The Construction Of Infinitesimal Hopf Algebra On The Sweedler Algebra

Hopf algebras are an integral part of the field of algebra.In many examples of Hopf algebras,besides group algebra and envelope algebra of Lie algebra,the Sweedler’s fourdimensional Hopf algebra is an important kind of Hopf algebra.Moreover,it is neither commutative nor cocommutative.Infinitesimal bialgebras were first introduced in 1979 by Joni and Rota in order to carve differentials from a generational mathematical point of view.The basic theory and framework of infinitesimal bialgebras were introduced by Aguiar in 2000.Aguiar pointed out that path algebras are infinitesimal bialgebras,and introduced the Drinfel’d double concept on the infinitesimal bialgebra for the first time.And he established the relationship between the quasi-triangular infinitesimal bialgebra and the associated Yang-Baxter equation.Besides,he extend many classic results of Hopf algebra to infinitesimal Hopf algebra.This article was based on the above analysis and considerations,constructed infinitesimal Hopf algebra and its quasi-triangle Hopf algebra by new construction of coalgebra on the Sweedler algebra.This thesis is divided into four chapters:In Chapter 1,we gave the research background of this thesis,introduced the development of Hopf algebra and infinitesimal Hopf algebra,and gave the main results of this paper.In Chapter 2,we introduced the examples and theoretical basis of Hopf algebra and infinitesimal Hopf algebra.In Chapter 3,we mainly gave the construction of infinitesimal Hopf algebra and its quasitriangle Hopf algebra on the 2-dimensional and 3-dimensional subalgebra of the Sweedler algebra.In Chapter 4,we mainly constructed infinitesimal Hopf algebra and its quasi-triangle Hopf algebra on the Sweedler algebra.