ON GROUP CROSSED PRODUCTS

Abstract

Let π be a group and let H = {Hα}α∈π be a Hopf π-coalgebra in the sense of Turaev . Let H act weakly on an algebra A and σ : H1⊗H1 → Aa k-linear map. Then we first introduce the notion of a π-crossed product A#πσH = {A#σHα}α∈π and find some sufficient and necessary conditions under which each A#σHα forms an algebra. Next we define a comultiplication, a counit and an antipode on A#πσH making it into a Hopf π-coalgebra. Finally, we obtain the duality theorem of π-crossed product A#πσH, generalizing Corollary 5.8 in the authors’ paper.

Keywords

• Hopf group-coalgebra, group-crossed product, duality theorem