Strictification of weakly equivariant Hopf algebras. (Q2376831)

Assume that \(A\) is a Hopf algebra over a field \(K\) and \(G\) is a finite group. A concept of a weak \(G\)-action of \(G\) on \(A\) is introduced and studied in the paper. By a weakly \(G\)-equivariant Hopf algebra the authors mean a Hopf algebra \(A\) with a \(G\)-action by inner automorphisms of \(A\). Main results of the paper are summarized in the abstract as follows. It is shown that every weakly \(G\)-equivariant Hopf algebra \(A\) can be replaced by a Morita equivalent \(K\)-algebra \(A^{str}\) (a stratification of \(A\)) with a strict action of \(G\) and with a coalgebra structure that leads to a tensor equivalent representation category. Moreover, the Morita equivalence \(A\text{-mod}\cong A^{str}\text{-mod}\) induces an equivalence of corresponding tensor categories. The paper is a supplement to the authors' article published in [Adv. Theor. Math. Phys. 16, No. 1, 289-358 (2012; Zbl 1273.81198)].