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Eight-dimensional non completely reducible symplectic Lie algebras

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Pith's one-line read This paper establishes a complete classification, up to symplectomorphism, of eight-dimensional non completely reducible symplectic Lie algebras, together with a complete description of symplectic Lie algebras admitting a one-dimensional…

desk verdict The oxidation framework is a genuinely useful new tool, but the completeness theorem rests on a false unimodularity inference and the classification is not proven as written. read the letter →

arxiv 2506.13699 v2 pith:IS6PAN34 submitted 2025-06-16 math.SG

T. Aït Aissa , S. El Bourkadi , M. W. Mansouri , SM. Sbai This is my paper · ORCID
classification math.SG MSC 22E6017B05
keywords SymplecticLiealgebrasreductiondoubleextensionIrreducibleGeneralizedoxidationD-extensionIsotropicidealsEight-dimensionalclassification
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 aims to complete the classification, up to symplectomorphism, of non completely reducible symplectic Lie algebras in dimension eight. A non completely reducible symplectic Lie algebra is one that cannot be dismantled to the trivial symplectic Lie algebra by repeated symplectic reduction through isotropic ideals. Theorem 1 states that every such eight-dimensional algebra is isomorphic to exactly one entry in two explicit tables, one for irreducible algebras and one for algebras obtained by a single symplectic reduction. The same methods are claimed to finish the classification for all dimensions $n\leq 8$ and to give a complete description of every symplectic Lie algebra with a one-dimensional isotropic ideal. A sympathetic reader should care because dimension six is the lowest dimension where irreducible symplectic Lie algebras exist, so these tables give the first nontrivial inventory of algebras that resist reduction to zero.

What carries the argument

The load-bearing construction is the generalized symplectic oxidation: given a symplectic Lie algebra $(\mathfrak{g},\omega)$ and a one-dimensional isotropic ideal generated by $\xi$, the oxidation rebuilds the larger algebra on $\langle\ell\rangle\oplus\mathfrak{g}\oplus\langle\xi\rangle$ with brackets $[x,y]=[x,y]+\omega_D(x,y)\xi$, $[\xi,x]=-\mu(x)\xi$, $[\xi,\ell]=t\xi$, and $[\ell,x]=\mu(x)\ell+D(x)+\lambda(x)\xi$, where $D$ is an endomorphism, $\mu,\lambda$ are linear forms, and $t$ is a scalar satisfying four compatibility equations. Proposition 6 shows that this construction reverses symplectic reduction by a one-dimensional ideal, so any such algebra is an oxidation of its reduction. Specializing to $\mu=0,t=0$ gives central symplectic oxidation, $\mu=0$ gives normal symplectic oxidation, and the remaining general case is classified through a new D-extension construction that builds Lie algebras from an endomorphism together with a one-dimensional extension. These constructions turn the classification problem into linear systems for the data $(D,\mu,\lambda,t)$ and isomorphism problems for the resulting extensions.

What would settle it

Check the paper's own reference [2], the classification of eight-dimensional non-solvable symplectic Lie algebras, for any member that admits a one-dimensional isotropic ideal; a single such algebra would contradict Proposition 9 and show that Tables 1 and 2 omit it.

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Extended reading notes

Core claim

The central claim is that every eight-dimensional non completely reducible symplectic Lie algebra $(\mathfrak{g},\omega)$ falls into one of two classes: irreducible algebras, which admit no nontrivial isotropic ideal, and reducible algebras, which reduce in one step to the unique six-dimensional irreducible symplectic Lie algebra. Theorem 1 lists the irreducible cases as three families with parameters $a,b,\lambda,\mu$, and the reducible cases as ten families distinguished by whether the one-dimensional isotropic ideal sits centrally, normally, or in the general position. The paper further claims that every symplectic Lie algebra with a one-dimensional isotropic ideal is a generalized symplectic oxidation of its reduction, that the central, normal, and generalized oxidation cases are all classified in dimension eight, and that the resulting enumeration is complete up to symplectomorphism.

Load-bearing premise

The load-bearing premise is that every eight-dimensional non completely reducible symplectic Lie algebra is solvable, in the sense that it has no semisimple Levi factor; the cited support establishes only the weaker fact that symplectic Lie algebras are unimodular, and if a non-solvable example exists the two tables are not exhaustive.

Editorial extensions

If this is right

  • In dimension eight, every non completely reducible symplectic Lie algebra is isomorphic to exactly one algebra in Tables 1 and 2, completing the classification up to symplectomorphism for all dimensions $n\leq 8$.
  • Every such eight-dimensional algebra is claimed to be solvable and in fact 2-step solvable, meaning the tables contain no semisimple Levi factor.
  • Every symplectic Lie algebra admitting a one-dimensional isotropic ideal is a generalized symplectic oxidation of its reduction, so the data $(D,\mu,\lambda,t)$ encode the complete structure of this class.
  • The reducible eight-dimensional cases split into central, normal, and generalized oxidations, and the table entries are distinguished by how the one-dimensional isotropic ideal sits inside the algebra.
  • Any symplectic form on these eight-dimensional non completely reducible algebras is itself non completely reducible, so the property survives changing the symplectic form on the same Lie algebra.

