Exercise 5, § 3

Suppose that K is a field of characteristic 0. Let V be a finite-dimensional vector K-space, g = sl(V), U the enveloping algebra of g, U_n the set of elements of filtration \leq n in U, U^n the set of images in U of homogeneous symmetric tensors of order n over g and (\alpha_1, \ldots, \alpha_m) a basis of the vector space g. Let W_s = \bigotimes^s V. The identity representation of g defines a representation of g on W_s which extends to a homomorphism \pi_s of U into the algebra M_s = \mathscr{L}(W_s).

(a) Let M'_s be the subspace of M_s generated by the elements of the form \beta_1 \otimes \beta_2 \otimes \cdots \otimes \beta_s, where \beta_i \in \mathscr{L}(V) and \beta_i = 1 for at least one i. Let z be an element of U_s and z_0 = \sum_{1 \leq i_1, \ldots, i_s \leq m} a_{i_1 \ldots i_s} \alpha_{i_1} \otimes \cdots \otimes \alpha_{i_s} its component in U^s, where the a_{i_1 \ldots i_s} are symmetric with respect to permutation of indices. Show that

(b) Deduce from (a) that for all z \in U there exists a finite-dimensional representation \pi of U such that \pi(z) \neq 0. (For a sequel to this exercise, cf. § 7, Exercise 3.)