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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