Free Reductive Lie Algebra Pairs of Lie-Yamaguti algebras

Friedrich Wagemann; Martin Bordemann; Sa\"id Benayadii

arxiv: 2606.06422 · v1 · pith:CC3CLKPYnew · submitted 2026-06-04 · 🧮 math.RA

Free Reductive Lie Algebra Pairs of Lie-Yamaguti algebras

Sa\"id Benayadii , Martin Bordemann , Friedrich Wagemann This is my paper

Pith reviewed 2026-06-27 22:32 UTC · model grok-4.3

classification 🧮 math.RA

keywords reductive Lie algebra pairsLie-Yamaguti algebrasenveloping algebrarestriction functorleft adjointright adjointsurjective morphismscategorical adjunctions

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The pith

A left adjoint to the restriction functor from reductive Lie algebra pairs to Lie-Yamaguti algebras exists, and the enveloping construction becomes a right adjoint on surjective morphisms.

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

The paper maps out categorical connections between reductive Lie algebra pairs and Lie-Yamaguti algebras. The usual enveloping construction that sends a Lie-Yamaguti algebra to a reductive Lie algebra pair does not respect arbitrary morphisms, so the authors build a left adjoint to the restriction functor from the first category to the second. Restricting both categories to surjective morphisms makes the enveloping construction functorial, at which point it serves as a right adjoint to the same restriction functor. A reader would care because these adjunctions turn an ad-hoc assignment into a systematic categorical link that preserves structure under controlled conditions.

Core claim

There exists a left adjoint to the natural restriction functor G from the category of reductive Lie algebra pairs to the category of Lie-Yamaguti algebras. When the morphisms in both categories are restricted to surjections, the standard enveloping-algebra construction becomes functorial and is a right adjoint to G.

What carries the argument

The restriction functor G: RLP → LY together with its left adjoint, whose value on a Lie-Yamaguti algebra is the free reductive Lie algebra pair generated by it.

If this is right

The two categories become linked by a pair of adjoint functors once surjective morphisms are imposed.
Free reductive Lie algebra pairs provide a universal way to lift Lie-Yamaguti structures.
Morphisms that are not surjective are the precise obstruction to functoriality of the enveloping construction.
The adjunctions allow transfer of algebraic constructions between the two settings in a controlled way.

Where Pith is reading between the lines

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

The free construction may be used to embed arbitrary Lie-Yamaguti algebras into reductive pairs while preserving universal properties.
Questions about representations or derivations on Lie-Yamaguti algebras could be translated to questions on the free reductive pairs.
The surjectivity condition suggests that the categories behave well when objects are presented by generators and relations without kernel issues.

Load-bearing premise

The standard enveloping construction from Lie-Yamaguti algebras to reductive Lie algebra pairs does not send every morphism to a morphism.

What would settle it

An explicit non-surjective morphism of Lie-Yamaguti algebras whose image under the enveloping construction fails to be a morphism of reductive Lie algebra pairs, or conversely a proof that every morphism is preserved.

read the original abstract

The goal of this article is to show the categorical links between on the one hand the category of reductive Lie algebra pairs $\mathcal{RLP}$ and on the other hand the category of Lie-Yamaguti algebras $\mathcal{LY}$. The fact that the well-known construction of an enveloping algebra associating to a Lie-Yamaguti algebra a reductive Lie algebra pair is not functorial leads us to the main construction of the article, namely a left adjoint to the natural restriction functor $G:\mathcal{RLP}\to\mathcal{LY}$. As a final result we observe that the construction of the enveloping algebra becomes functorial when one restricts the morphisms of the categories $\mathcal{RLP}$ and $\mathcal{LY}$ to the surjective ones. Then it becomes a right adjoint to the restriction functor.

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 constructs an explicit left adjoint to the restriction functor because the standard enveloping map is claimed not to be functorial, plus a clean observation that surjective morphisms make it a right adjoint.

read the letter

The main takeaway is that the authors build a left adjoint to the restriction functor G from the category of reductive Lie algebra pairs to Lie-Yamaguti algebras, motivated by the standard enveloping construction failing to be functorial on all morphisms. They then observe that restricting both categories to surjective morphisms turns the enveloping into a right adjoint.

What is actually new is the left-adjoint construction itself and the surjective-morphism observation. This extends the usual enveloping association in a categorical direction. The paper sets up the two categories and the functors in a straightforward way, and the final remark about surjections is a useful clarification of when the construction behaves well as an adjoint.

The soft spot is the premise that the enveloping construction is not functorial on the full categories. The abstract asserts this without exhibiting a specific morphism in LY that fails to induce a valid map in RLP or identifying the exact obstruction. The stress-test note flags this correctly. If the body of the paper supplies a concrete counterexample or a derivation showing where the axioms break, the motivation stands; otherwise the left adjoint addresses a claim that remains unverified in the visible material.

