Sympathetic Lie algebras and adjoint cohomology for Lie algebras

Dietrich Burde; Friedrich Wagemann

REVIEW 3 major objections 3 minor 1 cited by

Sympathetic Lie algebras and adjoint cohomology for Lie algebras

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper claims to settle Pirashhvili's weak conjecture: a nontrivial complex Lie algebra is semisimple exactly when its Leibniz homology vanishes.

desk verdict The advertised proof of Pirashvili's conjecture collapses on a step the authors themselves mark invalid; the auxiliary sympathetic-algebra results are solid enough to warrant a referee, but this version should not be accepted. read the letter →

arxiv 1908.05963 v3 pith:PQ76TCXG submitted 2019-08-16 math.RA

Dietrich Burde , Friedrich Wagemann This is my paper · ORCID

classification math.RA MSC 17A3217B56

keywords LiealgebracohomologyLeibnizhomologyPirashhviliconjecturesemisimplesympatheticperfectcompleteadjoint

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

The paper claims to prove Pirashhvili's weak conjecture: a nontrivial finite-dimensional complex Lie algebra is semisimple if and only if all its Leibniz homology groups with trivial coefficients vanish. One direction was already known; the paper's contribution is the converse. Assuming a non-semisimple algebra satisfied the vanishing conditions, the authors show its solvable radical would have to be nilpotent and non-abelian, then construct a nonzero first cohomology class with coefficients in the radical, contradicting an earlier vanishing result. If the proof stands, the equivalence gives a clean homological characterization of semisimplicity. The printed route depends on a cited isomorphism that the authors themselves annotate as seemingly incorrect.

What carries the argument

The central machinery is the Hochschild–Serre spectral sequence, applied to the Levi decomposition $\mathfrak{g}=\mathfrak{s}\ltimes\mathfrak{r}$ and then to the abelian ideal $Z(\mathfrak{n})$. The key identity identifies an invariant-cohomology piece with $\mathrm{Hom}_{\mathfrak{s}}(Z(\mathfrak{n}),Z(\mathfrak{n}))$, which contains the identity map, guaranteeing a nonzero $H^1(\mathfrak{g},\mathfrak{r})$. A cited isomorphism from [9], $\mathrm{HL}_p(\mathfrak{g})\cong \mathrm{HL}_{p-1}(\mathfrak{g},\mathfrak{r}^*)$, translates vanishing Leibniz homology into vanishing Lie algebra cohomology with coefficients in the radical, and that translation is the load-bearing bridge of the proof.

What would settle it

For a concrete low-dimensional perfect Lie algebra with a non-abelian nilpotent radical, compute both sides of the isomorphism $\mathrm{HL}_p(\mathfrak{g})\cong \mathrm{HL}_{p-1}(\mathfrak{g},\mathfrak{r}^*)$ for small $p$; if they differ, Proposition 3.1 is false and the printed proof cannot be repaired by this route.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes Theorem 4.1: every nontrivial finite-dimensional complex Lie algebra satisfying the Pirashhvili conditions—vanishing of all Leibniz homology groups $\mathrm{HL}_p(\mathfrak{g})=0$ for $p\ge 1$, equivalently vanishing adjoint cohomology $H^p(\mathfrak{g},\mathfrak{g})=0$ with $\mathfrak{g}$ perfect—is semisimple. The argument starts from the Levi decomposition $\mathfrak{g}=\mathfrak{s}\ltimes\mathfrak{r}$ and shows the radical $\mathfrak{r}$ must be nilpotent and non-abelian; then the center $Z(\mathfrak{n})$ of the nilradical is a nonzero abelian ideal, and the identity map on $Z(\mathfrak{n})$ yields a nonzero $\mathfrak{s}$-invariant element in $H^1(Z(\mathfrak{n}),\mathfrak{n})$, which by the Hochschild–Serre spectral sequence survives as a nonzero class in $H^1(\mathfrak{g},\mathfrak{r})$, contradicting $H^p(\mathfrak{g},\mathfrak{r})=0$ for all $p$. The paper notes that the isomorphism $\mathrm{HL}_p(\mathfrak{g})\cong \mathrm{HL}_{p-1}(\mathfrak{g},\mathfrak{r}^*)$ used to obtain $H^p(\mathfrak{g},\mathfrak{r})=0$ comes from [9] and 'seems to be incorrect,' which would invalidate the proof of Proposition 3.3.

