38 2. Basic notions of representation theory

(f) Show that for each N 0, there exists a unique up to isomor-

phism irreducible representation of sl(2) of dimension N. Compute

the matrices E, F, H in this representation using a convenient basis.

(For V finite dimensional irreducible take λ as in (a) and v ∈ V (λ)

an eigenvector of H. Show that v, Fv, . . . , F

λv

is a basis of V , and

compute the matrices of the operators E, F, H in this basis.)

Denote the (λ+1)-dimensional irreducible representation from (f)

by Vλ. Below you will show that any finite dimensional representation

is a direct sum of Vλ.

(g) Show that the operator C = EF + FE +

H2/2

(the so-called

Casimir operator) commutes with E, F, H and equals

λ(λ+2)

2

Id on

Vλ.

Now it is easy to prove the direct sum decomposition. Namely,

assume the contrary, and let V be a reducible representation of the

smallest dimension, which is not a direct sum of smaller representa-

tions.

(h) Show that C has only one eigenvalue on V , namely

λ(λ+2)

2

for some nonnegative integer λ (use the fact that the generalized

eigenspace decomposition of C must be a decomposition of represen-

tations).

(i) Show that V has a subrepresentation W = Vλ such that

V/W = nVλ for some n (use (h) and the fact that V is the smallest

reducible representation which cannot be decomposed).

(j) Deduce from (i) that the eigenspace V (λ) of H is (n + 1)-

dimensional. If v1,...,vn+1 is its basis, show that F

jvi,

1 ≤ i ≤ n+1,

0 ≤ j ≤ λ, are linearly independent and therefore form a basis of V

(establish that if Fx = 0 and Hx = μx for x = 0, then Cx =

μ(μ−2)

2

x

and hence μ = −λ).

(k) Define Wi = span(vi,Fvi,...,F λvi). Show that Wi are sub-

representations of V and derive a contradiction to the fact that V

cannot be decomposed.

(l) (Jacobson-Morozov lemma) Let V be a finite dimensional com-

plex vector space and A : V → V a nilpotent operator. Show that

there exists a unique, up to an isomorphism, representation of sl(2)