The Representations Of A Class Of Quantum Doubles And Related Topics

The quantum doubles, also called the Drinfeld doubles, are a class of Hopf al-gebras which are very important and typical. They were defined by Drinfeld in the context of finding solutions to quantum Yang-Baxter equations. Quantum doubles not only promote the development of the Hopf theory, but have remarkable applica-tions in the fields of theoretical physics, noncommutative geometry, low-dimensional topology and so on. The quantum double of a finite dimensional group algebra has recently attracted many mathematicians’interest for its simple structure and broad ap-plications. Cibils et al.[13,15,35,45] gave the characterization of its representations, respectively.In the dissertation, we mainly discuss the modular representations of quantum double of dihedral groups. We will describe its indecomposable matrix representa-tion, its Auslander-Reiten quiver and the ring structure of its Grothendieck group. In addition, we discuss the decomposition of the n-folds tensor product of the unique2-dimensional simple module of Uq(sl(2)). Meanwhile, we discuss the category of comodules over a coring. Our results provide a recipe for computing the representa-tions over quantum double of finite groups. In fact, our method here can be generated to more general cases.Given a finite group G, Witherspoon[45] proved that the quantum double of G is not semisimple if and only if the characteristic of the field k divides the order of G. In Chapters2,3,4, we discuss the representations of D(kDn), where k is a field of characteristic p, a odd prime, Dn is the dihedral group of order In, and n=pst with s≥1.In Chapter2, we describe the indecomposable representations of D(kDn) by con-structing the indecomposable representations of centralizer subgroups of the represen-tatives of conjugate classes of Dn. Hence, the main task of the chapter is to compute the indecomposable representations of Klein four group, cyclic group Cn and dihedral group Dn. We get the representations of these groups through induced ones.We compute AR-quiver of D(kDn) in Chapter3. We first prove that, for any finite group G, the almost split sequences of D(kG) can be induced from the ones of centralizer subgroups of the representatives of conjugate classes of G. Hence, the task of the chapter deduces to computing the almost split sequences of kCn and kDn. Fortunately, we prove that kCn and kDn are both Nakayama algebras. Hence, we obtain the AR-quiver of D(kDn) by the method introduced in [3].We describe the ring structure of Grothendieck group of D(kDn) in Chapter4. Let M be a Yetter-Drinfeld kG-module. Let D(N)=kG(?)kCG(gc) N be a simple Yetter-Drinfeld kG-module, where gc is a representative of some conjugate class C and N is a simple kCG(gc)-module. Then the multiplicity of D(N) in M as composition factors is equal to the multiplicity of N=D(N)gc in Mgc as composition factors of kCG(gc)-modules. Hence, the ring structure of Grothendieck group of D(kG) can be obtained by the Grothendieck ring of some finite groups, which can be described by the Brauer characters.In Chapter5, we discuss the decomposition of V(1)(?)n, where V(1) is the unique2-dimensional simple Uq(sl(2))-moddule. We first prove the standard basis theorem (The-orem5.1), which generalizes the ring structure of Grothendieck group of Uq(sl(2)). Then we obtain two combinatorial formulae, which are the coefficients in the decom-position of V(1)(?)n. Meanwhile, we obtain the unified proof of quantum Clebsch-Gordan formula and Clebsch-Gordan formula.In the last part of the dissertation, we discuss the category of comodules over a coring and category of modules over a ring. By Caenepeel et al.[10], we know that there are a pair of adjoint functors(F, G) between categories MB and MC. We mainly examine when F and G are mutual-inverse equivalences. We give some sufficient conditions for (F, G) to be a pair of mutual-inverse equivalences by using the properties of split forks and coseparable corings, which generalizes some results in [10].