18 2. Basic notions of representation theory This must be zero, so we have ∑r j=0 Qj(t)a(a−1) . . . (a−j+1)t−j = 0 in k[a][t, t−1]. Taking the leading term in a, we get Qr(t) = 0, a contradiction. (ii) Any word in x, y, x−1,y−1 can be ordered at the cost of mul- tiplying it by a power of q. This easily implies both the spanning property and the linear independence. Remark 2.7.2. The proof of (i) shows that the Weyl algebra A can be viewed as the algebra of polynomial differential operators in one variable t. The proof of (i) also brings up the notion of a faithful represen- tation. Definition 2.7.3. A representation ρ : A → End V of an algebra A is faithful if ρ is injective. For example, k[t] is a faithful representation of the Weyl algebra if k has characteristic zero (check it!), but not in characteristic p, where (d/dt)pQ = 0 for any polynomial Q. However, the representation E = tak[a][t, t−1], as we’ve seen, is faithful in any characteristic. Problem 2.7.4. Let A be the Weyl algebra. (a) If char k = 0, what are the finite dimensional representations of A? What are the two-sided ideals in A? Hint: For the first question, use the fact that for two square matrices B, C, Tr(BC) = Tr(CB). For the second question, show that any nonzero two-sided ideal in A contains a nonzero polynomial in x, and use this to characterize this ideal. Suppose for the rest of the problem that char k = p. (b) What is the center of A? Hint: Show that xp and yp are central elements. (c) Find all irreducible finite dimensional representations of A. Hint: Let V be an irreducible finite dimensional representation of A, and let v be an eigenvector of y in V . Show that the collection of vectors {v, xv, x2v,...,xp−1v} is a basis of V . Problem 2.7.5. Let q be a nonzero complex number, and let A be the q-Weyl algebra over C.

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