On Riemann-Poisson Lie groups

Ahmed Sid'Ahmed Lessiad; Brahim Alioune; Mohamed Boucetta

REVIEW 2 major objections 3 minor 13 references

On Riemann-Poisson Lie groups

T0 review · 2 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Riemann-Poisson Lie algebras are precisely Kähler Lie subalgebras with two skew-symmetric cross-actions, and this characterization yields the complete list through dimension 5.

desk verdict Useful structural theorem and low-dimensional catalog, but Theorem 3.1 as printed has a repeated-index typo that makes condition (14) force the cross-action to zero, contradicting the paper's own examples. read the letter →

arxiv 1908.05060 v1 pith:P33PD4Y5 submitted 2019-08-14 math.DG math.SG

Brahim Alioune , Mohamed Boucetta , Ahmed Sid'Ahmed Lessiad This is my paper · ORCID

classification math.DGmath.SG MSC 53D1753C3053C55

keywords Riemann-PoissonLiegroupsalgebrasKählerclassicalYang-Baxterequationsymplecticsubalgebrasleft-invariantmetricslow-dimensionalclassificationEuclidean

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 paper studies Riemann-Poisson Lie groups: Lie groups carrying a left-invariant Riemannian metric and a left-invariant Poisson tensor that are compatible in the sense introduced in [2], a condition weaker than requiring the Levi-Civita connection to kill the Poisson tensor. The authors aim to describe the infinitesimal counterparts, Riemann-Poisson Lie algebras, in structural terms. The main theorem says that such an algebra is exactly a Kähler Lie subalgebra $S$ (a Lie subalgebra with Euclidean metric and a nondegenerate 2-form parallel for its Levi-Civita product), together with an orthogonal complement $S^\perp$ whose mutual actions satisfy two explicit skew-symmetry equations. This structural description is then turned into a construction recipe, and the recipe is used to list all Riemann-Poisson Lie algebras up to dimension $5$.

What carries the argument

The load-bearing object is the splitting $\mathfrak{g}=S\oplus S^\perp$ determined by $r$, where $S=\operatorname{Im}r^\#$ carries the nondegenerate form $\omega_r$ and the induced metric. The proof transports the contravariant Levi-Civita connection on $(\mathfrak{g}^*,\rho^*)$ through the isomorphisms $r^\#$ and the metric duality $\#$ to obtain the Levi-Civita connection of $(S,\rho|_S)$; under this transport the compatibility condition $D\pi=0$ becomes exactly the Kähler condition on $S$ together with equations (13) and (14). The maps $\varphi_S$ and $\varphi_{S^\perp}$ are what the theorem uses to package the cross-action of each factor on the other.

What would settle it

Independently enumerate all solutions of Problem 1 in dimension 5 with $h$ a two-dimensional abelian Kähler subalgebra and $\varphi_h=0$ (the cases covered by Tables 7–8), and compare the resulting brackets with the listed families; a single bracket satisfying (19) that does not appear in the tables would disprove the claimed completeness. A second check would verify that every family listed in Tables 5–9 does satisfy the original equations (i) and (ii) of Proposition 2.1.

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Extended reading notes

Core claim

The central discovery is Theorem 3.1. Let $\mathfrak{g}$ be a real Lie algebra, $r\in\wedge^2\mathfrak{g}$, and $\rho$ a Euclidean metric on $\mathfrak{g}$. Put $S=\operatorname{Im}r^\#$ and let $\omega_r$ be the nondegenerate 2-form on $S$ induced by $r$. The theorem states that $(\mathfrak{g},r,\rho)$ is a Riemann-Poisson Lie algebra if and only if $(S,\rho|_S,\omega_r)$ is a Kähler Lie subalgebra and, with $\varphi_S(s)=\operatorname{pr}_{S^\perp}\circ\operatorname{ad}_s$ and $\varphi_{S^\perp}(u)=\operatorname{pr}_S\circ\operatorname{ad}_u$, the identities $\rho(\varphi_S(s)(u),v)+\rho(u,\varphi_S(s)(v))=0$ for $s\in S$, $u,v\in S^\perp$ and $\omega_r(\varphi_{S^\perp}(u)(s_1),s_2)+\omega_r(s_1,\varphi_{S^\perp}(u)(s_2))=0$ for $u\in S^\perp$, $s_1,s_2\in S$ hold. In words, the complement acts on $S$ preserving the Kähler form, while $S$ acts on the complement skew-symmetrically. The paper repackages this as a construction scheme and applies it, using the classification of three-dimensional Euclidean Lie algebras and of four-dimensional Kähler Lie algebras, to give the complete list of Riemann-Poisson Lie algebras up to dimension 5.

