Generalized Frobenius-schur Indicators For Quasitriangular Hopf Algebra

This paper mainly studies the generalized Frobenius-Schur indicators of finite-dimensional semisimple triangular Hopf algebra.We define a class of generalized Frobenius-Schur indicators in finite-dimensional semisimple triangular Hopf algebra and obtain the structural information of related quasitriangular Hopf algebra indirectly by studying some arithmetic conditions and properties of such indicators.This paper is divided into the following four parts:firstly,we give the definition of general-ized Frobenius-Schur indicators for finite-dimensional semisimple quasitriangular Hopf algebras.The classical Frobenius-Schur indicators and generalized Frobenius-Schur indicators of cyclic group algebra below order 4 are calculated according to the definition.It can be found that the generalized Frobenius-Schur indicators have better performance in the classification of mod-ules;Next,the initial definition is further simplified and studied.Another equivalent definition of generalized Frobenius-Schur indicators and the first,second formulas are obtained.Then,the properties and applications of generalized Frobenius-Schur indicators for finite-dimensional semisimple quasitriangular Hopf algebras are explored.Finally,the Drinfeld double correspond-ing to the algebra of the cyclic group below order 4 is taken as the calculation examples to calculate their classical and generalized Frobenius-Schur indicators,respectively.