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arxiv: 2606.24287 · v1 · pith:5YFJJWFDnew · submitted 2026-06-23 · 🧮 math.GR · math.RA

On Simply Connected Simple Lie Skew Braces with Nilpotent Multiplicative Group

Marco Damele , Andrea Loi This is my paper

Pith reviewed 2026-06-25 22:42 UTC · model grok-4.3

classification 🧮 math.GR math.RA
keywords Lie skew bracessimple structuresnilpotent groupspost-Lie algebrassimply connected Lie groupsabelian structuresmultiplicative nilpotencyLie group ideals
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The pith

A simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian.

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

The paper establishes that simplicity in a simply connected Lie skew brace is incompatible with nilpotency of the multiplicative group except in dimension one, where the structure must be abelian. The argument reduces the problem to the associated post-Lie algebra, where solvable additive algebras have their nilradical as an ideal and nilpotent pairs always produce a proper ideal with trivial quotient. These local ideals lift to connected closed ideals in the global skew brace through homomorphisms onto trivial abelian structures. A sympathetic reader would care because the result shows that a standard nilpotency condition on one operation rules out all higher-dimensional simple examples in the Lie setting.

Core claim

We prove that a simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian. Equivalently, if (G,·,∘) is a simply connected Lie skew brace with nilpotent multiplicative Lie group and dim G>1, then (G,·,∘) is not simple. The proof is carried out at the post-Lie algebra level. First, if the additive Lie algebra is solvable, then its nilradical is automatically an ideal of the associated post-Lie algebra. Second, when both Lie algebras underlying an integrable post-Lie structure are nilpotent, one always obtains a proper post-Lie ideal with trivial quotient. To pass from infinitesimal ideals to global ideals of the Lie skew brace, we show t

What carries the argument

The lifting of trivial post-Lie quotients to homomorphisms onto abelian trivial Lie skew braces whose kernels are connected closed ideals.

Load-bearing premise

Trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces whose kernels yield connected closed ideals.

What would settle it

A simply connected Lie skew brace of dimension two or higher that is simple and has nilpotent multiplicative group would disprove the claim.

read the original abstract

We prove that a simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian. Equivalently, if $(G,\cdot,\circ)$ is a simply connected Lie skew brace with nilpotent multiplicative Lie group and $\dim G>1$, then $(G,\cdot,\circ)$ is not simple. Thus, in the simply connected Lie setting, nilpotency of the multiplicative group is incompatible with simplicity in every dimension greater than one. The proof is carried out at the post-Lie algebra level. First, if the additive Lie algebra is solvable, then its nilradical is automatically an ideal of the associated post-Lie algebra. Second, when both Lie algebras underlying an integrable post-Lie structure are nilpotent, one always obtains a proper post-Lie ideal with trivial quotient. To pass from infinitesimal ideals to global ideals of the Lie skew brace, we show that trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces, whose kernels yield connected closed ideals.

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 / 0 minor

Summary. The manuscript proves that a simply connected simple Lie skew brace with nilpotent multiplicative Lie group must be one-dimensional and abelian. Equivalently, if (G,·,∘) is a simply connected Lie skew brace with nilpotent multiplicative Lie group and dim G>1, then (G,·,∘) is not simple. The argument is carried out at the post-Lie algebra level: the nilradical of a solvable additive Lie algebra is an ideal of the associated post-Lie algebra; when both underlying Lie algebras are nilpotent one obtains a proper post-Lie ideal with trivial quotient; trivial post-Lie quotients induce homomorphisms onto abelian trivial Lie skew braces whose kernels are connected closed ideals of the skew brace.

Significance. If the lifting from post-Lie ideals to skew-brace ideals holds, the result shows that nilpotency of the multiplicative group is incompatible with simplicity for simply connected Lie skew braces in every dimension greater than one. This supplies a clean classification-type restriction in the theory of Lie skew braces and integrable post-Lie structures, obtained directly from standard Lie-algebraic properties without free parameters or ad-hoc constructions.

major comments (1)
  1. [Abstract (final paragraph)] Abstract, final paragraph: the lifting claim that 'trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces, whose kernels yield connected closed ideals' is load-bearing for the global simplicity statement. It is not shown whether a post-Lie ideal I automatically satisfies the skew-brace ideal condition (i.e., that x·y − x∘y lies in I for x,y in the preimage) or whether the induced map preserves both operations while guaranteeing that the kernel is closed and connected under the simply-connected hypothesis. Without this explicit correspondence the infinitesimal proper ideal does not necessarily produce a proper global ideal, so the conclusion that the brace is not simple does not follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make the lifting from post-Lie ideals to skew-brace ideals fully explicit. We address the concern point by point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: Abstract, final paragraph: the lifting claim that 'trivial post-Lie quotients give rise to homomorphisms onto abelian trivial Lie skew braces, whose kernels yield connected closed ideals' is load-bearing for the global simplicity statement. It is not shown whether a post-Lie ideal I automatically satisfies the skew-brace ideal condition (i.e., that x·y − x∘y lies in I for x,y in the preimage) or whether the induced map preserves both operations while guaranteeing that the kernel is closed and connected under the simply-connected hypothesis. Without this explicit correspondence the infinitesimal proper ideal does not necessarily produce a proper global ideal, so the conclusion that the brace is not simple does not follow.

    Authors: We agree that the correspondence must be stated explicitly rather than summarized in the abstract. When the post-Lie quotient is trivial the two Lie bracket operations coincide, so the induced Lie-group homomorphism (which exists by simple connectedness of the domain) automatically intertwines both group operations · and ∘. Consequently x·y − x∘y lies in the kernel ideal by construction of the quotient map. The kernel is closed as the preimage of the identity and connected because the source is simply connected. We will add a short lemma in the body (and a clarifying sentence in the abstract) that records this correspondence in full detail. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct proof from Lie algebra properties

full rationale

The derivation proceeds by establishing two infinitesimal results on post-Lie ideals (nilradical ideal when additive algebra solvable; proper ideal with trivial quotient when both algebras nilpotent) and then proving that trivial post-Lie quotients induce homomorphisms to abelian trivial braces whose kernels are connected closed ideals. These steps are presented as explicit constructions and verifications internal to the paper using standard Lie algebra facts, with no equations reducing a claimed prediction to a fitted input, no self-definitional loops, and no load-bearing self-citations. The global simplicity conclusion follows from the lifting argument as stated, without renaming known results or smuggling ansatzes via prior work. The argument is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no free parameters, invented entities, or non-standard axioms are visible. The argument relies on background facts about Lie algebras and post-Lie structures.

axioms (1)
  • standard math Standard properties of Lie algebras, nilradicals, and post-Lie algebra ideals hold without additional assumptions.
    Invoked when stating that the nilradical is automatically an ideal and that nilpotent post-Lie structures yield proper ideals with trivial quotient.

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discussion (0)

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Reference graph

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10 extracted references · 1 linked inside Pith

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