2.16. Problems on Lie algebras 39

on V such that E = A. (Use the classification of the representations

and the Jordan normal form theorem.)

(m) (Clebsch-Gordan decomposition) Find the decomposition of

the representation Vλ ⊗ Vμ of sl(2) into irreducibles components.

Hint: For a finite dimensional representation V of sl(2) it is use-

ful to introduce the character χV (x) =

Tr(exH

), x ∈ C. Show that

χV

⊕W

(x) = χV (x) + χW (x) and χV

⊗W

(x) = χV (x)χW (x). Then

compute the character of Vλ and of Vλ ⊗ Vμ and derive the decompo-

sition. This decomposition is of fundamental importance in quantum

mechanics.

(n) Let V =

CM

⊗

CN

and A = J0,M ⊗ IdN + IdM ⊗J0,N , where

J0,n is the Jordan block of size n with eigenvalue zero (i.e., J0,nei =

ei−1, i = 2,...,n, and J0,ne1 = 0). Find the Jordan normal form of

A using (l) and (m).

2.16. Problems on Lie algebras

Problem 2.16.1 (Lie’s theorem). The commutant K(g) of a Lie

algebra g is the linear span of elements [x, y], x, y ∈ g. This is an ideal

in g (i.e., it is a subrepresentation of the adjoint representation). A

finite dimensional Lie algebra g over a field k is said to be solvable if

there exists n such that

Kn(g)

= 0. Prove the Lie theorem: if k = C

and V is a finite dimensional irreducible representation of a solvable

Lie algebra g, then V is 1-dimensional.

Hint: Prove the result by induction in dimension. By the in-

duction assumption, K(g) has a common eigenvector v in V ; that is,

there is a linear function χ : K(g) → C such that av = χ(a)v for any

a ∈ K(g). Show that g preserves common eigenspaces of K(g). (For

this you will need to show that χ([x, a]) = 0 for x ∈ g and a ∈ K(g).

To prove this, consider the smallest subspace U containing v and

invariant under x. This subspace is invariant under K(g) and any

a ∈ K(g) acts with trace dim(U)χ(a) in this subspace. In particular

0 = Tr([x, a]) = dim(U)χ([x, a]).)