MATH 531
Written Homework 1 Solution
Radford
10/17/07
1. (20 points)
(a) (3) First note that a2 = 0 for all a A implies ab = ba for all a, b A as
0 = (a + b)2 = a2 + ab + ba + b2 = ab + ba.
Thus
J (a, a, b) = a(ab) + a(ba) + b(aa) = a(ab) + a(ab) + a(0) =
MATH 531
Written Homework 2 Solution
Radford
10/18/07
1. (20 points) A is a Lie algebra over F , S is a subspace of L, S (0) , S (1) , S (2) , . . . dened
inductively by S (0) = S and S (i+1) = [S (i) S (i) ] for i 0.
(a) (4) (S (i) )(j ) = S (i+j ) for a
MATH 531
Written Homework 4 Solution
Radford
11/21/07
In this exercise set we begin a rather detailed study of sl (n, F ).
1. (20 points) For all 1 i, j, k, n note ei j ek = j,k ei ; thus [ei j ek ] = j,k ei ,i ek j .
(a) (5) This is boring but necessary.
MATH 531
Written Homework 5 Solution
Radford
11/20/07
We follow the notation of the text and that used in class. You may use results from
the course materials on the class homepage and the text. This version replaces the
previous one.
1. (35 points) Lk, =
MATH 531
Radford 01/14/03, revised 02/08/03, 09/06/07
Decomposition of Operators
Throughout V is a nite-dimensional vector space over a eld F and T is a linear endomorphism of V . Recall that A = End(V ) is an associative algebra with unity IV over F unde
MATH 531
Written Homework 3 Solution
Radford
11/05/07
In the following exercises F is a eld, algebraically closed and of characteristic zero. We
follow the notation of the text and that used in class.
1. (25 points) You may assume that L = L1 Lr is an alg
MATH 531
Notes on Simple sl (2, F )-modules
Radford 11/09/07
Throughout F is an algebraically closed eld of characteristic zero. Let F and Z () be
the vector space over F with basis cfw_v Z . We dene endomorphisms x, y, h of Z () by
x(v ) = ( + 1)v
for a