Calabi–Yau property under monoidal Morita–Takeuchi equivalence

Abstract

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.

Cohomological dimensions of universal cosovereign Hopf algebras

“…The above result also can be obtained as a direct consequence of [4, Corollary 2.5], as well. Indeed, the Hochschild cohomological dimension of a Hopf algebra coincides with its (right or left) global dimension (this is pointed out in [50]), hence the result follows from Corollary 2.5 in [4], having in mind that since A and B are non trivial, the free product algebra A * B is necessarily infinite-dimensional, so is not semisimple, and hence cd(A * B) ≥ 1.…”

“…To conclude, it is interesting to note that the question of a generalization of Corollary 6.5 (positive answer to Question 1.1 in [8]) is studied as well in the recent preprint [50], in the setting of Hopf algebras having an homological duality, with interesting partial positive answers.…”

On the monoidal invariance of the cohomological dimension of Hopf algebras

“…While the proof of the first statement (Theorem 8.1) is obtained by carefully inspecting arguments in [42,39] and the third one (Theorem 8.3) is a rather direct consequence of previous results [24,16,1], the main effort in this paper is in proving the second statement (Theorem 4.6). Removing the assumption S 4 = id from [7] (with instead the finiteness assumption on cohomological dimensions) enables us to compute cohomological dimension in a number of new situations, see Section 7 for examples regarding universal cosovereign Hopf algebras and free wreath products.…”

“…Partial positive answers to Question 1.1 were provided in [6,7] when A, B are cosemisimple with antipode satisfying S 4 = id, and by Wang, Yu and Zhang in [39], when A is twisted Calabi-Yau and B is homologically smooth.…”

A note on the bijectivity of the antipode of a Hopf algebra and its applications

“…In recent development, the study of infinite-dimensional Hopf algebras seems to be of growing importance, which reveals that some well-known results about finite-dimensional Hopf algebras surprisingly have incarnations in the realm of noetherian Hopf algebras (see, e.g., survey papers [3,6]). Among these progress, it is worthy to point out that the bijectivity of the antipode frequently plays an essential role in establishing these properties (see, e.g., [2,8,13,20]). Therefore, one prompts to ask for criterions concerning the bijectivity of the antipode of a Hopf algebra.…”