Load-bearing premise

The completeness of the dimension-5 list rests on the completeness of the quoted classification of three-dimensional Euclidean Lie algebras [10] and on the computer-algebra solution of the cocycle condition (20); if either missed a case, the tables would omit some Riemann-Poisson Lie algebras.

Editorial extensions

If this is right

Every Riemann-Poisson Lie algebra can be written as $\mathfrak{g}=h\oplus p$ with $h$ a Kähler Lie algebra, $p$ a Euclidean vector space, and data $\varphi_h,\varphi_p,\mu,[\ ,\ ]_p$ satisfying the Jacobi identities (19); the theorem reduces all examples to this algebraic problem.
The rank of a left-invariant Poisson tensor on a Riemann-Poisson Lie group is always even, because $S=\operatorname{Im}r^\#$ is symplectic.
The classification tables give all Riemann-Poisson Lie algebras in dimensions 3, 4, and 5; in dimension 3 there are exactly the two families displayed in Table 1.
Any Lie group integrating such an algebra becomes a Riemann-Poisson Lie group, and passing to a quotient by any discrete subgroup produces a Riemann-Poisson manifold.
Every even-dimensional flat Riemannian Lie group carries a left-invariant Kähler form, so flat Lie groups provide a large source of examples.

Reading between the lines

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

A natural extension not pursued in the paper is to read Theorem 3.1 as an extension theory: fix a Kähler Lie algebra $h$ and a Euclidean space $p$, and classify the pairs of actions satisfying (13)–(14) with the Jacobi equations; the low-dimensional tables suggest this obstruction is governed by the cocycle condition (20).
The same algebraic splitting should make sense for pseudo-Riemannian Poisson Lie algebras with an indefinite metric; the expected statement would replace 'Kähler' by 'pseudo-Kähler' and keep equations (13)–(14) verbatim, which is a testable extension.
Since the theorem is infinitesimal, it suggests a local normal-form picture for constant-rank Riemann-Poisson manifolds: neighbourhoods should look like a Kähler leaf factor with a transverse action obeying the same skew-symmetry identities, refining the foliation result quoted as Theorem 1.1 at the level of germs.

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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

What you should know: the paper delivers a genuinely useful structural characterization of Riemann-Poisson Lie algebras together with a low-dimensional catalog, but the central theorem has a typo that as printed makes it false. The fix is small and recoverable from the proof, so the paper is worth refereeing rather than dismissing.

The setup is clean. Problem 1 reduces the construction to a Kähler Lie subalgebra h and a Euclidean complement p with actions φh, φp and a cocycle μ, satisfying Jacobi-like equations. Theorem 3.1's intended statement—S is a Kähler Lie subalgebra and the two cross-actions are skew-adjoint in the appropriate senses—is the right reduction. The proof goes directly through Propositions 2.1 and 2.2, and the classification tables in Section 4 are the first complete-looking list up to dimension 5. Spot checks of the bracket equations are internally consistent.

The soft spots are real but proportionate. Condition (14) as printed is ωr(φ_{S⊥}(u)(s1), s2) + ωr(s1, φ_{S⊥}(u)(s1)) = 0. Since ωr is alternating, the second term is identically zero, so (14) forces φ_{S⊥}(u) = 0. That contradicts the paper's own examples in Propositions 4.2–4.7 and Tables 4–9, where φp is nonzero. The proof, however, derives the correct skew-adjointness with s2 in the second slot, so it is a typo, not a conceptual error. The same repeated-index slip appears in equation (10), where the second Auv should be Avw. These are the kind of typos a careful referee will catch and the authors can fix in a day.

Two smaller issues. The completeness of Tables 5–9 rests on a Maple computation that is not documented; the authors should post the code or state exactly which cases were verified symbolically. And one row of Table 7 just says "There are many cases, see [10]", which is a hole in the claimed catalog unless the authors delimit it. The paper also does not explicitly state what is new relative to Boucetta's earlier [5]; the tables and Theorem 3.1 go further, but the boundary should be drawn.

Overall: the structural theorem is sound after the typo fix, the classification is a useful resource, and the construction method is a good tool for building examples. I would send this to peer review with instructions to correct (14) and (10), document the Maple computation, and close or clearly mark the Table 7 gap.

