The Idempotents Of The Drinfeld Double Of Taft Algebras

Quantum groups are special cases of Hopf algebras.The starting point of the research is the quantum inverse scattering method proposed by Faddeev and "Leningrad school",which aims to solve some "integrable quantum systems".A key step of this method is to study the quantum Yang Baxter equation.Drinfeld introduced the concept of "quasitriangular Hopf algebra".The representation category of this kind of Hopf algebra is a braid tensor category,which can provide solutions for the Yang Baxter equation.Chen constructs an infinite dimensional non-commutative and non-cocommutative Hopf algebra H（p,q）for any two numbers p、q（p≠0）in the field k.When q is a root of the n th cyclotomic polynomial over Z,there is an n4-dimensional quotient Hopf algebra H_n（p,q）.When q is a nth root of a unity,for any p≠0,H_n（p,q）is isomorphic to the Drinfeld double D（A_n（ω））of Taft algebra A_n（ω）)as a Hopf algebra.Therefore,the study of D（A_n（ω））is equivalent to the study of H_n（p,q）,or equivalently,the study of H_n（1,q）.Chen studied the primitive orthogonal idempotents of H₃（1,q）,and obtained the structure and isomorphic classification of indecomposable projective modules.This paper studies the idempotents of H_n（p,q）if n takes other values,and gives the complete set of primitive orthogonal idempotents.The first chapter is preliminary knowledge,introduces some basic concepts such as Taft algebra,double cross product and quasi triangular Hopf algebra,and introduces Hopf algebra H（p,q）and H_n（p,q）and its basic structure and related conclusions.The first section of the second chapter mainly gives some general expressions of basic idempotents and introduces the formula of regular module decomposed into direct sum of indecomposable modules and the number of elements in the complete set of primitive orthogonal idempotents.The second section mainly introduces the expression formula of primitive idempotents of H₃（1,q）,and gives the complete set of primitive orthogonal idempotents at this time.The third chapter is the main body of this paper.It studies the idempotents of H_n（1,q）when taking n as 2,4,5 and 6 respectively.For each case,it gives the expressions of primitive idempotents,and then gives the complete set of primitive orthogonal idempotents.In the research,due to the large number of equations involved and the relatively complex operation process,we use the mathematical calculation software MATLAB and GAP for auxiliary solution and verification.While obtaining the solution,we manually screen all the solutions to obtain the qualified expressions.