The quantisation of Poisson structures arising in Chern-Simons theory with gauge group $G\ltimes \mathfrak{g}^*$

TL;DR

This paper develops a quantisation of a specific Poisson structure related to Chern-Simons theory with a semidirect product gauge group, constructing the quantum algebra and revealing the quantum double D(G) as a symmetry.

Contribution

It introduces a novel quantisation method for Poisson structures in Chern-Simons theory with gauge group G⋉g* and identifies the quantum double D(G) as a natural symmetry.

Findings

Constructed the quantum algebra for the Poisson structure.

Derived irreducible representations of the quantum algebra.

Identified the quantum double D(G) as a symmetry of the quantum algebra.

Abstract

We quantise a Poisson structure on H^{n+2g}, where H is a semidirect product group of the form $G\ltimes\mathfrak{g}^*$. This Poisson structure arises in the combinatorial description of the phase space of Chern-Simons theory with gauge group $G\ltimes\mathfrak{g}^*$ on $R \times S_{g,n}$, where S_{g,n} is a surface of genus g with n punctures. The quantisation of this Poisson structure is a key step in the quantisation of Chern-Simons theory with gauge group $G\ltimes\mathfrak{g}^*$. We construct the quantum algebra and its irreducible representations and show that the quantum double D(G) of the group G arises naturally as a symmetry of the quantum algebra.