A counterexample to a Proposition of Feldvoss-Wagemann and Burde-Wagemann

Teimuraz Pirashvili

REVIEW 2 major objections 4 minor 11 references

A counterexample to a Proposition of Feldvoss-Wagemann and Burde-Wagemann

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper constructs a six-dimensional Lie algebra for which a key isomorphism in a recent proof of the weak Leibniz homology conjecture fails, invalidating that proof.

desk verdict Pirashvili's six-dimensional counterexample likely refutes the key proposition in Burde-Wagemann's proof, but the note must pin down the unproven duality that makes the example bite. read the letter →

arxiv 1908.11596 v1 pith:GTYRG276 submitted 2019-08-30 math.KT

Teimuraz Pirashvili This is my paper · ORCID

classification math.KT MSC 17A3217B5617B20

keywords LeibnizhomologyLiealgebracohomologycounterexampleweakconjectureadjointrepresentationLevidecompositionHochschild-Serrespectralsequencesemi-simple

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

This note targets a recent proof of the weak Leibniz-homology conjecture, which says a finite-dimensional complex Lie algebra is semi-simple exactly when its Leibniz homology vanishes in all positive degrees. The author shows that the proof's key proposition—the claimed isomorphism between $HL_p(\mathfrak{g})$ and $HL_{p-1}(\mathfrak{g},\mathfrak{r}^\sharp)$ for every $p\ge 1$—is false. The counterexample is the six-dimensional Lie algebra $\mathfrak{g}=\mathfrak{sl}_2\ltimes\mathfrak{r}$, where $\mathfrak{r}$ is abelian and is the adjoint representation of $\mathfrak{sl}_2$. The failure is detected by the nontrivial coinvariant module $H_0(\mathfrak{s}, H_*(\mathfrak{r})\otimes\mathfrak{s})$, forced by the Killing form. If the note is correct, the claimed proof collapses, two theorems in a companion preprint are wrong, and the weak conjecture remains open.

What carries the argument

The load-bearing device is Lemma 1, which lists five equivalent conditions under which the restriction map $\mathfrak{g}^\sharp\to\mathfrak{r}^\sharp$ induces an isomorphism $HL_*(\mathfrak{g},\mathfrak{g}^\sharp)\cong HL_*(\mathfrak{g},\mathfrak{r}^\sharp)$. Among these is condition (v), $H_0(\mathfrak{s}, H_*(\mathfrak{r})\otimes\mathfrak{s})=0$, which is computed with the homological Hochschild-Serre spectral sequence for the extension $0\to\mathfrak{r}\to\mathfrak{g}\to\mathfrak{s}\to 0$. The counterexample works by showing condition (v) fails: the Killing form produces a nonzero element in $H_0(\mathfrak{sl}_2,\mathfrak{sl}_2\otimes\mathfrak{sl}_2)$, so the equivalent conditions cannot hold, and the claimed isomorphism of Proposition 3.1 fails.

What would settle it

One could compute the Leibniz homology groups $HL_2(\mathfrak{g})$ and $HL_1(\mathfrak{g},\mathfrak{r}^\sharp)$ directly for $\mathfrak{g}=\mathfrak{sl}_2\ltimes\mathfrak{sl}_2$; if they turn out to be isomorphic, the note's central claim that Proposition 3.1 fails would be incorrect.

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Extended reading notes

Core claim

The central claim is that Proposition 3.1 of [1] does not hold for all Lie algebras. The author exhibits the Lie algebra $\mathfrak{g}=\mathfrak{sl}_2\ltimes\mathfrak{r}$ with $\mathfrak{r}$ abelian and isomorphic to the adjoint representation of $\mathfrak{sl}_2$, and shows that condition (v) of Lemma 1 fails: $H_0(\mathfrak{s}, H_1(\mathfrak{r})\otimes\mathfrak{s})\neq 0$ because the Killing form gives a nonzero invariant pairing $H_1(\mathfrak{r})\otimes\mathfrak{s}\to\mathbb{C}$. Since Lemma 1 asserts that condition (v) is equivalent to the isomorphism $HL_p(\mathfrak{g})\cong HL_{p-1}(\mathfrak{g},\mathfrak{r}^\sharp)$ for all $p\ge 1$, this example refutes Proposition 3.1. Consequently the proof of the weak conjecture in [1] is invalid, Theorem 4.7 and Corollary 4.8 of [3] are false, and the weak conjecture stays open.

