2.7. Examples of algebras 17
2.7. Examples of algebras
The following two examples are among the simplest interesting ex-
amples of noncommutative associative algebras:
(1) the Weyl algebra, k x, y / yx xy 1 ;
(2) the q-Weyl algebra, generated by x,
x−1,y,y−1
with defin-
ing relations yx = qxy and
xx−1
=
x−1x
=
yy−1
=
y−1y
=
1.
Proposition 2.7.1. (i) A basis for the Weyl algebra A is
{xiyj,i,j

0}.
(ii) A basis for the q-Weyl algebra Aq is {xiyj,i,j Z}.
Proof. (i) First let us show that the elements xiyj are a spanning set
for A. To do this, note that any word in x, y can be ordered to have
all the x’s on the left of the y’s, at the cost of interchanging some x
and y. Since yx−xy = 1, this will lead to error terms, but these terms
will be sums of monomials that have a smaller number of letters x, y
than the original word. Therefore, continuing this process, we can
order everything and represent any word as a linear combination of
xiyj
.
The proof that
xiyj
are linearly independent is based on represen-
tation theory. Namely, let a be a variable, and let E =
tak[a][t, t−1]
(here
ta
is just a formal symbol, so really E = k[a][t,
t−1]).
Then E
is a representation of A with action given by xf = tf and yf =
df
dt
(where
d(ta+n)
dt
:= (a + n)ta+n−1). Suppose now that we have a non-
trivial linear relation

cijxiyj = 0. Then the operator
L =
cijti
d
dt
j
acts by zero in E. Let us write L as
L =
r
j=0
Qj (t)
d
dt
j
,
where Qr = 0. Then we have
Lta
=
r
j=0
Qj(t)a(a 1) . . . (a j +
1)ta−j.
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