On Poisson structures arising from a Lie group action

E. L. Mansfield; G. M. Beffa

arxiv: 1906.10789 · v1 · pith:BOZB5G5Nnew · submitted 2019-06-26 · 🧮 math.DG · math-ph· math.MP

On Poisson structures arising from a Lie group action

G. M. Beffa , E. L. Mansfield This is my paper

Pith reviewed 2026-05-25 15:40 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MP

keywords Poisson bracketsLie group actionsLie algebroidsinfinite-dimensional Lie algebrasHamiltonian flowsDarboux structureLie-Poisson brackets

0 comments

The pith

Lie group actions on manifolds induce Poisson brackets on functions over M times g star that form a class strictly larger than the Darboux symplectic and classical Lie-Poisson brackets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that a Lie group action on a manifold M endows the space of maps from M to the Lie algebra with two distinct Lie algebra structures. Both structures satisfy the conditions to be Lie algebroids. A construction then produces Poisson brackets on the space of functions on M cross the dual of the Lie algebra. These brackets properly include the usual Darboux and Lie-Poisson examples while forming a broader family. The authors compute the Hamiltonian flows, their invariants, and related structures in numerous explicit cases.

Core claim

A Lie group G acting on a manifold M induces two Lie algebra structures on the space of smooth maps from M to the Lie algebra g, each of which is a Lie algebroid. Applying the construction that yields a Poisson structure from such an algebroid gives a Poisson bracket on smooth functions on M times g star. This produces a family of Poisson brackets that includes the standard Darboux symplectic structure and the classical Lie Poisson brackets on g star but is strictly larger, as shown by explicit examples and computations of the associated Hamiltonian flows and invariants.

What carries the argument

The construction associating a Poisson bracket to each of the two Lie algebroids on the space of maps from M to g

If this is right

Hamiltonian flows and their invariants can be studied explicitly for the resulting brackets.
Canonical maps induced by the Lie group action preserve the Poisson structure.
Compatible Poisson structures arise on the same manifold.
Central extensions of the Lie algebras produce additional infinite-dimensional Poisson brackets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The broader class may allow modeling of systems where standard Poisson structures do not suffice.
The alternate derivation of the bracket could extend to settings without a metric.
The computational examples point toward applications in concrete problems in mechanics and geometry.

Load-bearing premise

Both Lie algebra structures on the space of smooth maps from the manifold to the Lie algebra must qualify as Lie algebroids.

What would settle it

Computing the Jacobi identity for the bracket in a specific example derived from a Lie group action and finding it does not hold would show that the construction does not always produce a Poisson bracket.

Figures

Figures reproduced from arXiv: 1906.10789 by E. L. Mansfield, G. M. Beffa.

**Figure 1.** Figure 1: For the Hamiltonian H = 1 5 (x 2 + y 2 ) + 2(ξ 1 ) 2 − (ξ 2 ) 2 + 3(ξ 3 ) 2 , the plots for ( ˙z, ˙ξ) T = Λ(∇zH, ∇ξH) T with Λ given in (35) the initial data x(0) = y(0) = ξ1(0) = ξ2(0) = ξ3(0) = 1 are shown. In (ii), the plot for the Lie Poisson system for so(3)∗ with H = 2(ξ 1 ) 2 − (xi2 ) 2 + 3(ξ 3 ) 2 , with the same initial data, is shown for comparison with the dashed line. (i) t 7→ (x(t), y(t)) (ii)… view at source ↗

**Figure 2.** Figure 2: If the action is free and regular on a domain [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗

