arxiv: 2604.26115 · v1 · submitted 2026-04-28 · 🧮 math.RA · math.RT

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On simple transposed Poisson algebras

Amir Fern\'andez Ouaridi

Pith reviewed 2026-05-07 13:42 UTC · model grok-4.3

classification 🧮 math.RA math.RT

keywords transposed Poisson algebrasZassenhaus algebrassimple algebrasfinite-dimensional algebrasclassificationmutationsLie algebrasrepresentations

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The pith

Finite-dimensional simple transposed Poisson algebras over algebraically closed fields of characteristic p > 3 are precisely the algebras W_n(q) whose Lie algebra is the Zassenhaus algebra W(1;n).

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

The paper develops a structure theory for transposed Poisson algebras over fields of characteristic not 2, proving that every finite-dimensional one over an algebraically closed field decomposes as the direct sum of a unital ideal and a nilpotent ideal. This decomposition yields strong restrictions on any simple examples. The author then classifies all simple finite-dimensional transposed Poisson algebras over algebraically closed fields of characteristic p > 3, showing each has underlying Lie algebra equal to a Zassenhaus algebra W(1;n) and is isomorphic to one of the explicit algebras W_n(q) obtained by mutating a natural associative commutative product on W(1;n). A reader would care because the result supplies a concrete list of all such simple objects, opening the way to compute their representations and trace connections to Jordan superalgebras and related structures.

Core claim

We prove that every finite-dimensional transposed Poisson algebra over an algebraically closed field decomposes as the direct sum of a unital ideal and a nilpotent ideal. As a consequence, we obtain restrictions on simple transposed Poisson algebras and classify the simple finite-dimensional ones over an algebraically closed field of characteristic p>3: every such algebra has as underlying Lie algebra a Zassenhaus algebra W(1;n) and is isomorphic to one of the algebras of the family W_n(q) arising from a mutation of a natural associative commutative structure on W(1;n). We then study the corresponding isomorphism problem for the family W_n(q) and determine the irreducible finite-dimensional

What carries the argument

The family of algebras W_n(q) obtained by mutating the natural associative commutative product on the Zassenhaus algebra W(1;n), which serve as the explicit models for all the simple transposed Poisson algebras in the stated setting.

If this is right

The unital-nilpotent decomposition imposes immediate restrictions on which algebras can be simple.
The isomorphism problem inside the family W_n(q) admits an explicit solution.
Irreducible finite-dimensional representations of the unital members of W_n(q) are completely determined.
The classification yields direct applications to the structure theory of Jordan superalgebras, weak-Leibniz algebras, and quasi-Poisson algebras.

Where Pith is reading between the lines

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

The explicit models W_n(q) may admit natural deformations or extensions that preserve transposed Poisson identity in other characteristics.
Similar decomposition arguments could be tested on infinite-dimensional or non-simple transposed Poisson algebras to see whether the unital-nilpotent pattern persists.
The representations of these algebras could be used to construct new examples of modules over related nonassociative structures.

Load-bearing premise

The algebra must be finite-dimensional over an algebraically closed field of characteristic p greater than 3.

What would settle it

Exhibiting even one simple finite-dimensional transposed Poisson algebra over such a field whose underlying Lie algebra is not a Zassenhaus algebra W(1;n) or which is not isomorphic to any W_n(q) would falsify the classification.

read the original abstract

We develop a structure theory for transposed Poisson algebras over fields of characteristic different from two. In particular, we prove that every finite-dimensional transposed Poisson algebra over an algebraically closed field decomposes as the direct sum of a unital ideal and a nilpotent ideal. As a consequence, we obtain restrictions on simple transposed Poisson algebras and use them to classify the simple finite-dimensional transposed Poisson algebras over an algebraically closed field of characteristic $p>3$. Precisely, we show that every such algebra has as underlying Lie algebra a Zassenhaus algebra $\mathcal{W}(1;n)$ and is isomorphic to one of the algebras of the family $\mathcal{W}_n(q)$ arising from a mutation of a natural associative commutative structure on $\mathcal{W}(1;n)$. We then study the corresponding isomorphism problem for the family $\mathcal{W}_n(q)$ and determine the irreducible finite-dimensional representations of these simple transposed Poisson algebras $\mathcal{W}_n(q)$ in the unital case. We conclude with some applications to Jordan superalgebras, weak-Leibniz algebras and quasi-Poisson algebras.

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.

