Returning Arrows for Self-injective Algebras and Artin-Schelter Regular Algebras

TL;DR

This paper explores the role of returning arrows in the quivers of self-injective and Artin-Schelter regular algebras, showing how extensions affect their structure and properties, including complexity and Calabi-Yau conditions.

Contribution

It demonstrates the appearance of returning arrows in quivers during extensions and constructs new Koszul Artin-Schelter regular algebras with increased dimensions using Koszul duality.

Findings

Returning arrows appear in quivers during trivial extensions.

Complexity increases by 1 in Koszul cases after extension.

Constructed algebras include central extensions and Calabi-Yau algebras.

Abstract

In this paper, we discuss returning arrows with respect to the Nakayama translation appearing in the quivers of some important algebras when we construct extensions. When constructing twisted trivial extensions for a graded self-injective algebra, we show that the returning arrows appear in the quiver, that the complexity increases by 1 in Koszul cases, and the representation dimension also increases by 1 under certain additional conditions. By applying Koszul duality, for each Koszul Artin-Schelter regular algebra of global dimension l and Gelfand-Kirilov dimension $c$, we construct a family of Koszul Artin-Schelter regular algebras of global dimension $l+1$ and Gelfand-Kirilov dimension $c+1$, among them one is central extension and one is $l+1$-Calabi-Yau.