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.2000 Mathematics Subject Classification. 16E65, 16E40, 16W30, 16W35.

Abstract. Let H and L be two Hopf algebras such that their comodule categories are monoidal equivalent. We prove that if H is a twisted Calabi-Yau (CY) Hopf algebra, then L is a twisted CY algebra when it is homologically smooth. Especially, if H is a Noetherian twisted CY Hopf algebra and L has finite global dimension, then L is a twisted CY algebra.

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Let k be an algebraically closed field. The main goal of this paper is to classify the finite-dimensional pointed Hopf algebras over k of finite corep-resentation type. To do so, we give a necessary and sufficient condition for a basic Hopf algebra over k to be of finite representation type firstly. Explicitly, we prove that a basic Hopf algebra over k is of finite representation type if and only if it is Nakayama. By this conclusion, we classify all finite-dimensional pointed Hopf algebras over k of finite corepresentation type.

It is proved that the Poisson enveloping algebra of a double Poisson-Ore extension is an iterated double Ore extension. As an application, properties that are preserved under iterated double Ore extensions are invariants of the Poisson enveloping algebra of a double Poisson-Ore extension.