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arxiv: 2604.15346 · v1 · submitted 2026-03-20 · 🧮 math.RT · math-ph· math.MP· math.QA· math.RA

D-bialgebras, dendrification and embeddings into AWB of almost Poisson algebras

Sami Mabrouk This is my paper

Pith reviewed 2026-05-15 07:00 UTC · model grok-4.3

classification 🧮 math.RT math-phmath.MPmath.QAmath.RA
keywords almost Poisson algebrasDrinfel'd bialgebrasManin triplesmatched pairsaveraging operatorsalgebras with bracketsRota-Baxter operatorstridendriform algebras
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The pith

Almost Poisson Drinfel'd bialgebras equate to matched pairs and Manin triples, while almost Poisson algebras embed into algebras with brackets via averaging operators.

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

The paper defines almost Poisson Drinfel'd bialgebras as the direct analogue of Poisson D-bialgebras without requiring commutativity or skew-symmetry. It proves these new bialgebras stand in one-to-one correspondence with matched pairs of almost Poisson algebras and with Manin triples built from them. The work also introduces almost tridendriform Poisson algebras as the algebraic objects that sit under relative Rota-Baxter operators on almost Poisson structures. The central embedding result shows that any almost Poisson algebra sits inside an algebra with bracket by means of an averaging operator, or more generally a relative averaging operator coming from any representation of the original algebra.

Core claim

We introduce almost Poisson Drinfel'd bialgebras and establish the equivalence between these objects, matched pairs of almost Poisson algebras, and Manin triples. We define almost tridendriform Poisson algebras as the structures associated with relative Rota-Baxter operators on almost Poisson algebras. Every almost Poisson algebra embeds into an algebra with bracket through averaging operators or relative averaging operators attached to a given representation.

What carries the argument

Almost Poisson Drinfel'd bialgebras, which carry a compatible pair of almost Poisson structures on a vector space and its dual together with the Leibniz-type compatibility that defines algebras with brackets.

If this is right

  • Matched pairs of almost Poisson algebras correspond bijectively to almost Poisson D-bialgebras.
  • Manin triples of almost Poisson algebras are in one-to-one correspondence with these D-bialgebras.
  • Relative Rota-Baxter operators on almost Poisson algebras produce almost tridendriform Poisson algebras.
  • Every almost Poisson algebra admits an embedding into an algebra with bracket via an averaging operator.
  • Relative averaging operators attached to any representation give a more general embedding construction.

Where Pith is reading between the lines

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

  • The embedding construction supplies a systematic way to realize almost Poisson structures inside noncommutative algebras with brackets, potentially allowing transfer of invariants.
  • The dendrification process via relative Rota-Baxter operators may extend to other Leibniz-type structures beyond the almost Poisson case.
  • Equivalence with Manin triples suggests that geometric constructions on Poisson manifolds could lift directly to the noncommutative setting.
  • Representations of almost Poisson algebras now yield concrete embeddings, opening a route to study deformations or representations of the resulting AWBs.

Load-bearing premise

Averaging operators exist for arbitrary representations of an almost Poisson algebra and the bracket in the target algebra with bracket satisfies the required Leibniz compatibility with the associative product.

What would settle it

An explicit almost Poisson algebra together with a representation for which no averaging operator exists, or a matched pair whose induced structure fails to satisfy the D-bialgebra axioms.

read the original abstract

An algebra with bracket ({\sf AWB} for short) is an associative algebra endowed with a bilinear bracket satisfying a Leibniz-type compatibility condition, as introduced in \cite{casas}. It can be viewed as a noncommutative generalization of an almost Poisson algebra; indeed, when the associative product is commutative and the bracket is skew-symmetric, one recovers the notion of an almost Poisson algebra. In this paper, we introduce the notion of {almost Poisson Drinfel'd bialgebras ($D$-bialgebras)} as an analogue of Poisson $D$-bialgebras, and we establish the equivalence between matched pairs, Manin triples, and almost Poisson $D$-bialgebras. Furthermore, we define a new algebraic structure, called {almost tridendriform Poisson algebras}, which can be regarded as the underlying algebraic structures associated with relative Rota-Baxter operators on almost Poisson algebras. Finally, we show that every almost Poisson algebra can be embedded into an algebra with bracket ({\sf AWB}) via averaging operators, and more generally via relative averaging operators associated to a given representation of the almost Poisson algebra.

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

1 major / 2 minor

Summary. The paper introduces almost Poisson Drinfel'd bialgebras (D-bialgebras) as an analogue of Poisson D-bialgebras and proves their equivalence to matched pairs and Manin triples. It defines almost tridendriform Poisson algebras as the structures underlying relative Rota-Baxter operators on almost Poisson algebras. It further claims that every almost Poisson algebra embeds into an algebra with bracket (AWB) via averaging operators, and more generally via relative averaging operators associated to any representation of the almost Poisson algebra, preserving the Leibniz compatibility condition.

