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 = 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)
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