A note on integrabiliy of Hamiltonian systems on the co-adjoint Lie groupoids

Ghorbanali Haghighatdoost; Rezvaneh Ayoubi

REVIEW 5 major objections 7 minor 16 references

A note on integrabiliy of Hamiltonian systems on the co-adjoint Lie groupoids

T0 review · 5 major / 7 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proves that, for co-adjoint Lie groupoids of product form, integrability of Hamiltonian systems on the groupoid side is equivalent to integrability on the corresponding co-adjoint orbit of a Lie group.

desk verdict The integrability reduction and symplectic inheritance claims are not proven; Lemma 23's proof contradicts the paper's own equations. read the letter →

arxiv 2411.16631 v1 pith:YESGXMMD submitted 2024-11-25 math.DS math-phmath.MP

Ghorbanali Haghighatdoost , Rezvaneh Ayoubi This is my paper · ORCID

classification math.DSmath-phmath.MP MSC 18B4053D1770H0837J35

keywords co-adjointLiegroupoidalgebroidHamiltoniansystemintegrablesymplecticlinearPoissonstructurefunctions

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 studies Hamiltonian systems on co-adjoint Lie groupoids, which are groupoids built from orbits of the co-adjoint action of a Lie groupoid on its isotropy Lie algebroid. Its central claim is that when such a co-adjoint groupoid has the product form O(ξ) = M × O(ξ′), where O(ξ′) is a co-adjoint orbit of a Lie group, integrability of the Hamiltonian system on the groupoid side is exactly the same as integrability of the corresponding system on the orbit O(ξ′). This matters because it reduces an apparently more complicated groupoid-level integrability question to the classical, well-studied integrability of Hamiltonian systems on co-adjoint orbits of Lie groups. The paper also shows that the structure functions of the co-adjoint Lie algebroid coincide with those of the original Lie algebroid, and that the co-adjoint groupoid of a symplectic groupoid is again a symplectic groupoid.

What carries the argument

The central device is the co-adjoint Lie groupoid O(ξ) = M × O(ξ′) and its co-adjoint Lie algebroid A*G = M × T*_{ξ′}O(ξ′), together with the linear Poisson structure determined by the original Lie algebroid's structure functions. The argument rides on the identity {F, H}_{A*G} = {f, h}, where the right-hand side is the canonical Lie–Poisson bracket on T*O(ξ′), for Hamiltonians of the form H(p, λ) = h(λ); this identity transfers Hamiltonian vector fields and first integrals from the product groupoid to the Lie-group orbit. Lemma 22's equality of anchor coefficients and bracket constants makes the transfer explicit in local coordinates and is what lets the paper compare the two Hamiltonian systems.

What would settle it

Construct a regular Lie groupoid whose co-adjoint orbit is not diffeomorphic to a product M × O(ξ′), choose a Hamiltonian on A*G that varies along M, and check whether its first integrals still correspond to those of a Hamiltonian on O(ξ′); a single such example with non-corresponding first integrals would break Corollary 27.

Watch

Extended reading notes

Core claim

For a co-adjoint Lie groupoid of the special product form O(ξ) = M × O(ξ′), with O(ξ′) ⊂ g* a co-adjoint orbit of a Lie group, the Hamiltonian dynamics on the dual of the co-adjoint Lie algebroid is exactly the Hamiltonian dynamics on T*O(ξ′) with the M-direction inert. The first integrals of the Hamiltonian vector field on A*G are in one-to-one correspondence with the first integrals of the corresponding vector field on T*O(ξ′), so the Hamiltonian system on the groupoid side is integrable exactly when the system on the Lie-group orbit is integrable. The proof hinges on the equality {F, H}_{A*G} = {f, h}, where F = (p, f), H(p, λ) = h(λ), and the bracket on the right is the canonical Lie–Poisson bracket on the orbit. The paper also establishes that the anchor coefficients and bracket constants of the co-adjoint Lie algebroid equal those of the original Lie algebroid, and that co-adjoint Lie groupoids inherit symplectic groupoid structures from symplectic groupoids.