Significance. If the equivalences hold and the embedding construction is made rigorous, the work supplies a noncommutative generalization of almost Poisson structures together with bialgebraic and dendriform-type companions. The Manin-triple and matched-pair correspondences could streamline constructions in deformation theory and integrable systems, while the embedding result would provide a systematic way to realize AWBs from given almost Poisson data.

major comments (1)
  1. [§5 (embedding via relative averaging operators)] The embedding theorem (the claim that every almost Poisson algebra embeds into an AWB via relative averaging operators for an arbitrary representation) lacks an existence proof or module-theoretic hypotheses. The required operator R must satisfy both R(x·y)=R(x)·y+x·R(y)−R(x)·R(y) and the Leibniz identity in the target AWB, yet no construction, splitting condition, or projectivity assumption is supplied; standard averaging-operator results in related structures (Rota-Baxter, dendriform) typically require such hypotheses, which may fail for arbitrary representations.
minor comments (2)
  1. [§3] The definition of almost tridendriform Poisson algebras should include an explicit comparison with the classical tridendriform Poisson case to clarify the precise weakening introduced by the almost-Poisson bracket.
  2. [§2] Notation for the bracket in AWBs and the compatibility condition with the associative product should be unified across sections to avoid confusion with the skew-symmetric almost-Poisson bracket.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. The positive evaluation of the paper's potential contributions to noncommutative generalizations of almost Poisson structures is appreciated. We address the single major comment below.

read point-by-point responses

  1. Referee: [§5 (embedding via relative averaging operators)] The embedding theorem (the claim that every almost Poisson algebra embeds into an AWB via relative averaging operators for an arbitrary representation) lacks an existence proof or module-theoretic hypotheses. The required operator R must satisfy both R(x·y)=R(x)·y+x·R(y)−R(x)·R(y) and the Leibniz identity in the target AWB, yet no construction, splitting condition, or projectivity assumption is supplied; standard averaging-operator results in related structures (Rota-Baxter, dendriform) typically require such hypotheses, which may fail for arbitrary representations.

    Authors: We agree that the manuscript as written does not supply a full existence proof or the requisite module-theoretic hypotheses for the general embedding via relative averaging operators on arbitrary representations. The referee correctly identifies that the operator R must simultaneously satisfy the averaging identity R(x·y)=R(x)·y+x·R(y)−R(x)·R(y) and ensure the Leibniz rule holds in the target AWB, and that standard results in the literature often impose projectivity or splitting conditions. In the revised manuscript we will explicitly state these hypotheses (projectivity of the underlying module together with the existence of a module splitting) and provide a rigorous construction: we embed the almost Poisson algebra into the free AWB on the module via the universal property of free algebras with bracket, then define the relative averaging operator by extending the given representation data. For the special case of ordinary averaging operators (no external representation), the embedding is realized concretely by adjoining a formal idempotent element satisfying the averaging relation and verifying the Leibniz identity directly. Section 5 will be expanded with these details, the corrected statement of the theorem, and the complete verification. revision: yes

Circularity Check

0 steps flagged

No circularity: new definitions and equivalences are independent constructions

full rationale

The paper introduces fresh notions (almost Poisson D-bialgebras, almost tridendriform Poisson algebras) and states that it proves equivalences among matched pairs, Manin triples, and these bialgebras, plus an embedding theorem for almost Poisson algebras into AWBs via averaging operators. These steps are presented as theorems to be established from the given definitions rather than identities that hold by construction or by renaming prior results. No fitted parameters are relabeled as predictions, no self-citations form load-bearing premises, and no ansatz is smuggled via prior work. The derivation chain therefore remains self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper relies on the standard definition of AWB (associative product plus Leibniz-compatible bracket) from the cited reference and extends it to the almost Poisson setting. No free parameters are introduced. The new structures are defined rather than postulated as physical entities.

axioms (1)
  • domain assumption The bracket satisfies a Leibniz-type compatibility condition with the associative product.
    This is the core defining property of an AWB as stated in the abstract and the cited work.
invented entities (2)
  • almost Poisson Drinfel'd bialgebra (D-bialgebra) no independent evidence
    purpose: To serve as an analogue of Poisson D-bialgebras for the almost Poisson case.
    Newly defined structure whose properties are then related to matched pairs and Manin triples.
  • almost tridendriform Poisson algebra no independent evidence
    purpose: To provide the underlying algebraic structure for relative Rota-Baxter operators on almost Poisson algebras.
    New definition introduced to link operators to the algebra.

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