Cleft extensions of Koszul twisted Calabi-Yau algebras

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 $A#_σH$ 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.

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References in corpus (1)

• Calabi-Yau algebras