Load-bearing premise

The reduction only works when the co-adjoint orbit splits as a product O(ξ) = M × O(ξ′) and the Hamiltonian depends only on the second factor; without that product structure the paper gives no route from co-adjoint groupoid integrability to Lie-group orbit integrability.

Editorial extensions

If this is right

If the central claim is correct, integrability of Hamiltonian systems on product-type co-adjoint Lie groupoids is fully equivalent to integrability on the corresponding co-adjoint orbit of a Lie group (Corollary 27).
First integrals on A*G for such groupoids are in bijection with first integrals on T*O(ξ′), so any integrability result for Lie-group orbits immediately carries over to the groupoid setting.
The equality of structure functions (Lemma 22) means the local Poisson data of the co-adjoint Lie algebroid are the same as those of the original Lie algebroid, so computations on either side are interchangeable in the product case.
For the examples treated in the paper—trivial groupoids, gauge groupoids, and transitive action groupoids—the co-adjoint orbits are of the product form, so the reduction applies to each of these families.
If the original Lie groupoid is symplectic, its co-adjoint Lie groupoid is symplectic, which extends the symplectic-groupoid property to this class of co-adjoint constructions.

Reading between the lines

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

The paper leaves open whether the reduction holds when the co-adjoint orbit is not a product O(ξ) = M × O(ξ′); a natural test is a regular Lie groupoid whose co-adjoint orbit is a nontrivial bundle over M, where the M-direction may no longer be inert.
The structure-function equality suggests that other Poisson-geometric data—such as bi-Hamiltonian chains or Casimir functions—might transfer between a Lie algebroid and its co-adjoint algebroid, though the paper does not pursue that transfer.
The symplectic groupoid result could be relevant to deformation quantization of co-adjoint groupoids, since a multiplicative symplectic form is the standard starting point for such quantizations, but the paper only establishes the symplectic structure itself.
A testable extension would be to weaken the product-form hypothesis by allowing O(ξ) to be a locally trivial fibration over M with typical fiber O(ξ′); the paper gives no evidence for whether Corollary 27 would survive in that generality.

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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

Here is my take. The two headline claims are not established. Lemma 23, which is the load-bearing step for the integrability reduction, has a proof that contradicts the paper's own general formula (7). The Hamiltonian equations (9) in Lemma 23 are not a specialization of (7) for any structure functions: the dx/dt equation should involve ∂H/∂y_α, not ∂H/∂x^i, and the dy/dt equation should contain the missing ∂H/∂x^i term. The Poisson bivector displayed in the proof includes '1/2 ∂/∂x^i ∧ ∂/∂x^i', which is identically zero. So the proof is internally inconsistent, and the conclusion that H is independent of the base coordinates is unsupported. Corollary 27 rests entirely on that lemma, so the claimed reduction is at best conditional on an assumption the paper fails to prove.

What is genuinely new: Proposition 30, stating that the co-adjoint groupoid of a symplectic groupoid is symplectic, is not in the cited references, as far as I can tell. The paper also frames the integrability reduction cleanly, and the worked examples for trivial, gauge, and action groupoids are standard but useful. The authors are honest that this is a continuation of [4], and they cite their earlier work clearly.

Where the paper falls short beyond Lemma 23: Proposition 30 defines ω'(η1,η2)=ω(X,Y) without checking well-definedness. The map X↦ad*_X ξ has a kernel in general, so if ad*_X ξ=ad*_{X'}ξ, one must verify ω(X,Y)=ω(X',Y) before calling ω' a form. No such check appears. The multiplicativity argument also mishandles the pullback notation, writing ω'(ad*_X ξ) as if ω' took one argument. So the proof of Prop 30 is missing a real argument. Nearly all foundational statements are delegated to [4], which is acceptable for a continuation but leaves this note with little independent content.

