Skew Calabi-Yau Algebras and Homological Identities

TL;DR

This paper investigates the properties of skew Calabi-Yau algebras, focusing on homological identities related to the Nakayama automorphism, and explores their implications in algebraic structures and automorphisms.

Contribution

It establishes new homological identities involving Nakayama automorphisms in skew Calabi-Yau algebras and their relations under various algebraic operations.

Findings

Nakayama automorphism of smash product relates to those of component algebras

Nakayama automorphism of graded twist relates to original algebra

Homological determinant of Nakayama automorphism is trivial under certain conditions

Abstract

A skew Calabi-Yau algebra is a generalization of a Calabi-Yau algebra which allows for a non-trivial Nakayama automorphism. We prove three homological identities about the Nakayama automorphism and give several applications. The identities we prove show (i) how the Nakayama automorphism of a smash product algebra A # H is related to the Nakayama automorphisms of a graded skew Calabi-Yau algebra A and a finite-dimensional Hopf algebra H that acts on it; (ii) how the Nakayama automorphism of a graded twist of A is related to the Nakayama automorphism of A; and (iii) that Nakayama automorphism of a skew Calabi-Yau algebra A has trivial homological determinant in case A is noetherian, connected graded, and Koszul.