Load-bearing premise

The proof depends on a cited formula that turns vanishing Leibniz homology into vanishing cohomology with coefficients in the largest solvable ideal; the paper's own note says that formula 'seems to be incorrect,' which breaks the final step as written.

Editorial extensions

If this is right

Pirashhvili's weak conjecture holds: for finite-dimensional complex Lie algebras, vanishing Leibniz homology with trivial coefficients characterizes semisimplicity.
A perfect Lie algebra with all adjoint cohomology groups $H^p(\mathfrak{g},\mathfrak{g})=0$ must be semisimple.
Sympathetic Lie algebras—perfect and complete—with vanishing adjoint cohomology are exactly the semisimple ones.
Non-perfect algebras can have vanishing adjoint cohomology without being semisimple, as the affine example shows; perfectness is the needed extra condition.

Reading between the lines

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

If the flagged isomorphism in Proposition 3.1 is genuinely false, the theorem may still be true but needs a different route to $H^p(\mathfrak{g},\mathfrak{r})=0$; checking that isomorphism on a low-dimensional perfect Lie algebra with nontrivial radical would settle the point.
The obstruction constructed from the identity map on $Z(\mathfrak{n})$ suggests a general principle: any nonzero abelian characteristic ideal of the radical obstructs vanishing adjoint cohomology of a perfect Lie algebra.
The same spectral-sequence argument may yield an analogous characterization over any algebraically closed field of characteristic zero, not just the complex numbers.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

The headline is simple: the paper announces a proof of Pirashvili's weak conjecture, and the proof is not there. The authors attach a dated note to Proposition 3.1 saying the result 'seems to be incorrect' and that the proof of Proposition 3.3 is therefore not valid. Proposition 3.3 is the engine of Theorem 4.1: it is what produces the vanishing H^p(g,r)=0 that the final contradiction relies on. Without it, the argument in Section 4 does not go through. So the central claim should be withdrawn or explicitly re-labeled as conditional.

That said, the paper is not without value. Lemmas 3.5 and 3.6 — perfect Lie algebras have nilpotent radical, and sympathetic Lie algebras with abelian radical are semisimple — are clean and believable, and Proposition 3.8 (a nonzero abelian radical forces H^1(g,g)≠0) is a useful contribution to the study of complete and sympathetic algebras. The Hochschild-Serre computations are standard but handled carefully. Section 2's translation of the Pirashvili conditions into adjoint cohomology vanishing is a helpful survey.

The soft spots, in order of severity: the main theorem is unsupported; the abstract overstates what is proved, since it does not mention the invalidation; and the paper's own internal note leaves the reader with no clue which parts of Section 3 and 4 survive if Proposition 3.1 is false. The authors should either repair the bridge or reframe the paper around the sympathetic-algebra results, which may be the lasting part. Incidentally, the self-citation to [9] is not a problem by itself; the problem is that the cited result is now suspect.

This is a paper that a serious referee should see, not because the main theorem is in any state to be accepted, but because the auxiliary results are worth checking and the conjecture is important enough to keep working on. Send it to a referee who knows Leibniz cohomology and let them sort out what is salvageable. But as written, it should be rejected.

Referee Report

3 major / 3 minor

Summary. The paper studies sympathetic Lie algebras (perfect and complete) and their relation to the vanishing of adjoint Lie algebra cohomology and Leibniz (co)homology. Its central result, Theorem 4.1, claims to prove Pirashvili's weak conjecture: a non-trivial finite-dimensional complex Lie algebra is semisimple if and only if its Leibniz homology with trivial coefficients vanishes. The proof strategy is to derive, from the Pirashvili conditions, the vanishing of H^p(g,r) for the solvable radical r using a result from the authors' companion paper [9], then use structural results on the radical plus a Hochschild-Serre spectral sequence argument to obtain a contradiction for a non-semisimple algebra. The paper also contains several auxiliary results on perfect/complete Lie algebras and on H^1(g,g) for abelian radicals.

