  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 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 $\Bbb R^{1,3}$. Irreps are labelled by pairs $(C, \pi)$,
where $C$ is a conjugacy class in the role of a mass-shell, and $\pi$ is a
representation of the isotropy group $C_G$ in the role of spin. The geometric
picture entails $D^\vee(G)\to \Bbb C(C_G)\blacktriangleright\!\!\!\!< \Bbb C G$
as a quantum homogeneous bundle where the base is $G/C_G$, and $D^\vee(G)\to
\Bbb C(G)$ 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 $\Omega^1_{C, \pi}$ on $D^\vee(G)$ and
these in turn correspond to braided-Lie algebras $\mathcal{L}_{C, \pi}$ in the
braided category of $G$-crossed modules, which we call `braided racks' and
study. We show under mild assumptions that $U(\mathcal{L}_{C,\pi})$ quotients
to a braided Hopf algebra $B_{C,\pi}$ related by transmutation to a
coquasitriangular Hopf algebra $H_{C,\pi}$.
