Hopf-Galois objects of Calabi-Yau Hopf algebras

TL;DR

This paper demonstrates that Hopf-Galois objects of twisted Calabi-Yau Hopf algebras retain the twisted Calabi-Yau property and explicitly describes their Nakayama automorphisms, with applications to cleft Galois objects and quantum automorphism groups.

Contribution

It establishes that Hopf-Galois objects of twisted Calabi-Yau Hopf algebras are also twisted Calabi-Yau and provides explicit formulas for their Nakayama automorphisms, extending the understanding of their structure.

Findings

Hopf-Galois objects of twisted Calabi-Yau Hopf algebras are twisted Calabi-Yau.

Explicit Nakayama automorphisms are given for these objects.

Cleft Galois objects and quantum automorphism groups also exhibit the twisted Calabi-Yau property.

Abstract

By using the language of cogroupoids, we show that Hopf-Galois objects of a twisted Calabi-Yau Hopf algebra with bijective antipode are still twisted Calabi-Yau, and give their Nakayama automorphism explicitly. As applications, cleft Galois objects of twisted Calabi-Yau Hopf algebras and Hopf-Galois objects of the quantum automorphism groups of non-degenerate bilinear forms are proved to be twisted Calabi-Yau.