《微分流形》课程教学资源（英文讲义）PSet5-2 LIE GROUPS ACTIONS

PROBLEMSET5,PART2:LIEGROUPSACTIONSDUE:NOV.28(1) [The Lie algebra so(3)](a) Prove: The Lie algebra so(3) has a basis7000)700A:001B 0000(0-101000Moreover, prove it is isomorphic to the Lie algebra (R3, ×), where × is the crossproduct(b) Consider the SO(3)-action on R3 by rotations.Prove: The induced vector fieldsassociated to A,B,C onR3are2123022023022.82100r2-0r3Ori0r20r3Orl(2) [Invariants of a Lie group action] [Not required]Suppose G acts on M smoothly.(a)Prove:Theformula(g .0)(r) := (g-1 . r)defines a linear action of G on Co(M).Definition 0.1. A fixed point of the above induced action of G on Coo(M) is calledan invariant of the G-action on M.The set of invariants is denoted by Coo(M)G(b) Prove:If G is connected, then f E Co(M)G if and only if Xmf =o for allXEg.(c) Suppose G acts on M properly and freely. Prove: Co(M)G ~ C(M/G).(d) What are the invariants for the standard O(n) action on Rn?(e) What are the invariants for the adjoint action of GL(n, C) on gl(n, C)?(3)[More Lie subgroups/subalgebras ]Let G is a Lie group, with Lie algebra g.(a) Define the center of G to be Z(G) = (g E G [ gg' = g'g for all g' E G). Showthat Z(G) is a Lie subgroup of G, and find its Lie algebra.(b) Let :G -→ G be a Lie group homomorphism. Let H be the set of y-fixedpoints, i.e. H = (g E G I (g) = g). Prove that H is a Lie subgroup of G, andfind its Lie algebra.1

PROBLEM SET 5, PART 2: LIE GROUPS ACTIONS DUE: NOV. 28 (1) [The Lie algebra so(3)] (a) Prove: The Lie algebra so(3) has a basis A =   0 0 0 0 0 1 0 −1 0   , B =   0 0 −1 0 0 0 1 0 0   , C =   0 1 0 −1 0 0 0 0 0   . Moreover, prove it is isomorphic to the Lie algebra (R 3 , ×), where × is the cross product. (b) Consider the SO(3)-action on R 3 by rotations. Prove: The induced vector fields associated to A, B, C on R 3 are x 3 ∂ ∂x2 − x 2 ∂ ∂x3 , x1 ∂ ∂x3 − x 3 ∂ ∂x1 , x2 ∂ ∂x1 − x 1 ∂ ∂x2 . (2) [Invariants of a Lie group action] [Not required] Suppose G acts on M smoothly. (a) Prove: The formula (g · φ)(x) := φ(g −1 · x) defines a linear action of G on C∞(M). Definition 0.1. A fixed point of the above induced action of G on C∞(M) is called an invariant of the G-action on M. The set of invariants is denoted by C∞(M) G. (b) Prove: If G is connected, then f ∈ C∞(M) G if and only if XMf = 0 for all X ∈ g. (c) Suppose G acts on M properly and freely. Prove: C∞(M) G ≃ C∞(M/G). (d) What are the invariants for the standard O(n) action on R n ? (e) What are the invariants for the adjoint action of GL(n, C) on gl(n, C)? (3) [More Lie subgroups/subalgebras ] Let G is a Lie group, with Lie algebra g. (a) Define the center of G to be Z(G) = {g ∈ G | gg′ = g ′ g for all g ′ ∈ G}. Show that Z(G) is a Lie subgroup of G, and find its Lie algebra. (b) Let φ : G → G be a Lie group homomorphism. Let H be the set of φ-fixed points, i.e. H = {g ∈ G | φ(g) = g}. Prove that H is a Lie subgroup of G, and find its Lie algebra. 1

2PROBLEMSET5.PART2:LIEGROUPSACTIONSDUE:NOV.28(4)[Equivariant maps][Notrequired]Let M, N be G-manifolds. A smooth map f : M → N is called G-equivariant ifg-f(m)=f(g-m),VgEG,mEMProve:(a) Prove: If G acts transitively on M, then f is of constant rank.(b) Suppose G is connected, then f is G-equivariant if and only if it is g-equivariantin the senseX(f(m))=dfm(Xm),VXEg,mEM

2 PROBLEM SET 5, PART 2: LIE GROUPS ACTIONS DUE: NOV. 28 (4) [Equivariant maps] [Not required] Let M, N be G-manifolds. A smooth map f : M → N is called G-equivariant if g · f(m) = f(g · m), ∀g ∈ G, m ∈ M. Prove: (a) Prove: If G acts transitively on M, then f is of constant rank. (b) Suppose G is connected, then f is G-equivariant if and only if it is g-equivariant in the sense XN (f(m)) = dfm(Xm), ∀X ∈ g, m ∈ M