**Figure 3.** Figure 3: For the Hamiltonian H, given in (42), some plots for ( ˙σ, ˙ξ) T = Λ(∇σH, ∇ξH) T with Λ given in (41) and the initial data σ a (0) = 1, σ b (0) = −1, σ c (0) = 1 2 , ξ1(0) = ξ2(0) = ξ3(0) = 1 are shown. In Plot (i), the dashed line is for H = ξ 2 1 + ξ 2 2 + ξ 2 3 , the Lie Poisson structure Λ(sl(2)∗ ), and the same initial data. (i) t 7→ t 7→ (ξ1(t), ξ2(t), ξ3(t)) (ii) t 7→ u(t) = −σ b (t)/σa (t) Recall t… view at source ↗

read the original abstract

We investigate some infinite dimensional Lie algebras and their associated Poisson structures which arise from a Lie group action on a manifold. If $G$ is a Lie group, $\g$ its Lie algebra and $M$ is a manifold on which $G$ acts, then the set of smooth maps from $M$ to $\g$ has at least two Lie algebra structures, both satisfying the required property to be a Lie algebroid. We may then apply a {construction} by Marle to obtain a Poisson bracket on the set of smooth real or complex valued functions on $M\times \g^*$. In this paper, we investigate these Poisson brackets. We show that the set of examples include the standard Darboux symplectic structure and the classical Lie Poisson brackets, but is a strictly larger class of Poisson brackets than these. Our study includes the associated Hamiltonian flows and their invariants, canonical maps induced by the Lie group action, and compatible Poisson structures. Our approach is mainly computational and we detail numerous examples. The Lie brackets from which our results derive, arose from the consideration of connections on bundles with zero curvature and constant torsion. We give an alternate derivation of the Lie bracket which will be suited to applications to Lie group actions for applications not involving a Riemannian metric. We also begin a study of the infinite dimensional Poisson brackets which may be obtained by considering a central extension of the Lie algebras.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs Poisson brackets on M × g* from two Lie algebroid structures on C^∞(M, g) induced by a group action, and claims via examples that the resulting family properly contains the Darboux and Lie-Poisson cases.

read the letter

The core contribution is a computational construction that starts from a Lie group action on M, equips the space of maps M to g with two Lie algebra structures that both satisfy the Lie algebroid axioms, and then applies Marle’s method to produce Poisson brackets on functions on M × g*. The authors recover the standard Darboux symplectic structure and the classical Lie-Poisson bracket as special cases, then exhibit further examples that they argue are distinct. They also track the associated Hamiltonian flows and invariants, note canonical maps coming from the group action, and check compatibility of the structures. An alternate derivation of one bracket is given that avoids Riemannian metrics, and a short section begins the study of central extensions. The work is mainly example-driven and stays close to explicit calculations rather than abstract classification theorems. This is useful for readers who want concrete instances of Poisson structures arising from group actions beyond the textbook cases. The evidence for the “strictly larger class” claim rests entirely on the detailed examples; if those examples hold up under direct verification, the claim is supported. The main soft spot is that the Lie algebroid property for both structures is asserted rather than derived in the abstract, so a referee would want to see the explicit bracket formulas and the verification that the Jacobi identity and anchor map conditions are satisfied. The central-extension discussion is only sketched, so it does not yet carry weight. The paper is aimed at differential geometers and Poisson geometers working with infinite-dimensional Lie algebras or with computational approaches to group actions. It is narrow but self-contained, and the claims are checkable. It deserves a serious referee because the method is reproducible and the examples are the load-bearing part of the argument.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs Poisson brackets on C^∞(M × g*) from a Lie group G acting on M by equipping C^∞(M, g) with two Lie algebra structures, each asserted to be a Lie algebroid, then applying Marle's construction. It recovers the standard Darboux symplectic structure and classical Lie-Poisson brackets as special cases, claims the resulting family is strictly larger, and studies the associated Hamiltonian flows, invariants, canonical maps induced by the action, and compatible Poisson structures. The approach is primarily computational with numerous examples; an alternate derivation of one Lie bracket is given from flat connections with constant torsion, and a preliminary study of central extensions is begun.

