Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Assignment #10 Solutions
11.2 Suppose (, ) = 0. Then:
2(, )
)
(, )
= (, ) 2(, )
= (, ) = 0.
(, s ( ) = (,
Now suppose that , are related as in the proof of Lemma 11.8. It is enough to show and
s ( ) are also related. If (, ) = 0 then = s (
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Week 2 Solutions
1.9 Suppose : L1 L2 is an isomorphism. Let = cfw_v1 , v2 , . . . , vn be a basis of L1 . Then
= cfw_ (v1 ), (v2 ), . . . , (vn ) is a basis of L2 , since is an invertible map. Suppose:
n
(1)
ck vk .
i,j
[vi , vj ] =
k=1
App
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Assignment #11 Solutions
13.2 Show that the root systems of type Bn and Cn are dual to each other.
Consider the roots for type Bn on page 131. For a root the coroot is equal to
this and the fact that ( i , j ) = ij , we can compute:
i = 2 i
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Assignment #9 Solutions
10.5 Let L be a complex semisimple Lie algebra with Cartan subalgebra H and root system .
The root space decomposition on p.97 immediately implies that:
dim L = dim H + .
Suppose H is onedimensional. Then must be of
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Assignment #5 Solutions
6.3 Let L be a complex Lie algebra. Show that L is nilpotent if and only if every 2dimensional
subalgebra of L is abelian.
Proof: Let L be nilpotent and suppose L has a 2dimensional nonabelian subalgebra. By Theorem
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Assignment #8 Solutions
9.9(i) Suppose L is a semisimple Lie algebra. Since L is an ideal of L, by Lemma 9.12 we have
L = L (L ) .
But (L ) L/L , so is abelian. But L is semisimple, so has no nonzero abelian ideals. Thus L = L .
=
(ii) Let L
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Assignment #7 Solutions
8.2
(i) Note that V0 is one dimensional, spanned by the constant polynomial, and e, f, h all act as zero
since the actions all involve rst taking a partial derivative. Thus V0 is indeed the onedimensional
trivial rep
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Assignment #6 Solutions
7.3: Suppose V is irreducible and 0 = v V . Then the submodule generated by v is nonzero, and
thus is all of V , since V is irreducible.
Conversely suppose V has the described property and suppose 0 U V is a submodule
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Assignment #4 Solutions
1. Let L be a Lie algebra, and let I and J be nilpotent ideals. Prove that I + J is nilpotent. Use
this to prove that L contains a unique maximal nilpotent ideal which contains all nilpotent ideals.
Proof: We claim by
Introduction to Lie Algebras and Representation Theory
MATH 461

Fall 2010
Math 461/561 Week 3 Solutions
2.10 Let F be a eld. Show the derived algebra of gl(n, F ) is sl(n, F ).
Proof: For any A, B gl(n, F ), the bracket [A, B ] = AB BA has trace 0. Thus gl(n, F )
sl(n, F ). To show the reverse containment we will show a basis