Reading between the lines

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

  • The completeness of the two tables rests on Proposition 9, whose cited support appears to prove only the weaker statement that symplectic Lie algebras are unimodular; until the solvability step is repaired, the tables are safest read as classifying the solvable members of the class.
  • The oxidation-plus-D-extension scheme is not tied to dimension eight: in any even dimension, choosing a reduction base with known automorphism group and known cohomology should produce an analogous central, normal, and generalized three-tier classification.
  • A direct consistency test is to check the published classification of eight-dimensional non-solvable symplectic Lie algebras for any entry with a one-dimensional isotropic ideal; a single such example would contradict Proposition 9 and require removing that algebra from the non completely reducible class or revising the theorem.
  • Because non completely reducible algebras are exactly those whose reduction chain stops before the zero algebra, the two tables can be viewed as a catalogue of 'stubborn' symplectic Lie algebras, which may be useful for understanding how isotropic ideals control the global structure of symplectic Lie algebras.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper claims a complete classification, up to symplectomorphism, of eight-dimensional non completely reducible symplectic Lie algebras, together with a description of symplectic Lie algebras admitting one-dimensional isotropic ideals. The main result, Theorem 1, is presented as two lists of normal forms: irreducible algebras (Table 1) and reducible algebras arising via central, normal, or generalized symplectic oxidation (Table 2). The technical machinery includes generalized symplectic oxidation, non-central extensions, and D-extensions, with isomorphism theorems for each construction. The proof strategy is to reduce by a one-dimensional isotropic ideal to the unique six-dimensional irreducible symplectic Lie algebra, classify the possible oxidation data, and then apply the isomorphism criteria. The paper also contains a classification of eight-dimensional irreducible symplectic Lie algebras and their symplectic forms, and a theorem asserting that every symplectic form on a non completely reducible eight-dimensional symplectic Lie algebra is again non completely reducible.

Significance. If the classification is correct, it would be a useful reference for low-dimensional symplectic Lie algebras and for the behavior of symplectic reduction in dimension eight. The systematic use of generalized symplectic oxidation and D-extensions provides a coherent framework for organizing the normal forms, and the explicit tables are concrete and checkable. The paper does not ship machine-checked proofs, but the isomorphism criteria in Sections 4 and 5 are substantial and potentially reusable. However, the completeness claim of Theorem 1 depends on the assertion in Proposition 9 that every non completely reducible eight-dimensional symplectic Lie algebra is solvable, and the proof of that proposition contains an invalid inference. Since the listed algebras are all solvable, an unsupported solvability reduction is a load-bearing gap in the central classification claim.

major comments (4)
  1. [§4, Proposition 9] The proof of Proposition 9 concludes that the eight-dimensional algebra is solvable from the sentence "since g is symplectic and unimodular, it must be solvable [8]". This is a non-sequitur: Chu's theorem in [8] proves that a symplectic Lie algebra is unimodular, not that an unimodular Lie algebra is solvable. Unimodular non-solvable Lie algebras such as sl(2,R) and so(3) are immediate counterexamples to the inference. Because Theorem 1 claims completeness over all eight-dimensional non completely reducible symplectic Lie algebras and Tables 1 and 2 contain only solvable algebras, this step is load-bearing. The proof should instead deduce solvability from the solvability of the six-dimensional reduction, for instance by invoking Proposition 7(2), but that proposition is not cited and itself has a defective proof (see the next comment).
  2. [§3, Proposition 7(2)] The derived-series computation in Proposition 7(2) is not valid as written. The displayed formula D^1(g)=⟨D^1(g), μ(g)ℓ ⊕ D(g), ξ⟩ implicitly replaces the subspace spanned by the vectors μ(x)ℓ + D(x) + λ(x)ξ with the larger sum μ(g)ℓ + D(g) + Rξ, and that replacement is not justified. The subsequent description of D^2(g) then brackets combinations of these elements as though μ(g)ℓ and D(g) were separately bracketable summands. Consequently the asserted equivalence that the oxidation is solvable if and only if the base g is solvable is not established by the given computation. Since this equivalence is the natural route to proving Proposition 9, it must be repaired with a correct derived-series or extension argument.
  3. [§2, Lemma 1] The proof of Lemma 1 assumes what it needs to prove. If j is an isotropic ideal of (g,ωα), the equation ωα(j,j)=0 gives ω(j,j)=α([j,j]) up to sign convention. Because [j,j] is contained in j, the right-hand side is not forced to vanish, so the displayed equality ω(j,j)=0 is unjustified. The argument therefore does not prove that (g,ωα) is irreducible. If this lemma is used in the classification of symplectic forms or in Proposition 4, a correct proof is required.
  4. [§5, Theorem 6] The proof of Theorem 6 is not sufficiently detailed to support the claim. The asserted "natural bijection" S(g)→S(g), ω↦ω+ξ*∧ℓ*, is stated without proof, and the notation alternates between (g,ω) and its reduction without a clear distinction. The theorem's conclusion that every symplectic form on an eight-dimensional non completely reducible symplectic Lie algebra is again non completely reducible therefore remains unsubstantiated. Either a proof of the bijection and of the reduction statement should be supplied, or the claim should be formulated as a conjecture.
minor comments (4)
  1. [Notation] The symbols g8,1, g8,2, and g8,3 are used both for irreducible algebras in Table 1 and for central oxidation algebras in Proposition 11, which is confusing; please introduce distinct labels for the two families.
  2. [Throughout] There are numerous typos, including "non triavial" in Proposition 9, "eigh-dimensional" in the Introduction, and "algberas" in Section 5.1; these should be corrected.
  3. [Appendix, Lemma 10] The displayed solution in Lemma 10 contains evident errors, such as D1e2=-d21e2, which is inconsistent with the derivation matrix and with D1e1=d21e2 and should presumably read D1e2=-d21e1; there is also a duplicated D2e3 line. These errors make the deferred computations hard to check.
  4. [Appendix, §6.2] The proof of Proposition 4 repeatedly switches to French with "Finalement et le meme raisonement precedent", and some block matrices in Proposition 3 are displayed with irregular zero-block placement; the appendix would benefit from a careful rewrite.

