The quasi-Poisson Goldman formula

Nie, Xin

doi:10.1016/j.geomphys.2013.06.010

Cited by 9 publications

(12 citation statements)

References 10 publications

(12 reference statements)

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“…Our construction of brackets yields as special cases the Poisson structures on moduli spaces of representations of surface groups introduced by Atiyah-Bott [3] and studied by Goldman [5,6]. Our construction also yields the quasi-Poisson refinements of those structures due to Alekseev, Kosmann-Schwarzbach and Meinrenken, see [1,10,9,12].…”

Section: Introductionmentioning

confidence: 88%

“…Indeed, formula (12.4) is the algebraic analogue of Goldman's formula [6,Theorem 3.5], where the operation (12.2) appears implicitly; for instance, the formulas (11.9) correspond to [6,. The quasi-Poisson bracket {−, −} in A B is an algebraic version of the quasi-Poisson refinement of the Atiyah-Bott-Goldman bracket introduced in [1] and studied in [9,12]. Indeed, formula (12.3) is the algebraic analogue of the quasi-Poisson refinement of Goldman's formulas obtained by Li-Bland & Ševera [9, Theorem 3] and Nie [12,Theorem 2.5].…”

mentioning

confidence: 99%

“…The quasi-Poisson bracket {−, −} in A B is an algebraic version of the quasi-Poisson refinement of the Atiyah-Bott-Goldman bracket introduced in [1] and studied in [9,12]. Indeed, formula (12.3) is the algebraic analogue of the quasi-Poisson refinement of Goldman's formulas obtained by Li-Bland & Ševera [9, Theorem 3] and Nie [12,Theorem 2.5]. For G = GL N , formula (12.3) was obtained in [10] using Van den Bergh's theory (see Appendix B).…”

mentioning

confidence: 99%

“…For G = GL N , formula (12.3) was obtained in [10] using Van den Bergh's theory (see Appendix B). Note that our proof of the quasi-Jacobi identity in A B (and, consequently, of the Jacobi identity in (A B ) inv ) is purely algebraic and involves neither infinitedimensional methods of [3,5,6] nor the inductive "fusion" method of [1,9,12].…”

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confidence: 99%

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Brackets in representation algebras of Hopf algebras

Massuyeau¹,

Turaev²

2015

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For any graded bialgebras A and B, we define a commutative graded algebra AB representing the functor of B-representations of A. When A is a cocommutative graded Hopf algebra and B is a commutative ungraded Hopf algebra, we introduce a method deriving a Gerstenhaber bracket in AB from a Fox pairing in A and a balanced biderivation in B. Our construction is inspired by Van den Bergh's non-commutative Poisson geometry, and may be viewed as an algebraic generalization of the Atiyah-Bott-Goldman Poisson structures on moduli spaces of representations of surface groups.

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Section: Introductionmentioning

confidence: 88%

mentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Brackets in representation algebras of Hopf algebras

Massuyeau¹,

Turaev²

2015

Preprint

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“…The following lemma is originally due to [AKSM02, AMM98] (see also §6 in [VdB08] for equivalent operation on quasi Poisson brackets for quiver path algebras, and [Nie13] for a good recent review of fusion in the context of surface brackets).…”

Section: 1mentioning

confidence: 99%

Noncommutative Networks on a Cylinder

Ovenhouse¹,

Shapiro

Arthamonov

2020

Preprint

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In this paper a double quasi Poisson bracket in the sense of Van den Bergh is constructed on the space of noncommutative weights of arcs of a directed graph embedded in a disk or cylinder Σ, which gives rise to the quasi Poisson bracket of G.Massuyeau and V.Turaev on the group algebra kπ 1 (Σ, p) of the fundamental group of a surface based at p ∈ ∂Σ. This bracket also induces a noncommutative Goldman Poisson bracket on the cyclic space C ♮ , which is a k-linear space of unbased loops. We show that the induced double quasi Poisson bracket between boundary measurements can be described via noncommutative r-matrix formalism. This gives a more conceptual proof of the result of [Ove20] that traces of powers of Lax operator form an infinite collection of noncommutative Hamiltonians in involution with respect to noncommutative Goldman bracket on C ♮ .

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