Cleft extensions of Koszul twisted Calabi-Yau algebras
Xiaolan Yu, Fred Van Oystaeyen, Yinhuo Zhang

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 be a twisted Calabi-Yau (CY) algebra and a 2-cocycle on . Let be an -Koszul twisted CY algebra such that is a graded -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 -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 , a class of pointed Hopf algebras introduced by Andruskiewitsch and Schneider, are Calabi-Yau algebras.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Advanced Operator Algebra Research