Circularity Check

0 steps flagged · score 1.0 of 10

No constructional circularity: the classification is built from symplectic reduction and oxidation constructions over an external uniqueness theorem; the flagged Proposition 9 solvability gap is a correctness concern, not a circular reduction.

full rationale

The central claim (Theorem 1) is a classification generated by oxidation constructions, not by assuming the target list. The one-dimensional reduction base is fixed by Proposition 1, which is proved internally from the external Baues–Cortés characterization (Theorem 2 in the paper). Propositions 11, 13 and 20 list isomorphism classes obtained by solving the generalized symplectic oxidation data (Proposition 5) and by explicit isomorphism criteria; the listed algebras are outputs, not inputs. No parameter is fitted to a subset of data and then renamed a prediction, and no invariants are defined in terms of the classification they are supposed to prove. The self-citations ([1], [2]) appear only as literature context and are not load-bearing for the main derivation. The manuscript does contain a serious rigor gap: Proposition 9 concludes solvability from 'since g is symplectic and unimodular, it must be solvable [8]', where the cited Chu theorem gives symplectic implies unimodular, not unimodular implies solvable; also the derived-series computation in Proposition 7(2) appears to conflate spans. These issues bear on the completeness of the classification, but they are not circularity: the theorem is not assumed in its own proof, and the gap is an invalid inference rather than a reduction of the conclusion to the premises.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on standard classification theorems and on lengthy algebraic computations. No parameters are fitted to data. Generalized symplectic oxidation and D-extension are mathematical constructions, not invented physical entities. The main unstated reliance is that all computations in Lemmas 3, 7, 10, 11 and Appendix 6 are correct; this is not independently verified here.

assumptions (4)
  • standard math Baues-Cortes characterization of irreducible symplectic Lie algebras (Theorem 2 in the paper, [4])
    Used as the starting point for irreducible cases and to identify the metabelian semidirect product structure of the 6-dimensional building block.
  • standard math Symplectic Lie algebras are unimodular (Chu, [8])
    Invoked in Proposition 9. The theorem is true in this direction, but the paper inverts it, and that inversion is the identified gap.
  • domain assumption The 6-dimensional irreducible symplectic Lie algebra is unique up to isomorphism and its symplectic forms are parameterized by η > 0 (Proposition 1)
    The classification of oxidations fixes a base g(1,0,0,1); if Proposition 1 is incorrect, the 8-dimensional list built from it misses other reductions.
  • domain assumption Symplectic reduction by a one-dimensional ideal in dimension 8 yields a 6-dimensional irreducible base when the algebra is non-completely reducible
    Used to reduce all non-irreducible cases to oxidations of g(1,0,0,1); this follows from the reduction framework and the final base being irreducible.

