Cluster Structure on Generalized Weyl Algebras

TL;DR

This paper introduces a non-commutative cluster structure on generalized Weyl algebras, providing conditions for finiteness of cluster variables and using combinatorial data to construct irreducible representations.

Contribution

It presents a new non-commutative cluster structure on generalized Weyl algebras and links combinatorial data to representation theory.

Findings

Conditions for finiteness of cluster variables

Construction of irreducible representations from cluster strands

Introduction of a non-commutative geometric cluster structure

Abstract

We introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the cluster variables of these cluster structures are provided. Some combinatorial data, called \textit{cluster strands,} arising from the cluster structure are used to construct irreducible representations of generalized Weyl algebras.