Some cohomologically rigid solvable Leibniz algebras

Bakhrom Omirov; Gulkhayo Solijanova; Ivan Kaygorodov; Luisa M. Camacho

REVIEW 3 major objections 5 minor 26 references

Some cohomologically rigid solvable Leibniz algebras

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Pith's one-line read This paper proves that for every characteristic sequence of the nilpotent radical, the solvable Leibniz algebra with maximal complementary subspace is unique, centerless, and cohomologically rigid.

desk verdict New Leibniz analogue of cohomological rigidity with a genuine gap: the constructed algebra R is plausible and checkable in the k=2 case, but the uniqueness claim is not justified because the family L(α_i,β_i) is not shown to exhaust the fiber over n_c. read the letter →

arxiv 1908.02360 v1 pith:VQGGSDPO submitted 2019-08-06 math.RA

Luisa M. Camacho , Ivan Kaygorodov , Bakhrom Omirov , Gulkhayo Solijanova This is my paper · ORCID

classification math.RA MSC 17A3217A6017B1017B20

keywords Leibnizalgebrassolvablenilpotentradicalcharacteristicsequencecohomologicallyrigidcompletealgebracohomology

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Solvable Leibniz algebras are a non-antisymmetric generalization of Lie algebras, and this paper asks when a solvable one is rigid in the strongest sense: no nontrivial deformations and no outer derivations. For each decreasing sequence $(n_1,\dots,n_k,1)$—the characteristic sequence, recording Jordan block sizes of a right multiplication operator—it considers nilpotent radicals $L(\alpha_i,\beta_i)$ whose liezation (the quotient by the ideal spanned by squares) is the model algebra $n_c$, together with a complementary subspace of dimension $k+1$, the largest allowed by the number of nil-independent derivations. The paper proves that the resulting solvable Leibniz algebra $R$ is unique up to isomorphism and centerless, that all its derivations are inner (so $R$ is complete), and that $HL^1(R,R)=HL^2(R,R)=0$. Consequently $R$ is cohomologically rigid, and by known deformation theory it is rigid as a point in the variety of Leibniz algebra laws. The upshot is a supply of rigid Leibniz algebras indexed by integer partitions, one for each characteristic sequence.

What carries the argument

The load-bearing object is the pair consisting of the family $L(\alpha_i,\beta_i)$ of nilpotent Leibniz algebras and the solvable extension $R=L(\alpha_i,\beta_i)\oplus Q$ built from the $(k+1)$-dimensional space $Q$ spanned by nil-independent derivations $d_1,\dots,d_{k+1}$. The maximality of $\dim Q$ forces these derivations to act with distinct diagonal weights, and applying the Leibniz identity to triples such as $(e_1,e_1,x_1)$ and $(e_1,e_2,x_1)$ eliminates all parameters $\alpha_i,\beta_i$, leaving the unique table of Theorem 20. The cohomology argument uses the decomposition $R=r_c\oplus J$, where $J=\langle h\rangle$ is a one-dimensional ideal and $r_c$ is the model Lie algebra whose self-cohomology is already known to vanish; the candidate 2-cocycles on the complementary pieces are listed explicitly and shown to be 2-coboundaries.

What would settle it

Work out the smallest case, say characteristic sequence $(2,1,1)$: check whether every nilpotent Leibniz algebra whose liezation is $n_c$ is isomorphic to one of the listed $L(\alpha_i,\beta_i)$, and compute $HL^2(R,R)$ directly from the table of Theorem 20. A single nilpotent radical outside the family that still admits a $(k+1)$-dimensional complementary subspace, or a non-coboundary 2-cocycle on $R$, would disprove the uniqueness or rigidity claim.

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Extended reading notes

Core claim

The central claim is that rigidity is forced by the shape of the nilpotent radical together with maximality of the complementary subspace. Fix a decreasing sequence $(n_1,\dots,n_k,1)$ and let $L(\alpha_i,\beta_i)$ be a nilpotent Leibniz algebra whose liezation is the model nilpotent Lie algebra $n_c$. Any solvable Leibniz algebra $R$ with nilpotent radical $L(\alpha_i,\beta_i)$ and a $(k+1)$-dimensional complementary subspace $Q$ is isomorphic to the explicit algebra of Theorem 20: the parameters $\alpha_i,\beta_i$ are killed by the Leibniz identity once the $k+1$ nil-independent derivations act diagonally, so no choice remains in the multiplication. The paper then shows $Z(R)=0$, $\operatorname{Der}R=\operatorname{Inn}R$, and $HL^1(R,R)=HL^2(R,R)=0$. The proof of the cohomology vanishing reduces the calculation to the known vanishing for the quotient Lie algebra $r_c=R/\langle h\rangle$ and a finite list of candidate 2-cocycles that turn out to be coboundaries.

