Crossed The Left H-¦Ð-mode, The Right H-¦Ð-comodule With Braids Tensor Category

Recently, Turaev in [19, Section11.2] introduced the notions ofπ-coalgebras and Hopfπ-coalgebras. Let k be a fixed field andπbe a discrete group. Aπ-coalgebra over k is a family C = {Cα}α∈πof k -spaces endowed with a comultiplicationΔand a counitε, whereΔ= {Δα,β:Cαβ→Cα-Cβ}α,β∈π,ε: C1→k. AndΔis coassociative,εis counitary. A Hopfπ-coalgebra H = ({ Hα}α∈π,Δ,ε)is aπ-coalgebra H = { Hα}α∈πendowed with an antipode 1S = {Sα: Hα→Hα- }α∈πwhich satisfies some compatibility conditions . In [19] Turaev also gave the definitions of crossed Hopfπ-coalgebras and quasitriangular Hopfπ-coalgebras. In [1], Alexis Virelizier studied algebraic properities of Hopf group-coalgebras and gave the definition ofπ- comodules. Meanwhile, He also generalized the main properties of quasitriangular Hopfπ-coalgebras. Upon the background above, in this paper, we first give the definitions of left H -π-modules and crossed left H -π-modules. Then we show that the category of left H -π- modules is a tensor category and the category of crossed left H-π-modules is a tensor category too. And then we discuss the relationships between the quasitriangular Hopfπ- coalgebras and the category of crossed left H-π-modules being a braided tensor category. Then, we define right H -π-comodules and coquasitriangular Hopfπ-coalgebras ,and similarly we show that the category of right H-π-comodules is a tensor category. We also investigate the relationships between the coquasitriangular Hopfπ-coalgebr -as and the category of right H-π-comodules being a braided tensor category.The paper is organized as follows. In Section 1, we introduce some basic notions aboutπ- coalgebras, Hopfπ- coalgebras, crossed Hopfπ- coalgebras, quasitriangular Hopfπ- coalgebras and so on. And we also recall the definition of braided tensor categories. In Section 2, First of all, we define left H-π-modules, and we prove that the category of left H -π- modules is a tensor category. Secondly, we give the definition of crossed left H-π-modules. Moreover, we show that the category of crossed left H-π-modules is a tensor category. Finally, we obtain one of the main results in this paper, as follows: Theorem 2.9. Let H = ({ Hα}α∈π,Δ,ε, S , -, R) be a quasitriangular Hopfπ-coalgebras. Then the category of crossed left H-π-modules H M crossed is a braided tensor category.In Section 3, we first give the definitions of right H-π-comodules and coquasitriangular Hopfπ-coalgebras. Then we show that the tensor products of two right H-π-comodules is also a right H -π-comodule, and that the category of right H-π-comodules is a tensor category. Finally, we get another main result in this paper, as follows:Theorem 3.9. Let H = ({ Hα}α∈π,Δ,ε, S,σ) be a coquasitriangular Hopfπ-coalgebras, whereσis the coquasitriangular structure of H. Then the category of right H-π-comodules M H is a braided tensor category.