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.