The Antipodes Of Coquasitriangular CoquasiTuraev Group Algebra

As an important generalisation of the quasi-Hopf algebra and the Hopf group coalgebra,this paper discusses the coquasitriangular coquasi-Turaev group algebra.First of all,two basic definitions of the coquasitri angular coquasi-Turaev group algebra are given.The first is the coquasitriangular coquasi-Turaev group algebra with the matrix R invertible and the antipode S bijective;the second is coquasitriangular coquasi-Turaev group algebra without the matrix R invertible and the antipode S bijective.Secondly,analyses the relationship between the coquasitriangular coquasiTuraev group algebra and the corepresentation category,and discusses some important properties of the antipode of the coquasitriangular coquasi-Turaev group algebra.Finally discusses the relationship between the coquasitriangular coquasi-Turaev group algebras.The main results of the paper are as follows:(1)It is proved that H={Hα}α∈π is a coquasi group bialgebra if and only if its corepresentation category is a tensor group category;(2)it is proved that H={Hα}α∈π is a coquasitriangular coquasi-Turaev group algebra if and only if its corepresentation category is a Turaev-braided group category;(3)if H={Hα}α∈π is a coquasitriangular coquasi-Turaev group algebra,it is shown that the antipode of H is inner,i.e.,Sα-1Sα(h)=∑ξαh(1,α)h(2,α)ξα-1(h(3,α));(4)it is proved that two coquasitriangular coquasi-Turaev group algebra definitions are equivalent and the S4 formula on the coquasitriangular coquasi-Turaev group algebra are given,i.e.,(Sα-1Sα)2(a)=uα-1-1(Sα(a(1,α)(1,α))uα(a(1,α),(2,α))a(2,α)uα-1(a(3,α)(1,α))uα-1(Sα(a(3,α)(2,α))).