Bicrossproducts Structures

The aim of this dissertation is to carry out a series of further research on weak monoidal Hom-bialgebras,Hopf(co)quasigroups,and the relationships between Hom-bialgebras and monoidal Hom-bialgebras from the following partsFirstly,we give the definitions of weak monoidal Hom-bialgebras,as weak case of monoidal Hom-bialgebras,we follow it with an non-trivial example.Base on it,we define the weak(?,?)-Yetter-Drinfeld modules over weak monoidal Hom-bialgebras,and we then prove the category of such kinds of modules HWMHyDH(?,?)is equal to the weak monoidal entwined Hom-module category HMH(?(?,?)),where ? and ? are automorphisms of weak monoidal Hom-bialgebras H.Furthermore,we prove HWMHyDH DH is a monoidal category,and we give a new kind of braided T-categories by constructing the braiding of it.Secondly,we study the relationships between monoidal Hom-bialgebras and Hom-bialgebras Generally,we get the one to one correspondence between the collection of the monoidal Hom-bialgebras,and the collection of the unital Hom-bialgebras which admit proper conditions.Moreover,we discover the gauge equivalence between the monoidal Hom-bialgebras and Hom-bialgebras.Thirdly,we discuss some properties of Galois linear maps,and as applications,we also consider some equivalent conditions of Hopf algebras and Hopf quasigroups.On the other side,we also invest the conditions of quasimodules and comodules to be generalized Yetter-Drinfeld modules over Hopf quasigroups,and we also construct a new kind of braided group categories.Finally,we consider the comodule coalgebras and the smash coproducts over regular weak multiplier Hopf algebras and discuss some examples.We also consider the*-cases.