Quantum geometric Wigner construction for D (G) and braided racks

The quantum double D (G) = C (G) C G of a finite group plays an important role in the Kitaev model for quantum computing, as well as in associated TQFT's, as a kind of Poincar\'e group. We interpret the known construction of its irreps, which are quasiparticles for the model, in a geometric manner strictly analogous to the Wigner construction for the usual Poincar\'e group of R^1, 3. Irreps are labelled by pairs (C, ), where C is a conjugacy class in the role of a mass-shell, and is a representation of the isotropy group CG in the role of spin. The geometric picture entails D^ (G) C (CG) \!\!\!\!< C G as a quantum homogeneous bundle where the base is G/CG, and D^ (G) C (G) as another homogeneous bundle where the base is the group algebra C G as noncommutative spacetime. Analysis of the latter leads to a duality whereby the differential calculus and solutions of the wave equation on C G are governed by irreps and conjugacy classes of G respectively, while the same picture on C (G) is governed by the reversed data. Quasiparticles as irreps of D (G) also turn out to classify irreducible bicovariant differential structures ¹₂, on D^ (G) and these in turn correspond to braided-Lie algebras L₂, in the braided category of G-crossed modules, which we call `braided racks' and study. We show under mild assumptions that U (L₂, ) quotients to a braided Hopf algebra B₂, related by transmutation to a coquasitriangular Hopf algebra H₂,.