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arxiv: 2605.23369 · v1 · pith:BGCLV4ZBnew · submitted 2026-05-22 · 🧮 math.RT · math.RA· math.SG

Quasi-Poisson varieties from double quasi-Poisson algebras in types B,C,D

Semeon Arthamonov , Maxime Fairon This is my paper

Pith reviewed 2026-05-25 03:01 UTC · model grok-4.3

classification 🧮 math.RT math.RAmath.SG
keywords double quasi-Poisson algebrastwisted representation spacesquiver varietiesPoisson structurestypes B C Dinvolutive anti-automorphismHopf algebrascharacter varieties
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The pith

Double quasi-Poisson brackets on associative algebras induce Poisson structures on twisted representation spaces of types B, C and D when a compatible involutive anti-automorphism is present.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a double quasi-Poisson bracket on an associative algebra, when equipped with a compatible involutive anti-automorphism, produces a Poisson bracket on the corresponding twisted representation spaces. This construction works over an arbitrary semisimple base and extends the type-A case to the orthogonal, symplectic and other classical settings. The same formalism is applied to quivers, yielding Poisson structures on twisted localised multiplicative quiver varieties where vertices may carry different types. It recovers the Poisson geometry of character varieties for orthogonal and symplectic groups from Fox pairings on Hopf algebras and defines a modified Kontsevich system.

Core claim

If an associative algebra carries both a double quasi-Poisson bracket and a compatible involutive anti-automorphism, the bracket descends to a Poisson structure on the twisted representation spaces associated with the corresponding classical groups, and the same data produces Poisson structures on the associated twisted quiver varieties.

What carries the argument

The double quasi-Poisson bracket on an associative algebra together with a compatible involutive anti-automorphism that upgrades the natural GL-action to an action of an orthogonal or symplectic group.

If this is right

  • Twisted quiver varieties carry natural Poisson structures that admit multiplicative localisation.
  • A single quiver can mix vertices of different classical types while retaining a global Poisson structure.
  • The Poisson geometry of character varieties for orthogonal and symplectic groups arises uniformly from Fox pairings on Hopf algebras.
  • A modified Kontsevich system of differential equations can be defined on the resulting Poisson varieties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same upgrade technique may apply to other algebraic structures equipped with involutive anti-automorphisms beyond associative algebras and quivers.
  • Poisson structures on representation spaces of all classical groups could be treated in a single framework once the anti-automorphism condition is verified.
  • Deformation quantisation or integrability questions attached to the modified Kontsevich system become accessible through the same algebraic data.

Load-bearing premise

The associative algebra admits a compatible involutive anti-automorphism that interacts with the double quasi-Poisson bracket in the required way.

What would settle it

An explicit double quasi-Poisson algebra with involutive anti-automorphism for which the induced bracket on a twisted representation space violates the Jacobi identity.

Figures

Figures reproduced from arXiv: 2605.23369 by Maxime Fairon, Semeon Arthamonov.

Figure 1
Figure 1. Figure 1: Example of construction of the quivers QΥ,Λ and QΥ,Λ from a quiver Υ and a choice of loops Λ = {c} [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
read the original abstract

Double (quasi-)Poisson brackets were introduced on associative algebras by Van den Bergh to induce a (quasi-)Poisson structure on their representation spaces naturally equipped with a $\mathrm{GL}$-action (type $\mathtt{A}$). If there exists a compatible involutive anti-automorphism on the underlying associative algebras, Olshanski and Safonkin proved that this construction can be upgraded to induce a Poisson structure on twisted representation spaces (types $\mathtt{B},\mathtt{C},\mathtt{D}$). We provide an analogous result for double quasi-Poisson brackets, and over an arbitrary semisimple base. We also apply our theory to quivers in order to understand the Poisson structure on twisted (localised multiplicative) quiver varieties. The formalism permits that different vertices are assigned different types. As a first application, we recover the framework of Massuyeau and Turaev for Hopf algebras with a Fox pairing, which induces in particular the Poisson structure of character varieties for the orthogonal or symplectic groups. As a second application, we introduce a modified Kontsevich system.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper extends Van den Bergh's double (quasi-)Poisson bracket construction on associative algebras (type A) to types B, C, D by assuming a compatible involutive anti-automorphism, working over arbitrary semisimple bases. It applies the resulting quasi-Poisson structures to twisted representation spaces of quivers (allowing mixed types per vertex) to obtain Poisson structures on twisted localized multiplicative quiver varieties. Two applications are given: recovery of the Massuyeau-Turaev framework for Hopf algebras equipped with Fox pairings (hence Poisson structures on orthogonal/symplectic character varieties), and introduction of a modified Kontsevich system.