Load-bearing premise

The classification assumes that the family $L(\alpha_i,\beta_i)$ displayed in Section 3 contains every nilpotent Leibniz algebra whose liezation is the model Lie algebra $n_c$; the paper gives no proof of exhaustiveness, so if other nilpotent Leibniz algebras share that liezation, the uniqueness and rigidity results would cover only a proper subclass.

Editorial extensions

If this is right

For each decreasing sequence $(n_1,\dots,n_k,1)$ there is exactly one solvable Leibniz algebra, up to isomorphism, in the class described; it has zero center and only inner derivations.
The algebra $R$ satisfies $HL^1(R,R)=HL^2(R,R)=0$, so it is cohomologically rigid; by the deformation-theoretic criterion of [4], it is rigid in the variety of Leibniz algebra laws.
The quotient $R/\langle h\rangle$ is the cohomologically rigid Lie algebra $r_c$, so $R$ is a one-dimensional extension of a rigid Lie algebra that preserves rigidity.
The count of characteristic sequences gives at least $p(n)$ distinct irreducible components of the variety of Leibniz algebras of dimension $n+k+3$, where $p(n)$ is the partition number of $n$.
Completeness (centerless and all derivations inner) is established alongside rigidity, so the algebra has no nontrivial automorphisms beyond the inner ones.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The proof leaves open whether every nilpotent Leibniz algebra with liezation $n_c$ appears in the family $L(\alpha_i,\beta_i)$; a natural next step is to classify those nilpotent algebras and, if new ones exist, test whether they admit solvable extensions with the same rigidity.
The mechanism suggests a broader principle: a maximal torus of nil-independent derivations may be what forces the Leibniz law to be rigid. One could test this by varying the characteristic sequence or allowing several 'generator' elements and checking whether the parameters still die.
Because the multiplication tables are explicit, the same algebras can be used to compute higher cohomology groups $HL^q(R,R)$ for $q\ge 3$, which the paper does not do, and to study degenerations between the irreducible components they define.
The asymptotic count of irreducible components via $p(n)$ could be sharpened for concrete small dimensions using the explicit tables, which might reveal component structure beyond mere existence.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

My take: this is the first Leibniz version of the Ancochea–Campoamor rigidity results for solvable Lie algebras with model nilpotent radicals, and for the k=2 case it is concrete and checkable. The construction of R, the proof of completeness, and the claim HL^2(R,R)=0 for that particular case are the real content. The paper also gives a sensible lower bound on irreducible components, which is a nice payoff. I credit the authors for being honest that the general case is a sketch. The soft spots are real, and the stress-test note lands. At the start of Section 3 the family L(α_i,β_i) is introduced as if it were the class of nilpotent Leibniz algebras whose liezation is the model Lie algebra n_c. No normal form or exhaustion argument is given. That is load-bearing for the uniqueness statement: if there are other nilpotent Leibniz algebras with the same liezation, then Theorems 15 and 20 and the abstract's "such Leibniz algebra is unique" only describe a proper subclass. The rigidity result for the specific R does not depend on exhaustiveness, so the gap is in the classification scope, not in the construction itself. The second soft spot is Proposition 17. The listed 2-cochains are claimed to form a basis of ZL^2(R,R) and BL^2(R,R), but the proof is only a sketch and the actual calculations are not shown. For a table this large, that is asking a lot of the reader. Section 4 then extends the same pattern to arbitrary k by saying the routine computations are omitted. That is thinner than I would like for a classification theorem, though the k=2 case gives a plausible template. None of this is a load-bearing flaw in the central algebra. The construction likely works and the rigidity claim for R is probably correct. The authors need to either prove exhaustiveness of the family or explicitly frame the results as conditional on the family, and they should provide fuller verification for Proposition 17. I would send this to a serious referee: the topic is important enough, the k=2 case is a solid proof of concept, and the gaps are fixable. A specialist in Leibniz algebra cohomology should be able to tell whether the general-case computations really follow.

Referee Report

3 major / 5 minor

Summary. The paper studies finite-dimensional solvable complex Leibniz algebras whose quotient by a one-dimensional ideal is a Lie algebra with characteristic sequence (n1,...,nk,1) and whose complementary subspace to the nilpotent radical has dimension k+1. For a family of nilpotent Leibniz algebras L(α_i,β_i) whose liezation is the model nilpotent Lie algebra n_c, the authors construct a solvable Leibniz algebra R, prove it is centerless and complete, and prove that HL^1(R,R)=HL^2(R,R)=0, using the vanishing theorem for the quotient Lie algebra r_c from [2]. The case k=2 is worked out with explicit derivations, multiplication tables, and Leibniz-identity constraints; the general case is presented as a sketch in Section 4.