Significance. If the Lie algebroid axioms hold and the examples confirm genuinely new structures, the work supplies a systematic, action-based method for generating Poisson brackets that properly contains the Darboux and Lie-Poisson cases. The computational verification, alternate derivation suited to applications without a Riemannian metric, and examination of flows and invariants constitute concrete strengths that would be useful in geometric mechanics and symmetric dynamical systems.

major comments (2)

[Construction of the Lie algebra structures (near the definition of the two brackets)] The assertion that both Lie algebra structures on C^∞(M, g) are Lie algebroids (anchor map, Leibniz rule, and Jacobi identity) is load-bearing for the application of Marle's construction, yet the manuscript only states the property without an explicit verification of the axioms in the main text; this verification must be supplied before the central claim can be accepted.
[Examples and comparison with Darboux/Lie-Poisson] § on examples: the claim that the family is strictly larger than Darboux and Lie-Poisson requires at least one concrete example whose Poisson tensor is shown to be inequivalent (e.g., by rank, Casimir functions, or cohomology class) to both standard cases; the current computational survey must isolate such an instance explicitly.

minor comments (2)

Notation for the two distinct Lie brackets on C^∞(M, g) should be introduced with separate symbols (e.g., [·,·]_1 and [·,·]_2) rather than relying on context alone.
The statement that the Lie brackets 'arose from the consideration of connections on bundles with zero curvature and constant torsion' would benefit from a one-sentence reminder of the precise relation before the alternate derivation is presented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and for highlighting areas where the manuscript can be strengthened. We respond to the major comments point by point below.

read point-by-point responses

Referee: [Construction of the Lie algebra structures (near the definition of the two brackets)] The assertion that both Lie algebra structures on C^∞(M, g) are Lie algebroids (anchor map, Leibniz rule, and Jacobi identity) is load-bearing for the application of Marle's construction, yet the manuscript only states the property without an explicit verification of the axioms in the main text; this verification must be supplied before the central claim can be accepted.

Authors: We agree with the referee that an explicit verification of the Lie algebroid axioms is essential. Although the structures were constructed to satisfy these properties (as indicated by the alternate derivation from flat connections with constant torsion), the main text does not include the full check. In the revised version, we will add a subsection providing the detailed verification of the anchor, Leibniz identity, and Jacobi identity for both brackets. revision: yes
Referee: [Examples and comparison with Darboux/Lie-Poisson] § on examples: the claim that the family is strictly larger than Darboux and Lie-Poisson requires at least one concrete example whose Poisson tensor is shown to be inequivalent (e.g., by rank, Casimir functions, or cohomology class) to both standard cases; the current computational survey must isolate such an instance explicitly.

Authors: We accept this point. While the manuscript includes numerous examples and asserts that the family is strictly larger, it does not isolate a single instance with an explicit comparison of invariants. We will revise the examples section to include a dedicated example (such as the action on a sphere or a specific matrix group action) where we compute the Poisson tensor, its rank, and identify Casimir functions that differ from those in the Darboux and Lie-Poisson cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper constructs Poisson brackets on M × g* by applying Marle's construction to two Lie algebra structures on C^∞(M, g) that are asserted to be Lie algebroids. It recovers the Darboux and Lie-Poisson cases explicitly and exhibits additional structures via direct computation and examples. An alternate derivation of the brackets is supplied without dependence on prior self-citations for the core claims. No equation or result is shown to reduce to its inputs by definition, fitted parameters renamed as predictions, or load-bearing self-citation chains; the central claim of a strictly larger class rests on explicit verification rather than tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction builds on standard Lie theory and Marle's known method; no new free parameters or entities are introduced in the abstract.

axioms (1)

domain assumption Maps from M to g admit two Lie algebra structures that are Lie algebroids.
This is the starting point stated in the abstract for deriving the Poisson structures via Marle's construction.

pith-pipeline@v0.9.0 · 5783 in / 1055 out tokens · 24186 ms · 2026-05-25T15:40:07.579068+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages · 1 internal anchor

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Cannas da Silva, A

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Coste, P

A. Coste, P. Dazord, A. Weinstein, (1987) Groupoides symplectiques, Pub. Dep. Math. Lyon, 2/A, 1-62

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Dufour and N.-T

J.-P. Dufour and N.-T. Zung, (2005) Poisson structures and their normal forms, Progress in Mathematics, 242 Birkhauser Verlag, Basel

