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# Advances in Theoretical and Mathematical Physics

## Volume 7 (2003)

### Number 6

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

Pages: 1003 – 1043

DOI: http://dx.doi.org/10.4310/ATMP.2003.v7.n6.a3

#### Authors

#### 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.