Significance. If the announced results could be fully proved, the paper would provide an explicit infinite family of cohomologically and geometrically rigid solvable Leibniz algebras parameterized by decreasing sequences, together with a lower bound on the number of irreducible components of the relevant variety. The k=2 portion is a concrete, checkable construction: the derivation matrices, the nil-independence count, and the Leibniz-identity constraint tables are explicit, and the use of the external vanishing theorem for r_c is legitimate. However, the general-k theorems are not actually proved in the submitted text, and the exhaustion of the nilpotent-radical family is assumed rather than established. As it stands, the significance is limited to the k=2 case plus a plausibility argument for arbitrary k.

major comments (3)

[Section 3, family L(α_i,β_j)] The displayed family is introduced as the family of nilpotent Leibniz algebras whose corresponding Lie algebra is n_c, but no argument is given that every nilpotent Leibniz algebra with liezation n_c is isomorphic to one of the algebras in this family. The normalization argument only shows that α1 can be made nonzero inside the displayed family; it does not prove exhaustiveness. Since Theorem 20 and, through it, Corollary 22 are stated only for solvable Leibniz algebras whose nilpotent radical is one of the L(α_i,β_i), the abstract's claim that 'such Leibniz algebra is unique' is stronger than what the proofs establish unless exhaustiveness is supplied.
[Section 4, Theorems 20 and 21] The general case is explicitly a sketch: step (1) says 'we compute' Der(L(α_i,β_i)) and 'indicate' k+1 nil-independent derivations, while step (4) reduces the triviality of HL^2 to 'computations of dimensions' without presenting those computations. No proof of Theorem 20 or Theorem 21 for k>2 is actually given. Because these theorems are the main generalization advertised in the abstract and introduction, the manuscript as submitted does not establish the stated results for arbitrary characteristic sequence; either the omitted calculations must be included or the claims must be restricted to the case k=2.
[Section 3.1, Proposition 17] Proposition 17 lists fifteen families of 2-cochains and asserts that they form a basis of ZL^2(R,R) and BL^2(R,R), but the proof only says this follows by straightforward calculations using Theorem 8 and by 'identifying the basis of complementary subspace.' Since Theorem 19, the main rigidity statement for the k=2 case, is exactly the equality of the dimensions of these spaces, the proposition needs at least explicit dimension counts or a reproducible verification of the cocycle and coboundary conditions and of linear independence. As written, the key cohomological claim is asserted rather than demonstrated.

minor comments (5)

[Section 3.1, Proposition 17] In the displayed list for φ5, the expression 'φ5(f1,x3) = -φ11(x3,f1) = h' uses the subscript 11, which is inconsistent with the label φ5; this appears to be a typo and should be corrected.
[Section 3.1, opening paragraph] The family is named L(α1,α2,β1,β2), but the multiplication table also contains the parameter α3 for [f1,f1]; the notation should be made uniform, for instance by renaming the family L(α2,α3,β1,β2) after the normalization α1=1.
[Lemma 13] The formula for d(h) contains the term α3ν1, while the preceding derivation results in Lemma 12 use only α1 and α2; the indexing should be clarified so that the reader can track the parameters through the proof.
[Proposition 14, proof] The equation marked (*) is difficult to parse as printed; the authors should spell out the intermediate steps showing how [f1,[x2,x4]] expands to -e2 plus an element of L(αi,βj)^2.
[Remark 23] The definition of p(n) uses the condition nk ≥ 0, whereas the characteristic sequences in the paper are required to have nk ≥ 1; this convention should be stated explicitly to avoid ambiguity.

Circularity Check

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No significant circularity; the derivation is self-contained modulo standard external theorems.

full rationale

Walking the derivation chain, the central construction starts from an explicit family L(alpha_i,beta_j) of nilpotent Leibniz algebras whose liezation is the model Lie algebra n_c (Section 3). The complement dimension bound is derived from a computation of nil-independent derivations (Lemma 13, Proposition 14), and the algebra R is obtained by enforcing the Leibniz identity on a generic lift (Theorem 15). No parameter is fitted to the quantity being predicted; the alpha_i,beta_j parameters are either normalized by basis change or forced to zero by the derivation and cocycle equations. The cohomological rigidity argument imports the vanishing theorem H^a(r_c,r_c)=0 (a<=3) from [2], an external source not authored by the current authors, and then computes the complementary subspaces to ZL^2(r_c,r_c) and BL^2(r_c,r_c) directly in Proposition 17; this is a standard reduction to an external benchmark, not a circular equation. The use of [10] for the dimension bound is prior background by an overlapping author, but it is a general structural lemma for solvable Leibniz algebras and does not encode the target rigidity result. The one substantive weakness, absence of a proof that the displayed family L(alpha_i,beta_j) exhausts all nilpotent Leibniz algebras with liezation n_c, is a completeness/scoping issue: it affects the claimed scope of the classification, but no displayed equation in the paper defines the theorem into its hypotheses. No circular step is present.

