Grothendieck rings of towers of generalized Weyl algebras in the finite orbit case

TL;DR

This paper computes tensor products of weight modules over generalized Weyl algebras supported on finite orbits, leading to a complete description of the Grothendieck ring for these categories, extending previous work on infinite orbits.

Contribution

It provides a full presentation of the Grothendieck ring for towers of GWAs with finite orbits, including generators, relations, and partial results on the split ring.

Findings

Tensor products of simple and indecomposable modules are explicitly computed.

A complete presentation of the Grothendieck ring is obtained.

Partial results are provided for the split Grothendieck ring.

Abstract

Previously we showed that the tensor product of a weight module over a generalized Weyl algebra (GWA) with a weight module over another GWA is a weight module over a third GWA. In this paper we compute tensor products of simple and indecomposable weight modules over generalized Weyl algebras supported on a finite orbit. This allows us to give a complete presentation by generators and relations of the Grothendieck ring of the categories of weight modules over a tower of generalized Weyl algebras in this setting. We also obtain partial results about the split Grothendieck ring. We described the case of infinite orbits in previous work.