MATH 6333, HOMEWORK 1, DUE FRIDAY AUGUST 30, SOLUTIONS
T. PRZEBINDA
1. Let i be the denote the block diagonal matrix with the diagonal blocks equal to the
quaternion i. In other words
1
0
0 0 .
0
0
0
0
0
1 0 0 .
.
.
.
. . . .
.
i=
0
0
0 0 .
1
0
0
0
0 0
MATH 6333, HOMEWORK 9, DUE FRIDAY NOVEMBER 1,
SOLUTIONS
T. PRZEBINDA
1. Construct a Lie algebra isomorphism
sl2 (C) so3 (C).
ab
c a
0
b c i(b + c)
.
0
2ia
cb
i(b + c) 2ia 0
2. Construct a Lie algebra isomorphism
su2 (C) so3 (R).
0
b c i(b + c)
.
0
2ia
c
MATH 6333, HOMEWORK 8, DUE FRIDAY OCTOBER 25,
SOLUTIONS
T. PRZEBINDA
1. Find the reference to the statement and the proof of Jacobson Morozov theorem mentioned in the previous homework.
[Jac62, Theorem 17, page 100].
2. Find an automorphism of the group G
MATH 6333, HOMEWORK 7, DUE FRIDAY OCTOBER 18,
SOLUTIONS
T. PRZEBINDA
1. Find the reference to the statement and the proof of Jordans theorem regarding the
decomposition of a complex square matrix into the sum of a semisimple part and a nilpotent part.
[La
MATH 6333, HOMEWORK 6, DUE FRIDAY OCTOBER 4,
SOLUTIONS
T. PRZEBINDA
Problems 1 - 6 clarify Problem 7 of homework 1 in terms of the representation theory of
the circle group U1 , also known as the theory of Fourier series.
1. Prove that for z C \ cfw_1,
1
MATH 6333, HOMEWORK 5, DUE FRIDAY SEPTEMBER 27,
SOLUTIONS
T. PRZEBINDA
1. For two matrices X, Y Mn (C) and k = 0, 1, . . . , n let sn,k (X, Y ) denote the sum
of all the possible products of n k X and k Y . Thus sn,0 (X, Y ) = X n , sn,1 (X, Y ) =
X n1 Y
MATH 6333, HOMEWORK 4, DUE FRIDAY SEPTEMBER 20,
SOLUTIONS
T. PRZEBINDA
1. Let V be a nite dimensional vector space over D = R, C or H and let ( , ) be
a positive denite hermitian form on V. Show that the isometry group of this form,
H = cfw_g End(V); g g
MATH 6333, HOMEWORK 3, DUE FRIDAY SEPTEMBER 13,
SOLUTIONS
T. PRZEBINDA
1. For a symmetric matrix A of size n with real entries let Ak
submatrix of size k :
a1,1 a1,2
a1,1 a1,2
A1 = a1,1 , A2 =
, A2 = a2,1 a2,2
a2,1 a2,2
a3,1 a3,2
denote the upper left
a1,
MATH 6333, HOMEWORK 1, DUE FRIDAY SEPTEMBER 6,
SOLUTIONS
T. PRZEBINDA
1. Let F be a eld and let V be a nite dimensional vector space over F. An alternate
form on V is a function
V V u, v (u, v ) F
such that for all u, u , v V and a F
(u, u) = 0
(u + u , v
MATH 6333, HOMEWORK 10, DUE FRIDAY NOVEMBER 8
T. PRZEBINDA
Let S (R) denote the Schwartz space of R. This is the complex vector space of all the
smooth functions f : R C such that for any two non-negative integers n, k ,
sup |xn k f (x)| < .
(1)
xR
(Here