Cohomology of Graded Twisting of Hopf Algebras

Abstract

Let A be a Hopf algebra and B a graded twisting of A by a finite abelian group Γ. Then, categories of comodules over A and B are equivalent (but they are not necessarily monoidally equivalent). We show the relation between the Hochschild cohomology of A and B explicitly. This partially answer a question raised by Bichon. As an application, we prove that A is a twisted Calabi–Yau Hopf algebra if and only if B is a twisted Calabi–Yau algebra, and give the relation between their Nakayama automorphisms.