## Advances in Theoretical and Mathematical Physics

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

#### 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 $\mathbb{R}\times S_{g,n}$, 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.

#### Article information

Source
Adv. Theor. Math. Phys., Volume 7, Number 6 (2003), 1003-1043.

Dates
First available in Project Euclid: 21 June 2004

Meusburger, C.; Schroers, B. J. The quantisation of Poisson structures arising in Chern-Simons theory with gauge group $G \ltimes \mathfrak{g}^*$. Adv. Theor. Math. Phys. 7 (2003), no. 6, 1003--1043. https://projecteuclid.org/euclid.atmp/1087840803