Desk Editor's Note private letter to a colleague

The paper classifies all simple finite-dimensional transposed Poisson algebras over algebraically closed fields of char p>3 as a specific family W_n(q) built from Zassenhaus Lie algebras.

read the letter

The core result is a classification: every simple finite-dimensional transposed Poisson algebra over an algebraically closed field of characteristic p>3 has underlying Lie algebra a Zassenhaus algebra W(1;n) and is isomorphic to one of the W_n(q) obtained by mutating the natural commutative associative structure on it. They also solve the isomorphism problem inside that family and describe the irreducible representations in the unital case, with some side applications to Jordan superalgebras and weak-Leibniz algebras at the end. A supporting decomposition theorem says any finite-dimensional transposed Poisson algebra over an algebraically closed field (char not 2) splits as a unital ideal direct sum a nilpotent ideal, which immediately limits what the simple ones can look like. That decomposition looks like the most reusable piece for other work in the area. The argument rests on standard facts about Zassenhaus algebras rather than new ad-hoc constructions, which keeps the circularity burden low and makes the claims easier to check. The restrictions to finite dimension, algebraically closed base field, and p>3 are stated up front, so there is no hidden generality claim. The only real softness is that the abstract gives the logical outline but not the full proofs, so a referee would still need to verify the details of how the decomposition forces the Zassenhaus form and how the mutation produces exactly the simple objects. No obvious gaps or invented entities show up in the stated chain. This is for people already working on nonassociative algebras, Poisson-type structures, or their links to Jordan superalgebras. If you care about classification results that organize a variety and open the door to representation theory, it is worth reading. I would send it to peer review; the result is concrete enough and the approach is grounded enough to deserve referee time even if some technical points need tightening.

Referee Report

0 major / 3 minor

Summary. The paper develops structure theory for transposed Poisson algebras over fields of characteristic different from 2. It proves that every finite-dimensional transposed Poisson algebra over an algebraically closed field decomposes as the direct sum of a unital ideal and a nilpotent ideal. As a consequence, it classifies the simple finite-dimensional transposed Poisson algebras over an algebraically closed field of characteristic p>3, showing that the underlying Lie algebra is a Zassenhaus algebra W(1;n) and the algebra is isomorphic to one of the family W_n(q) obtained by mutation of the natural associative commutative structure on W(1;n). The work also solves the isomorphism problem within the family W_n(q), determines the irreducible finite-dimensional representations of the unital members, and derives applications to Jordan superalgebras, weak-Leibniz algebras, and quasi-Poisson algebras.

Significance. If the proofs hold, the decomposition theorem is a useful general structural result for transposed Poisson algebras, while the classification of the simple finite-dimensional objects in characteristic p>3 supplies a concrete and complete description that was previously unavailable. The explicit construction of the W_n(q) family via mutation, together with the isomorphism classification and the representation theory, provides concrete tools for further work. The applications to related structures broaden the reach of the results.

minor comments (3)

The abstract states that the algebras arise 'from a mutation of a natural associative commutative structure' but does not display the mutated multiplication; adding a brief displayed formula (with equation number) in the introduction or §3 would improve immediate readability.
In the discussion of the isomorphism problem for the family W_n(q), the criterion distinguishing non-isomorphic members could be stated more explicitly as a numbered proposition or corollary for easier reference.
The applications section sketches connections to Jordan superalgebras and quasi-Poisson algebras; a single concrete low-dimensional example illustrating one of these links would strengthen the exposition without lengthening the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough summary of our work and for recommending acceptance. We are pleased that the referee finds the decomposition theorem, the classification of simple finite-dimensional transposed Poisson algebras in characteristic p>3, the isomorphism problem for the W_n(q) family, the representation theory, and the applications to be of value.

Circularity Check

0 steps flagged

No significant circularity in classification derivation

full rationale

The paper proves a decomposition of any finite-dimensional transposed Poisson algebra over an algebraically closed field of characteristic ≠2 into a unital ideal plus a nilpotent ideal using standard structure theory. This decomposition directly implies restrictions on simple algebras, which are then classified in characteristic p>3 by matching the underlying Lie algebra to the externally known Zassenhaus algebra W(1;n) and exhibiting isomorphism to the mutated family W_n(q). No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the argument chain is independent of the target classification and rests on prior Lie-algebra results that are not reproduced or assumed within the paper itself.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claims rest on standard assumptions about the base field and finite-dimensionality, with no free parameters or new invented entities.

axioms (3)

domain assumption Characteristic of the field is different from 2
Used throughout the structure theory for transposed Poisson algebras.
domain assumption The field is algebraically closed
Required for the decomposition into unital and nilpotent ideals and the classification.
domain assumption Characteristic p > 3
Required for the classification of simple finite-dimensional algebras.

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discussion (0)

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Poisson $n$-Lie algebras: constructions and the structure of solvable algebras
math.RA 2026-05 unverdicted novelty 6.0

Poisson n-Lie algebras are constructed from Jacobians and tensor quotients with a bidirectional correspondence to Poisson algebras, plus characterizations of solvable versions via hypo-nilpotent ideals and eigenspaces.
Poisson $n$-Lie algebras: constructions and the structure of solvable algebras
math.RA 2026-05 unverdicted novelty 5.0

Develops constructions for Poisson n-Lie algebras from Jacobians and tensor products, establishes correspondence with Poisson algebras, and characterizes solvable/nilpotent structures with hypo-nilpotent ideals.

Reference graph

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