Representation Of Quantum Double Of Quantum Double Dihedral Group Algebra And Its Application

The dissertation mainly focuses on the representation of the quantum double of the quantum double group algebras. We provide its first orthogonality relation for characters, the formula of the central primitive idempotents and the decompo-sition formula of the tensor product of any two irreducible modules. Especially we calculate and determine all the irreducible modules of the quantum double of quantum double dihedral group algebra D(D(CDm)) and the corresponding cen-tral primitive idempotents. As application, we obtain three new classes of Hopf algebras which take the quantum double dihedral algebras D(CD3) as coradical, one possible finite dimensional Nichols algebras of non-diagonal type, and one class solution of set-theoretical Yang-Baxter equation. All the solutions are not of group type.Firstly, we transform the representations of D(D(CG)) into the study of Yetter-Drinfel’d modules of D(CG) via the equivalence of categories, so we can avoid complicated computation in D(D(CG)). Then we demonstrate all modules of D(D(CG)) are unitary. We provide its first orthogonality relation for char-acters, the formula of the central primitive idempotents and the decomposition formula of the tensor product of any two irreducible modules.Secondly, we determine all the irreducible Yetter-Drinfel’d modules of D(CDm) by making use of the2-cocycle twist of the Quantum Double, and give the cor-responding central primitive idempotents of those irreducible modules. Then we carry out the procedure of Bosonization of Majid, and obtain three classes new Hopf algebras which take D(CD3) as the coradical.Historically, it’s difficult to look for the non-diagonal type finite dimensional Nichols algebras, and all the known examples belong to the Yetter-Drinfel’d cat-egory of group algebras. We define a class solution of the set-theoretical Yang-Baxter equation. All the solutions are not group type and they will provide an approach to finding new non-diagonal type finite dimensional Nichols algebras.