Smash Products of Calabi-Yau Algebras by Hopf Algebras

TL;DR

This paper explores conditions under which smash products of Calabi-Yau algebras with Hopf algebras preserve Calabi-Yau properties, providing explicit formulas for Nakayama automorphisms and dualising complexes.

Contribution

It establishes that the smash product of skew Calabi-Yau algebras with Hopf algebras remains skew Calabi-Yau and describes the Nakayama automorphism explicitly.

Findings

A#H is skew Calabi-Yau if A and H are skew Calabi-Yau.

The Nakayama automorphism of A#H is expressed via those of A and H.

The inverse dualising complex of A#H is characterized when A is a homologically smooth dg algebra.

Abstract

Let H be a Hopf algebra and A be an H-module algebra. This article investigates when the smash product A#H is (skew) Calabi-Yau, has Van den Bergh duality or is Artin-Schelter regular or Gorenstein. In particular, if A and H are skew Calabi-Yau, then so is A#H and its Nakayama automorphism is expressed using the ones of A and H. This is based on a description of the inverse dualising complex of A#H when A is a homologically smooth dg algebra and H is homologically smooth and with invertible antipode. This description is also used to explain the compatibility of standard constructions of Calabi-Yau dg algebras with taking smash products.