Significance. If Theorem 4.1 were valid, it would settle a long-standing conjecture of Pirashvili and would be a significant contribution to the cohomological characterization of semisimple Lie algebras. The auxiliary structural results, especially Lemmas 3.5 and 3.6 and Proposition 3.8, are clean and of independent interest, and the paper is careful in its use of Hochschild-Serre spectral sequences. However, the main theorem is not established in the submitted version: the authors themselves state that the key bridge result, Proposition 3.1 from [9], appears incorrect and that the proof of Proposition 3.3 is therefore invalid. Since Proposition 3.3 is the only place where the Pirashvili conditions are converted into the vanishing H^p(g,r)=0 that drives the final contradiction, the central claim is unsupported.

major comments (3)

[Section 3, Proposition 3.3] The proof of Proposition 3.3 relies on Proposition 3.1 to pass from HL_p(g)=0 (the Pirashvili conditions) to HL_{p-1}(g,r*)=0, and then via duality and Proposition 3.2 to H^{p-1}(g,r)=0. Immediately after Proposition 3.1, the authors insert the note: 'This result seems to be incorrect. Therefore the proof of Proposition 3.3 is not valid (04.09.2019).' This is an explicit, in-manuscript admission that the proof of Proposition 3.3 is invalid. Since Proposition 3.3 is the only step that yields H^p(g,r)=0 for all p under the Pirashvili conditions, the rest of the paper cannot rely on it.
[Section 4, Theorem 4.1] In the proof of Theorem 4.1, after showing that H^1(g,r) is nonzero for a non-semisimple algebra satisfying the Pirashvili conditions, the final contradiction is obtained by invoking Proposition 3.3. Because Proposition 3.3 is not established (see the previous comment and the authors' own note), the contradiction has no foundation. The structural results used earlier, such as Proposition 3.7 and the non-vanishing argument for H^1(g,g), only constrain the radical to be nilpotent and non-abelian; they do not alone rule out the existence of a non-semisimple Lie algebra satisfying the Pirashvili conditions. Thus Theorem 4.1 is not proven in this manuscript.
[Section 3, Proposition 3.4] In the proof of the converse direction of Proposition 3.4, the authors write 'For all ℓ ≥ 3 we obtain H^3(s) ⊗ H^{ℓ-3}(r,r)^s = 0. Since s is semisimple, we have H^3(s) ≠ 0', and from this conclude H^{ℓ-3}(r,r)^s = 0. This step fails when the Levi factor s is zero, i.e. when g is solvable, because H^3(0) = 0. The statement of the proposition may still hold in that case by a simpler argument, but the proof as written has a gap. This issue is secondary to the failure of Proposition 3.3, but it is a concrete defect in the exposition.

minor comments (3)

[Section 3, after Proposition 3.1] The note 'This result seems to be incorrect...' is a self-correction marker that should not appear in a submitted version; it must be resolved either by proving a corrected version, removing the dependence on Proposition 3.1, or withdrawing the claim.
[Throughout] There are several typographical issues: 'Poicar´e' should be 'Poincaré', 'Propo sition' has an extra space, and 'Pirashivili' appears in the arXiv title while 'Pirashvili' is used elsewhere. These should be fixed in revision.
[Section 3, Proposition 3.2] The statement of Proposition 3.2 is fine, but the proof says the homology part is 'analogous' without details; given that the paper's main argument later uses the homology version implicitly, a short indication of the proof would improve clarity.