Who is this for? Readers already working with the authors' co-adjoint groupoids who want the hoped-for reduction stated compactly. As written, I would desk-reject it. The authors should fix Lemma 23 by deriving the Hamiltonian equations correctly from (7), restate Corollary 27 under the explicit hypothesis H(p,λ)=h(λ) if that is what they need, and supply a genuine well-definedness proof for the symplectic form in Prop 30. If those repairs are made, the note could become useful, but it does not deserve referee time in its current form.

Referee Report

5 major / 7 minor

Summary. The manuscript continues the authors' earlier work on co-adjoint Lie groupoids. It compares the structure functions of a Lie algebroid and its co-adjoint algebroid (Lemma 22), then, for co-adjoint groupoids of the special product form O(\xi)=M\times O(\xi'), attempts to prove that the Hamiltonian is independent of the base M (Lemma 23), that first integrals correspond to those on O(\xi') (Proposition 26), and that integrability reduces accordingly (Corollary 27). It also claims that the co-adjoint Lie groupoid of a symplectic groupoid is a symplectic groupoid (Proposition 30).

Significance. If the reduction were correct, it would connect Hamiltonian dynamics on co-adjoint Lie groupoids to the classical Kirillov orbit picture, which would be a useful result. The comparison of structural functions in Section 4 and the three worked examples are potentially helpful. However, the proof of the central Lemma 23 is internally inconsistent with the paper's own general formula, Proposition 26 only treats a restricted class of functions, and Proposition 30 is not proved. The paper also relies heavily on [4] for the groupoid structure on O(\xi), the bracket identity, and the Poisson bracket equality used in the reduction. As it stands, the announced results are not established.

major comments (5)

[Section 5.1, Lemma 23 (Eqs. (7), (9))] The Hamiltonian equations displayed in Lemma 23 do not follow from the paper's own general formula (7). In (7) the x-component is (\partial H/\partial y_\alpha)\rho^i_\alpha, whereas (9) states dx^i/dt = \partial H/\partial x^i. The proof also writes the Poisson tensor as (1/2)\partial/\partial x^i \wedge \partial/\partial x^i - (1/2)C^\gamma_{\alpha\beta} y_\gamma \partial/\partial y_\alpha \wedge \partial/\partial y_\beta; the first term is identically zero and is incompatible with the first term \rho^i_\alpha \partial/\partial x^i \wedge \partial/\partial y_\alpha of the general bivector (5). The y-equation in (9) omits the term -(\partial H/\partial x^i)\rho^i_\alpha that appears in (7). Hence the conclusion \partial H/\partial x^i=0 is not proved.
[Section 5.1, Remark 25 and Proposition 26] Proposition 26 concerns only functions of the special form F=(p,f), meaning functions constant along the M factor. Remark 25 asserts without proof that every smooth function on M\times T^*_{\xi'}O(\xi') has this form, which is false in general. Consequently the converse direction of Proposition 26 applies only to M-independent functions; the paper does not rule out first integrals that genuinely depend on the base coordinates x^i, and for such functions the equality {F,H}_{A^*G}={f,h}_{K.K} used in the proof is not justified.
[Section 5, Corollary 27] Corollary 27 is not established because it assumes H(p,\lambda)=h(\lambda), which was supposed to follow from Lemma 23. If H depends on x^i, then the fiber equation from (7) contains the additional coupling term -(\partial H/\partial x^i)\rho^i_\alpha, so integrability of the orbit Hamiltonian h would not control the full system. Thus the reduction to O(\xi') is conditional on an unproved and, as written, incorrectly derived independence statement.
[Section 6, Proposition 30] The 2-form \omega'(\eta_1,\eta_2)=\omega(X,Y) is not shown to be well-defined: the map X\mapsto ad^*_X\xi has kernel in general, and no argument is given that \omega(X,Y) is independent of the choice of representatives X,Y. The proof of multiplicativity is dimensionally incorrect, since it writes m'^*\omega'(\eta_1,\eta_2)=\omega'(Tm'(\eta_1,\eta_2)), evaluating a 2-form on a single tangent vector; a pullback 2-form must be evaluated on a pair of tangent vectors. Closedness and nondegeneracy of \omega' are never verified, so the proposition does not prove that O(\xi) is a symplectic groupoid.
[Sections 3 and 5, reliance on [4]] Several load-bearing statements are quoted from [4] without proof: the groupoid structure on O(\xi) (Theorem 13), the bracket identity [|X',Y'|]'=ad^*_{[|X,Y|]}\xi (Lemma 18), and the equality {F,H}_{A^*G}={f,h}_{K.K} used in Lemma 23. The last equality, in particular, is used to prove the x-independence of H, but the hypotheses under which it holds are not stated in this manuscript. A continuation note may legitimately cite previous work, but here the quoted result carries the full weight of the main reduction.

