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Symplectic structure of the moduli space of flat connection on a Riemann surface. (English) Zbl 0829.53028
The authors consider the canonical symplectic structure on the moduli space $ℳ$ of flat $𝒢$-connections on a Riemann surface of genus $g$ with $n$ marked points. A combinatorial description of this symplectic structure is given and its efficient formula is obtained for the case of a surface with marked points. The relation of the symplectic structure on $ℳ$ and Poisson-Lie symplectic structures are studied. It is shown that for $𝒢$ being a semisimple Lie algebra, the symplectic form may be represented as a sum of $n$ copies of the Kirillov symplectic form on the orbit of dressing transformations in the Poisson- Lie group ${G}^{*}$ (corresponding to the Lie algebra $𝒢$) and $g$ copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group $G$.

##### MSC:
 53C15 Differential geometric structures on manifolds 58D27 Moduli problems for differential geometric structures on spaces of mappings
##### References:
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