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

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arxiv math/9802054 v2 pith:RCPCX5OW submitted 1998-02-10 math.QA hep-thmath.GRmath.RT

V.V.Fock , A.A.Rosly This is my paper · ORCID
classification math.QAhep-thmath.GRmath.RT
keywords graphpoissonspacestructureconnectionsmoduliciliatedendowed
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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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Cited by 3 Pith papers

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  1. On the structure and representations of quantum graph algebras at roots of unity

    math.QA 2026-01 accept novelty 7.0 of 10

    Root-of-unity quantum graph algebras have irreducible representations of maximal dimension l^{g·dim(g)+n·N}, and their small-quantum-group invariants have maximal dimension l^{g·dim(g)+N(n−1)−m}, with centers describe...

  2. Integrable systems from Poisson reductions of generalized Hamiltonian torus actions

    math-ph 2025-07 unverdicted novelty 7.0 of 10

    Develops sufficient conditions for Poisson reduction of generalized Hamiltonian torus actions to preserve integrability and applies them to open problems on Lie group doubles and flat-connection moduli spaces.

  3. Entanglement and geometric transitions in topological string theory

    hep-th 2026-07 conditional novelty 5.5 of 10

    Open-string anyonic entanglement on entanglement branes equals gravitational replica entropy of the dual closed-string Calabi–Yau via geometric transitions in the A-model TQFT.

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