Representations of the quantum double of quasitriangular Hopf algebras

Conjugacy classes for groups were generalized in previous works of the authors to semisimple Hopf algebras over algebraically closed fields of characteristic zero. An essential property of the set of conjugacy classes {C_j} is that they are irreducible representations of the quantum double D(H) of H. We show that more is true. When (H, R) is quasitriangular, then any irreducible representations of D(H) is a direct summand of a tensor product of the form of {C_j} V_i, where V_i is an irreducible representation of H. The proof follows by applying a partition defined in our 2011 paper on the indices of the irreducible characters of D(H) and identifying the partition with the equivalence classes obtained from Nichols-Richmond equivalence relations on the set of the simple subcoalgebra of D(H)^∗. For any block of this partition, there is a unique C_j whose character as a D(H)-representation belongs to it. When V is a 1-dimensional representation of H then C_j ⊗ V is an irreducible representation of D(H), isomorphic to C_j as a vector space endowed with a twisted action of D(H). We apply the above results to H=kG, where G are the groups S₃ and D₄, and decompose all C_j ⊗ V_i into their irreducible components.