| In this thesis,we mainly study the copivotal structures on finite-dimensional weak Hopf algebras and on finite-dimensional coquasi-Hopf algebras,and give a necessary and sufficient condition for the category of corepresentations becomes a pivotal category.Moreover,we also discuss the relations between the copivotal structures and coribbon structures,and give the realization of Deligne type theorem on finite-dimensional weak Hopf algebras and on finite-dimensional coquasi-Hopf algebras.The thesis is organized as follows:The first and the second chapters are introduction and preliminary respectively.In the third chapter,we first study the definition of copivotal forms on a weak Hopf algebra H,and then discuss the corepresentations on it,and prove that H is copivotal if and only if Corep(H)is a pivotal category.Finally we give a necessary and sufficient condition to give coribbon structures on a finite-dimensional coquasitriangular weak Hopf algebra(H,a)with bijective antipode.In the fourth chapter,we first study the definition of the copivotal coquasi-Hopf algebras,and then discuss their corepresentations.We prove that(A,ω)is copivotal coquasi-Hopf algebra if and only if Corep(A)is a pivotal category.At last we give a necessary and sufficient condition to define coribbon structure on a finite-dimensional coquasitriangular coquasi-Hopf algebra(A,σ)with bijective antipode. |
