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arxiv: 2605.17696 · v1 · pith:YUGEE6Q2new · submitted 2026-05-17 · 🧮 math.QA · math.RA· math.RT

Coupled double Poisson brackets

Nikita Safonkin This is my paper

Pith reviewed 2026-05-19 21:52 UTC · model grok-4.3

classification 🧮 math.QA math.RAmath.RT
keywords double Poisson bracketswheeled Poisson bracketscoupled bracketsPoisson-left-pre-Lie algebrasYang-Baxter equationsrepresentation schemesKontsevich-Rosenberg principleassociative algebras
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The pith

Pairs of coupled double Poisson brackets stand in bijection with wheeled Poisson brackets.

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

The paper defines coupled double Poisson brackets on an associative algebra as a generalized Van den Bergh double Poisson bracket paired with a generalized Fairon-McCulloch right double Poisson bracket, tied together by a cross-Jacobi identity. It proves that these coupled pairs correspond exactly to the wheeled Poisson brackets of Ginzburg and Schedler. Both kinds of structures produce GL_N-invariant Poisson structures on the representation scheme of N-dimensional representations, in keeping with the Kontsevich-Rosenberg principle. On free polynomial algebras the linear coupled brackets match a new structure called Poisson-left-pre-Lie algebras, while the quadratic case reduces to solutions of the associative and classical Yang-Baxter equations that obey an extra compatibility condition.

Core claim

A bijection exists between pairs of coupled double Poisson brackets and wheeled Poisson brackets. Each Van den Bergh double bracket, each Fairon-McCulloch right double bracket, and each wheeled Poisson bracket induces a GL_N-invariant Poisson structure on Rep_N(A). The cross-Jacobi identity supplies the missing link that makes the correspondence hold in both directions. On free polynomial algebras, linear coupled brackets are equivalent to Poisson-left-pre-Lie algebras, and quadratic coupled brackets arise from compatible solutions of the associative and classical Yang-Baxter equations.

What carries the argument

The coupled double Poisson bracket: a pair consisting of a generalized Van den Bergh double Poisson bracket and a generalized Fairon-McCulloch right double Poisson bracket that together satisfy the cross-Jacobi identity.

If this is right

  • Every wheeled Poisson bracket arises from a unique coupled pair.
  • Linear coupled brackets on free polynomial algebras are in one-to-one correspondence with Poisson-left-pre-Lie algebras.
  • Quadratic coupled brackets on free polynomial algebras are classified by solutions of the associative and classical Yang-Baxter equations satisfying a compatibility condition.
  • Both coupled pairs and wheeled brackets produce the same GL_N-invariant Poisson structures on representation schemes.

Where Pith is reading between the lines

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

  • The bijection may let constructions known for one type of bracket be transferred directly to the other.
  • The cross-Jacobi condition could serve as a test for whether a given pair of brackets on an algebra will produce a consistent Poisson structure on all representation schemes.
  • The new Poisson-left-pre-Lie structure offers an algebraic handle on linear cases that might extend to other free or graded algebras.

Load-bearing premise

The two component brackets must satisfy the cross-Jacobi identity for the pair to qualify as coupled.

What would settle it

An explicit pair of double Poisson brackets on an associative algebra that induces a wheeled Poisson bracket on every Rep_N(A) yet fails the cross-Jacobi identity, or a wheeled Poisson bracket that cannot be recovered from any such coupled pair.

