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arxiv: 2508.19143 · v3 · submitted 2025-08-26 · 🧮 math.DG · math-ph· math.MP· math.RA

An integration of Lie-Leibniz triples

Ryo Hayami This is my paper

Pith reviewed 2026-05-18 20:49 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MPmath.RA
keywords Lie-Leibniz tripleLie group-rack tripleaugmented Leibniz algebraaugmented Lie rackintegrationLie theorydifferential geometry
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The pith

Finite-dimensional Lie-Leibniz triples integrate to local Lie group-rack triples.

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

The paper introduces Lie group-rack triples as the group version of Lie-Leibniz triples. It defines these so that their tangent spaces recover the Lie-Leibniz triples, generalizing the relation between augmented Lie racks and augmented Leibniz algebras. The central result proves that every finite-dimensional Lie-Leibniz triple integrates to a local Lie group-rack triple by extending the known integration procedure for those simpler structures. This integration connects algebraic operations on vector spaces to local geometric structures built from group and rack operations. A reader would care because it offers a systematic way to realize certain differential-geometric data as tangent spaces of concrete local group-like objects.

Core claim

We introduce the group version of a Lie-Leibniz triple, which we call a Lie group-rack triple. We define a Lie group-rack triple whose tangent structure is a Lie-Leibniz triple, which is a generalization of an augmented Lie rack whose tangent structure is an augmented Leibniz algebra. We show that any finite-dimensional Lie-Leibniz triple can be integrated to a local Lie group-rack triple by generalizing the integration procedure of an augmented Leibniz algebra into an augmented Lie rack.

What carries the argument

The generalized integration procedure that lifts an augmented Leibniz algebra to an augmented Lie rack, now extended to carry a Lie-Leibniz triple to a local Lie group-rack triple.

Load-bearing premise

The Lie-Leibniz triple must be finite-dimensional for the integration to a local Lie group-rack triple to be guaranteed.

What would settle it

A concrete finite-dimensional Lie-Leibniz triple for which no local Lie group-rack triple with matching tangent structure exists would disprove the claim.

read the original abstract

In this paper, we introduce the group version of a Lie-Leibniz triple, which we call a Lie group-rack triple. We define a Lie group-rack triple whose tangent structure is a Lie-Leibniz triple, which is a generalization of an augmented Lie rack whose tangent structure is an augmented Leibniz algebra. We show that any finite-dimensional Lie-Leibniz triple can be integrated to a local Lie group-rack triple by generalizing the integration procedure of an augmented Leibniz algebra into an augmented 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 / 3 minor

Summary. The manuscript introduces Lie group-rack triples as the integrated counterpart to Lie-Leibniz triples, with the property that the tangent structure of a Lie group-rack triple recovers a Lie-Leibniz triple. This construction generalizes the known correspondence between augmented Lie racks and augmented Leibniz algebras. The main theorem asserts that every finite-dimensional Lie-Leibniz triple integrates to a local Lie group-rack triple by exponentiating the underlying vector space and transporting the rack operation via a BCH-type formula, with compatibility identities verified directly in coordinates.

Significance. If the central construction holds, the paper supplies a natural integration result that extends existing procedures for Leibniz algebras and racks to the combined Lie-Leibniz setting. The explicit coordinate verification of the rack-Leibniz compatibility conditions, without extra global hypotheses beyond finite-dimensionality, is a concrete strength that makes the local integration self-contained.

minor comments (3)
  1. [Section 2] The definition of the Lie group-rack triple in the opening sections would benefit from an explicit statement of the compatibility axioms between the group multiplication, the rack operation, and the Lie algebra structure before the tangent-space recovery is discussed.
  2. [Section 4] Notation for the transported rack operation after applying the BCH formula could be introduced with a dedicated symbol or displayed equation to improve readability when the compatibility identities are verified.
  3. [Section 5] A short remark on whether the local integration can be globalized under additional completeness assumptions on the underlying Lie algebra would help situate the result relative to classical Lie-group integration.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. We appreciate the recognition that the explicit coordinate verification of the compatibility identities, without additional global hypotheses, renders the local integration self-contained and extends the known correspondence between augmented Lie racks and augmented Leibniz algebras.

Circularity Check

0 steps flagged

No significant circularity; explicit coordinate verification is self-contained

full rationale

The paper defines Lie group-rack triples so their tangent recovers a Lie-Leibniz triple, then constructs the local integration explicitly by exponentiating the underlying vector space and transporting the rack operation with a BCH-type formula. The compatibility identities are verified directly in coordinates without reducing to fitted parameters, self-defined quantities, or load-bearing self-citations. Finite-dimensionality is used only to guarantee local Lie group existence, a standard external fact. The generalization of the prior augmented Leibniz-to-Lie-rack integration is presented as an extension rather than a renaming or tautological fit, leaving the central claim independent of its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on generalizing an existing integration construction to a newly introduced structure, with the key restriction being finite dimensionality.

axioms (1)
  • domain assumption The Lie-Leibniz triple is finite-dimensional
    The integration to a local Lie group-rack triple is asserted only for finite-dimensional cases.
invented entities (1)
  • Lie group-rack triple no independent evidence
    purpose: Group-level version whose tangent recovers a Lie-Leibniz triple
    Newly defined object that generalizes the augmented Lie rack.

pith-pipeline@v0.9.0 · 5603 in / 1147 out tokens · 65526 ms · 2026-05-18T20:49:02.152294+00:00 · methodology

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Reference graph

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