# Smash products of Calabi–Yau algebras by Hopf algebras

### Patrick Le Meur

Université Clermont Auvergne, Aubière, France and Université Paris Diderot - Paris 7, France

## 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.

## Cite this article

Patrick Le Meur, Smash products of Calabi–Yau algebras by Hopf algebras. J. Noncommut. Geom. 13 (2019), no. 3, pp. 887–961

DOI 10.4171/JNCG/341