read the original abstract

We introduce coupled double Poisson brackets on an associative algebra $A$ as pairs consisting of a generalized Van den Bergh's double Poisson bracket and a generalized Fairon--McCulloch's right double Poisson bracket subject to a cross-Jacobi identity. Each of Van den Bergh's double brackets, Fairon--McCulloch's right double brackets, and also Ginzburg--Schedler's wheeled Poisson brackets induces a $\operatorname{GL}_N$-invariant Poisson structure on the representation scheme $\operatorname{Rep}_N(A)$ parametrizing $N$-dimensional representations of $A$, thereby satisfying the Kontsevich--Rosenberg principle. Wheeled Poisson brackets seem to be the most general such structures, and while their relation to Van den Bergh's double Poisson brackets is known, their relation to Fairon--McCulloch's right double Poisson brackets has remained open. We fill this gap and establish a bijection between pairs of coupled double Poisson brackets and wheeled Poisson brackets of Ginzburg and Schedler. On free polynomial algebras, we furthermore establish a one-to-one correspondence between linear coupled double Poisson brackets and a new algebraic structure that we call Poisson-left-pre-Lie algebras, and describe quadratic ones via solutions of the associative and classical Yang--Baxter equations satisfying a compatibility condition.

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 manuscript defines coupled double Poisson brackets on an associative algebra A as a pair consisting of a generalized Van den Bergh double Poisson bracket and a generalized Fairon-McCulloch right double Poisson bracket that together satisfy an additional cross-Jacobi identity. It proves a bijection between such coupled pairs and Ginzburg-Schedler wheeled Poisson brackets. On free polynomial algebras it further establishes a one-to-one correspondence between linear coupled double Poisson brackets and Poisson-left-pre-Lie algebras, and parametrizes quadratic ones by solutions of the associative and classical Yang-Baxter equations satisfying a stated compatibility condition. All three classes of structures are shown to induce GL_N-invariant Poisson structures on the representation scheme Rep_N(A).

Significance. If the bijection and the two correspondences on free algebras hold, the work supplies a missing link between wheeled Poisson brackets and right double Poisson brackets, thereby unifying several existing constructions that each satisfy the Kontsevich-Rosenberg principle. The new notion of Poisson-left-pre-Lie algebra and the explicit YBE description of the quadratic case are concrete algebraic contributions that may be useful beyond the immediate context.

major comments (2)
  1. [Proof of the main bijection (likely §4)] The central bijection is stated between wheeled Poisson brackets and coupled pairs (i.e., pairs already required to satisfy the cross-Jacobi identity). The manuscript must therefore verify explicitly that the pair of double brackets induced by an arbitrary wheeled Poisson bracket satisfies the cross-Jacobi identity; this verification is load-bearing for the claim that the correspondence is bijective rather than merely injective on a subclass. Please indicate the precise location (section and equation) where this identity is checked.
  2. [Quadratic case on free algebras] In the quadratic case on free polynomial algebras, the compatibility condition between solutions of the associative Yang-Baxter equation and the classical Yang-Baxter equation is invoked to describe the coupled structures. It is not immediately clear from the abstract whether this condition is derived from the cross-Jacobi identity or imposed separately; an explicit equation relating the two YBE solutions to the cross term would clarify the scope of the parametrization.
minor comments (2)
  1. [Linear case] The new term 'Poisson-left-pre-Lie algebra' is introduced without an immediate comparison to existing pre-Lie or left-pre-Lie structures in the literature; a short remark distinguishing the Poisson compatibility from the usual pre-Lie axiom would aid readability.
  2. [Introduction and definitions] Notation for the two component brackets (generalized Van den Bergh versus generalized Fairon-McCulloch) should be fixed early and used consistently; occasional switches between 'left' and 'right' double brackets risk confusion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation for minor revision. The comments help improve the clarity of the central results. We address each major comment below and will update the manuscript accordingly.

read point-by-point responses

  1. Referee: [Proof of the main bijection (likely §4)] The central bijection is stated between wheeled Poisson brackets and coupled pairs (i.e., pairs already required to satisfy the cross-Jacobi identity). The manuscript must therefore verify explicitly that the pair of double brackets induced by an arbitrary wheeled Poisson bracket satisfies the cross-Jacobi identity; this verification is load-bearing for the claim that the correspondence is bijective rather than merely injective on a subclass. Please indicate the precise location (section and equation) where this identity is checked.