minor comments (7)

[Title] The word 'integrabiliy' in the title should be 'integrability'.
[Abstract] The abstract contains a spacing typo, 'co-adjo int Lie groupoids'.
[Corollary 27] The phrase 'if only if' should be 'if and only if'.
[Lemma 23 statement] The statement that H is 'equal to Hamiltonian h' is imprecise; the intended claim is H(p,\lambda)=h(\lambda) for all p\in M and \lambda\in T^*_{\xi'}O(\xi').
[Section 4, Examples] In the first example, the basis {e^\alpha_i} and the formula [|e^\alpha_i,e^\beta_j|]=\theta^\gamma_{\alpha\beta}\vartheta_\gamma are not written as an expansion in the same basis; the index ranges and the role of the tangent-bundle summand should be clarified.
[Section 5, Corollary 27] The notion of 'integrable' is not defined in this paper; the reader must infer the definition from references [1,6,14].
[Remark 9] The standing assumption that G is regular is not explicitly used in the statements of Lemma 23 or Proposition 30, so its role should be clarified.

Circularity Check

3 steps flagged · score 7.0 of 10

The main reduction (Corollary 27) is built into the chosen ansatz H(p,λ)=h(λ) and depends on a load-bearing self-citation to [4]; Lemma 23's attempted derivation is not a valid deduction.

self definitional [Section 5.1, paragraph after Lemma 23 (before Definition 24)]
"Therefore, according to the lemma mentioned above (lemma 23), we consider the Hamiltonian function H : M×T^*_{\xi'}O(\xi') → R such that H = (p,h), i.e. for δ = (p,λ)∈M×T^*_{\xi'}O(\xi'), we have that H(p,λ) = h(λ), where h : T^*O(\xi') → R is Hamiltonian function on T^*O(\xi')."

The conclusion that H is independent of the M-coordinates is precisely what Lemma 23 was supposed to prove, but the text then adopts it as the standing ansatz H(p,λ)=h(λ). Proposition 26 and Corollary 27 only compare systems of this restricted form. Any Hamiltonian with genuine x-dependence would have extra dynamics (and typically extra first integrals) on the product M×T^*_{\xi'}O(\xi'), so the claimed 'if and only if' is not a derived prediction; it is already contained in the chosen form of H.
self citation load bearing [Lemma 23 proof, Section 5.1 (also used in Proposition 26 proof)]
"Furthermore, as we proved in [4], Hamiltonian H :M×T^*_{\xi'}O(\xi')→ R, {F,H}_{A^*G} = {f,h}_{K.K}"

This equality between the co-adjoint Lie groupoid Poisson bracket and the Kirillov-Kostant bracket is the entire mechanism of Proposition 26 and hence of Corollary 27. It is not re-derived in this paper; it is quoted from the authors' own previous paper [4]. The cited prior work is not machine-checked or otherwise independently verified here, so the central integrability reduction rests on a self-citation whose content is exactly the identification needed for the conclusion.

