A More General Way Of Constructing Braided Monoidal Categories Based On Hopf Quasigroups And Related Action Research

The main research objects of this paper are the category of two-sided two-cosided Hopf modules,Hopf quasi(co)modules in Yetter-Drinfeld structure category LLyDQ(C),Sweedler's dual of Hopf algebras in HHyDQC.The main contents are arranged as follows:Firstly,we introduce the notions of IIopf quasigroup and Yetter-Drinfeld quasimod-ule over a Hopf quasigroup,Let H be a Hopf quasigroup over k possessing an adjoint quasiaction.Then we first show that if M is any right H-comodule and N is any right H-quasimodule such that ?N,M2=idN(?)M,where ?N,M:N(?)M?M(?)N is a favourable map,then we have H=k as a generalization of Pareigis' Theorem.Moreover,we will s-tudy the notion of a generalized Long quasimodule and construct a new braided monoidal category induced by that of the category of Yetter-Drinfeld quasimodules over a Hopf quasigroup.Furthermore,we will construct classes of new braided categories inside the framework of two-sided two-cosided Hopf modules over a Hopf quasigroup.Secondly,we give the notion of the category of Yetter-Drinfeld quasimodules LLyDQ.By using dual properties,we first prove that if H is a finite dimensional Hopf quasigroup in Yetter-Drinfled quasimodule category LLyDQ,then its linear dual space H*is a Hopf coquasigroup in LLyDQ and we have Pontryagin dual:H**(?)H as a Hopf quasigroup.Furthermore,H*has a right H-Hopf quasimodule structure in LLyDQ.Again,we introduce the definitions of Hopf coquasigroup and the category of Yetter-Drinfeld quasicomodules LLDQC.we will show that if H denotes a finite dimensional Hopf coquasigroup in Yetter-Drinfeld quasicomodule category LLyVQC,then its linear dual space H*is a Hopf quasigroup in LLyDQC and also H*has a right H-Hopf quasicomodule structure in LLyDQC.furthermore,we study the Sweedler's dual of infinite-dimensional Hopf algebras in LLyDQC.Finally,let H be a Hopf quasigroup in a symmetric monoidal category,we will obtain some generalization results inside the framework of chapter two.