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

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