Irreducible Representation Of D(kS₃) And Ring Structures Of Its Grothendieck Group

Let k be an algebraically closed field of characteristic 2 and S₃ be the symmetric group on three elements .In this thesis, we will examine the irreducible representations of the Drinfeld double D ( kS₃) of Hopf algebra kS₃.The ring structure of the Grothendieck group G₀ ( D(kS₃)) is discussed also.In Chapter 1, we recall the background of Hopf algebras, and review some concepts and conclusions which are used later in this paper. In particular, we recall the quasitriangular Hopf algebras, the Drinfeld double D ( H )of a finite dimensional Hopf algebra H and its structure. The relation between the category of D ( H )-modules and the category of Yetter-Drinfeld H -modules is also reviewed.In Chapter 2, we first describe the structure of the Drinfeld double D ( kS₃) and study the irreducible representations of D ( kS₃). We show that there're exactly 6 simple modules over D ( kS₃) up to isomorphism. The 6 simple D ( kS₃)-modules are denoted by V₁ , V2 , V3 , V4 ,V5 and V6 respectively. The structures of these simple modules are described.In Chapter 3, we consider the ring structure of Grothendieck group G₀ ( D(kS₃)). Note that G₀ ( D(kS₃))is a commutative ring since D ( kS₃) is a quasitriangular Hopf algebra. As an additive group, G₀ ( D(kS₃))is a free Abel group with aΖ-basis {[V₁ ],[V₂ ],[V₃ ],[V₄ ],[V₅ ],[V₆ ]}. The main object of this chapter is to find the tensor product's structures of any two simple modules. When V_i (?) V_j is semi-simple, we decompose the tensor product as a direct sum of simple modules. Otherwise we obtain the decomposition of the socle of V_i (?) V_j and V_i (?) V_j Soc (V_i (?)V_j) which must be semi-simple. Thus we get the structure of multiplication of G₀ ( D(kS₃)).