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The (2+1)-dimensional analog self-dual gravity which is obtained viadimension reduction of the (3+1)-dimensional Holst action without reducing theinternal gauge group is studied. A Chern-Simons formulation for this theory isconstructed based on the gauge group $SL(2,\CC)_\RR\rcross \Rsix$ and maps the3d complex self-dual dynamical variable and connection to $6d$ real variableswhich combines into a $12d$ Cartan connection. Quantization is given by theapplication of the combinatorial quantisation program of Chern-Simons theory.The Poisson structure for the moduli space of flat connections on$(SL(2,\CC)_\RR\rcross \Rsix)^{n+2g}$ which emerges in the combinatorialdescription of the phase space on $\RR \times \Sigma_{g,n},$ where$\Sigma_{g,n}$ is a genus $g$ surface with $n$ punctures is given in terms ofthe classical $r$-matrix for the quantum double $D(SL(2,\CC)_\RR)$ viewed asthe double of a double $ D(SU(2)\dcross AN(2))$. This quantum double provides afeature for quantum symmetries of the quantum theory for the model.

A Chern-Simons approach to self-dual gravity in (2+1)-dimensions and quantisation of Poisson structure