The Calabi-Yau property of Hopf algebras and braided Hopf algebras

TL;DR

This paper investigates conditions under which the smash product of a braided Hopf algebra and a semisimple Hopf algebra is Calabi-Yau, providing necessary and sufficient criteria in various algebraic contexts.

Contribution

It establishes necessary and sufficient conditions for the Calabi-Yau property of smash products involving Hopf algebras and braided Hopf algebras.

Findings

Characterization of when $R#H$ is Calabi-Yau based on $R$ and $H$ properties

Conditions for $R$ to be Calabi-Yau when $H$ is a group algebra

Link between Calabi-Yau properties of $R$ and $R#H$

Abstract

Let $H$ be a finite dimensional semisimple Hopf algebra and $R$ a braided Hopf algebra in the category of Yetter-Drinfeld modules over $H$. When $R$ is a Calabi-Yau algebra, a necessary and sufficient condition for $R#H$ to be a Calabi-Yau Hopf algebra is given. Conversely, when $H$ is the group algebra of a finite group and the smash product $R#H$ is a Calabi-Yau algebra, we give a necessary and sufficient condition for the algebra $R$ to be a Calabi-Yau algebra.