Nilpotency and Frattini theory for transposed Poisson algebras

Jiarou Jin, Yanyong Hong

Pith reviewed 2026-05-07 13:50 UTC · model grok-4.3

classification 🧮 math.RA

keywords transposed Poisson algebrasnilpotencyEngel's theoremFrattini theorynilpotent radicalLie-nilpotent algebrasmaximal subalgebraszero socle

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The pith

A finite-dimensional transposed Poisson algebra is nilpotent precisely when the left multiplication operators in both the associative and Lie structures are nilpotent.

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

The paper develops nilpotency and Frattini theory for transposed Poisson algebras. It proves an analogue of Engel's theorem that reduces the question of nilpotency to the nilpotency of left multiplications by elements under the associative product and under the Lie bracket. It further shows that for Lie-nilpotent algebras the derived Lie subalgebra is a nilpotent ideal, so the nilpotent radical equals the associative radical, and establishes several results on the Frattini subalgebra, Frattini ideal, maximal subalgebras, and zero socle. A sympathetic reader would care because these statements supply concrete criteria and decomposition tools that mirror classical results in Lie and associative algebra theory and therefore make it easier to recognize, construct, and classify finite-dimensional examples.

Core claim

A finite-dimensional transposed Poisson algebra is nilpotent if and only if the left multiplication operators in both the associative and the Lie structures are nilpotent. The lower central series admits a simplified form. For a Lie-nilpotent transposed Poisson algebra the derived Lie subalgebra is a nilpotent ideal, which implies that the nilpotent radical coincides with the associative radical. The Frattini subalgebra is always contained in the derived algebra and the Frattini ideal is associative nilpotent. When the algebra is nilpotent all maximal subalgebras are ideals and the Frattini subalgebra equals the derived algebra. Conversely, for a Lie-nilpotent algebra, if all maximal subalgs

What carries the argument

The Engel-type criterion that equates nilpotency of the transposed Poisson algebra with simultaneous nilpotency of left multiplications under the associative product and under the Lie bracket.

Load-bearing premise

The transposed Poisson algebra is finite-dimensional over a field and satisfies the compatibility condition between its associative multiplication and Lie bracket.

What would settle it

A concrete finite-dimensional transposed Poisson algebra in which every left multiplication operator is nilpotent yet the lower central series never reaches zero would falsify the main nilpotency theorem.

read the original abstract

We develop the theory of nilpotency and the Frattini theory for transposed Poisson algebras. The lower central series is shown to admit a simplified form, and an analogue of Engel's theorem is established: a finite-dimensional transposed Poisson algebra is nilpotent precisely when the left multiplication operators in both the associative and the Lie structures are nilpotent. Constructions of nilpotent and solvable algebras via tensor products and derivations are given. For a finite-dimensional Lie-nilpotent transposed Poisson algebra, we prove that the derived Lie subalgebra is a nilpotent ideal, which implies that the nilpotent radical coincides with the associative radical. In the framework of Frattini theory, we show that the Frattini subalgebra is always contained in the derived algebra and the Frattini ideal is associative nilpotent. When the algebra is nilpotent, all maximal subalgebras are ideals and the Frattini subalgebra equals the derived algebra. Conversely, for a Lie-nilpotent transposed Poisson algebra, if all maximal subalgebras are ideals, the algebra either is nilpotent or decomposes as a direct sum of a one-dimensional algebra generated by an idempotent and the nilpotent radical; if the Frattini subalgebra equals the derived algebra, the algebra is necessarily nilpotent. We also prove that the zero socle coincides with the nilpotent radical, and when the Frattini ideal is zero, the algebra splits into a subalgebra and its zero socle; in the Lie-nilpotent case this subalgebra is abelian as a Lie 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.

Desk Editor's Note private letter to a colleague

The paper gives a workable Engel-type criterion for nilpotency in finite-dimensional transposed Poisson algebras plus some standard Frattini consequences, but stays inside the usual incremental extension of Lie/associative results.

