Quiver down-up algebras of type A

TL;DR

This paper introduces quiver down-up algebras of type A, generalizing previous down-up algebras, and explores their algebraic properties including noetherianity, Calabi--Yau structure, and isomorphism classification.

Contribution

It extends down-up algebra theory to quiver-based structures, establishing their key properties and addressing the isomorphism problem for graded cases.

Findings

Quiver down-up algebras are noetherian piecewise domains.

They are twisted Calabi--Yau under certain conditions.

The paper discusses the isomorphism problem for graded quiver down-up algebras.

Abstract

We present a generalization of down-up algebras, originally defined by Benkart and Roby. These quiver down-up algebras arise as quotients of the double of the extended Dynkin quiver of type A. Under a certain non-degeneracy condition, we show that quiver down-up algebras are noetherian piecewise domains, and that they are twisted Calabi--Yau. Finally, we consider the isomorphism problem for graded quiver down-up algebras.