1 more flagged steps

other [Lemma 23, Section 5.1, equations after X^{Π_{A^*Gξ}}_H]
"X^{Π_{A^*Gξ}}_H = ∂H/∂x^i ∂/∂x^i − C^γ_{αβ} yγ ∂H/∂yβ ∂/∂yα. Thus, the corresponding Hamiltonian equations are as follows: dx^i/dt = ∂H/∂x^i, dyα/dt = −C^γ_{αβ} yγ ∂H/∂yβ. (9)"

This part of Lemma 23 is the attempted derivation of ∂H/∂x^i=0, but equation (9) is not the Hamiltonian equation of the displayed Poisson tensor: the x-term in the displayed tensor is (1/2)∂/∂x^i∧∂/∂x^i, which vanishes identically, and equation (7) would give dx^i/dt=(∂H/∂y_α)ρ^i_α rather than dx^i/dt=∂H/∂x^i. The comparison of (9) with (10) therefore does not force ∂H/∂x^i=0; the x-independence used later is an extra assumption, not a consequence of the Lie algebroid structure.

full rationale

Corollary 27 is the paper's headline claim: integrability of the Hamiltonian vector field on the co-adjoint Lie groupoid O(ξ)=M×O(ξ') is equivalent to integrability on the co-adjoint orbit O(ξ') of a Lie group. The proof chain for that claim consists of (i) Lemma 23, which purports to show H cannot depend on M; (ii) the choice H(p,λ)=h(λ) immediately after Lemma 23; (iii) Proposition 26, which proves the first-integral correspondence using the identity {F,H}_{A*G}={f,h}_{K.K} quoted from [4]. Each stage is problematic: the x-independence of H is both the lemma's conclusion and the standing ansatz adopted in the next sentence; the key bracket identity is taken verbatim from the authors' own earlier publication; and the attempted proof of Lemma 23 compares inconsistent equations, since the x-part of the displayed Poisson tensor vanishes while equation (9) contains ∂H/∂x^i as a velocity. Thus the integrability reduction is effectively assumed in the input data (product orbit plus M-independent Hamiltonian plus self-cited bracket identity), not independently derived. This is not a case of harmless self-citation: without [4]'s bracket identity and without the H=h∘pr_2 ansatz, Corollary 27 has no remaining content. Score 7 reflects that the central claim reduces by construction and by a load-bearing self-citation, though some surrounding material (e.g., Lemma 22 on structural functions, Proposition 30 on symplectic groupoids) is derivative rather than circular.

Assumptions & free parameters 1 free parameters · 6 assumptions · 0 invented entities

The paper's new claims rest almost entirely on structures and identities imported from the authors' earlier paper [4]: the Lie groupoid structure on co-adjoint orbits (Definition 12-13), the bracket identity for sections (Lemma 18), and the Poisson bracket equality used in Lemma 23. In addition, Section 5 restricts to product-type orbits and chooses M-independent Hamiltonians. There are no numerical fits, but the choice of H is a functional restriction that makes the reduction true.

free parameters (1)

Hamiltonian H chosen independent of M = H(p,λ)=h(λ)
Imposed before Lemma 23; the claimed equivalence (Prop. 26, Cor. 27) holds only for this restricted class of Hamiltonians.

assumptions (6)

domain assumption The co-adjoint orbit O(ξ) admits a Lie groupoid structure when the stabilizer G_ξ is a normal Lie subgroupoid (Definitions 11-13).
This is the foundation of the entire paper; the proof is cited to [4], not reproduced.
domain assumption G is assumed to be a regular Lie groupoid throughout (Remark 9).
Regularity is needed for the isotropy groupoid to be a Lie groupoid (Lemma 10, cited from [13]).
ad hoc to paper The co-adjoint algebroid bracket satisfies [|X′,Y′|]′ = ad*_{[|X,Y|]}ξ (Lemma 18).
This identity is used in Lemma 22 and in the Poisson bracket computation; its proof is deferred to [4].
ad hoc to paper The linear Poisson bracket on A^*G satisfies {F,H}_{A^*G}={f,h}_{K.K} for F=(p,f), H=(p,h).
Used in Lemma 23 and Proposition 26; stated as proved in [4], not shown here.
ad hoc to paper The co-adjoint Lie groupoid is of product form O(ξ)=M×O(ξ′).
Section 5 assumes this special case for the integrability statements; it is not proven for general regular Lie groupoids.
ad hoc to paper The 2-form ω′(η1,η2)=ω(X,Y) is a well-defined symplectic form on O(ξ) (Proposition 30).
Well-definedness and nondegeneracy are asserted, not verified; the map (X,Y)↦(η1,η2) may have kernel.

