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Zbl 0829.53028
Alekseev, A.Yu.; Malkin, A.Z.
Symplectic structure of the moduli space of flat connection on a Riemann surface. (English)
[J] Commun. Math. Phys. 169, No.1, 99-119 (1995). ISSN 0010-3616; ISSN 1432-0916/e

The authors consider the canonical symplectic structure on the moduli space $\cal M$ of flat $\cal G$-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 $\cal M$ and Poisson-Lie symplectic structures are studied. It is shown that for $\cal G$ 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 $\cal G$) and $g$ copies of the symplectic structure on the Heisenberg double of the Poisson-Lie group $G$.
[St.Janeczko (Warszawa)]

MSC 2000:: *53C15 Geometric structures on manifolds
58D27 Moduli problems for diff.geometric structures on spaces of mappings

Keywords: moduli space; Kirillov symplectic form; Poisson-Lie group; Heisenberg double

Cited in: Zbl 0835.58015

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