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arXiv · 1405.6040

Cleft extensions of Koszul twisted Calabi-Yau algebras

Xiaolan Yu·Fred Van Oystaeyen·Yinhuo Zhang
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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.

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Xiaolan Yu, Fred Van Oystaeyen, Yinhuo Zhang. 2014-05-23. Cleft extensions of Koszul twisted Calabi-Yau algebras. https://arxiv.org/abs/1405.6040

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