Partial group (co)actions of Hopf group coalgebras

Abstract

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

Recent developments around partial actions

“…Furthermore, Birget [56] applied the partial action definition of Thompson's groups to study algorithmic problems for them. Since then the algebraic approach is being developed in diverse directions in various levels of generality, including partial actions of Hopf (or, more generally, weak Hopf) algebras [15,[17][18][19][20][21][22]48,69,72,[78][79][80][81][82][83][86][87][88]165,250,282], semigroups [68,97,132,[197][198][199]209,213,220,227,233,234,242,255], inductive constellations [198], groupoids [37,[40][41][42][43]178], and, more generally, categories [244]. In particular, further algebraic applications have been found to graded algebras [117,121], to Hecke algebras…”

“…Recent works around partial actions include also the study of the category of the partial Doi-Hopf modules in [86], of partial actions on power sets in [34], of the category of partial G-sets for a fixed group G in [28], of partial orbits and n-transitivity in [31], of partial group entwining structures and partial group (co)actions of a Hopf group coalgebra on a family of algebras in [87], of generalized partial smash products in [143], and of twisted partial Hopf coactions and corresponding partial crossed coproducts in [88], as well as a note on sums of ideals [33]. More information around partial actions may be found in the surveys [45,115,116,170,250,251,260].…”

Recent developments around partial actions

“…Recent works around partial actions include also the study of the category of the partial Doi-Hopf modules in [86], of partial actions on power sets in [34], of the category of partial G-sets for a fixed group G in [28], of partial orbits and n-transitivity in [31], of partial group entwining structures and partial group (co)actions of a Hopf group coalgebra on a family of algebras in [87], of generalized partial smash products in [143], and of twisted partial Hopf coactions and corresponding partial crossed coproducts in [88], as well as a note on sums of ideals [33]. More information around partial actions may be found in the surveys [45], [115], [116], [170], [250], [251] and [260].…”

“…Furthermore, J.-C. Birget [56] applied the partial action definition of Thompson's groups to study algorithmic problems for them. Since then the algebraic approach is being developed in diverse directions in various levels of generality, including partial actions of Hopf (or, more generally, weak Hopf) algebras [15], [17], [18], [19], [20], [21], [22], [48], [69], [72], [78], [79], [80], [81], [82], [83], [86], [87], [88], [165], [250], [282], semigroups [68], [97], [132], [197], [198], [199], [209] [213], [220], [227], [233], [234], [242], [255], inductive constellations [198], groupoids [37], [40], [41], [42], [43], [178], and, more generally, categories [244]. In particular, further algebraic applications have been found to graded algebras …”