# New Artin-Schelter regular and Calabi-Yau algebras via normal extensions

## TL;DR

This paper presents a novel method for constructing 4-dimensional Artin-Schelter regular and Calabi-Yau algebras from 3-dimensional ones using normal extensions, expanding the class of known regular algebras.

## Contribution

Introduces a new technique to generate 4D Artin-Schelter regular and Calabi-Yau algebras via normal extensions of 3D algebras, including explicit relations and automorphism analysis.

## Key findings

- Constructs large classes of new 4D Artin-Schelter regular algebras.
- Produces flat families of 4-Calabi-Yau extensions from 3-Calabi-Yau algebras.
- Shows the homological determinant of the Nakayama automorphism is 1 for these algebras.

## Abstract

We introduce a new method to construct 4-dimensional Artin-Schelter regular algebras as normal extensions of (not necessarily noetherian) 3-dimensional ones. The method produces large classes of new 4-dimensional Artin-Schelter regular algebras. When applied to a 3-Calabi-Yau algebra our method produces a flat family of central extensions of it that are 4-Calabi-Yau, and all 4-Calabi-Yau central extensions having the same generating set as the original 3-Calabi-Yau algebra arise in this way. Each normal extension has the same generators as the original 3-dimensional algebra, and its relations consist of all but one of the relations for the original algebra and an equal number of new relations determined by "the missing one" and a tuple of scalars satisfying some numerical conditions. We determine the Nakayama automorphisms of the 4-dimensional algebras obtained by our method and as a consequence show that their homological determinant is 1. This supports the conjecture by Mori-Smith that the homological determinant of the Nakayama automorphism is 1 for all Artin-Schelter regular connected graded algebras. Reyes-Rogalski-Zhang proved this is true in the noetherian case.