Written Homework 1 Solution
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.
J (a, a, b) = a(ab) + a(ba) + b(aa) = a(ab) + a(ab) + a(0) =
Written Homework 2 Solution
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
Written Homework 4 Solution
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.
Written Homework 5 Solution
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
1. (35 points) Lk, =
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
Written Homework 3 Solution
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
Notes on Simple sl (2, F )-modules
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