pith. sign in

arxiv: 1907.07401 · v1 · pith:WQMB2NRYnew · submitted 2019-07-17 · 🧮 math.RA

Lie-central derivations, Lie-centroids and Lie-stem Leibniz algebras

G. R. Biyogmam , J. M. Casas , N. Pacheco Rego This is my paper

Pith reviewed 2026-05-24 20:03 UTC · model grok-4.3

classification 🧮 math.RA
keywords Leibniz algebrasLie-central derivationsLie-stem algebrasLie-derivationsLie-isoclinismnilpotent Leibniz algebrasderivations
0
0 comments X

The pith

Lie-stem Leibniz algebras are characterized by their Lie-central derivations.

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

The paper defines Lie-derivations on Leibniz algebras as maps that generalize ordinary derivations while respecting the Leibniz identity in a Lie-specific way. It isolates the subclass of Lie-central derivations whose images sit inside the Lie-center and shows that these derivations alone determine whether a Leibniz algebra is Lie-stem. For the subclass of Lie-nilpotent Leibniz algebras of class 2 the paper records several structural properties of the resulting Lie algebra of derivations. It further introduces ID_*-Lie-derivations, which vanish on the Lie-center and land inside the second term of the lower Lie-central series, and proves both a dimension bound and invariance of the set of such derivations under Lie-isoclinism.

Core claim

A Leibniz algebra is Lie-stem precisely when its Lie-central derivations satisfy the characterizing condition that identifies the stem property; for Lie-nilpotent Leibniz algebras of class 2 the Lie algebra formed by these derivations obeys additional relations; the Lie algebra ID_*^{Lie}(G) of ID_*-Lie-derivations admits an explicit upper bound on dimension, and any two Lie-isoclinic Leibniz algebras have isomorphic sets ID_*^{Lie}(G) and ID_*^{Lie}(Q).

What carries the argument

Lie-central derivations (Lie-derivations whose image lies inside the Lie-center) together with the auxiliary ID_*-Lie-derivations (those that also vanish on Lie-central elements and land in the second term of the lower Lie-central series).

If this is right

  • Lie-stem Leibniz algebras can be recognized directly from the Lie algebra of their Lie-central derivations.
  • The dimension of the Lie algebra of ID_*-Lie-derivations is bounded above for every Leibniz algebra.
  • Lie-isoclinism preserves the set of ID_*-Lie-derivations up to isomorphism.
  • The Lie algebra of Lie-central derivations carries extra structure when the Leibniz algebra is Lie-nilpotent of class 2.

Where Pith is reading between the lines

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

  • The characterization may reduce the problem of listing Lie-stem Leibniz algebras to a computation inside the derivation algebra.
  • The invariance result suggests that Lie-isoclinism is a coarser equivalence than ordinary isomorphism yet still respects derivation data.
  • Similar dimension bounds or isomorphisms could be sought for other filtered series in Leibniz algebras.

Load-bearing premise

The definitions of Lie-center, lower Lie-central series, and Lie-isoclinism remain consistent when the new Lie-derivation maps are substituted into them.

What would settle it

An explicit Leibniz algebra that is Lie-stem yet fails the stated characterization by its Lie-central derivations, or a pair of Lie-isoclinic Leibniz algebras whose ID_*^{Lie} sets are not isomorphic.

read the original abstract

In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ${\sf ID}_*-Lie$-derivations. A ${\sf ID}_*-Lie$-derivation of a Leibniz algebra G is a Lie-derivation of G in which the image is contained in the second term of the lower Lie-central series of G, and that vanishes on Lie-central elements. We provide an upperbound for the dimension of the Lie algebra $ID_*^{Lie}(G)$ of $ID_*Lie$-derivation of G, and prove that the sets $ID_*^{Lie}(G)$ and $ID_*^{Lie}(G)$ are isomorphic for any two Lie-isoclinic Leibniz algebras G and Q.

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 / 2 minor

Summary. The manuscript introduces Lie-derivations on Leibniz algebras as a generalization of ordinary derivations. It defines Lie-central derivations (those whose image lies in the Lie-center) and claims a characterization of Lie-stem Leibniz algebras in terms of these maps. For Lie-nilpotent Leibniz algebras of class 2 it proves several properties of the Lie algebra of Lie-central derivations. It further defines ID_*-Lie-derivations (Lie-derivations whose image lies in the second term of the lower Lie-central series and that vanish on Lie-central elements), supplies an upper bound on the dimension of the space ID_*^{Lie}(G), and shows that ID_*^{Lie}(G) ≅ ID_*^{Lie}(Q) whenever G and Q are Lie-isoclinic.

