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

The quantum double D(G)=Bbb C(G)rtimes Bbb 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 Poincaré 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 Poincaré group of Bbb R^1,3. Irreps are labelled by pairs (C, π), where C is a conjugacy class in the role of a mass-shell, and π is a representation of the isotropy group C_G in the role of spin. The geometric picture entails D^vee(G)→ Bbb C\(C_G\)blacktriangleright< Bbb C G as a quantum homogeneous bundle where the base is G/C_G, and D^vee(G)→ Bbb C(G) as another homogeneous bundle where the base is the group algebra Bbb C G as noncommutative spacetime. Analysis of the latter leads to a duality whereby the differential calculus and solutions of the wave equation on Bbb C G are governed by irreps and conjugacy classes of G respectively, while the same picture on Bbb C(G) is governed by the reversed data. Quasiparticles as irreps of D(G) also turn out to classify irreducible bicovariant differential structures Ω¹_{C, π} on D^vee(G) and these in turn correspond to braided-Lie algebras mathcal{L}_{C, π} in the braided category of G-crossed modules, which we call `braided racks' and study. We show under mild assumptions that U\(mathcal{L}_C,π\) quotients to a braided Hopf algebra B_C,π related by transmutation to a coquasitriangular Hopf algebra H_C,π.