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arxiv: 2307.09102 · v2 · pith:TGEHQAWAnew · submitted 2023-07-18 · 🧮 math.RA

Non-nilpotent Leibniz algebras with one-dimensional derived subalgebra

Alfonso Di Bartolo , Gianmarco La Rosa , Manuel Mancini This is my paper

Pith reviewed 2026-05-24 08:13 UTC · model grok-4.3

classification 🧮 math.RA MSC 17A32
keywords Leibniz algebraderived subalgebranon-nilpotentderivationsautomorphismsbiderivationsLie rackcoquecigrue problem
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The pith

Every non-nilpotent non-Lie Leibniz algebra with one-dimensional derived subalgebra is isomorphic to L_n, the direct sum of a fixed two-dimensional algebra and an abelian algebra.

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

The paper classifies non-nilpotent non-Lie Leibniz algebras over fields of characteristic not 2 that have a one-dimensional derived subalgebra. It proves that every such algebra is isomorphic to L_n, formed by taking the two-dimensional non-nilpotent non-Lie Leibniz algebra and adding an abelian algebra of dimension n minus 2. This extends an earlier classification that held only over the complex numbers. The explicit form of L_n then permits direct computation of its derivations, automorphisms, and biderivations, and allows the algebra to be integrated into a Lie rack.

Core claim

We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by L_n, where n equals the dimension. We also determine the Lie algebra of derivations, the Lie group of automorphisms, and the Leibniz algebra of biderivations of L_n. Finally we solve the coquecigrue problem for L_n by integrating it into a Lie rack.

What carries the argument

The algebra L_n, defined as the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra with an abelian algebra of dimension n-2.

If this is right

  • The derivation algebra of L_n can be described explicitly as a Lie algebra.
  • The automorphism group of L_n is a Lie group whose structure follows from the direct-sum decomposition.
  • The biderivation algebra of L_n forms a Leibniz algebra that can be computed directly from the form of L_n.
  • L_n admits an integration into a Lie rack that solves the coquecigrue problem for these algebras.

Where Pith is reading between the lines

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

  • The reduction to a standard non-abelian summand plus abelian part may extend to Leibniz algebras whose derived subalgebra has dimension two or three.
  • The explicit Lie-rack integration supplies a geometric model that could be used to study representations or actions associated with L_n.
  • Similar direct-sum classifications might simplify the study of cohomology or deformations for this family.

Load-bearing premise

The algebra is non-nilpotent, non-Lie, and has derived subalgebra of dimension exactly one over a field of characteristic not 2.

What would settle it

An explicit example of a non-nilpotent non-Lie Leibniz algebra with one-dimensional derived subalgebra that is not isomorphic to L_n for its dimension would disprove the classification.

read the original abstract

In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with $\operatorname{char}(\mathbb{F}) \neq 2$. We prove that such an algebra is isomorphic to the direct sum of the two-dimensional non-nilpotent non-Lie Leibniz algebra and an abelian algebra. We denote it by $L_n$, where $n=\dim_{\mathbb{F}} L_n$. This generalizes the result found in [11], which is only valid when $\mathbb{F}=\mathbb{C}$. Moreover, we find the Lie algebra of derivations, its Lie group of automorphisms and the Leibniz algebra of biderivations of $L_n$. Eventually, we solve the coquecigrue problem for $L_n$ by integrating it into a Lie rack.

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

0 major / 0 minor

Summary. The manuscript proves a classification theorem stating that every non-nilpotent non-Lie Leibniz algebra L over a field F with char(F) ≠ 2 and dim L' = 1 is isomorphic to L_n, the direct sum of the unique 2-dimensional non-nilpotent non-Lie Leibniz algebra and an (n-2)-dimensional abelian algebra. It additionally computes the derivation algebra, automorphism group, biderivation algebra of L_n and integrates L_n into a Lie rack to solve the coquecigrue problem.

Significance. If the classification holds, the result extends the known classification over ℂ to arbitrary fields of characteristic not 2 and supplies explicit structural data (derivations, automorphisms, biderivations, and rack integration) that can be used directly in further work on Leibniz algebras and their geometric realizations.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript, the recognition of its significance in extending the classification from ℂ to arbitrary fields of characteristic not 2, and the recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper establishes a classification theorem by direct algebraic case analysis on the multiplication table and derived subalgebra of non-nilpotent non-Lie Leibniz algebras over fields of characteristic not 2. The isomorphism to the direct sum of the fixed 2-dimensional algebra and an abelian summand follows from exhaustive casework on basis elements and the one-dimensionality of L', using only the Leibniz identity and the stated assumptions; no parameter fitting, self-referential definitions, or load-bearing self-citations reduce the central claim to its own inputs. The cited prior result for ℂ is external and the present proof extends it independently.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard axioms of vector spaces and bilinear products together with the explicit hypothesis that the base field has characteristic not equal to 2; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption The base field has characteristic not equal to 2.
    Explicitly required in the statement of the main theorem.
  • standard math Leibniz algebra axioms: bilinear product satisfying the Leibniz identity.
    Background definition used throughout the classification.

pith-pipeline@v0.9.0 · 5675 in / 1249 out tokens · 68369 ms · 2026-05-24T08:13:45.869860+00:00 · methodology

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