read the letter

The main result is that a finite-dimensional transposed Poisson algebra is nilpotent exactly when the left multiplications for the associative product and the adjoint operators for the Lie bracket are all nilpotent. They first simplify the lower central series with the compatibility identity, then prove both directions of the equivalence. The forward direction is immediate from the definitions. The converse uses finite-dimensionality to produce a common eigenvector or simultaneous triangularization that respects the transposed Poisson condition. They also construct nilpotent and solvable examples via tensor products and derivations, show that the derived Lie subalgebra is a nilpotent ideal when the algebra is Lie-nilpotent, and conclude that the nilpotent radical equals the associative radical. The Frattini section proves the subalgebra sits inside the derived algebra, the Frattini ideal is associative nilpotent, and when the algebra is nilpotent every maximal subalgebra is an ideal with the Frattini subalgebra coinciding with the derived algebra. Some converses and zero-socle statements round it out. All of this follows from the axioms and standard series arguments without circularity or extra field assumptions beyond finite-dimensionality. The arguments look internally consistent and the stress-test confirms no hidden obstructions on the base field. The work is incremental rather than foundational; it organizes known techniques for one more compatible structure but does not introduce new methods or resolve open questions outside this niche. No counterexamples or explicit low-dimensional checks appear in the abstract, which would have helped gauge sharpness. This is for specialists already working on transposed Poisson algebras or nearby non-associative classes. A reader who needs classification tools or radical theory in this setting will find usable statements. The claims are specific enough and the logic tight enough that it deserves a serious referee to check the details of the triangularization step and the Frattini converses. I would send it to peer review.

Referee Report

0 major / 4 minor

Summary. The manuscript develops nilpotency and Frattini theory for transposed Poisson algebras. It simplifies the lower central series via the transposed Poisson identity and proves an Engel-type theorem: a finite-dimensional transposed Poisson algebra is nilpotent precisely when all left multiplication operators L_x (associative) and ad_x (Lie) are nilpotent. It gives constructions of nilpotent/solvable examples via tensor products and derivations, shows that the derived Lie subalgebra is a nilpotent ideal (hence nilpotent radical coincides with associative radical), and establishes Frattini results including containment of the Frattini subalgebra in the derived algebra, nilpotency of the Frattini ideal, and characterizations of nilpotency via maximal subalgebras or equality of Frattini and derived algebras. Additional results address the zero socle coinciding with the nilpotent radical and splitting properties when the Frattini ideal vanishes.

Significance. If the results hold, the work supplies a coherent structural theory for transposed Poisson algebras that directly extends classical Engel and Frattini theorems from Lie and associative algebras. The central equivalence is load-bearing and useful for classification, as it reduces nilpotency checks to operator nilpotency while respecting the compatibility condition; the Frattini and radical results then follow without circularity. The derivations are grounded in the algebra axioms and standard series definitions, with no free parameters or invented entities.

minor comments (4)

The preliminaries section should explicitly recall the transposed Poisson identity (associator and Lie compatibility) to make the simplification of the lower central series self-contained.
In the statement of the Engel analogue, clarify the precise definition of 'nilpotent algebra' (associative nilpotency, Lie nilpotency, or joint) before the equivalence is proved.
Cross-references from the abstract claims to the corresponding theorem numbers in the body would improve readability.
The constructions via tensor products and derivations would benefit from a short example verifying the transposed Poisson condition holds.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary and recommendation of minor revision. The assessment that the work provides a coherent structural theory extending classical Engel and Frattini theorems is appreciated. No specific major comments were provided in the report, so we interpret the minor revision as addressing any small presentational or typographical issues that may arise during copyediting.

Circularity Check

0 steps flagged

Derivation self-contained from axioms and finite-dimensional linear algebra

full rationale

The central Engel-type equivalence and Frattini results are derived directly from the transposed Poisson compatibility identity, standard definitions of lower central series and nilpotency, and finite-dimensionality arguments (common eigenvectors or simultaneous triangularization). No step reduces a claimed result to a fitted parameter, self-defined term, or unverified self-citation chain. All implications follow from the algebra axioms without circular renaming or ansatz smuggling. This is the normal case of an independent structural theorem.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of a transposed Poisson algebra (associative algebra with compatible Lie bracket) and finite-dimensionality over a field; no free parameters or invented entities are introduced.

axioms (2)

domain assumption The structure is a finite-dimensional vector space over a field equipped with an associative product and a Lie bracket satisfying the transposed Poisson compatibility condition.
Invoked throughout as the ambient category for all theorems.
standard math Standard properties of lower central series, derived series, and radicals hold in this setting.
Used to define nilpotency and Frattini objects.

pith-pipeline@v0.9.0 · 5576 in / 1353 out tokens · 51741 ms · 2026-05-07T13:50:40.997056+00:00 · methodology

discussion (0)

Reference graph

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