We will develop partial group (co)actions of a Hopf group coalgebra on a family of algebras by introducing partial group entwining structure. Then we give necessary and sufficient conditions for a family of functors from the category of partial group entwining modules to the category of modules over a suitable algebra to be separable. Also, the applications of our results are considered.
132In Section 2, we recall some definitions of group coalgebras, Hopf group coalgebra and separable functors.In Section 3, partial π-entwined structures(modules) are introduced, and for all α ∈ π, we show that the functor F (α) : U π−C A (ψ) → M A α which forgets the partial π-C-coaction has an adjoint. In Section 4, we will develop a theory of partial (co)actions of Hopf group coalgebras and introduce the concepts of partial group co(module) (co)algebra, partial Doi-Hopf group structures(modules).In Section 5, we will characterize the separability of the forgetful functor F (α) from the category of socalled partial π-entwined modules U π−C A (ψ) to the category of all A α -modules (fixed α ∈ π) which leads to a generalized notion of integral of a partial π-entwined structure. Finally, the main applications of our results are considered in Section 6.
Group Coalgebras, Hopf Group coalgebras and Separable FunctorsThroughout this paper, we always let π be a discrete group with a neutral element e and k a field. All (co)algebras and comodules are all over k. Let M and N be vectors, τ M,N : M ⊗ N → N ⊗ M : m ⊗ n → n ⊗ m denotes the flip map.
Group CoalgebrasRecall from [23] that π-coalgebra is a family of k-spaces C = {C α } α∈π together with a family of k-linear maps ∆ = {∆ α,β : C αβ → C α ⊗ C β } α,β∈π (called a comultiplication ) and a k-linear map ε : C e → k (called a counit) such that ∆ is coassociative in the sense thatfor any α, β, γ ∈ π andfor all α ∈ π.Remark 2.1. (C e , ∆ e,e , ε) is an ordinary coalgebra in the sense of Sweedler.Following the Sweedler's notation for π-coalgebras, for any α, β ∈ π and c ∈ C αβ , one writes ∆ α,β (c) = c (1,α) ⊗ c (2,β) .
Hopf Group CoalgebrasRecall from [7] that a semi-Hopf π-coalgebra is a π-coalgebra H = ({H α } α∈π , ∆ = {∆ α,β }, ε) such that the following conditions hold: (SH1) Each H α is an algebra with multiplication m α and unit 1 α ∈ H α , (SH2) For all α, β ∈ π, ∆ α,β and ε : H e → k are algebra maps.A semi-Hopf π-coalgebra H = ({H α , m α , 1 α } α∈π , ∆ = {∆ α,β }, ε) is called a Hopf π-coalgebra, if there exists a family of k-linear maps S = {S α : H α → H α −1 } α∈π (called an antipode) such that
