Winter 2014
Lie Algebras (MATH 6324)
February 28
Assignment 4
All vector spaces on this assignment are assumed to be nite-dimensional.
1. Let A and B be linear operators on a vector space V over an algebraically closed eld.
(a) Prove that if A and B are s
Winter 2014
Lie Algebras (MATH 6324)
January 24
Solutions to Assignment 1
1. (a) By denition, (x) = x + I, so in order to have f = f we must set f (x + I) =
f (x), for all x L. This denition actually makes sense because if x + I = y + I
then x y I ker f ,
Winter 2014
Lie Algebras (MATH 6324)
January 24
Assignment 2
1. For any Lie algebra L, prove that ad(L) is an ideal of Der(L).
2. Let L and M be Lie algebras and let : M gl(L), b b , be a linear map. Dene
L M to be the vector space L M with the following
Winter 2014
Lie Algebras (MATH 6324)
February 12
Solutions to Assignment 2
1. First of all, for any a L, ad(a) is a derivation: ad(a)([x, y]) = [a, x], y] + [x, [a, y]
for all x, y L (an equivalent form of Jacobi identity). If is any derivation of L
then
Winter 2014
Lie Algebras (MATH 6324)
January 15
Assignment 1
1. Let f : L M be a homomorphism of Lie algebras. Suppose that I
L and I ker f .
(a) Show that there exists a unique homomorphism f : L/I M such that f = f ,
where : L L/I is the canonical map.
Winter 2014
Lie Algebras (MATH 6324)
February 23
Solutions to Assignment 3
1. A pair (L, ) where L is a Lie algebra and : X L is a map, is called the free Lie
algebra on X if it satises the following universal property: for any Lie algebra M
and a map f :
Winter 2014
Lie Algebras (MATH 6324)
February 5
Assignment 3
1. Dene the free Lie algebra on a set X by an appropriate universal property. Prove
that if A(X) is the free unital associative algebra on X and L(X) is the subalgebra of
A(X)() generated by X t
Winter 2014
Lie Algebras (MATH 6324)
March 24
Assignment 6 (bonus)
All vector spaces on this assignment (including algebras and modules) are assumed to be
nite-dimensional unless stated otherwise. The ground eld F is assumed to have characteristic 0.
1. L
Winter 2014
Lie Algebras (MATH 6324)
March 24
Solutions to Assignment 5
1. Let R = Rad L. We already know that R0 := I R is the radical of I (problem 6(d)
on Assignment 4). We have to prove that S0 := I S satises the denition of a Levi
subalgebra of I, i.
Winter 2014
Lie Algebras (MATH 6324)
March 12
Assignment 5
All vector spaces on this assignment (including algebras and modules) are assumed to be
nite-dimensional unless stated otherwise. The ground eld F is assumed to have characteristic 0.
1. Prove tha
Winter 2014
Lie Algebras (MATH 6324)
March 12
Solutions to Assignment 4
1. (a) Since A is semisimple, we have V = V1 (A) Vk (A) where 1 , . . . , k are
the distinct eigenvalues of A and Vi (A) are the corresponding eigenspaces. Since
AB = BA, each of the