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M. Fels and P.J. Olver, (1999) Moving coframes II , Acta Appl. Math., 55, 127–208

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de Graaf, (2000) Lie Algebras: Theory and Algorithms , North-Holland Mathematical Library, Elsevier, Amsterdam

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Hirsch, (1976) Diﬀerential Topology, Graduate Texts in Mathematics 33, Springer Verlag, New York

M.W. Hirsch, (1976) Diﬀerential Topology, Graduate Texts in Mathematics 33, Springer Verlag, New York

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Quantization, Poisson brackets and beyond

Y. Kosmann-Shwarzbach and K. C. H. Mackenzie, (2002) Diﬀerential operators and actions of Lie algebroids , Contemporary Mathematics 315, 213–234. Pro- ceedings of “Quantization, Poisson brackets and beyond”, UMIST, UK 2001

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Kosmann-Shwarzbach and F

Y. Kosmann-Shwarzbach and F. Magri, (1990) Poisson-Nijenhuis strutures , Ann. Inst. H. Poincar´ e Phys. T´ eor.,53(1):3581

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Mackenzie, (2005) General Theory of Lie Groupoids and Lie Algebroids , London Mathematical Society Lecture Note Series 213, Cambridge University Press

K. Mackenzie, (2005) General Theory of Lie Groupoids and Lie Algebroids , London Mathematical Society Lecture Note Series 213, Cambridge University Press

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Mansﬁeld, (2010) A practical guide to the invariant calculus , Cambridge University Press, Cambridge, UK

E.L. Mansﬁeld, (2010) A practical guide to the invariant calculus , Cambridge University Press, Cambridge, UK

work page 2010
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Differential calculus on a Lie algebroid and Poisson manifolds

C–M. Marle, (2002) Diﬀerential calculus on a Lie algebroid and Poisson man- ifolds. In: The J.A. Pereira Birthday Schrift, Textos de matematica 32, De- partamento de matematica da Universidade de Coimbra, Portugal, 83–149. Available from https://arxiv.org/abs/0804.2451

work page internal anchor Pith review Pith/arXiv arXiv 2002
[16]

Munthe–Kaas and A

H. Munthe–Kaas and A. Lundervold, (2013), On post-Lie algeras, Lie–Butcher series and Moving Frames , Foundations of Computational Mathematics, 13 583–613

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[17]

Olver, (1993) Applications of Lie groups to diﬀerential equations , Second edition, Graduate Texts in Mathematics 107, Springer Verlag, New York

P.J. Olver, (1993) Applications of Lie groups to diﬀerential equations , Second edition, Graduate Texts in Mathematics 107, Springer Verlag, New York

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Schmeding and C

A. Schmeding and C. Wockerl, (2014) The Lie group of bisections of a Lie groupoid, Ann. Global Analysis and Geometry, 48 (1) 87–123. 37

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[1] [1]

Cannas da Silva, A

A. Cannas da Silva, A. Weinstein, (1999) Geometric models for noncommuta- tive algebras, Berkeley Mathematics Lecture Notes, 10, AMS

work page 1999

[2] [2]

Choquet–Bruhat, C, deWitt–Morette with M

Y. Choquet–Bruhat, C, deWitt–Morette with M. Dillard–Bleick, (1982) Anal- ysis, Manifolds and Physics , North-Holland, Amsterdam

work page 1982

[3] [3]

Coste, P

A. Coste, P. Dazord, A. Weinstein, (1987) Groupoides symplectiques, Pub. Dep. Math. Lyon, 2/A, 1-62

work page 1987

[4] [4]

Dufour and N.-T

J.-P. Dufour and N.-T. Zung, (2005) Poisson structures and their normal forms, Progress in Mathematics, 242 Birkhauser Verlag, Basel

work page 2005

[5] [5]

