Grothendieck rings of towers of twisted generalized Weyl algebras

Abstract

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 $A(R,\sigma ,t{t}^{\prime })\to A(R,\sigma ,t){\otimes }_{R}A(R,\sigma ,t^{\prime })$. 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 $R=\mathbb{C}[z]$. 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 ${\mathfrak{sl}}_2$ is a tensor product of two Weyl algebra modules.

31 pages, 3 figures