The Representation Ring Of Drinfeld Even D (KA₄)

The quantum doubles are a class of important Hopf algebras.The concept was introduced by Drinfeld when he studied the quantum Yang-Baxter equations.Then it attracted widely algebra researchers' attention.It has greatly promoted the development of the Hopf algebras and their representation theory.As a special case of quantum doubles,the quantum doubles of finite dimensional group algebras have recently attracted many mathematicians' interests for their relatively simple structures and broad applications.In this dissertation,we mainly study the representations and representation rings of Hopf algebra kA4 and its Drinfeld double D(kA4).Because the order of group A4 is 12,we work in three cases that 12 is not divisible by the characteristic of k,the characteristic of k is 2 and the characteristic of k is 3.The paper is organizes as follows.In Chapter 1,we introduce some basic concepts and conclusions needed in this paper.In particular,we recall the concepts of algebras,coalgebras,the Hopf algebras,quasitriangular Hopf algebras and the Drinfeld double D(H)of a finite dimensional Hopf algebra H,and the structures of Drinfeld doubles.The relation between the category of D(H)-modules and the category of Yetter-Drinfeld H-modules is also introduced.We also recall the notions of the Grothendieck ring and representation ring of a finite dimensional Hopf algebras.In Chapter 2,we first give the structure of the group A4.Then we study the representations of the centralizer subgroups of representatives of the conjugate classes of A4 in three cases of the characteristic of the ground field k,by which we construct the renpresentations of D(kA4)for each case.Hence,the main task of this chapter is to compute the representations of the group A4,Klein four group K4 and the cyclic group Z3 of order 3 in three cases as stated above.Up to isomorphism,we give all simple kA4-modules in the cases that 12 is not divisible by the characteristic of k and the characteristic of k is 2,and all indecomposable kA4-modules in the case that the characteristic of k is 3.We describe the structures of all these simple kA4-modules and indecomposable kA4-modules,by which we derive the classification of the corresponding simple D(kA4)-modules and indecomposable D(kA4)-Modules.Upon the previous chapter,we study the structures of the representation rings of the Hopf algebra kA4 and its Drinfeld double D(kA4)in Chapter 3.When 12 is not divisible by the characteristic of k,kAa and D(kA4)are both semisimple Hopf algebras,and hence their representation rings are the same as their Grothendieck rings.In this case,we describe the structures of the representation rings of kA4 and D(kA4)by generators with relations,respectively.When the characteristic of k is 2,we only consider the Grothendieck rings of kA4 and D(kA4).We describe these two Grothendieck rings as some quotient rings of polynomial rings.In case that the characteristic of k is 3,we first study the representation rings of kA4 and D(kA4).The generators and the relations satisfied by them are given.Then using a general result that the Grothendieck ring of a Hopf algebra is a quotient ring of its representation ring,we characterize the Grothendieck rings of kA4 and D(kA4)by their representation rings,respectively.