The Study Of Hopf Nonassociative Group Coalgebras And Related Theories Thereof

This Ph.D.dissertation primarily focuses on how to construct new braided tensor categories,and thus studies the constructions of Hopf nonassociative group-coalgebras and Hopf noncoassociative group-algebras and their group-(co)module actions,group-integrals and(co)quasitriangular structures,with the following main work.First of all,in order to provide a unified framework for the study of classical concepts such as Hopf algebras,Hopf(co)quasigroups,and Hopf group-(co)algebras,the concepts of Hopf nonassociative group-coalgebras and Hopf noncoassociative group-algebras are introduced and discussed in this dissertation.The corresponding examples are constructed by Hopf(co)quasigroup theory.Creatively using nonassociative convolution algebras,this dissertation elucidates some key properties of Hopf nonassociative group-coalgebras and Hopf noncoassociative group-algebras.We also construct associators and coassociators for Hopf nonassociative group-coalgebras and Hopf nonassociative group-algebras,respectively.Secondly,by introducing the concept of Galois map,the equivalent characterizations of Hopf nonassociative group-coalgebras and Hopf noncoassociative group-algebras are given.In other words,for a nonassociative group-cobialgebra H,the left and right Galois maps can be defined,then H is a Hopf nonassociative group-coalgebra if and only if the left and right Galois maps of H are almost H-linearly invertible.For a nonassociative noncounital group-coalgebra H,the left and right Galois maps can still be defined.When both left and right Galois maps are bijective,H is a group-coalgebra,but it is not necessarily a nonassociative group-cobialgebra in that,due to the loss of the associativity of the multiplication of H,the counit constructed is not necessarily an algebra homomorphism.On the other hand,for a noncoassociative group-bialgebra H,the left and right Galois maps can be defined,then H is a Hopf noncoassociative group-algebra if and only if the left and right Galois maps of H are almost H-colinearly invertible.Thirdly,the concepts of Hopf nonassociative group-comodules and Hopf noncoassociative groupmodules are introduced for Hopf nonassociative group-coalgebras and Hopf noncoassociative groupalgebras,respectively.The fundamental theorem of Hopf nonassociative group-comodules over Hopf nonassociative group-coalgebras is given by using the coinvariants,and a conjecture on the fundamental theorem of Hopf noncoassociative group-modules over Hopf noncoassociative group-algebras is proposed by using the invariants.Then,the concepts of group-integrals on Hopf nonassociative group-coalgebras and group-integrals in Hopf noncoassociative group-algebras are introduced.For any Hopf nonassociative group-coalgebra,its faithful group-integral is unique up to scalar.For finitetype Hopf nonassociative group-coalgebras and finite-type Hopf noncoassociative group-algebras,their spaces of group-integrals are one-dimensional,their antipodes are bijective,and both have distinguished group-grouplike elements.In addition,the dual relations between Hopf nonassociative groupcoalgebras of finite type and Hopf noncoassociative group-algebras of finite type,the Fourier transformations on Hopf nonassociative group-coalgebras,Frobenius Hopf nonassociative group-coalgebras,separable Hopf noncoassociative group-algebras,co-Hopf modules over Hopf(co)quasigroups and their duals,equivalent characterizations of Hopf group-algebras and more are also discussed.Finally,the concepts of crossed Hopf nonassociative group-coalgebras,crossed Hopf noncoassociative group-algebras,(co)quasitriangular Hopf nonassociative group-coalgebras and(co)quasitriangular Hopf noncoassociative group-algebras are introduced.The equivalent characterizations of coquasitriangular Hopf nonassociative group-coalgebras and quasitriangular Hopf noncoassociative group-algebra are then given.The generalized quantum Yang-Baxter(-like)equations are obtained and the representation category is also discussed.For example,the group-corepresentation category on any coquasitriangular Hopf nonassociative group-coalgebra is a noncoassociative braided tensor category,the group-representation category on any quasitriangular Hopf noncoassociative groupalgebra is a noncoassociative braided tensor category,the quasirepresentation category on any crossed Hopf nonassociative group-coalgebra is a crossed group-category,and the R-quasirepresentation category on any quasitriangular Hopf nonassociative group-coalgebra(H,R)is a braided group-category.