Grothendieck Rings of Towers of Twisted Generalized Weyl Algebras

Twisted generalized Weyl algebras (TGWAs) A(R, sigma, t) are defined over a base ring R by parameters sigma and t, where sigma is an n-tuple of automorphisms, and t is an n-tuple of elements in the center of R. We show that, for fixed R and sigma, there is a natural algebra map A(R, sigma, tt ') -> A(R, sigma, t) circle times(R) A(R, sigma, t '). This gives a tensor product operation on modules, inducing a ring structure on the direct sum (over all t) of the Grothendieck groups of the categories of weight modules for A(R,sigma,t). We give presentations of these Grothendieck rings for n = 1,2, when R = C[z]. As a consequence, for n = 1, any indecomposable module for a TGWA can be written as a tensor product of indecomposable modules over the usual Weyl algebra. In particular, any finite-dimensional simple module over sl(2) is a tensor product of two Weyl algebra modules.