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]).)

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