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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