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.