Load-bearing premise

The argument relies on the standing duality $HL_{*+1}(\mathfrak{g})\cong HL_*(\mathfrak{g},\mathfrak{g}^\sharp)$, which translates Proposition 3.1 into condition (i) of Lemma 1; if that duality has hidden hypotheses, the counterexample would only refute condition (v) without directly contradicting Proposition 3.1.

Editorial extensions

If this is right

The proof of the weak Leibniz-homology conjecture given in [1] is invalid; the conjecture itself is not settled by that argument.
The isomorphism $HL_p(\mathfrak{g})\cong HL_{p-1}(\mathfrak{g},\mathfrak{r}^\sharp)$ is not a general fact; any future proof must handle the radicals that admit invariant pairings with the semisimple part.
Theorems 4.7 and 4.8 of [3] are false as stated, since they rely on the same proposition.
For the six-dimensional example, the nontrivial action of $\mathfrak{sl}_2$ on $H_1(\mathfrak{r})$ through the adjoint representation is the obstruction; trivial or suitably constrained actions would satisfy condition (v).

Reading between the lines

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

A natural next step is to classify the Lie algebras for which condition (v) holds; the example suggests that the vanishing of $H_0(\mathfrak{s},H_*(\mathfrak{r})\otimes\mathfrak{s})$ is a genuine restriction, not an automatic consequence of semisimplicity.
The same construction should work for any semisimple $\mathfrak{s}$ with a nondegenerate invariant bilinear form and $\mathfrak{r}$ a non-trivial module: as long as an invariant map $\mathfrak{r}\otimes\mathfrak{s}\to\mathbb{C}$ exists, condition (v) fails.
A direct computation of the Leibniz homology of the six-dimensional algebra would provide an independent check of the paper's conclusion and would clarify whether the duality $HL_{*+1}(-)\cong HL_*(-,\mathfrak{g}^\sharp)$ itself holds in this case.

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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

Bottom line: this note looks right to me, and it does what it claims. The six-dimensional example—sl2 acting on an abelian radical by the adjoint representation—is a clean falsification of Proposition 3.1 in the Burde-Wagemann preprint. If it stands, the claimed proof of the weak Leibniz homology conjecture is invalid, and the conjecture is open again. That is a meaningful thing to publish, even though the note proves no positive theorem.

What the paper does well: Lemma 1 is a neat reduction. The proof uses the Hochschild-Serre spectral sequence and the author's earlier theorems, and I did not find a gap there. The Killing form argument for condition (v) is correct and short: H1(r) is isomorphic to s, and the Killing form gives a nonzero s-invariant map s⊗s → C, so the coinvariants do not vanish.

The soft spot is the bridge. The line 'Since HL_{*+1}(g) ≅ HL_*(g,g♯) holds always' appears without proof or reference. That isomorphism is what converts the violation of condition (v) into a counterexample to Proposition 3.1. If it is true, everything works. If it has unstated hypotheses, the example only shows conditions (i)–(iv) of Lemma 1 fail, which is not directly a contradiction. I believe this duality is a known result, possibly in [9] or Loday-Pirashvili's papers, but the note should have cited it. A referee should ask for that citation or a one-paragraph proof. This is a presentation gap, not a fatal mathematical flaw.

Two smaller comments. The paper says Theorem 4.7 and Corollary 4.8 of [3] are wrong, but it never shows the dependency; that is worth a sentence. And the reference list has a typo ('Bunde' for 'Burde'), which is harmless but sloppy.

Who this is for: anyone working on Leibniz homology or on the Pirashvili conjecture. The note is short, readable, and the counterexample is easy to verify. It deserves a serious referee and, once the duality question is resolved, publication. I would send it to review.

Referee Report

2 major / 4 minor

Summary. The paper claims to refute Proposition 3.1 of Burde–Wagemann [1] (and consequently Theorem 4.7 and Corollary 4.8 of [3]) by constructing a six-dimensional Lie algebra g = r ⋊ s, where s = sl2 and r is the abelian radical given by the adjoint representation of s. The author proves a lemma giving equivalent conditions for a certain Leibniz-homology isomorphism to hold, shows that the example violates condition (v) of that lemma via the Killing form, and then uses the asserted duality HL_{*+1}(g) ≅ HL_*(g, g^♯) to conclude that Proposition 3.1 of [1] is false. If correct, the note invalidates the claimed proof of the weak Leibniz-homology conjecture, which remains open.