Significance. If the stated results hold, the work supplies new structural invariants for Leibniz algebras, especially nilpotent ones, by extending classical derivation and isoclinism techniques from Lie-algebra theory. The dimension bound and the invariance under Lie-isoclinism are concrete, potentially useful contributions to the classification and comparison of Leibniz algebras.

major comments (1)
  1. [Abstract and section introducing Lie-stem Leibniz algebras] The central characterization of Lie-stem Leibniz algebras by their Lie-central derivations (abstract) rests on definitions that already incorporate the image-restriction and vanishing conditions. The manuscript must exhibit an independent definition of “Lie-stem” (presumably in the preliminary section on Lie-center and lower Lie-central series) and verify that the equivalence is not tautological by construction.
minor comments (2)
  1. [Abstract] Abstract, final sentence: the isomorphism statement repeats “ID_*^{Lie}(G)” instead of “ID_*^{Lie}(Q)”.
  2. [Abstract] The phrase “upperbound” should be written as two words throughout.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation for major revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract and section introducing Lie-stem Leibniz algebras] The central characterization of Lie-stem Leibniz algebras by their Lie-central derivations (abstract) rests on definitions that already incorporate the image-restriction and vanishing conditions. The manuscript must exhibit an independent definition of “Lie-stem” (presumably in the preliminary section on Lie-center and lower Lie-central series) and verify that the equivalence is not tautological by construction.

    Authors: We agree that clarity requires an explicit, independent definition of Lie-stem Leibniz algebras. In the revised manuscript we will state, in the preliminaries, that a Leibniz algebra G is Lie-stem when its Lie-center is contained in its Lie-derived algebra: Z_Lie(G) ⊆ [G,G]_Lie. We will then prove the stated characterization as a non-trivial equivalence between this condition and the corresponding property of Lie-central derivations, and we will include a concrete example of a non-stem algebra to illustrate that the equivalence is not tautological. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper introduces Lie-derivations and ID_*-Lie-derivations as direct extensions of standard Leibniz algebra notions (Lie-center, lower Lie-central series, Lie-isoclinism) taken from prior literature. All stated results—a characterization of Lie-stem algebras, dimension bounds, and isomorphisms under Lie-isoclinism—are obtained by algebraic definition and proof from these extensions. No equation reduces to a fitted parameter renamed as prediction, no self-citation is load-bearing for a uniqueness claim, and no ansatz is smuggled via citation. The derivation chain is self-contained against external algebraic benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The work rests on the standard definition of Leibniz algebras and their Lie-center from prior literature; new entities are the introduced derivation maps.

axioms (1)
  • domain assumption Leibniz algebras satisfy the Leibniz identity and possess a well-defined Lie-center and lower Lie-central series.
    Invoked when defining Lie-central derivations and the series used in ID_*-Lie-derivations.
invented entities (2)
  • Lie-derivation no independent evidence
    purpose: Generalizes ordinary derivations to respect only the Lie bracket in a Leibniz algebra.
    New map introduced to study central properties; no independent evidence outside the definitions.
  • Lie-central derivation no independent evidence
    purpose: Lie-derivation whose image lies inside the Lie-center.
    Core new object used for the characterization of Lie-stem algebras.

pith-pipeline@v0.9.0 · 5732 in / 1407 out tokens · 22741 ms · 2026-05-24T20:03:58.086216+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Albeverio, B

    S. Albeverio, B. A. Omirov and I. S. Rakhimov: Classification of 4- dimensional nilpotent complex Leibniz algebras, Extracta Math. 21 (3) (2006), 197–210

    work page 2006

  2. [2]

    Benkart and E

    G. Benkart and E. Neher: The centroid of extended affine and root graded Lie algebras, J. Pure Appl. Algebra 205 (1) (2006), 117–145

    work page 2006

  3. [3]

    G. R. Biyogmam and J. M. Casas: On Lie-isoclinic Leibniz algebras , J. Alge- bra 499 (2018), 337–357. DOI: 10.1016/j.jalgebra.2017.01.034

    work page doi:10.1016/j.jalgebra.2017.01.034 2018

  4. [4]

    G. R. Biyogmam and J. M. Casas: The c-nilpotent Schur Lie-multiplier of Leibniz algebras, J. Geom. Phys. 138 (2019), 55–69

    work page 2019

  5. [5]

    G. R. Biyogmam and J. M. Casas: A Study of n-Lie-isoclinic Leibniz Algebras, J. Algebra Appl. (2019) (to appear). DOI: 10.1142/S0219498820 500139

    work page doi:10.1142/s0219498820 2019

  6. [6]