Circularity Check

1 steps flagged · score 8.0 of 10

The proof of Theorem 4.1 is carried by the authors' own Proposition 3.1 from [9], which the paper's note withdraws; this is a load-bearing self-citation, not an independent derivation.

self citation load bearing [Section 3, Proposition 3.1 and the note after it; Proposition 3.3 proof; Theorem 4.1 proof.]
"0 = HL_p(g) ∼= HL_{p−1}(g, r∗) ∼= (HL_{p−1}(g, r))∗. ... Note: This result seems to be incorrect. Therefore the proof of Proposition 3.3 is not valid (04.09.2019)."

The contradiction in Theorem 4.1 is H^1(g,r) ≠ 0 (from Hom_s(Z(n),Z(n)) ≠ 0) versus H^1(g,r) = 0 from Proposition 3.3. Proposition 3.3 is obtained by the chain 0 = HL_p(g) ≅ HL_{p−1}(g, r*) ≅ (HL_{p−1}(g, r))*, and the first isomorphism is Proposition 3.1, quoted as Corollary 4.8 of [9] (Feldvoss–Wagemann; Wagemann is a present author). The paper's own note after Proposition 3.1 says the result 'seems to be incorrect' and that therefore the proof of Proposition 3.3 is not valid. Hence the only bridge from the Pirashvili input to the vanishing H^p(g,r) = 0 that Theorem 4.1 needs is an unverified, explicitly doubted self-citation. The theorem is forced through this self-citation chain, not derived from independent premises.

full rationale

Walking the derivation chain: Pirashvili's conditions are translated by Proposition 2.5 (from Pirashvili [15]) into vanishing adjoint (co)homology plus perfection; this is an external equivalence and not circular. The sympathetic-Lie-algebra results (Lemmas 3.5, 3.6, Propositions 3.7 and 3.8) are proved from Hochschild–Serre and elementary complete-Lie-algebra arguments; they do not assume semisimplicity. The decisive step is Proposition 3.3: H^p(g,r) = 0 for all p, proved via the chain 0 = HL_p(g) ≅ HL_{p−1}(g, r*) ≅ (HL_{p−1}(g, r))*, where the first isomorphism is Proposition 3.1 quoted from [9], a paper coauthored by the present second author. Immediately after Proposition 3.1 the authors insert: 'This result seems to be incorrect. Therefore the proof of Proposition 3.3 is not valid (04.09.2019).' Theorem 4.1 then relies solely on Proposition 3.3 for the contradiction H^1(g,r) ≠ 0 versus H^1(g,r) = 0. Thus the main claim is not independently derived: it is carried by a self-citation that the paper itself withdraws. There is no fitted parameter or definitional identity, so the circularity is of the self-citation-load-bearing kind rather than a self-definitional reduction; the score is set at 8 because the central conclusion is forced by the self-citation chain once that chain is accepted.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new objects or fitted parameters. Its central claim depends on standard cohomology tools plus one cited lemma that the authors themselves suspect is false, and that lemma is the main reason the proof fails.

assumptions (6)

standard math Chevalley-Eilenberg and Leibniz (co)homology standard definitions, duality isomorphisms, and long exact sequences
Used throughout, e.g. Lemma 2.1 and Proposition 2.5.
standard math Whitehead's first and second lemmas for finite-dimensional modules over a semisimple Lie algebra
Used in Proposition 3.4 and Proposition 3.8 to kill H^1 and H^2 terms.
standard math Hochschild-Serre spectral sequence for Lie algebras
Used in Proposition 3.4, Proposition 3.8, and Theorem 4.1.
standard math Poincaré duality for unimodular Lie algebras (Lemma 2.2)
Used in the proof of Proposition 3.3 to pass from homology to cohomology.
ad hoc to paper Proposition 3.1 from Feldvoss-Wagemann [9]: HL_p(g) ≅ HL_{p-1}(g, r^*)
This is the load-bearing cited result; the authors annotate it as seemingly incorrect, and it is used to prove Proposition 3.3, which is needed for Theorem 4.1.
standard math For nonzero semisimple Lie algebra s, H^3(s) is nonzero
Used in the converse direction of Proposition 3.4.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Sympathetic Lie algebras and adjoint cohomology for Lie algebras." pith.science (2026). https://pith.science/paper/PQ76TCXG