Significance. If the counterexample is valid, it is a significant contribution: it shows that the published proof of the weak conjecture in [1] and [3] is flawed and that the conjecture remains open. The example itself is simple and elegant, and the use of the Killing form to detect the invariant map H_1(r) ⊗ s → C is convincing. The proof of Lemma 1 is largely supported by standard spectral sequence results and the author's earlier work [9]. The main load-bearing gap is the unproven duality in Section 3, which, if supplied, makes the argument sound.

major comments (2)

[Section 3, first paragraph] The assertion 'Since HL_{*+1}(g) ≅ HL_*(g,g^♯) holds always' is the key link between failure of condition (i) of Lemma 1 and falsity of Proposition 3.1 of [1], but no proof or reference is given. If this duality is false or has additional hypotheses, the example only shows that H_0(s, H_*(r) ⊗ s) ≠ 0 and does not by itself contradict Proposition 3.1. Please supply a proof or a precise citation for this duality, or alternatively verify directly for the stated example that HL_p(g) ≇ HL_{p-1}(g,r^♯) for some p (for instance p = 1).
[Section 2, Lemma 1 proof, (i)⇔(ii)] The sentence 'the cohomologies are dual vector spaces of homologies' is too terse and appears to conflate Leibniz cohomology with Leibniz homology with dual coefficients. Condition (i) is a statement about HL_*(g,g^♯) → HL_*(g,r^♯) while condition (ii) concerns HL_*(g,r) → HL_*(g,g); the claimed duality needs a precise formulation, including the relevant module structures, or an explicit reference.

minor comments (4)

[Section 1 and references] There are several typos: 'the the Cheva lley-Eilenberg', 'S. Bunde' in reference [1] should be 'S. Burde', and 'sc F. W agemann' in reference [3] should be 'F. Wagemann'.
[Section 1, displayed diagram] The commutative diagram is difficult to read as typeset; please ensure all arrows and labels are clearly printed.
[Section 3, final sentence] 'This also imply' should be 'This also implies'.
[Section 2, condition (v)] It would be helpful to state explicitly that H_*(r) is regarded as an s-module via the action induced by the section, not merely as a graded vector space.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning detected: the counterexample is an independent computation that uses external theorems as lemmas, not as restatements of the target proposition.

full rationale

The paper's central claim is that a six-dimensional semidirect product sl2 ⋉ adjoint fails condition (v) of Lemma 1 and therefore, via the equivalences in Lemma 1 and the always-true duality HL_{*+1}(g) ≅ HL_*(g, g^♯), refutes Proposition 3.1 of [1]. No step is circular. The target proposition is not used as an input: the example is constructed and checked directly through the Killing form invariant, and the bridge from condition (v) to Proposition 3.1 consists of separately stated equivalences (Lemma 1) and a separately stated duality. The proof of Lemma 1 invokes Theorem A and Proposition 1 from the author's earlier paper [9] and the Hochschild–Serre spectral sequence [4]; these are external published theorems with stated hypotheses (s semi-simple or zero), and they are not equivalent to the proposition being refuted. The duality 'HL_{*+1}(g) ≅ HL_*(g, g^♯)' is asserted without proof or citation, which is a correctness gap in the paper's argument chain, but it is not a circularity: it does not define the target result in terms of itself, nor is it a fitted parameter renamed as a prediction, nor does it reduce to the counterexample by construction. The self-citations are load-bearing but as independent published mathematical facts, not as unverified restatements. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's central argument rests on standard theorems in Lie algebra cohomology (Hochschild-Serre spectral sequence, Killing form, abelian Lie algebra homology) and on two prior results of the author [9]. The most notable unseen premise is the duality between Leibniz homology and Leibniz homology with coefficients in the dual of the adjoint representation, which is asserted without a citation. No free parameters or invented entities appear.

assumptions (5)

domain assumption The duality HL_{*+1}(g) ≅ HL_*(g, g^♯) holds for all finite-dimensional complex Lie algebras g.
Stated without proof or reference in Section 3, first paragraph; it is the bridge that connects Proposition 3.1 of [1] to Lemma 1's condition (i).
domain assumption Theorem A and Proposition 1 of [9] (Pirashvili, On Leibniz homology) are valid.
Used in the proof of Lemma 1 to relate HL_*(g,s) to H_*(g,s). These are prior results by the same author, likely established in the literature.
standard math The homological version of Theorem 13 of Hochschild-Serre [4] applies to the extension 0 -> r -> g -> s -> 0 with s semisimple or zero, giving H_n(g,s) = ⊕_{p+q=n} H_p(s) ⊗ H_0(s, H_q(r,s)).
Invoked in Lemma 1's proof of (iv) <=> (v); standard spectral sequence collapse for semisimple quotient.
standard math The Killing form of sl2 is a nondegenerate invariant bilinear form, so it defines a nontrivial s-module map H_1(r) ⊗ s -> C when r is the adjoint representation.
Used in Section 3 to show condition (v) fails; standard fact about semisimple Lie algebras.
standard math For an abelian Lie algebra r, H_1(r) = r_ab = r, and the action of s on H_1(r) is the adjoint action.
Used implicitly in Section 3; standard computation of Lie algebra homology of an abelian algebra.

