Skew polynomial algebras with coefficients in Koszul Artin-Schelter regular algebras

TL;DR

This paper investigates skew polynomial algebras over Koszul Artin-Schelter regular algebras, showing they are Calabi-Yau and related to Frobenius algebras, with constructions involving superpotentials and derivation quotients.

Contribution

It demonstrates that skew polynomial algebras with coefficients in Koszul Artin-Schelter regular algebras are Calabi-Yau and constructs associated superpotentials, extending the understanding of their algebraic structure.

Findings

Yoneda Ext-algebra of $A[z;\xi]$ is a trivial extension of a Frobenius algebra

$A[z;\xi]$ is Calabi-Yau

Construction of a superpotential $ ilde{w}$ for derivation quotients

Abstract

Let $A$ be a Koszul Artin-Schelter regular algebra with Nakayama automorphism $\xi$. We show that the Yoneda Ext-algebra of the skew polynomial algebra $A[z;\xi]$ is a trivial extension of a Frobenius algebra. Then we prove that $A[z;\xi]$ is Calabi-Yau; and hence each Koszul Artin Schelter regular algebra is a subalgebra of a Koszul Calabi-Yau algebra. A superpotential $\hat{w}$ is also constructed so that the Calabi-Yau algebra $A[z;\xi]$ is isomorphic to the derivation quotient of $\hat{w}$. The Calabi-Yau property of a skew polynomial algebra with coefficients in a PBW-deformation of a Koszul Artin-Schelter regular algebra is also discussed.