Grothendieck rings of towers of twisted generalized Weyl algebras

DOI10.1007/S10468-021-10070-WzbMATH Open1508.16029arXiv2003.00957MaRDI QIDQ6335972

Jonas T. Hartwig, Daniele Rosso

Publication date: 2 March 2020

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 $A(R,sigma,t{t}^{\prime })oA(R,sigma,t)otime{s}_{R}A(R,sigma,t^{\prime })$. 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 $R=mathbbC[z]$. 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.

Mathematics Subject Classification ID

Module categories in associative algebras (16D90) Twisted and skew group rings, crossed products (16S35)

This page was built for publication: Grothendieck rings of towers of twisted generalized Weyl algebras