Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix

Details Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix Poisson structure on moduli of flat connections on Riemann surfaces and $r$-matrix

1998-02-10

arxiv.org/abs/math/9802054v2

Am.Math.Soc.Transl. 191 (1999) 67-86

Mathematics

Quantum Algebra

LaTeX, 26 pages 12 .eps figures; a reference added

Scientific paper

We consider the space of graph connections (lattice gauge fields) which can be endowed with a Poisson structure in terms of a ciliated fat graph. (A ciliated fat graph is a graph with a fixed linear order of ends of edges at each vertex.) Our aim is however to study the Poisson structure on the moduli space of locally flat vector bundles on a Riemann surface with holes (i.e. with boundary). It is shown that this moduli space can be obtained as a quotient of the space of graph connections by the Poisson action of a lattice gauge group endowed with a Poisson-Lie structure. The present paper contains as a part an updated version of a 1992 preprint ITEP-72-92 which we decided still deserves publishing. We have removed some obsolete inessential remarks and added some newer ones.

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