PROBLEM SET 5,PART 1:LIEGROUPS AND LIE ALGEBRASDUE:NOV.28(1)[Left invariant vector fields]Let X be a left invariant vector field on a Lie group G.Prove:(a) X is smooth.(b) X is complete.(2) [The exponential map for matrix Lie groups]Define the norm of a matrix A = (aii) to be llAll = (Zi, lai;l2)/2. It is easy to checkA+ BL≤ IA + BI and IAB|≤AIL|BI(a) Show that the exponential map exp : M(n, R) → GL(n, R) is given byA2A3exp(A) = I + A +2!+3!(youneedtoprovethattheseriesisabsolutelyconvergent)(b) [Not required] Let V = [A E M(n, R) I IIn - All<1]. Define81Xklog : V→ M(n, R), log A= -(Ink=1Prove: log is a smooth map form V to GL(n, R) and is the local inverse of exp.(c) [Not required] Prove: For any A, B E M(n, R),eA+B = lim (eA/keB/k)ke[A,B) = lim (eA/keB/ke-A/ke-B/k)k.(3)[Liegroup structure on s3]Identify s3 with the set of quaternions of norm 1.(a) With the help of the multiplication of quaternions, show that 3 is a Lie group,and determine its Lie algebra.(b) (Not required) Show that as Lie groups, s3 is isomorphic tozSU(2) = ^[z,C,+[=1]-元(4)[Abelian Lie groups](a) Let G be a Lie group. Prove: G is abelian (i.e. g192 = 9291 for all g1,92 E G) ifand only if g is abelian (i.e. [Xi, X2] = 0 for all X1, X2 E g).(b) (Not required) Prove: Any connected abelian Lie group of dimension n is iso-morphic to Rr × Tn-r for some r.1
PROBLEM SET 5, PART 1: LIE GROUPS AND LIE ALGEBRAS DUE: NOV. 28 (1) [Left invariant vector fields] Let X be a left invariant vector field on a Lie group G. Prove: (a) X is smooth. (b) X is complete. (2) [The exponential map for matrix Lie groups] Define the norm of a matrix A = (aij ) to be kAk = (P i,j |aij | 2 ) 1/2 . It is easy to check kA + Bk ≤ kAk + kBk and kABk ≤ kAkkBk. (a) Show that the exponential map exp : M(n, R) → GL(n, R) is given by exp(A) = I + A + A2 2! + A3 3! + · · · . (you need to prove that the series is absolutely convergent) (b) [Not required] Let V = {A ∈ M(n, R) | kIn − Ak < 1}. Define log : V → M(n, R), log A = − X∞ k=1 1 k (In − X) k . Prove: log is a smooth map form V to GL(n, R) and is the local inverse of exp. (c) [Not required] Prove: For any A, B ∈ M(n, R), e A+B = lim k→∞ (e A/ke B/k) k e [A,B] = lim k→∞ (e A/ke B/ke −A/ke −B/k) k . (3) [Lie group structure on S 3 ] Identify S 3 with the set of quaternions of norm 1. (a) With the help of the multiplication of quaternions, show that S 3 is a Lie group, and determine its Lie algebra. (b) (Not required) Show that as Lie groups, S 3 is isomorphic to SU(2) = z w −w¯ z¯ | z, w ∈ C, |z| 2 + |w| 2 = 1 . (4) [Abelian Lie groups] (a) Let G be a Lie group. Prove: G is abelian (i.e. g1g2 = g2g1 for all g1, g2 ∈ G) if and only if g is abelian (i.e. [X1, X2] = 0 for all X1, X2 ∈ g). (b) (Not required) Prove: Any connected abelian Lie group of dimension n is isomorphic to R r × T n−r for some r. 1
2PROBLEMSET5,PART1:LIEGROUPSANDLIEALGEBRASDUE:NOV.28(5)[Simply connected Lie groups] [Not required]Show that the universal covering space of a connected Lie group admits a Lie groupstructuresuchthatthecoveringmapisaLiegrouphomomorphism(6)[[Surjectivity of the exponential map](a) Show that exp : so(2) → SO(2) is surjective, and thus SO(2) is connected.(b) Show that exp : s(2, R) → SL(2, R) is not surjective, but SL(2, R) is connected.[Hint: Show that tr(A2)≥-2for AE SL(2,R)](7)[The affine group]Consider the smooth manifold G = GL(n, R) × Rn. Define a multiplication operationonGvia(X,a) - (Y,y) := (XY, Xy +a).(a) Prove: G is a Lie group with respect to this multiplication.(the afine group of R")(b) Consider the vector space g = g(n, R) × Rn. Define a bracket on g via[(A,a),(B,b)] := (AB - BA, Ab - Ba).Prove: (g, [, J) is a Lie algebra.(c) Definea mape:g→ G bye(A,a) :=For any (A, a) E g, let h(t) := e(tA,ta). Prove: [h(t) [ t e R) is a one-parametersubgroup of G.(d) [Not required] Find a subgroup of GL(n + 1, R) that is isomorphic to G, andexplain how do we get g and e above.(8)[Lie algebras of more linear Lie groups]For any n × n matrix B, we defineGLB(n,R) = (X GL(n,R) XTBX = B),(a) Prove: GLB(n, R) is a Lie subgroup of GL(n, R),(b) Find the Lie algebra gl(n, R) of GLB(n, R).(9) [No small subgroup] [Not required]Show that any Lie group G does not have small subgroup, i.e. prove that there existsan open neighborhood U of e e G such that [e] is the only subgroup of G that isentirely contained in U
2 PROBLEM SET 5, PART 1: LIE GROUPS AND LIE ALGEBRAS DUE: NOV. 28 (5) [Simply connected Lie groups] [Not required] Show that the universal covering space of a connected Lie group admits a Lie group structure such that the covering map is a Lie group homomorphism. (6) [Surjectivity of the exponential map] (a) Show that exp : so(2) → SO(2) is surjective, and thus SO(2) is connected. (b) Show that exp : sl(2, R) → SL(2, R) is not surjective, but SL(2, R) is connected. [Hint: Show that tr(A2 ) ≥ −2 for A ∈ SL(2, R).] (7) [The affine group] Consider the smooth manifold G = GL(n, R) × R n . Define a multiplication operation on G via (X, x) · (Y, y) := (XY, Xy + x). (a) Prove: G is a Lie group with respect to this multiplication.(the affine group of Rn) (b) Consider the vector space g = gl(n, R) × R n . Define a bracket on g via [(A, a),(B, b)] := (AB − BA, Ab − Ba). Prove: (g, [·, ·]) is a Lie algebra. (c) Define a map e : g → G by e(A, a) := e A, X∞ n=1 1 n! A n−1 a ! . For any (A, a) ∈ g, let h(t) := e(tA, ta). Prove: {h(t) | t ∈ R} is a one-parameter subgroup of G. (d) [Not required] Find a subgroup of GL(n + 1, R) that is isomorphic to G, and explain how do we get g and e above. (8) [Lie algebras of more linear Lie groups] For any n × n matrix B, we define GLB(n, R) = {X ∈ GL(n, R) | XT BX = B}. (a) Prove: GLB(n, R) is a Lie subgroup of GL(n, R). (b) Find the Lie algebra glB(n, R) of GLB(n, R). (9) [No small subgroup] [Not required] Show that any Lie group G does not have small subgroup, i.e. prove that there exists an open neighborhood U of e ∈ G such that {e} is the only subgroup of G that is entirely contained in U