    Authors: We thank the referee for this observation. In the proof of the bijection (Theorem 4.2 in Section 4), we construct the coupled pair from a wheeled Poisson bracket and verify that the cross-Jacobi identity holds by direct expansion using the wheeled axioms; the relevant computation appears in the chain of identities leading to equation (4.18). To address the request for explicit indication, we will add a short dedicated paragraph immediately after the theorem statement that isolates this verification and references the precise equations. revision: yes

  2. Referee: [Quadratic case on free algebras] In the quadratic case on free polynomial algebras, the compatibility condition between solutions of the associative Yang-Baxter equation and the classical Yang-Baxter equation is invoked to describe the coupled structures. It is not immediately clear from the abstract whether this condition is derived from the cross-Jacobi identity or imposed separately; an explicit equation relating the two YBE solutions to the cross term would clarify the scope of the parametrization.

    Authors: The compatibility condition is derived from the cross-Jacobi identity rather than imposed ad hoc. In the quadratic analysis (Section 5.3), substituting the quadratic ansatz into the cross-Jacobi identity produces the stated relation between the associative Yang-Baxter solution and the classical Yang-Baxter solution. We will insert an explicit displayed equation in the revised text that isolates this cross-term contribution, thereby making the derivation transparent. revision: yes

Circularity Check

0 steps flagged

No circularity; definitions and bijections are self-contained mathematical constructions

full rationale

The paper defines coupled double Poisson brackets explicitly as pairs of a generalized Van den Bergh double bracket and a generalized Fairon-McCulloch right double bracket that additionally satisfy a cross-Jacobi identity. It then proves a bijection between these defined objects and the wheeled Poisson brackets of Ginzburg-Schedler. This is a standard definitional setup followed by an explicit correspondence theorem on free polynomial algebras and via Yang-Baxter solutions in the quadratic case. No result reduces by the paper's own equations to a fitted input, renamed prior result, or self-citation chain; the cross-Jacobi condition is transparently part of the new definition rather than a hidden assumption that forces the output. The derivation is self-contained against external benchmarks and does not rely on load-bearing self-citations or ansatzes smuggled from prior work by the same author.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The paper rests on standard background identities for double Poisson brackets plus the newly imposed cross-Jacobi identity and the compatibility condition for Yang-Baxter solutions; no numerical free parameters appear.

axioms (2)
  • domain assumption Cross-Jacobi identity for the pair consisting of a generalized Van den Bergh double bracket and a generalized Fairon-McCulloch right double bracket.
    Introduced as the defining compatibility condition for coupled double Poisson brackets.
  • domain assumption Compatibility condition between solutions of the associative Yang-Baxter equation and the classical Yang-Baxter equation.
    Required to describe quadratic coupled brackets on free polynomial algebras.
invented entities (2)
  • Coupled double Poisson bracket no independent evidence
    purpose: Pair of generalized double brackets satisfying the cross-Jacobi identity.
    New structure defined to establish the bijection with wheeled Poisson brackets.
  • Poisson-left-pre-Lie algebra no independent evidence
    purpose: New algebraic structure placed in one-to-one correspondence with linear coupled double Poisson brackets on free polynomial algebras.
    Introduced as the target of the linear correspondence.

pith-pipeline@v0.9.0 · 5750 in / 1635 out tokens · 47539 ms · 2026-05-19T21:52:44.995520+00:00 · methodology

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

  • IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We introduce coupled double Poisson brackets on an associative algebra A as pairs consisting of a generalized Van den Bergh's double Poisson bracket and a generalized Fairon–McCulloch's right double Poisson bracket subject to a cross-Jacobi identity... establish a bijection between pairs of coupled double Poisson brackets and wheeled Poisson brackets of Ginzburg and Schedler.

  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    On free polynomial algebras, we furthermore establish a one-to-one correspondence between linear coupled double Poisson brackets and Poisson-left-pre-Lie algebras, and describe quadratic ones via solutions of the associative and classical Yang–Baxter equations satisfying a compatibility condition.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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