Referee Report

2 major / 3 minor

Summary. The paper defines Riemann-Poisson Lie groups as Lie groups equipped with a left-invariant Riemannian metric and a left-invariant Poisson tensor satisfying the compatibility condition Dπ=0. It gives an infinitesimal characterization (Proposition 2.1) in terms of a Lie algebra g, an element r∈∧²g satisfying the classical Yang-Baxter equation, and a Euclidean product ρ. The main structural result (Theorem 3.1) states that (g,r,ρ) is Riemann-Poisson exactly when S=Im r# is a Kähler Lie subalgebra for the induced metric and symplectic form, and two compatibility conditions on the cross-actions hold. From this it formulates a construction problem (Problem 1) and solves it in low dimensions, presenting tables of Riemann-Poisson Lie algebras in dimensions 3, 4, and 5. It also proves that every even-dimensional flat Riemannian Lie group admits a left-invariant Kähler form.

Significance. If the corrected version of Theorem 3.1 holds, the paper gives a clean reduction of a combined metric-Poisson condition to a Kähler Lie subalgebra plus controlled orthogonal cross-terms, which is a useful structural tool. The construction method in Section 4 is systematic, and the extensive tables of low-dimensional examples are valuable for testing conjectures and for constructing further examples via discrete quotients. The direct proofs of Propositions 2.2, 3.1, and the derivation of Theorem 3.1 are checkable and do not rely on hidden assumptions. The paper is a solid contribution to the geometry of Riemann-Poisson manifolds, provided the displayed conditions (10) and (14) are corrected and the completeness of the classification is documented.

major comments (2)

[Theorem 3.1, Eq. (14), and Section 2, Eq. (10)] Condition (14) is printed with a repeated index: the second term is ωr(s1, φ_{S⊥}(u)(s1)), which vanishes identically because ωr is alternating, so the condition as printed reduces to ωr(φ_{S⊥}(u)(s1), s2)=0 for all s1,s2, forcing φ_{S⊥}(u)=0. This contradicts the nonzero φp maps in Propositions 4.2–4.7 and Tables 4–9, so the theorem as stated is internally inconsistent with the classification it supports. The proof around Eq. (17) derives the correct skew-adjointness condition ωr(φ_{S⊥}(u)(s1), s2)+ωr(s1, φ_{S⊥}(u)(s2))=0, so the statement is recoverable but must be corrected. The same repeated-index error occurs in Eq. (10), where the Kähler condition should read ω(Auv, w)+ω(v, Auw)=0. Please fix both and re-check that no subsequent formula relies on the printed versions.
[Section 4, cases (c53)–(c55), Tables 5–9] The paper claims to give all Riemann-Poisson Lie algebras up to dimension 5, but the completeness of Tables 5–9 rests on a Maple computation that is not documented and on the classification of 3-dimensional Euclidean Lie algebras from [10]. Table 7's final row explicitly defers to [10] with 'There are many cases', so it is unclear how the listed families exhaust the solutions of Problem 1 in that case. Please provide the Maple code or a human-readable proof of exhaustiveness, and explain how the deferred cases are accounted for. Without this, the completeness claim in the abstract is not fully supported.

minor comments (3)

[Proposition 4.6, proof of case 3] The claim 'we must have ξ = 0 in order to have the Jacobi identity' is stated without proof; please justify it or spell out the Jacobi identity check.
[Before Theorem 4.1] The sentence 'This theorem unknown to our knowledge can be used to build examples of Riemann-Poisson Lie algebras' appears garbled; please rewrite it, for example as 'The following theorem, apparently not previously known, can be used to build examples...'.
[Throughout] There are numerous typos and OCR artifacts (e.g., 'let invariant' for 'left invariant', 'diﬀerential', 'sekw-symmetric', 'h /nelementg'), and the paper would benefit from a careful proofreading pass.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 3.1 is derived from the defining equations and the classification is built on external benchmarks.

full rationale

The paper's main derivation chain is self-contained. Proposition 2.1 reduces the Riemann-Poisson condition Dπ = 0 for a left-invariant structure on a Lie group to the algebraic conditions (i) and (ii), and Definition 2.1 simply records these conditions as the definition of a Riemann-Poisson Lie algebra. Proposition 2.2 translates the classical Yang-Baxter equation into the statement that S = Im r# is a symplectic subalgebra; this is a standard equivalence proved from the definitions. Proposition 3.1 is proved by splitting g* = I ⊕ I⊥ and using the equations in (11), which come directly from Proposition 2.1. Theorem 3.1 is then proved from Proposition 3.1 by identifying S with I⊥ through r#, so no equation in Theorem 3.1 is assumed as an input. Problem 1 is not a prediction but a construction scheme: it lists exactly the data (Kähler subalgebra h, complement p, maps φh and φp, brackets and cocycles) that make Theorem 3.1 hold, and Proposition 4.1 verifies the Jacobi identity. The classification in Section 4 applies this scheme with dim h = 2 or 4; it does not feed the target list into the characterization. The cited works [2]–[5] are background for the notion of Riemann-Poisson manifolds and are not used as the proof of Theorem 3.1; Theorem 1.1 from [4] is motivational only. The completeness of the low-dimensional tables relies on the external classifications [10], [11] and on Maple computations, which is a support and reproducibility concern rather than a circular dependency. There appears to be a typographical error in condition (14) — and similarly in equation (10) — where the second argument repeats s1 instead of s2; this is a correctness issue, not a circular reduction, since the surrounding proof derives the intended skew-symmetry condition. Therefore no circular step is exhibited, and the score is 0.

