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

TL;DR

This paper introduces a geometric interpretation of the irreducible representations of the quantum double of a finite group, linking them to braided-Lie algebras called braided racks, with implications for quantum computing and topological quantum field theories.

Contribution

It develops a geometric framework for understanding irreps of D(G), connecting them to braided racks and braided Hopf algebras, and explores their role in differential structures and wave equations.

Findings

Irreps labeled by conjugacy classes and representations of isotropy groups.

Duality between differential calculus on C(G) and C(G) on the quantum double.

Classification of differential structures and braided-Lie algebras via irreps.

Abstract

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 $H_{C,\pi}$.