2.15. Representations of sl(2) 37 Definition 2.14.2. The dual representation V to a representa- tion V of a Lie algebra g is the dual space V to V with ρV (x) = −ρV (x)∗. It is easy to check that these are indeed representations. Problem 2.14.3. Let V, W, U be finite dimensional representations of a Lie algebra g. Show that the space Homg(V ⊗W, U) is isomorphic to Homg(V, U W ). (Here Homg := HomU(g).) 2.15. Representations of sl(2) This subsection is devoted to the representation theory of sl(2), which is of central importance in many areas of mathematics. It is useful to study this topic by solving the following sequence of exercises, which every mathematician should do, in one form or another. Problem 2.15.1. According to the above, a representation of sl(2) is just a vector space V with a triple of operators E, F, H such that HE −EH = 2E, HF −FH = −2F, EF −FE = H (the correspond- ing map ρ is given by ρ(e) = E, ρ(f) = F , ρ(h) = H). Let V be a finite dimensional representation of sl(2) (the ground field in this problem is C). (a) Take eigenvalues of H and pick one with the biggest real part. Call it λ. Let ¯ (λ) be the generalized eigenspace corresponding to λ. Show that E| ¯ (λ) = 0. (b) Let W be any representation of sl(2) and let w W be a nonzero vector such that Ew = 0. For any k 0 find a polynomial Pk(x) of degree k such that EkF k w = Pk(H)w. (First compute EF k w then use induction in k.) (c) Let v ¯ (λ) be a generalized eigenvector of H with eigenvalue λ. Show that there exists N 0 such that F N v = 0. (d) Show that H is diagonalizable on ¯ (λ). (Take N to be such that F N = 0 on ¯ (λ), and compute ENF N v, v ¯ (λ), by (b). Use the fact that Pk(x) does not have multiple roots.) (e) Let Nv be the smallest N satisfying (c). Show that λ = Nv−1.
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