MAT 552. HOMEWORK 3
DUE TU FEB 18
1. Prove that the dierential, assuming it exists, of the exponential map exp : Te (G) G
at 0 Te (G) is equal to identity map. Hint: use the denition of exp through the theory
of one-parametric subgroups.
MAT 552. HOMEWORK 10
DUE TU APR 29
Denote by [A, B] the commutator of two operators.
Let C (M ) be a linear space of smooth functions on a manifold M . Multiplication
on a smooth function f denes an operator f : C (M ) C (M ). A linear
MAT 552. HOMEWORK 7
DUE TH MAR 6
1. Let G be a closed subgroup of the unitary group U(n). Use Stone-Weierstrass theorem
to show that any continuous function f C(G) can be uniformly approximated by linear
combinations of matrix coecients of G-r
MAT 552. HOMEWORK 8
DUE TH APR 3
A Hausdor topological space X is a complex or holomorphic manifold if it has an
holomorphic atlas, that is X = U such that open U are such that exist local
homeomorphisms : R2n Cn V U that have holomorphic com
MAT 552. HOMEWORK 9
DUE TU APR 15
1. Dene Sp(n) as a group of H-linear automorphisms of Hn , which preserve an Hhermitian form < , > on Hn . The form is dened by the formula < x, y >= n xi yi .
(1) Use this data to show that Sp(n) Sp(2n, C
MAT 552. HOMEWORK 6
DUE TU MAR 4
We are going to use notations and denitions of HW 5.
Denition 1. Let M be a subset of a topological group G, and f (x) a real valued function
dened on . The function f (x) is called uniformly continuous if > 0
MAT 552. HOMEWORK 4
DUE TH FEB 20
1. Suppose X is a topological space, R is equivalence relation. Show that
(1) if the quotient space X/R is Hausdor, then R is closed in the product space X X.
(2) if the projection p of a space X onto the quot
MAT 552. HOMEWORK 5
DUE TH FEB 27
We say that an invariant integration is dened over a compact topological group G if
the following conditions are satised.
(1) To every real continuous function f (x) dened on G corresponds a real n
MAT 552. HOMEWORK 2
DUE TU FEB 11
1. The group Aut(Rn ) = GL(n) is an open subset in M atn and admits a single chart.
In this chart the tangent bundle T (GL(n) is a product GL(n) M atn . Write an explicit
formula for a left-invariant vector el
MAT 552. HOMEWORK 1
DUE TH FEB 6
(1) Prove that M is Hausdor space the diagonal M M is closed.
(2) Show that a topological group G is Hausdor one point set cfw_e G where e is
the unit is closed.
(3) Recall that Lie group is a topological gr
MAT 552. HOMEWORK 11
DUE TH MAY 1
Suppose that A is a symmetric nn matrix with Ai,i = 2 for all i and Ai,j cfw_1, 0
for i = j. To this matrix we can draw a non-oriented graph by connecting i and
j with an edge i Ai,j = 1. The graph completely