On Hopf-Galois extensions of linear categories

We continue the investigation of H-Galois extensions of linear categories, where H is a Hopf algebra. In our main result, the Theorem 2.2, we characterize this class of extensions in the case when H is finite dimensional. As an application, we prove a version of the Duality Theorem for crossed products with invertible cocycle. Introduction The duality theorems for actions and coactions originated in the work of Nagakami and Tagasaki on operator algebras, see [12]. A first purely algebraic version of the duality theorem was proved for actions of finite groups in [5]. Blattner and Montgomer showed that a similar result can be obtained for the actions and the coactions of a finite dimensional Hopf algebra, see [2]. A variant of the duality theorem for k-linear graded categories was considered in [3] in order to investigate the Galois coverings of a linear category. In this paper we continue our work on Hopf-Galois extensions of linear categories, started in [14]. Our main aim now is to prove a duality theorem for the crossed product of linear category with a finite dimensional Hopf algebra. Almost all of the contents of the article are the adaptation of the results known for algebras to the case of linear categories. In the first part of the article we fix the terminology and the notation that we use. All the definitions of small linear categories and modules over them in Subsection 1.1, as well as the results on projective modules in Subsection 1.5