Automorphisms Of Hopf Algebras And The Study Of Tensor Categories

Hopf algebra is an important branch of algebras and the automorphism group is an important invariant,which are important subjects in mathematical research.This dissertation will study the automorphism groups of some Hopf algebra biproducts.Since it is usually very difficult to determine the full group of algebra automorphisms,this thesis mainly studies certain subgroups of Hopf algebra automorphism group of biproducts.Since the rapid and vigorous development of tensor(or monoidal)category research,the automorphism group of twisted tensor biproducts will be further studied in the strict braided tensor category.In the follow-up research,the necessary and sufficient conditions for the relative Hom-Hopf module category to be a tensor category are discussed.This dissertation consists of five chapters.The first chapter mainly introduce.s the research background and deve.lopment status and the organizational structure and main conclusions of this thesis.In chapter two,we study certain subgroups of the full group of Hopf algebra automor-phisms of twisted tensor biproducts.Firstly,we recall some knowledge such as Hopf algebras,Yetter-Drinfel'd modules,Radford's biproducts,and twisted tensor biproducts,which need later.Secondly,we show that the endomorphism of twisted tensor biproducts has a factoriza-tion,and characterize the endomorphism monoid EndHopf(AT(?)?H,p)and automorphism group AUtHopf(AT(?)?H,p).Thirdly,we discuss the relationship between our conclusions and Radford's conclusions under certain conditions.Lastly,we generalize the main conclusions obt,ained to the strict braided tensor categories and describe the automorphism group of twisted tensor biproducts in the language of categories.In chapter three,we study certain subgroups of the Hopf group-coalgebra automor-phism group of Radford's 7r-biproduct.First,we review some concepts and conclusions related to Hopf algebras and ?-smash(co)products,and give the conditions for Radford's 7r-biproduct to be the Hopf group-coalgebra.Second,we discuss the endomorphism monoid End?-Hopf(A×H,p)and the automorphism group Aut?-Hopf(A×H,p)of Radford's ?-biproduct A x H={A×H?}???,and prove that the endomorphism(automorphism)has a factorization.What's more interesting is that a pair of maps(FL,FR)can be used to describe a family of mappings F=?F?}???.Third,we consider the relationship between the automorphism group Aut?-Hopf(A × H,p)and the automorphism group Aut?-yD(A)of A,and a normal subgroup of the automorphism group Aut?-Hopf(A×H,p).Finally,we specifically describe the automorphism group of an example.In chapter four,we study the Hom-Hopf algebra automorphism group of(?,?)-twisted Radford's Hom-biproducts.First,we review some definitions and basic results related to Hom-Hopf algebras and(?,?)-twisted Radford's Hom-biproducts.Then,we study the endomorphism monoid and automorphism group of(?,?)-twisted Radford's Hom-biproducts and show that the endomorphism has a factorization closely related to the factors(A,?)and(H,?).Next,we consider(?,?)-twisted Radford's Horn-biproduct automorphism group AutHom-Hopf(A×?? H,p)as a subgroup of some semi direct product U(C,A)op× g(A),Finally,we characterize the automorphisms of a concrete example.In chapter five,we study the tensor structure in the category of relative Hom-Hopf modules.First,let(H,?)be a Hom-bialgebra,introduce the definition of Hom-Yetter-Drinfel'd module of the form we need,and its module category HHyD,Then prove that the category HHyD is a pre-braided tensor category.Then define the concept of Hol-bialgebra in the pre-braided tensor category HHyD Then,we assume that(H,?)is a Hom-bialgebra,but no longer assume that(A,?)is a Hom-bialgebra:it will be sufficient that(A,?)is at the same time a left(H,?)-comodule Hom-algebra and a Hom-coalgebra,and we assume that there is left(H,?)-action(?):H(?)A? A,which is not assumed to be Hom-associative or Hom-unital from the beginning.These additional structures on(A,?)allow us to define a right(A,?)-action on the tensor product of two relative Hom-Hopf modules,and the unit object Ik of C.Thus the main conclusion of this chapter is obtained:The category of relative Hom-Hopf modules with this additional structure is a tensor category if and only if(A,a)is a braided Hom-bialgebra,that is a Hom-bialgebra in the pre-braided tensor category HHyD of Hom-Yetter-Drinfel'd modules.Finally,some examples and applications are given through the quasi-triangular Hom-Hopf algebras.