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 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 and of 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]).)
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