Decorated quantum character variety

We introduce the notion of decorated character variety to generalize the Betti moduli space. This decorated character variety is the quotient of the space of representations of the fundamental groupoid of arcs by a product of unipotent Borel sub-groups (one per each bordered cusp). We demonstrate that this representation space is endowed with a Poisson structure induced by the Fock{Rosly-type bracket and show that the quotient by unipotent Borel subgroups giving rise to the decorated character variety is a Poisson reduction. We present a construction of quantum decorated character varieties for $SL(k;C)$ on any Riemann surface $\Sigma_{g;s;n}$ of genus $g, s > 0$ holes, and with $n > 0$ bordered cusps endowed with Borel unipotent radicals, based on elementary "quantum triangle relation’’ $M_1^{(1)} M_2^{(2)} = M_2^{(2)} M_1^{(1)} R_{12}(q)$