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 R1,3. 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 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 Ω¹C, π on D⁽G) and these in turn correspond to braided-Lie algebras LC, π in the braided category of G-crossed modules, which we call `braided racks' and study. We show under mild assumptions that U(LC,π) quotients to a braided Hopf algebra BC,π related by transmutation to a coquasitriangular Hopf algebra HC,π.