@misc{pith2026190805963,
  author       = {Pith},
  title        = {Pith review of: Sympathetic Lie algebras and adjoint cohomology for Lie algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PQ76TCXG}},
  note         = {Machine review of arXiv:1908.05963}
}

read the original abstract

We study sympathetic Lie algebras, namely perfect and complete Lie algebras. They arise among other things in the study of adjoint Lie algebra cohomology. This is motivated by a conjecture of Pirashvili, which says that a non-trivial finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several results on sympathetic Lie algebras and the adjoint Lie algebra cohomology of Lie algebras in general, using the Hochschild-Serre formula. For certain semidirect products we obtain explicit results for the adjoint cohomology.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

A counterexample to a Proposition of Feldvoss-Wagemann and Burde-Wagemann
math.KT 2019-08 conditional novelty 6.0 of 10

A six-dimensional semidirect product example shows Proposition 3.1 of Burde-Wagemann (arXiv:1908.05963) is false, invalidating their proposed proof of the weak Leibniz homology conjecture.

Reference graph

Works this paper leans on

16 extracted references · 11 canonical work pages · cited by 1 Pith paper

[9]

Feldvoss, F

J. Feldvoss, F. Wagemann: On Leibniz cohomology . arXiv:1902.06128

arXiv 1902
[1]

Angelopoulos: Alg` ebres de Lie g satisfaisant [g,g] = g, Der(g) = ad( g)

E. Angelopoulos: Alg` ebres de Lie g satisfaisant [g,g] = g, Der(g) = ad( g). C. R. Acad. Sci. Paris, t. 306, S´ erie I (1988), 523–525

work page 1988
[2]

Angelopoulos: Construction d’alg` ebres de Lie sympathiques non semi-sim ples munies de produits scalaires invariants

E. Angelopoulos: Construction d’alg` ebres de Lie sympathiques non semi-sim ples munies de produits scalaires invariants. C. R. Acad. Sci. Paris, t. 317, S´ erie I (1993), 741–744

work page 1993
[3]

Arnal, H

D. Arnal, H. Benamor, S. Benayadi, G. Pinczon: Une alg` ebre de Lie non semi-simple v´ eriﬁant H 1(g) = H 2(g) = H 0(g,g) = H 1(g,g) = H 2(g,g) = 0. 8 D. BURDE AND F. W AGEMANN

work page
[4]

Benayadi: Certaines propri´ et´ es d’une classe d’alg` ebres de Lie quig´ en´ eralisent les alg` ebres de Lie semi- simples

S. Benayadi: Certaines propri´ et´ es d’une classe d’alg` ebres de Lie quig´ en´ eralisent les alg` ebres de Lie semi- simples. Ann. Fac. Sci. Toulouse, 5 e s´ erie, t.12, 1 (1991), 29–35

1991
[5]

Benayadi: Structure of perfect Lie algebras without center and outer d erivations

S. Benayadi: Structure of perfect Lie algebras without center and outer d erivations. Ann. Fac. Sci. Toulouse, 6e s´ erie, t.5, 2 (1996), 203–231

1996
[6]

Carles: Sur la structure des alg` ebres de Lie rigides

R. Carles: Sur la structure des alg` ebres de Lie rigides . Ann. Inst. Fourier Grenoble, t. 34, 3 (1984), 65–82

work page 1984
[7]

Carles: Sur certaines classes d’alg` ebres de Lie rigides

R. Carles: Sur certaines classes d’alg` ebres de Lie rigides . Math. Ann. 272 (1985), 477–488

work page 1985

Show all 16 references

[8]

Feldvoss: Leibniz algebras as non-associative algebras , in Nonassociative Mathematics and Its Applica- tions, Denver, CO, 2017 (eds

J. Feldvoss: Leibniz algebras as non-associative algebras , in Nonassociative Mathematics and Its Applica- tions, Denver, CO, 2017 (eds. P. Vojtˇ echovsk´ y, M. R. Bremner, J. S. Carter, A. B. Evans, J. Huerta, M. K. Kinyon, G. E. Moorhouse, J. D. H. Smith), Contemp. Math. vol...

