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

Abstract

We quantise a Poisson structure on H^{n+2g}, where H is a semidirect product group of the form $G⋉{\mathfrak{g}}^{\ast }$. This Poisson structure arises in the combinatorial description of the phase space of Chern-Simons theory with gauge group $G⋉{\mathfrak{g}}^{\ast }$ 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 $G⋉{\mathfrak{g}}^{\ast }$. 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.

35 pages, one eps figure; minor corrections