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Cite this review

Pith. "Pith review of A note on integrabiliy of Hamiltonian systems on the co-adjoint Lie groupoids." pith.science (2026). https://pith.science/paper/YESGXMMD

@misc{pith2026241116631,
  author       = {Pith},
  title        = {Pith review of: A note on integrabiliy of Hamiltonian systems on the co-adjoint Lie groupoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YESGXMMD}},
  note         = {Machine review of arXiv:2411.16631}
}

read the original abstract

As we said in our previous work [4], the main idea of our research is to introduce a class of Lie groupoids by means of co-adjoint representation of a Lie groupoid on its isotropy Lie algebroid, which we called coadjoint Lie groupoids. In this paper, we will examine the relationship between structural mappings of the Lie algebroid associated to Lie groupoid and co-adjoint Lie algebroid. Also, we try to construct and define integrabiliy of Hamiltonian system on the co-adjoint Lie groupoids. In addition, we show that co-adjoint Lie groupoid associated to a symplectic Lie groupoid is a symplectic Lie groupoid.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

16 extracted references · 15 canonical work pages

[4]

Haghighatdoost and R

Gh. Haghighatdoost and R. Ayoubi, Hamiltonian systems on co-adjoint Lie groupoids , Journal of Lie Theory 31, 2 (2021), 493-516

work page 2021
[1]

Bolsinov and A.T

A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian System: Geometry, Topology, Class iﬁcation, Boca Raton: CRC Press, (2004)

work page 2004
[2]

Bos, Geometric quantization of Hamiltonian actions of Lie algeb roids and Lie groupoids , Int

R. Bos, Geometric quantization of Hamiltonian actions of Lie algeb roids and Lie groupoids , Int. J. Geom. Methods Mod. Phys, 4 (2007), 389-436

work page 2007
[3]

Generalized geometric Hamilton-Jacobi theorem on Lie algebroids

Gh. Haghighatdoost and R. Ayoubi, Generalized geometric Hamilton-Jacobi theorem on Lie alge broids, arXiv: 1902.06969v1, (2019)

work page Pith review arXiv 2019
[5]

Haghighatdoost and F

Gh. Haghighatdoost and F. Hasani, Lie group and Lie algebra 1 , Payame noor university, 1395. 14

work page
[6]

Haghighatdoost and A.A Oshemkov, The topology of Liouville foliation for the Sokolov integra ble case on the Lie algebra so(4) , Sbornik: Mathematics, 200(6), 899 – 921, (2009)

Gh. Haghighatdoost and A.A Oshemkov, The topology of Liouville foliation for the Sokolov integra ble case on the Lie algebra so(4) , Sbornik: Mathematics, 200(6), 899 – 921, (2009)

work page 2009
[7]

Haghighatdoost Optimal control problems on the co-adjoint Lie groupoids

Gh. Haghighatdoost Optimal control problems on the co-adjoint Lie groupoids. , Archives of control sciences, 34(4), (2024)

work page 2024
[8]

Kirillov, Lectures on the Orbit Method , Graduate Studies in Mathematics, vol

A.A. Kirillov, Lectures on the Orbit Method , Graduate Studies in Mathematics, vol. 64, American Mathem atical Society, Providence, (2004)

work page 2004

Show all 16 references

[9]

Lang and Zh

H. Lang and Zh. Liu, Co-adjoint orbits of Lie groupoids , arXiv:1802.09923v2, (2018)

2018 arXiv
[10]

Mackenzie, General theory of Lie groupoids and Lie algebroids , London Math

K.C.H. Mackenzie, General theory of Lie groupoids and Lie algebroids , London Math. Soc. Lecture notes series 213, Cambridge University Press, Cambridge, (2005)

2005
[11]

Marle, Lie, symplectic and Poisson groupoids and their Lie algebro ids, arXiv preprint arXiv:1402.0059, (2014)

