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 λ v 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 j vi, 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 λ vi). 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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