Math 8700. Lie Groups.
Problem Set 1. Due on Friday, September 13.
1. Prove that if G and H are Lie groups, then their direct product
G H is also a Lie group.
2. Prove Proposition 2.1 (basic properties of the exponential map).
3. Prove that SLn (R) is a L
Math 8700. Lie Groups.
Problem Set 3. Due on Thursday, September 26.
1. Let F be a eld and A a (not necessarily associative) F -algebra.
Prove that Der(A), the set of F -derivations of A, is a Lie algebra over
F with Lie bracket given by [D1 , D2 ] = D1 D
Math 8700. Lie Groups.
Problem Set 5. Due on Thursday, October 17.
1. Recall that Theorem 9.3 (= Theorem 8.2 from the book) asserts that
for any Lie group G and any X, Y T1 G we have (adX )(Y ) = [X, Y ]
where by denition ad = Ad . The proof of that theor
Math 8700. Lie Groups.
Problem Set 4. Due on Thursday, October 3.
1. Let g : M N and f : N P be smooth maps between smooth
manifolds. Fix m M and let n = g (m).
(a) Prove that (f g ),m = f,n g,m
(b) Prove that for any smooth curve : (a, b) M and any
(a,
Math 8700. Lie Groups.
Problem Set 2. Due on Thursday, September 19.
1. The goal of this problem is to ll in the details of the proof of
Corollary 4.1 that were skipped in class. Let G be a topological group.
(a) Suppose that G is generated (as an abstrac
Math 8700. Lie Groups.
Problem Set 6. Due on Thursday, October 24.
1. Let pn denote the space of n n Hermitian matrices in M atn (C)
+
and Pn the set of positive-denite Hermitian matrices in GLn (C). In
+
class we veried that exp : pn Pn is a continuous b
Math 8700. Lie Groups.
Problem Set 7. Due on Friday, November 1.
1. In class we proved that every torus is topologically generated by one
element. Use this result to nd the (minimal) number of topological
generators of Rn .
2. Let G be a compact Lie group
Math 8700. Lie Groups.
Problem Set 8. Due on Thursday, November 21.
1. Show by direct computation that
(a) the root system of SO2n+1 R is isomorphic to Bn
(b) the root system of Sp2n := Sp2n (C) Un (C) is isomorphic to
Cn (see page 30 of the book for the