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The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular $r$-Matrix

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Victor Mouquin

Differential Geometry · Submitted 6 Oct 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular $r$-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs of Poisson actions, which were studied by J.-H. Lu and the author.

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arXiv ID1610.01782 (version 2)
Pith IDNSUOUFYV, Pith's permanent record of this paper and its review
CitationVictor Mouquin. "The Fock-Rosly Poisson Structure as Defined by a Quasi-Triangular $r$-Matrix." arXiv:1610.01782 (2016). https://pith.science/paper/NSUOUFYV
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,488 input tokens.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.02 · compares with prior results 0.90 · mentions code or data availability 0.02.

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We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular $r$-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs of Poisson actions, which were studied by J.-H. Lu and the author. The Fock-Rosly Poisson structure corresponds to the quasi-Poisson structure studied by Massuyeau, Turaev, Li-Bland, and Severa under an equivalence of categories between Poisson and quasi-Poisson spaces.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular $r$-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs of Poisson actions, which were studied by J.-H. Lu and the author. contribution 0.97 0.93 0.75 asserted 1.00
s1 The Fock-Rosly Poisson structure corresponds to the quasi-Poisson structure studied by Massuyeau, Turaev, Li-Bland, and Severa under an equivalence of categories between Poisson and quasi-Poisson spaces. result 0.74 0.40 0.24 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.94.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

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Original abstract

We reformulate the Poisson structure discovered by Fock and Rosly on moduli spaces of flat connections over marked surfaces in the framework of Poisson structures defined by Lie algebra actions and quasitriangular $r$-matrices, and we show that it is an example of a mixed product Poisson structure associated to pairs of Poisson actions, which were studied by J.-H. Lu and the author. The Fock-Rosly Poisson structure corresponds to the quasi-Poisson structure studied by Massuyeau, Turaev, Li-Bland, and Severa under an equivalence of categories between Poisson and quasi-Poisson spaces.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 18 sentences without a confident label

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Cites as background 6 papers
  • We recall in this section basic facts about quasitriangular $r$-matrices and refer to [ref] for a detailed exposition on Poisson Lie groups and Lie bialgebras.

    Mixed product Poisson structures associated to Poisson Lie groups and Lie bialgebras · Poisson structures defined by quasitriangular $r$-matrices · 1.00

  • If one “marks” finitely many points $V ⊂Σ$ in the boundary of $Σ$ and consider only gauge transformations which are trivial over $V$, Fock and Rosly have constructed in [ref] a Poisson structure $π_r$ on the corresponding moduli space $(Σ, V)$, which depends on a quasitriangular $r$-matrix $r_v$ for every $v ∈V$ such that all $r_v$'s have symmetric part $s$.

    math.QA/9802054 · Introduction · 0.85

  • Our main references are [ref] .

    Moduli spaces for quilted surfaces and Poisson structures · Ciliated graphs and marked surfaces · 0.99

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