Fels and P.J

M. Fels and P.J. Olver, (1999) Moving coframes II , Acta Appl. Math., 55, 127–208

work page 1999

[6] [6]

de Graaf, (2000) Lie Algebras: Theory and Algorithms , North-Holland Mathematical Library, Elsevier, Amsterdam

W.A. de Graaf, (2000) Lie Algebras: Theory and Algorithms , North-Holland Mathematical Library, Elsevier, Amsterdam. 36

work page 2000

[7] [7]

Hirsch, (1976) Diﬀerential Topology, Graduate Texts in Mathematics 33, Springer Verlag, New York

M.W. Hirsch, (1976) Diﬀerential Topology, Graduate Texts in Mathematics 33, Springer Verlag, New York

work page 1976

[8] [8]

Holm, (2011) Geometric Mechanics Part I and Part II , Second edition, Imperial College Press, London

D. Holm, (2011) Geometric Mechanics Part I and Part II , Second edition, Imperial College Press, London

work page 2011

[9] [9]

Hydon, (2000) Symmetry Methods for Diﬀerential Equations: A Begin- ner’s Guide, Cambridge University Press

P.E. Hydon, (2000) Symmetry Methods for Diﬀerential Equations: A Begin- ner’s Guide, Cambridge University Press

work page 2000

[10] [10]

Iserles, H.Z

A. Iserles, H.Z. Munthe–Kaas, S.P. Nørsett and A. Zanna, (2001) Lie-group methods, Acta Numerica, 9 215–365

work page 2001

[11] [11]

Quantization, Poisson brackets and beyond

Y. Kosmann-Shwarzbach and K. C. H. Mackenzie, (2002) Diﬀerential operators and actions of Lie algebroids , Contemporary Mathematics 315, 213–234. Pro- ceedings of “Quantization, Poisson brackets and beyond”, UMIST, UK 2001

work page 2002

[12] [12]

Kosmann-Shwarzbach and F

Y. Kosmann-Shwarzbach and F. Magri, (1990) Poisson-Nijenhuis strutures , Ann. Inst. H. Poincar´ e Phys. T´ eor.,53(1):3581

work page 1990

[13] [13]

Mackenzie, (2005) General Theory of Lie Groupoids and Lie Algebroids , London Mathematical Society Lecture Note Series 213, Cambridge University Press

K. Mackenzie, (2005) General Theory of Lie Groupoids and Lie Algebroids , London Mathematical Society Lecture Note Series 213, Cambridge University Press

work page 2005

[14] [14]

Mansﬁeld, (2010) A practical guide to the invariant calculus , Cambridge University Press, Cambridge, UK

E.L. Mansﬁeld, (2010) A practical guide to the invariant calculus , Cambridge University Press, Cambridge, UK

work page 2010

[15] [15]

Differential calculus on a Lie algebroid and Poisson manifolds

C–M. Marle, (2002) Diﬀerential calculus on a Lie algebroid and Poisson man- ifolds. In: The J.A. Pereira Birthday Schrift, Textos de matematica 32, De- partamento de matematica da Universidade de Coimbra, Portugal, 83–149. Available from https://arxiv.org/abs/0804.2451

work page internal anchor Pith review Pith/arXiv arXiv 2002

[16] [16]

Munthe–Kaas and A

H. Munthe–Kaas and A. Lundervold, (2013), On post-Lie algeras, Lie–Butcher series and Moving Frames , Foundations of Computational Mathematics, 13 583–613

work page 2013

[17] [17]

Olver, (1993) Applications of Lie groups to diﬀerential equations , Second edition, Graduate Texts in Mathematics 107, Springer Verlag, New York

P.J. Olver, (1993) Applications of Lie groups to diﬀerential equations , Second edition, Graduate Texts in Mathematics 107, Springer Verlag, New York

work page 1993

[18] [18]

Schmeding and C

A. Schmeding and C. Wockerl, (2014) The Lie group of bisections of a Lie groupoid, Ann. Global Analysis and Geometry, 48 (1) 87–123. 37

work page 2014