverse) • A is a set with nice (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, ) 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 −1 ca : G → G, g 7→ aga . 1 2 LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS

Example. Any is a Lie group under vector . Example. The S1 ⊂ C and the punctured C∗ ⊂ C are Lie groups under complex multiplication. Example. The real Tn = S1 × · · · × S1 ⊂ Cn and the (C∗)n are Lie groups. Example. The general GL(n, R) of all nonsingular matrices is a Lie group under multiplication. Definition 1.2. A Lie group 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 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 ι is a smooth . Note that we don’t require H to be embedded 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 H of G that is also an embedded submanifold. One can show that any closed Lie subgroup must be closed 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 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. 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 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 ∈ 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

Adg = (dcg)e : TeG → TeG. In other words, we get a map (the 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 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 d left-invariant vector field on G through the vector Xe = 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

∂ ∂ 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 . 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 dϕe : g = TeG → TeH = h of any Lie 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 !] • 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 homo- morphism 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 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 ). 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 ∈ GL(n, R) and X ∈ 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 ∈ 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 G- actions, 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 multiplica- tions) 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. 6 LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS

Definition 2.5. An action τ of a Lie group G on M is proper if the action map F : G × M → M × M, (g, m) = (g · m, m) is proper, i.e. the pre- of any compact set is compact.

¶ Infinitesimal actions. Now let τ : G → Diff(M) be a Lie . For any X ∈ g, one can define a vector field XM on M via

d XM (m) = exp(tX) · m. dt t=0 This gives us a map dτ : g → Vect(M),X 7→ XM . Definition 2.6. We call dτ the infinitesimal action of g on M associated to the G-action τ, and call XM the induced vector field on M of X ∈ g. One can show that the infinitesimal action dτ is a Lie algebra anti-homomorphism, i.e. [XM ,YM ] = −[X,Y ]M . Moreover, it is natural in the sense

τ(exp(tX)) = exp(tXM ). ¶ Important facts. Again we list some important facts for Lie group actions. • G · m is an immerse submanifold. • Gm is a Lie subgroup of G with Lie algebra

gm = {X ∈ g | XM (m) = 0.} • If G is compact, then any G-action is proper. • If G acts on M properly, then Gm is compact. • If G acts on M properly, then the quotient M/G is Hausdorff. • If G acts on M properly, then each orbit G · m is an embedded submanifold of M, with tangent space

Tm(G · m) = {XM (m) | X ∈ g}. • If G acts on M properly and freely, then the orbit space is a manifold and the quotient map π : M → M/G is a submersion. • In particular, for any closed subgroup H ⊂ G, acting on G via h · g = gh−1, the quotient G/H is a smooth manifold. • If G acts on M smoothly, then the map

φ : G/Gm → G · m, Gm · m 7→ g · m

is a diffeomorphism between the quotient manifold G/Gm and the orbit G·m. LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS 7

• Suppose G acts on M properly. Then for any subgroup H ⊂ G, the H-fixed point set M H = {m ∈ M | h · m = m for all h ∈ H} is a disjoint union of closed of M.

¶ The slice theorem.

Now suppose G acts properly on M. Then for any g ∈ Gm, g · m = m. Taking derivative one gets the isotropy action of Gm on TmM via

g · v = dτg(v).

Obviously the tangent space Tm(G · m) is a subspace of TmM which is invariant under the isotropy action of Gm. By choosing a Gm-invariant inner product on TmM one can get a Gm-invariant decomposition

TmM = Tm(G · m) ⊕ W.

Choose D to be a sufficiently small disc (with respect to some Gm-invariant inner product, which always exists because Gm is a compact Lie group) in W so that Gm also acts on D. Theorem 2.7 (The slice theorem). Let G be a Lie group acts properly on a smooth manifold M. Then for each m ∈ M, there exists a G-equivariant diffeomorphism from the disc bundle G ×Gm D onto a G-invariant neighborhood of the orbit G · m in

M, whose restriction to the zero section G ×Gm {0} = G/Gm is the diffeomorphism F : G/Gm → G · m above.

