The Jacobson Radicals Of The Drinfeld Doubles Of Low Dimensional Taft Algebras

The quantum doubles(or Drinfeld doubles)of finite dimensional Hopf algebras are important quasitriangular Hopf algebras whose representation categories are braided tensor categories.The braiding structure of a braided tensor category can supply solutions to the quantum Yang-Baxter equations.It is well known that the Jacobson radical of a finite dimensional algebra is both the largest nilpotent ideal and the smallest ideal that makes the corresponding quotient algebra into a semisimple algebra.It reflects the "distance" from the finite dimensional algebra to a semisimple algebra.The structures of semisimple algebras are well known.Therefore,a finite dimensional algebra can be better understood through the Jacobson radical.Taft algebras is a family of finite dimensional pointed Hopf algebras that have attracted attention widely.Many scholars construct new Hopf algebras by using Taft algebras as basic blocks.Therefore,the Jacobson radical of Drinfled double D(An(ω))of Taft algebra is a significant research topic.Chen constructed a class of n4-dimensional Hopf algebra Hn(p,q),where n≥2 is an integer,p,q are scalars in the ground field k and q is a primitive nth root of unity.When p≠0 and q=ω-1n(p,q)is isomorphic to D(An(ω))as a Hopf algebra.In particular,Hn(1,q)≌ D(An(ω))as Hopf algebras.Therefore,the study of D(An(ω))is equivalent to the study of Hn(1,q).In this thesis,we mainly study the Jacobson radical of Drinfled doubles Hn(1,q)of Taft algebras for n=2 and 3.In Chapter 1,firstly,we introduce the structures of Taft algebra An(ω)and Hopf algebra Hn(p,q).Then we introduce the structures and classifications of simple modules over Hn(1,q).There are n2 non-isomorphic simple modules over Hn(1,q).Finally,we introduce some basic structures and results about the finite dimensional algebras over an algebraically closed field and their representation theory.In Chapter 2,we discuss the Jacobson radical of H2(1,q)in case n=2.Firstly,we compute the annihilators for each simple module.Thereby,all four maximal ideals of H2(1,q)are obtained.For each maximal ideal,a set of generators and a basis are given respectively.Then we discuss the intersection of these maximal ideals.The Jacobson radical of H2(1,q)is described by a set of generators and a k-basis,respectively.In Chapter 3,we study the Jacobson radical of H3(1,q)for n=2.Up to isomorphism,H3(1,q)has nine simple modules.Firstly,for each simple module,we find a set of generators of its annihilator which is viewed as an ideal.Secondly,a basis of such any ideal is given by using the generating set.Thus,all maximal ideals of H3(1,q)are characterized.Finally,using the generators and bases of these maximum ideals,a set of generators of the Jacobson radical of H3(1,q)as an ideal is obtained.A k-bases of Jacobson radical of H3(1,q)is also obtained.