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Pith. "Pith review of Eight-dimensional non completely reducible symplectic Lie algebras." pith.science (2026). https://pith.science/paper/IS6PAN34

@misc{pith2026250613699,
  author       = {Pith},
  title        = {Pith review of: Eight-dimensional non completely reducible symplectic Lie algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IS6PAN34}},
  note         = {Machine review of arXiv:2506.13699}
}
abstract

A non completely reducible symplectic Lie algebra is a symplectic Lie algebra which cannot be symplectically reduced to the trivial symplectic Lie algebra. Our aim is to provide a complete classification, up to symplectomorphism of non completely reducible symplectic Lie algebras in dimensions $n \leq 8$ and, furthermore, to provide a complete description of symplectic Lie algebras admitting one-dimensional isotropic ideals.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

18 extracted references · 18 canonical work pages

  1. [8]

    Chu, Symplectic homogeneous spaces, Trans

    B. Chu, Symplectic homogeneous spaces, Trans. Amer. Math. Soc. 197, (1974) 145-159. 65

    work page 1974

  2. [1]

    A ¨ıt Aissa, T., Mansouri, M. W. Symplectic Novikov Lie algebras. Commun. Algebra 50(7), (2022) 2921-2933

    work page 2022

  3. [2]

    W.Eight-dimensional symplectic non-solvable Lie algebras

    A ¨ıt Aissa, T., Mansouri, M. W.Eight-dimensional symplectic non-solvable Lie algebras. Commun. Algebra, 52(3), (2023) 1196-1218

    work page 2023

  4. [3]

    A ¨ıt Aissa, T., El Bourkadi, S., Mansouri, M. W. A complete classification of symplectic forms on six-dimensional Frobeniusian real Lie algebras. Commun. Algebra, 53(5), (2024) 2123-2153

    work page 2024

  5. [4]

    (2016).Symplectic Lie groups

    Baues, O., Cort ´es, V . (2016).Symplectic Lie groups. Symplectic Reduction, Lagrangian Extensions, and Existence of Lagrangian Normal Subgroups. Ast`erisque, 379. Paris: Societ´e Math´ematique de France (SMF), vi, 90 p

    work page 2016

  6. [5]

    Burde, D. (2006). Characteristically nilpotent Lie algebras and symplectic structures. Forum Math. 18:769-787

    work page 2006

  7. [6]

    Campoamor-Stursberg, R. (2009). Symplectic forms on six-dimensional real solvable Lie algebras I . Algebra Colloq. 16(2):253-266

    work page 2009

  8. [7]

    Campoamor-Stursberg, R. (2004). A characterization of exact symplectic Lie algebras in terms of generalized Casimir invariants. In: Bentom, Ch. V ., ed. New Developments in Mathematical Physics Research. Hauppauge: Nova Science Publishers, pp. 55-84

    work page 2004

Show all 18 references
  1. [9]

    Cordero, M

    L.A. Cordero, M. Fern ´andez, L. Ugarte, Pseudo-K¨ ahler metrics on six-dimensional nilpotent Lie algebras, J. Geom. Phys. 50 (2004) 115-137

    2004

  2. [10]

    Dardi ´e, A

    J.M. Dardi ´e, A. Medina, Alg` ebres de Lie k¨ ahl´ eriennes et double extension, J. Algebra 185 (1996) 774-795

    1996

  3. [11]

    Dardi ´e and A

    J.M. Dardi ´e and A. Medina, Double extension symplectique d’un groupe de Lie symplectique, Adv. Math. 117 (2) (1996) 208-227

    1996

  4. [12]

    Fischer, M. (2019). Symplectic Lie algebras with degenerate center. J. Algebra 521:257-283

    2019

  5. [13]

    R., Jim´enez-Merch´an, A., Khakimdjanov, Y

    G ´omez, J. R., Jim´enez-Merch´an, A., Khakimdjanov, Y . (2001).Symplectic structures on the filiform Lie algebras. J. Pure Appl. Algebra 156(1):15-31

    2001

  6. [14]

    Khakimdjanov, Y ., Goze, M., Medina, A. (2004). Symplectic or contact structures on Lie groups . Differ. Geom. Appl. 21(1):41-54

    2004

  7. [15]

    Lichnerowicz A

    A. Lichnerowicz A. Medina. On Lie groups with left-invariant symplectic or K¨ ahlerian structures, Lett. Math. Phys. 16 (1988), no. 3, p. 225-235

    1988

  8. [16]

    Medina and P

    A. Medina and P. Revoy, Groupes de Lie ` a structure symplectique invariante, Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), Springer, New York, (1991) 247-266

    1991

  9. [17]

    Milnor, Curvatures of left invariant metrics on lie groups, Advances in Mathematics, V olume 21, Issue 3, (1976) 293-329

    J. Milnor, Curvatures of left invariant metrics on lie groups, Advances in Mathematics, V olume 21, Issue 3, (1976) 293-329

    1976

  10. [18]

    Ovando, Four dimensional symplectic Lie algebras, Contribution to Algebra and Geometry

    G. Ovando, Four dimensional symplectic Lie algebras, Contribution to Algebra and Geometry. V ol. 47, N 2, (2006) 419-434. 66

    2006

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