m-Koszul Artin-Schelter regular algebras

TL;DR

This paper investigates the structure and automorphisms of m-Koszul Artin-Schelter regular algebras, revealing their automorphism groups, homological determinants, and applications to quadratic noetherian cases.

Contribution

It identifies the automorphism group of these algebras, relates homological determinants to scalar actions, and applies results to quadratic noetherian Artin-Schelter regular algebras.

Findings

Homological determinant of automorphisms is scalar hdet(σ)

Homological determinant of Nakayama automorphism is 1

Homological and usual determinants coincide in most quadratic cases

Abstract

This paper studies the homological determinants and Nakayama automorphisms of not-necessarily-noetherian $m$-Koszul twisted Calabi-Yau or, equivalently, $m$-Koszul Artin-Schelter regular, algebras. Dubois-Violette showed that such an algebra is isomorphic to a derivation quotient algebra D(w,i) for a unique-up-to-scalar-multiples twisted superpotential w in a tensor power of some vector space V. By definition, D(w,i) is the quotient of the tensor algebra TV by the ideal generated by all i-th order left partial derivatives of w. We identify the group of graded algebra automorphisms of D(w,i) with a subgroup of GL(V). We show that the homological determinant of a graded algebra automorphism $\sigma$ of an $m$-Koszul Artin-Schelter regular algebra D(w,i) is the scalar hdet($\sigma$) given by the formula hdet($\sigma$) w =$\sigma^{\otimes m+i}$(w). It follows from this that the homological determinant of the Nakayama automorphism of an $m$-Koszul Artin-Schelter regular algebra is 1. As an application, we prove that the homological determinant and the usual determinant coincide for most quadratic noetherian Artin-Schelter regular algebras of dimension 3.