The quantisation of Poisson structures arising in Chern - Simons theory with gauge group G ⋉ g ∗

We quantise a Poisson structure on Hn+2g, where H is a semidirect product group of the form G ⋉ g. This Poisson structure arises in the combinatorial description of the phase space of Chern-Simons theory with gauge group G ⋉ g on R × Sg,n, where Sg,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 ⋉ 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.