C.M. Marle, Lie, symplectic and Poisson groupoids and their Lie algebro ids, arXiv preprint arXiv:1402.0059, (2014)

2014 arXiv
[12]

Marrero , Hamiltonian dynamics on Lie algebroids, unimodularity and preservation of volumes , arXiv preprint arXiv:0905.0123v1, (2009)

J.C. Marrero , Hamiltonian dynamics on Lie algebroids, unimodularity and preservation of volumes , arXiv preprint arXiv:0905.0123v1, (2009)

2009 arXiv
[13]

Schmeding, The Lie group of vertical bisections of a regular Lie groupoi d, Forum Mathematicum, 32, 479–489, (2019)

A. Schmeding, The Lie group of vertical bisections of a regular Lie groupoi d, Forum Mathematicum, 32, 479–489, (2019)

2019
[14]

Sokolov, One class of quadratic so(4) Hamiltonians , Dokl

H. Sokolov, One class of quadratic so(4) Hamiltonians , Dokl. Ross. Akad. Nauk, 394(5), 602–605, English transl. i n Dokl. Math., 69(1), 108–111, (2004)

2004
[15]

Troﬁmov and A.T

V.V. Troﬁmov and A.T. Fomenko, Dynamic systems on orbits of linear representations of Lie g roups and the complate integrability of some hydrodynamic systems , Functional Anal. Appl., 17 (1983), 31-39

1983
[16]

Voronov, Vector bundles , personalpages.manchester.ac.uk,Theodore.voronov,Te aching Diﬀerential Geometry, lecture2, (2009)

Th. Voronov, Vector bundles , personalpages.manchester.ac.uk,Theodore.voronov,Te aching Diﬀerential Geometry, lecture2, (2009). 15

2009

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [4]

Haghighatdoost and R

Gh. Haghighatdoost and R. Ayoubi, Hamiltonian systems on co-adjoint Lie groupoids , Journal of Lie Theory 31, 2 (2021), 493-516

work page 2021

[2] [1]

Bolsinov and A.T

A.V. Bolsinov and A.T. Fomenko, Integrable Hamiltonian System: Geometry, Topology, Class iﬁcation, Boca Raton: CRC Press, (2004)

work page 2004

[3] [2]

Bos, Geometric quantization of Hamiltonian actions of Lie algeb roids and Lie groupoids , Int

R. Bos, Geometric quantization of Hamiltonian actions of Lie algeb roids and Lie groupoids , Int. J. Geom. Methods Mod. Phys, 4 (2007), 389-436

work page 2007

[4] [3]

Generalized geometric Hamilton-Jacobi theorem on Lie algebroids

Gh. Haghighatdoost and R. Ayoubi, Generalized geometric Hamilton-Jacobi theorem on Lie alge broids, arXiv: 1902.06969v1, (2019)

work page Pith review arXiv 2019

[5] [5]

Haghighatdoost and F

Gh. Haghighatdoost and F. Hasani, Lie group and Lie algebra 1 , Payame noor university, 1395. 14

work page

[6] [6]

Haghighatdoost and A.A Oshemkov, The topology of Liouville foliation for the Sokolov integra ble case on the Lie algebra so(4) , Sbornik: Mathematics, 200(6), 899 – 921, (2009)

Gh. Haghighatdoost and A.A Oshemkov, The topology of Liouville foliation for the Sokolov integra ble case on the Lie algebra so(4) , Sbornik: Mathematics, 200(6), 899 – 921, (2009)

work page 2009

[7] [7]

Haghighatdoost Optimal control problems on the co-adjoint Lie groupoids

Gh. Haghighatdoost Optimal control problems on the co-adjoint Lie groupoids. , Archives of control sciences, 34(4), (2024)

work page 2024

[8] [8]

Kirillov, Lectures on the Orbit Method , Graduate Studies in Mathematics, vol

A.A. Kirillov, Lectures on the Orbit Method , Graduate Studies in Mathematics, vol. 64, American Mathem atical Society, Providence, (2004)

work page 2004