Significance. If the central construction holds, the work supplies a uniform algebraic mechanism for inducing Poisson structures on representation varieties attached to classical groups and mixed-type quivers. The extension to arbitrary semisimple bases and the explicit recovery of the Massuyeau-Turaev setting are concrete strengths; the mixed-type quiver application and modified Kontsevich system are new and potentially useful for further geometric study.

major comments (2)
  1. [§3] §3 (main theorem): the proof that the involutive anti-automorphism upgrades the double quasi-Poisson bracket to a quasi-Poisson structure on the twisted representation space is asserted by analogy with Olshanski-Safonkin; an explicit verification that the quasi-Poisson identity is preserved under the twisted GL-action is needed, as the quasi-Poisson case introduces an extra cocycle term absent in the Poisson setting.
  2. [§4.2] §4.2 (quiver application): the claim that the formalism permits arbitrary assignment of types B/C/D to vertices relies on the base being semisimple; it is unclear whether the resulting bracket remains non-degenerate or satisfies the quasi-Poisson identity when the type assignment is not constant across connected components of the quiver.
minor comments (2)
  1. [§2] Notation for the twisted representation space (e.g., the precise definition of the involution-induced action) should be introduced before the statement of the main theorem rather than in the applications section.
  2. [§5] The modified Kontsevich system in §5 is introduced without a comparison table to the classical Kontsevich bracket; adding such a table would clarify the modification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each major comment below and plan to incorporate revisions as indicated.

read point-by-point responses

  1. Referee: [§3] §3 (main theorem): the proof that the involutive anti-automorphism upgrades the double quasi-Poisson bracket to a quasi-Poisson structure on the twisted representation space is asserted by analogy with Olshanski-Safonkin; an explicit verification that the quasi-Poisson identity is preserved under the twisted GL-action is needed, as the quasi-Poisson case introduces an extra cocycle term absent in the Poisson setting.

    Authors: We agree that the current proof sketch relies on analogy and that the presence of the cocycle term in the quasi-Poisson setting necessitates an explicit check. We will revise §3 to include a detailed verification of the quasi-Poisson identity under the twisted action, adapting the Olshanski-Safonkin approach to account for this term explicitly. revision: yes

  2. Referee: [§4.2] §4.2 (quiver application): the claim that the formalism permits arbitrary assignment of types B/C/D to vertices relies on the base being semisimple; it is unclear whether the resulting bracket remains non-degenerate or satisfies the quasi-Poisson identity when the type assignment is not constant across connected components of the quiver.

    Authors: The semi-simplicity of the base allows the twisted representation space to be treated as a product over vertices, with the bracket defined independently at each vertex. Thus, the quasi-Poisson identity holds regardless of whether types are constant across connected components. Non-degeneracy is not asserted in general and may require additional assumptions; we will add a clarifying paragraph in §4.2 to make this explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extends the Olshanski-Safonkin result on upgrading Poisson structures via involutive anti-automorphisms to the setting of double quasi-Poisson brackets over arbitrary semisimple bases, then applies the construction to twisted quiver varieties (including mixed types). The central claim is a structural theorem whose hypotheses (existence of the anti-automorphism) and conclusions are stated independently of any fitted parameters, self-referential equations, or load-bearing self-citations. Applications to Hopf algebras with Fox pairings and a modified Kontsevich system are presented as consequences rather than redefinitions of inputs. No derivation step reduces by construction to its own inputs, and the result is self-contained against the referenced external construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The construction rests on the standard definition of double quasi-Poisson brackets and the domain assumption of a compatible involution; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Double quasi-Poisson brackets satisfy the axioms introduced by Van den Bergh
    The paper builds directly on this prior definition.
  • domain assumption Existence of a compatible involutive anti-automorphism on the algebra
    Required to upgrade the construction to twisted spaces in types B,C,D.