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Pith. "Pith review of A counterexample to a Proposition of Feldvoss-Wagemann and Burde-Wagemann." pith.science (2026). https://pith.science/paper/GTYRG276

@misc{pith2026190811596,
  author       = {Pith},
  title        = {Pith review of: A counterexample to a Proposition of Feldvoss-Wagemann and Burde-Wagemann},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GTYRG276}},
  note         = {Machine review of arXiv:1908.11596}
}

abstract

Our (weak) conjecture claims that a finite dimensional Lie algebra ${\bf g}$ over the field of complex numbers is semi-simple iff the Leibniz homology vanishes in positive dimensions $HL_i({\bf g})=0$, $i>0$. We will indicate a mistake in the recent proof of this conjecture due to Burde and Wagemann.

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Works this paper leans on

11 extracted references · 11 canonical work pages

[9]

Pirashvili

T. Pirashvili . On Leibniz homology. Ann. Inst. Fourier. 44 (1994), no. 2, 4 01–411

work page 1994
[3]

Feldvoss and sc F

J. Feldvoss and sc F. W agemann. On Leibniz cohomology. arXiv:1902.0612 8

work page arXiv 1902
[1]

Sympathetic Lie algebras and adjoint cohomology for Lie algebras

S. Bunde and W agemann. Pirashvili’s conjecture on the Leibniz homology for Lie algebras. arXiv:1908.05963

work page Pith review arXiv 1908
[2]

Chevalley and S

C. Chevalley and S. Eilenberg Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63, (1948). 85–124

work page 1948
[4]

Hochschild and J.-P

G. Hochschild and J.-P. Serre . Cohomology of Lie algebras. Ann. Math. 57(1959). 591-603

work page 1959
[5]

J.-L. Loday . Cyclic homology. Grundlehren der Mathematischen Wissens chaften

work page
[6]

Loday and T

J.-L. Loday and T. Pirashvili . Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296 (1993), no. 1, 139–158

work page 1993
[7]

Loday and T

J.-L. Loday and T. Pirashvili . Leibniz representations of Lie algebras. J. Algebra 181 (1996), 414–425

work page 1996

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[8]

P. Ntolo . Homologie de Leibniz d’alg´ ebres de Lie semi-simples. C. R . Acad Sci. Paris, S´ er. I. 318(1994), 707-710

1994
[10]

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T. Pirashvili . On strongly perfect Lie algebras. Communications in Algeb ra. 41(2013),1619-1625

2013
[301]

xviii+454 pp

Springer-Verlag, Berlin, 1992. xviii+454 pp. ISBN: 3- 540-53339

1992

Pith tools

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

[1] [9]

Pirashvili

T. Pirashvili . On Leibniz homology. Ann. Inst. Fourier. 44 (1994), no. 2, 4 01–411

work page 1994

[2] [3]

Feldvoss and sc F

J. Feldvoss and sc F. W agemann. On Leibniz cohomology. arXiv:1902.0612 8

work page arXiv 1902

[3] [1]

Sympathetic Lie algebras and adjoint cohomology for Lie algebras

S. Bunde and W agemann. Pirashvili’s conjecture on the Leibniz homology for Lie algebras. arXiv:1908.05963

work page Pith review arXiv 1908

[4] [2]

Chevalley and S

C. Chevalley and S. Eilenberg Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63, (1948). 85–124

work page 1948

[5] [4]

Hochschild and J.-P

G. Hochschild and J.-P. Serre . Cohomology of Lie algebras. Ann. Math. 57(1959). 591-603

work page 1959

[6] [5]

J.-L. Loday . Cyclic homology. Grundlehren der Mathematischen Wissens chaften

work page

[7] [6]

Loday and T

J.-L. Loday and T. Pirashvili . Universal enveloping algebras of Leibniz algebras and (co)homology. Math. Ann. 296 (1993), no. 1, 139–158

work page 1993

[8] [7]

Loday and T

J.-L. Loday and T. Pirashvili . Leibniz representations of Lie algebras. J. Algebra 181 (1996), 414–425

work page 1996