Assumptions & free parameters 1 free parameters · 7 assumptions · 0 invented entities

The paper relies on external theorems from [2], [3], [10], [14], and [21] as benchmarks. The only ad hoc premise is the unproved exhaustiveness of the model family L(alpha_i,beta_i) among nilpotent Leibniz algebras with liezation n_c. The final algebra R has no fitted constants.

free parameters (1)

characteristic sequence (n1,...,nk) = arbitrary nonincreasing sequence of positive integers
The construction and all theorems depend on choosing such a sequence; each sequence defines a distinct algebra R. It is an input parameter, not fitted to data.

assumptions (7)

standard math Leibniz identity defines the algebras under study.
Used throughout as the defining identity for Leibniz algebras.
domain assumption Theorem 8 from [2]: for the model solvable Lie algebra r_c, H^a(r_c,r_c)=0 for 0 <= a <= 3.
External benchmark used to establish vanishing of the quotient part of the Leibniz cohomology.
domain assumption Theorem 6 from [21]: a solvable Lie algebra R=N⊕Q with dimQ=dimN/N^2=k admits a basis with the given multiplication table.
Used to justify the structure of the complementary subspace in the Lie case, extended to Leibniz in the paper.
domain assumption Theorem 4 from [3]: a finite-dimensional Leibniz algebra over C is solvable iff L^2 is nilpotent.
Used to identify solvability and to restrict the structure of R^2.
ad hoc to paper The family L(alpha_i,beta_i) is presented as the family of nilpotent Leibniz algebras whose corresponding Lie algebra is n_c; exhaustiveness is not proved.
Load-bearing for the uniqueness claim in the abstract; if false, the classification covers only a subclass.
domain assumption Dimension of Q is bounded by the number of nil-independent derivations of the nilradical (from [10]).
Used in Proposition 14 to bound dim Q by 3 in the particular case and by k+1 in general.
standard math Remark 10 from [14]: for a centerless Lie algebra G, H^2(G,G) equals HL^2(G,G).
Used to translate the vanishing of Lie algebra cohomology for r_c into Leibniz cohomology language.

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Cite this review

Pith. "Pith review of Some cohomologically rigid solvable Leibniz algebras." pith.science (2026). https://pith.science/paper/VQGGSDPO

@misc{pith2026190802360,
  author       = {Pith},
  title        = {Pith review of: Some cohomologically rigid solvable Leibniz algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VQGGSDPO}},
  note         = {Machine review of arXiv:1908.02360}
}

read the original abstract

In this paper we describe solvable Leibniz algebras whose quotient algebra by one-dimensional ideal is a Lie algebra with rank equal to the length of the characteristic sequence of its nilpotent radical. We prove that such Leibniz algebra is unique and centerless. Also it is proved that the first and the second cohomology groups of the algebra with coefficients in itself is trivial.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [2]

Ancochea Berm´ udez J. M., Campoamor-Stursberg R., Cohomologically rigid solvable Lie algebras with a nilradi cal of arbitrary characteristic sequence, Linear Algebra and its Applications, 488 (2016), 135–147

work page 2016

[2] [1]

Adashev J., Camacho L., Omirov B., Central extensions of null-ﬁliform and naturally graded ﬁl iform non-Lie Leibniz algebras, Journal of Algebra, 479 (2017), 461–486

work page 2017

[3] [3]

A., Omirov B

A yupov Sh. A., Omirov B. A., On Leibniz algebras , Algebra and operator theory, Proceedings of the Colloquiu m in Tashkent 1997. Kluwer Academic Publishers, 1998, 1–12

work page 1997

[4] [4]

Balavoine D., D´eformations et rigidit ´e g ´eom´etrique des alg `ebres de Leibniz , Communications in Algebra, 24 (1996), 3, 1017–1034

work page 1996

[5] [5]

D., Hird J

Bosko-Dunbar L., Dunbar J. D., Hird J. T., Stagg K., Solvable Leibniz Algebras with Heisenberg Nilradical , Communica- tions in Algebra, 43 (2015), 6, 2272–2281

work page 2015

[6] [6]

Boyko V ., Patera J., Popovych R., Invariants of solvable Lie algebras with triangular nilrad icals and diagonal nilindepen- dent elements, Linear Algebra and its Applications, 428 (2008), 834–854

work page 2008

[7] [7]

W ., On Levi’s theorem for Leibniz algebras , Bulletin of the Australian Mathematical Society, 86 (2012 ), 2, 184–185

Barnes D. W ., On Levi’s theorem for Leibniz algebras , Bulletin of the Australian Mathematical Society, 86 (2012 ), 2, 184–185

work page 2012

[8] [8]

J., Camacho L

Calder´ on A. J., Camacho L. M., Omirov B. A., Leibniz algebras of Heisenberg type , Journal of Algebra, 452 (2016), 427–447

work page 2016