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Works this paper leans on

33 extracted references · 27 canonical work pages · 15 internal anchors

  1. [1]

    Preprint,arXiv:1804.09566

    Alekseev, A.; Kawazumi, N.; Kuno, Y.; Naef, F.:The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera. Preprint,arXiv:1804.09566

    work page arXiv

  2. [2]

    Alekseev, A.; Kosmann-Schwarzbach, Y.; Meinrenken, E.:Quasi-Poisson manifolds. Canad. J. Math. 54, 3–29 (2002);arXiv:math/0006168

    work page internal anchor Pith review Pith/arXiv arXiv 2002

  3. [3]

    Arthamonov, S.:Noncommutative inverse scattering method for the Kontsevich system. Lett. Math. Phys. 105, 1223–1251 (2015);arXiv:1410.5493

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  4. [4]

    Arthamonov, S.:Modified double Poisson brackets. J. Algebra 492, 212–233 (2017);arXiv:1608.08287

    work page arXiv 2017

  5. [5]

    Bocklandt, R.: A slice theorem for quivers with an involution. J. Algebra Appl. 9, 339–363 (2010);arXiv:0711.0165

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  6. [6]

    Crawley-Boevey, W.:Poisson structures on moduli spaces of representations. J. Algebra 325, 205–215 (2011)

    2011

  7. [7]

    Crawley-Boevey, W.; Shaw, P.:Multiplicative preprojective algebras, middle convolution and the Deligne-Simpson problem. Adv. Math. 201, no. 1, 180–208 (2006);arXiv:math/0404186

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  8. [8]

    Preprint,arXiv:2012.04451

    D’Alesio, S.:Noncommutative derived Poisson reduction. Preprint,arXiv:2012.04451

    work page arXiv 2012

  9. [9]

    Derksen, H.; Weyman, J.:Generalized quivers associated to reductive groups. Colloq. Math. 94, 151-173 (2002). QUASI-POISSON VARIETIES FROM DOUBLE QUASI-POISSON ALGEBRAS IN TYPESB, C, D43

    2002

  10. [10]

    Efimovskaya, O.; Wolf, T.:On integrability of the Kontsevich non-Abelian ODE system. Lett. Math. Phys. 100, 161–170 (2012);arXiv:1108.4208

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  11. [11]

    Fairon, M.:Double quasi-Poisson brackets : fusion and new examples. Algebr. Represent. Theory 24, no. 4, 911–958 (2021);arXiv:1905.11273

    work page arXiv 2021

  12. [12]

    Fairon, M.:Morphisms of double (quasi-)Poisson algebras and action-angle duality of integrable systems. Ann. H. Lebesgue 5, 179–262 (2022);arXiv:2008.01409

    work page arXiv 2022

  13. [13]

    Fairon, M.; McCulloch, C.:Around Van den Bergh’s double brackets for different bimodule structures. Comm. Algebra 51, no.4, 1673–1706 (2023); arXiv:2204.03298

    work page arXiv 2023

  14. [14]

    In: Moscow Seminar in Mathematical Physics, volume 191 of Amer

    Fock, V.; Rosly, A.:Poisson structure on moduli of flat connections on Riemann surfaces and ther-matrix. In: Moscow Seminar in Mathematical Physics, volume 191 of Amer. Math. Soc. Transl. Ser. 2, 67–86. Amer. Math. Soc., Providence, RI (1999);arXiv:math/9802054

    work page arXiv 1999

  15. [15]

    Gaiotto, D.; Witten, E.:S-duality of boundary conditions in N=4 super Yang-Mills theory. Adv. Theor. Math. Phys. 13, No. 3, 721–896 (2009);arXiv:0807.3720

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  16. [16]

    Goldman, W.:The symplectic nature of fundamental groups of surfaces. Adv. Math. 54, 200–225 (1984)

    1984

  17. [17]