2017
[10]

D. B. Fuchs: Cohomology of inﬁnite dimensional Lie algebras . Consultants Bureau, New York and London (1986)

1986
[11]

Hochschild, J-P

G. Hochschild, J-P. Serre: Cohomology of Lie algebras . Ann. Math. (2) 57 (1953), no. 3, 591–603

1953
[12]

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

J.-L. Loday: Une version non commutative des alg` ebres de Lie: les alg` eb res de Leibniz . Enseign. Math. (2) 39 (1993), no. 3-4, 269–293

1993
[13]

Loday, T

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

1993
[14]

Pirashvili: On Leibniz homology

T. Pirashvili: On Leibniz homology . Ann. Inst. Fourier Grenoble 44 (1994), no. 2, 401–411

1994
[15]

Pirashvili: On strongly perfect Lie algebras

T. Pirashvili: On strongly perfect Lie algebras . Communications in Algebra 41 (2013), no. 5, 1619–1625

2013
[16]

Zusmanovich: A converse to the second Whitehead lemma

P. Zusmanovich: A converse to the second Whitehead lemma . J. Lie Theory 18 (2008), no. 2, 295-299. F akult¨at f ¨ur Mathematik, Universit ¨at Wien, Oskar-Morgenstern-Platz 1, 1090 Wien, Aus- tria E-mail address : dietrich.burde@univie.ac.at Laboratoire de math ´ ematiques Jea...

2008

Pith tools

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

[1] [9]

Feldvoss, F

J. Feldvoss, F. Wagemann: On Leibniz cohomology . arXiv:1902.06128

arXiv 1902

[2] [1]

Angelopoulos: Alg` ebres de Lie g satisfaisant [g,g] = g, Der(g) = ad( g)

E. Angelopoulos: Alg` ebres de Lie g satisfaisant [g,g] = g, Der(g) = ad( g). C. R. Acad. Sci. Paris, t. 306, S´ erie I (1988), 523–525

work page 1988

[3] [2]

Angelopoulos: Construction d’alg` ebres de Lie sympathiques non semi-sim ples munies de produits scalaires invariants

E. Angelopoulos: Construction d’alg` ebres de Lie sympathiques non semi-sim ples munies de produits scalaires invariants. C. R. Acad. Sci. Paris, t. 317, S´ erie I (1993), 741–744

work page 1993

[4] [3]

Arnal, H

D. Arnal, H. Benamor, S. Benayadi, G. Pinczon: Une alg` ebre de Lie non semi-simple v´ eriﬁant H 1(g) = H 2(g) = H 0(g,g) = H 1(g,g) = H 2(g,g) = 0. 8 D. BURDE AND F. W AGEMANN

work page

[5] [4]

Benayadi: Certaines propri´ et´ es d’une classe d’alg` ebres de Lie quig´ en´ eralisent les alg` ebres de Lie semi- simples

S. Benayadi: Certaines propri´ et´ es d’une classe d’alg` ebres de Lie quig´ en´ eralisent les alg` ebres de Lie semi- simples. Ann. Fac. Sci. Toulouse, 5 e s´ erie, t.12, 1 (1991), 29–35

1991

[6] [5]

Benayadi: Structure of perfect Lie algebras without center and outer d erivations

S. Benayadi: Structure of perfect Lie algebras without center and outer d erivations. Ann. Fac. Sci. Toulouse, 6e s´ erie, t.5, 2 (1996), 203–231

1996

[7] [6]

Carles: Sur la structure des alg` ebres de Lie rigides

R. Carles: Sur la structure des alg` ebres de Lie rigides . Ann. Inst. Fourier Grenoble, t. 34, 3 (1984), 65–82

work page 1984

[8] [7]

Carles: Sur certaines classes d’alg` ebres de Lie rigides

R. Carles: Sur certaines classes d’alg` ebres de Lie rigides . Math. Ann. 272 (1985), 477–488

work page 1985