    Burde, K

    D. Burde, K. Dekimpe and B. Verbeke: Almost inner derivations of Lie al- gebras, J. Algebra Appl. 17 (1) (2018), 1850214, 26 pp

    work page 2018

  7. [7]

    E. M. Ca˜ nete and A. Kh. Khudoyberdiyev:The classification of 4-dimensional Leibniz algebras, Linear Algebra Appl. 439 (1) (2013), 273–288

    work page 2013

  8. [8]

    J. M. Casas and M. A. Insua: The Schur Lie-multiplier of Leibniz algebras , Quaest. Math. 41 (7) (2018), 917–936

    work page 2018

  9. [9]

    J. M. Casas, M. A. Insua, M. Ladra and S. Ladra: An algorithm for the classification of 3-dimensional complex Leibniz algebras , Linear Algebra Appl. 436 (9) (2012), 3747–3756

    work page 2012

  10. [10]

    J. M. Casas and E. Khmaladze: On Lie-central extensions of Leibniz alge- bras, Rev. R. Acad. Cienc. Exactas F ´ ıs. Nat. Ser. A Math. RACSAM 111 (1) (2017). 36–56. DOI 10.1007/s13398-016-0274-6

    work page doi:10.1007/s13398-016-0274-6 2017

  11. [11]

    J. M. Casas and T. Van der Linden: Universal central extensions in semi- abelian categories, Appl. Categor. Struct. 22 (1) (2014), 253–268

    work page 2014

  12. [12]

    Cuvier: Alg` ebres de Leibnitz: d´ efinitions, propri´ et´ es, Ann

    C. Cuvier: Alg` ebres de Leibnitz: d´ efinitions, propri´ et´ es, Ann. Sci. ´Ecol. Norm. Sup. 27 (4) (1994), 1–45. 21

    work page 1994

  13. [13]

    Dixmier and W

    J. Dixmier and W. J. Lixter: Derivations of Nilpotent Lie algebras , Proc. Amer. Math. Soc. 4 (1957), 155–158

    work page 1957

  14. [14]

    Everaert and T

    T. Everaert and T. Van der Linden: Baer invariants in semi-abelian cate- gories I: General theory , Theory Appl. Categ. 12 (1) (2004), 1–33

    work page 2004

  15. [15]

    C. S. Gordon and E. N. Wilson: Isospectral deformations of compact solv- manifolds, J. Differential Geom. 19 (1) (1984), 214–256

    work page 1984

  16. [16]

    Leger: A Note on the derivations of Lie algebras , Proc

    G. Leger: A Note on the derivations of Lie algebras , Proc. Amer. Math. Soc. 4 (1953), 511–514

    work page 1953

  17. [17]

    Loday: Cyclic homology , Grundl

    J.-L. Loday: Cyclic homology , Grundl. Math. Wiss. Bd. 301, Springer, Berlin (1992)

    work page 1992

  18. [18]

    Loday: Une version non commutative des alg` ebres de Lie: les alg` eb res de Leibniz , Enseign

    J.-L. Loday: Une version non commutative des alg` ebres de Lie: les alg` eb res de Leibniz , Enseign. Math. 39 (1993), 269–292

    work page 1993

  19. [19]

    Riyahi and J

    Z. Riyahi and J. M. Casas: Lie-isoclinism of pairs of Leibniz algebras , Bull. Malays. Math. Sci. Soc. (2018) (to appear). DOI: 10.1007/s4084 0-018-0681-2

    work page doi:10.1007/s4084 2018

  20. [20]

    Tˆ ogˆ o:On the derivations of Lie algebras , J

    S. Tˆ ogˆ o:On the derivations of Lie algebras , J. Sci. Math. Hiroshima Univ. Ser. 4, 19 (1955), 71–77

    work page 1955

  21. [21]

    Tˆ ogˆ o:On the derivation algebras of Lie algebras , Canad

    S. Tˆ ogˆ o:On the derivation algebras of Lie algebras , Canad. J. Math. 13 (1961), 201–216

    work page 1961

  22. [22]

    Tˆ ogˆ o:Derivations of Lie algebras , J

    S. Tˆ ogˆ o:Derivations of Lie algebras , J. Sci. Math. Hiroshima Univ., Ser. A-I Math. 28 (1964), 133–158

    work page 1964

  23. [23]

    Tˆ ogˆ o:Outer derivations of Lie algebras , Trans

    S. Tˆ ogˆ o:Outer derivations of Lie algebras , Trans. Amer. Math. Soc. 128 (1967), 264–276. 22

    work page 1967