    The Gelfand Mathematical Seminars, 1996–1999, Gelfand Math

    Kontsevich, M.; Rosenberg, A.L.:Noncommutative smooth spaces. The Gelfand Mathematical Seminars, 1996–1999, Gelfand Math. Sem., Birkh¨ auser, Boston, pp. 85–108 (2000)

    1996

  18. [18]

    Lawton, S.; Sikora, A.S.: Varieties of characters. Algebr. Represent. Theory 20, No. 5, 1133–1141 (2017); arXiv:1604.02164

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  19. [19]

    Quiver varieties and symmetric pairs

    Li, Y.:Quiver varieties and symmetric pairs. Represent. Theory 23, 1–56 (2019);arXiv:1801.06071

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  20. [20]

    Li-Bland, D.; ˇSevera, P.:Moduli spaces for quilted surfaces and Poisson structures. Doc. Math. 20, 1071–1135 (2015);arXiv:1212.2097

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  21. [21]

    Preprint; arXiv:2601.13455

    Maiza, M.M.:From multiplicative to additive geometry: deformation theory and 2D TQFT. Preprint; arXiv:2601.13455

    work page arXiv

  22. [22]

    Preprint;arXiv:2510.23567

    Maiza, M.M.; Mayrand, M.:Lax-Kirchhoff moduli spaces and Hamiltonian 2D TQFT. Preprint;arXiv:2510.23567

    work page arXiv

  23. [23]

    Massuyeau, G.; Turaev, V.:Quasi-Poisson structures on representation spaces of surfaces. Int. Math. Res. Not. IMRN, no. 1, 1–64 (2014);arXiv:1205.4898

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  24. [24]

    Massuyeau, G.; Turaev, V.:Brackets in representation algebras of Hopf algebras. J. Noncommut. Geom. 12, No. 2, 577–636 (2018);arXiv:1508.07566. [MS] Mikhailov, A.V.; Sokolov, V.V.:Integrable ordinary differential equations on free associative algebras. Theoret. Math. Phys. 122, 72–83 (2000);arXiv:solv-int/9908004

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  25. [25]

    Duke Math

    Nakajima, H.:Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras. Duke Math. J. 76, no.2, 365–416, (1994)

    1994

  26. [26]

    Preprint;arXiv:2510.13007

    Nakajima, H.:Instantons on ALE spaces for classical groups, involutions on quiver varieties, and quantum sym- metric pairs. Preprint;arXiv:2510.13007

    work page arXiv

  27. [27]

    :The quasi-Poisson Goldman formula

    Nie, X. :The quasi-Poisson Goldman formula. J. Geom. Phys. 74, 1–17 (2013);arXiv:1301.5231

    work page arXiv 2013

  28. [28]

    Olshanski, G.; Safonkin, N.:Double Poisson brackets and involutive representation spaces. Lett. Math. Phys. 114, 33 (2024);arXiv:2310.01086

    work page arXiv 2024

  29. [29]

    Safonkin, N.:private communication

  30. [30]

    Turaev, V.:Poisson-Gerstenhaber brackets in representation algebras. J. Algebra 402, 435–478 (2014); arXiv:1306.3859

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  31. [31]

    Van den Bergh, M.:Double Poisson algebras. Trans. Amer. Math. Soc., 360, no. 11, 5711–5769 (2008); arXiv:math/0410528

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  32. [32]

    Non-commutative quasi-Hamiltonian spaces

    Van den Bergh, M.:Non-commutative quasi-Hamiltonian spaces. In: Poisson geometry in mathematics and physics. Contemp. Math. 450, Amer. Math. Soc., Providence, RI, pp. 273–299 (2008);arXiv:math/0703293

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  33. [33]

    Zubkov, A.N.:Invariants of mixed representations of quivers I. J. Algebra Appl. 4, 245–285 (2005); arXiv:math/0305386. (Maxime Fairon)Universit ´e Bourgogne Europe, CNRS, IMB UMR 5584, F-21000 Dijon, France Email address:Maxime.Fairon@u-bourgogne.fr (Semeon Arthamonov)Beijing Institute of Mathematical Sciences and Applications, China Email address:arthamo...

    work page internal anchor Pith review Pith/arXiv arXiv 2005