This is narrow work aimed at people already working with Lie-Yamaguti algebras, reductive pairs, and adjoint functors in algebra. A reader focused on enveloping constructions or categorical aspects of non-associative structures would find the explicit construction and the surjective case worth checking. The thinking is clear and the claims are specific enough to be verified.

I would send it to peer review so referees can check the adjoint properties and the non-functoriality statement against the definitions.

Referee Report

2 major / 1 minor

Summary. The paper establishes categorical links between the category of reductive Lie algebra pairs (RLP) and Lie-Yamaguti algebras (LY). It asserts that the standard enveloping construction from a Lie-Yamaguti algebra to a reductive pair is not functorial on the full morphism categories, motivating the construction of a left adjoint to the restriction functor G: RLP → LY. It further claims that restricting morphisms in both categories to surjective ones renders the enveloping construction functorial, making it a right adjoint to G.

Significance. If the non-functoriality premise holds and the adjunctions are correctly established, the work supplies a categorical framework for free objects and universal properties relating these structures, which could aid in studying enveloping constructions and restrictions in nonassociative algebra. The explicit construction of the left adjoint and the observation on surjective morphisms represent potential strengths if supported by detailed verification.

major comments (2)

[Abstract] Abstract: The assertion that the well-known enveloping construction 'is not functorial' is presented as a fact motivating the left-adjoint construction, yet no explicit morphism f in LY is exhibited for which the induced map on enveloping pairs fails to satisfy the RLP axioms or preserve the Lie-Yamaguti relations. This omission makes the premise load-bearing but unsupported.
[Introduction (or equivalent section stating the motivation)] The central construction of the left adjoint to G relies on the claimed non-functoriality of the standard enveloping map on full categories; without a counterexample or obstruction analysis (e.g., specific failure to preserve reductivity or bracket compatibility), the necessity and correctness of the new left adjoint cannot be assessed.

minor comments (1)

Notation for the categories RLP and LY and the functor G should be introduced with explicit definitions of objects and morphisms early in the text for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for explicit support of the non-functoriality claim. We address the major comments below.

read point-by-point responses

Referee: [Abstract] Abstract: The assertion that the well-known enveloping construction 'is not functorial' is presented as a fact motivating the left-adjoint construction, yet no explicit morphism f in LY is exhibited for which the induced map on enveloping pairs fails to satisfy the RLP axioms or preserve the Lie-Yamaguti relations. This omission makes the premise load-bearing but unsupported.

Authors: We agree that the motivation would be strengthened by an explicit counterexample. In the revised manuscript we will exhibit a concrete morphism f in the category of Lie-Yamaguti algebras for which the standard enveloping construction fails to produce a morphism of reductive Lie algebra pairs (specifically, the induced map does not preserve the reductivity condition). revision: yes
Referee: [Introduction (or equivalent section stating the motivation)] The central construction of the left adjoint to G relies on the claimed non-functoriality of the standard enveloping map on full categories; without a counterexample or obstruction analysis (e.g., specific failure to preserve reductivity or bracket compatibility), the necessity and correctness of the new left adjoint cannot be assessed.

Authors: We accept the referee's point. The revised introduction will contain both an explicit counterexample and a short obstruction analysis showing the precise failure (non-preservation of reductivity under the induced map). This will justify the construction of the left adjoint to G on the full categories. revision: yes

Circularity Check

0 steps flagged

No circularity; standard categorical construction independent of inputs.

full rationale

The paper states that the well-known enveloping construction from Lie-Yamaguti algebras to reductive Lie algebra pairs is not functorial on full morphism categories, motivating a left adjoint to the restriction functor G. This premise is presented as external fact rather than derived from self-citation or fitted parameters within the paper. The subsequent constructions of adjoints (left adjoint in general, right adjoint on surjective morphisms) follow standard category-theoretic existence arguments without equations or definitions that reduce the output to the input by construction. No self-definitional steps, fitted predictions, or load-bearing self-citations appear. The derivation chain is self-contained against external category theory benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, ad-hoc axioms, or invented entities; all content rests on standard definitions of the two algebraic categories and the enveloping construction.

axioms (1)

standard math Standard axioms and definitions of categories, Lie-Yamaguti algebras, and reductive Lie algebra pairs as used in the literature.
The paper invokes these background structures without re-deriving them.

pith-pipeline@v0.9.1-grok · 5670 in / 1082 out tokens · 28683 ms · 2026-06-27T22:32:43.167963+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Benito, P., Elduque, A., Mart´ ın-Herce, F.:Irreducible Lie-Yamaguti algebras.J. Pure Appl. Algebra 213(2009) 795–808

2009

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Pure Appl

Benito, P., Elduque, A., Mart´ ın-Herce, F.:Irreducible Lie-Yamaguti algebras of generic type.J. Pure Appl. Algebra215(2011), 108–130

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(ed.) et al., Nonassociative mathematics and its applications

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Yamaguti, K.:On the Lie triple system and its generalization, J. Sci. Hiroshima Univ., Ser. A,21 (1958), 155–160. 13

1958