Cleft extensions of Koszul twisted Calabi-Yau algebras

TL;DR

This paper proves that cleft extensions of Koszul twisted Calabi-Yau algebras remain twisted CY, expanding understanding of their structure and identifying conditions under which cleft objects are Calabi-Yau.

Contribution

It demonstrates that cleft extensions of Koszul twisted CY algebras preserve the twisted CY property and characterizes cleft objects of certain pointed Hopf algebras as Calabi-Yau.

Findings

Cleft extensions of Koszul twisted CY algebras are also twisted CY.

The smash product of an N-Koszul twisted CY algebra with a twisted CY Hopf algebra remains twisted CY.

Cleft objects of specific pointed Hopf algebras are Calabi-Yau.

Abstract

Let $H$ be a twisted Calabi-Yau (CY) algebra and $\sigma$ a 2-cocycle on $H$. Let $A$ be an $N$-Koszul twisted CY algebra such that $A$ is a graded $H^\sigma$-module algebra. We show that the cleft extension $A#_\sigma H$ is also a twisted CY algebra. This result has two consequences. Firstly, the smash product of an $N$-Koszul twisted CY algebra with a twisted CY Hopf algebra is still a twisted CY algebra. Secondly, the cleft objects of a twisted CY Hopf algebra are all twisted CY algebras. As an application of this property, we determine which cleft objects of $U(\mathcal{D},\lambda)$, a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi-Yau algebras.