Note that if the G-action at m is free, then G ×Gm D = G × D. Remark. In general if P is a principal G-bundle over M so that G acts from right on P properly and freely, and suppose G also acts linearly on a vector space W , then G acts (from left) on the product P × W by g · (p, w) = (p · g−1, g · w). This action is proper and free, and the quotient P × W/G is a vector bundle over M = P/G. As a consequence, locally above a small open set

U ⊂ G/Gm, the disc bundle G ×Gm D looks like U × D.

3. Symplectic actions ¶ Symplectic actions. Now let (M, ω) be a , G a Lie group, and τ : G → Diff(M) a smooth action of G on M. The following definition is natural: Definition 3.1. The action τ is called a symplectic action if τ(G) ⊂ Symp(M, ω). 8 LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS

In other words, for any g ∈ G, τ(g) ∈ Symp(M, ω) is a . Example. Consider the S1 action on (S2, dθ ∧ dz) by rotations around the z-axis. In coordinates (θ, z) this action is explicitly given by t · (θ, z) = (θ + t, z). This is a symplectic action since for any fixed t, ∗ τt ω = d(θ + t) ∧ dz = dθ ∧ dz. 1 2 Example. Similarly the S action on (T , dθ1 ∧ dθ2) via

t · (θ1, θ2) = (θ1 + t, θ2) is a symplectic action.

2n Example. By definition the natural action of Sp(2n) on (R , Ω0) is symplectic. Proposition 3.2. A smooth action τ : G → Symp(M, ω) is symplectic if and only if for each X ∈ g, the induced vector field XM is a symplectic vector field. Proof. If the action is symplectic, then for any X ∈ g,

d ∗ d ∗ LXM ω = exp(tXM ) ω = τ(exp(tX)) ω = 0, dt t=0 dt t=0 i.e. XM is a symplectic vector field.

Conversely, suppose XM is a symplectic vector field for any X ∈ g. Then by definition exp(tXM ) are . In other words, for any t and X, τ(exp(tX)) ∈ Symp(M, ω). Since any element in G is a product of elements in this form, the conclusion follows.  ¶ Hamiltonian actions. It is natural to define Hamiltonian actions as those symplectic actions whose generating vector fields are Hamiltonian functions. Since for each X ∈ g there is a generating vector field XM , it is natural to suitable choose Hamiltonian functions µX for these vector fields so that there depends nicely on X, e.g. are linear in X ∈ g. Gluing these together, one gets a map µ∗ : g → C∞(M),X → µX , or alternately, a map µ from M to g∗ so that µX (m) = hµ(m),Xi. Aiming at the applications in the next several lectures, we would like the maps µ∗ and µ be nice, for example the µ∗ should be a Lie algebra homomorphism, and µ should be a G-equivariant map. (And, we will prove next time that in fact these two conditions are equivalent.) LECTURE 7: LIE GROUP ACTIONS ON SYMPLECTIC MANIFOLDS 9

Definition 3.3. A symplectic action τ of G on a symplectic manifold (M, ω) is called weakly-Hamiltonian if there exists a map µ : M → g∗ so that for any X ∈ g, X dµ = ιXM ω, where µX (·) = hµ(·),Xi is the component of µ along X. A weakly Hamiltonian action is called Hamiltonian if the map µ : G → g∗ is equivariant with respect to the G-action on M and the adjoint G-action on g∗, i.e. ∗ µ ◦ τg = Adg ◦ µ. We shall call µ the , and call (M, ω, G, µ) a Hamiltonian G-space. Remark. Consider the simple case G is an . Then g = Rn and g∗ = Rn, ∗ ∗ with trivial Lie bracket. Then the coadjoint action of G on g is trivial: Adg = Id for all g. It follows that the equivariance condition on µ is reduced to the simple condition: µ is G-invariant. Example. The S1-action on S2 by rotations described above is a Hamiltonian action with moment map µ(θ, z) = z, while the S1-action on T2 by rotations in the first variable is not a Hamiltonian action.

4. Student presentation