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

The central characterization theorem rests on standard Poisson-Lie theory, which the paper proves. The low-dimensional classification additionally relies on two prior classifications ([10] for 3D Euclidean Lie algebras, [11] for 4D Kähler Lie algebras) and on a Maple computation for equation (20). No new entities or fitted parameters are introduced.

assumptions (4)

domain assumption Classification of 3-dimensional Euclidean Lie algebras from [10] is exhaustive and correct.
Used in cases (c53)-(c55) to enumerate all possible complements p; completeness of Tables 5-9 depends on it.
domain assumption Classification of 4-dimensional Kähler Lie algebras and their symplectic derivations from [11] is exhaustive and correct.
Used in cases (c43) and (c52) to list h and Der_s(h); Tables 3-4 depend on it.
domain assumption Milnor's characterization of flat left-invariant metrics and the improved version from [1] are correct.
Theorem 4.1 uses the structure of [g,g] and [g,g]⊥ for flat Lie groups.
standard math sp(h,ω) ≅ sl(2,R) for a 2-dimensional symplectic vector space h.
Used repeatedly to identify possible φ_p images in Propositions 4.4, 4.6, 4.7.

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Cite this review

Pith. "Pith review of On Riemann-Poisson Lie groups." pith.science (2026). https://pith.science/paper/P33PD4Y5

@misc{pith2026190805060,
  author       = {Pith},
  title        = {Pith review of: On Riemann-Poisson Lie groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/P33PD4Y5}},
  note         = {Machine review of arXiv:1908.05060}
}

read the original abstract

A Riemann-Poisson Lie group is a Lie group endowed with a left invariant Riemannian metric and a left invariant Poisson tensor which are compatible in the sense introduced in C.R. Acad. Sci. Paris s\'er. {\bf I 333} (2001) 763-768. We study these Lie groups and we give a characterization of their Lie algebras. We give also a way of building these Lie algebras and we give the list of such Lie algebras up to dimension 5.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

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Pith tools

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

[1] [10]

Y ., Lee, J

Ha, K. Y ., Lee, J. B. Left invariant metrics and curvatur es on simply connected three dimensional Lie groups. Math. Nachr. 282 (2009), 868-898

work page 2009

[2] [5]

Boucetta, On the Riemann-Lie algebras and Riemann-Po isson Lie groups, Journal of Lie Theory

M. Boucetta, On the Riemann-Lie algebras and Riemann-Po isson Lie groups, Journal of Lie Theory. 15. 1 (2005) 183-195

work page 2005

[3] [1]

Ait Haddou, M

M. Ait Haddou, M. Boucetta, H. Lebzioui, Left-invariant Lorentzian ﬂat metrics on Lie groups, Journal of Lie Theory 22 (2012), No. 1, 269-289

work page 2012

[4] [2]

Boucetta, Compatibilit´ e des structures pseudo-rie manniennes et des structures de Poisson, C.R

M. Boucetta, Compatibilit´ e des structures pseudo-rie manniennes et des structures de Poisson, C.R. Acad. Sci. Par is s´ er.I 333 (2001) 763-768

work page 2001

[5] [3]

Boucetta, Riemann-Poisson manifolds and K¨ ahler-Ri emann foliations, C

M. Boucetta, Riemann-Poisson manifolds and K¨ ahler-Ri emann foliations, C. R. Acad. Sci. Paris, Ser. I 336 (2003) 423-428

work page 2003

[6] [4]

Boucetta, Poisson manifolds with compatible pseudo- metric and pseudo-Riemannian Lie algebras

M. Boucetta, Poisson manifolds with compatible pseudo- metric and pseudo-Riemannian Lie algebras. Di ﬀerantial Geometry and its applications. S´ er.20 (2004) 279-291

work page 2004

[7] [6]

Deninger, W

C. Deninger, W. Singhof, Real polarizable hodge structu res arising from foliations, Ann. Global Anal. Geom. 21 (2002) 377-399. 22

work page 2002

[8] [7]

Dufour and N

J.P . Dufour and N. T. Zung, Poisson Structures and Their N ormal Forms, Progress in Mathematics, V ol. 242, Birkh¨ auser V erlag 2005

work page 2005