Arnold Mathematical Journal
Volume 11, Issue 4, 2025

Research Contribution DOI:10.56994/ARMJ.011.004.007
Received: 27 Dec 2024; Accepted: 24 Aug 2025

On symplectic linearizable actions

Eva Miranda

Abstact.

1 Introduction

A classical result due to Bochner [1] establishes that a compact Lie group action on a smooth manifold is locally equivalent, in the neighbourhood of a fixed point, to its linearization. This result holds in the ${C}^{k}$ category. It is worth exploring if similar results hold in the non-compact case.

As observed in [8], if the Lie group is connected, the linearization problem can be formulated in the following terms: find a linear system of coordinates for the vector fields corresponding to the one-parameter subgroups of $G$ ; or more generally, consider the representation of a Lie algebra and find coordinates on the manifold that simultaneously linearize the vector fields in the image of the representation vanishing at a point. This is the perspective we adopt in this note when referring to linearization.

In the formal and analytic cases, the existence of coordinates that linearize the action is related to a cohomological equation that can always be solved when the Lie group under consideration is semisimple [9], [8]. Guillemin and Sternberg also studied the problem in the ${C}^{\infty}$ setting. At the end of [8], they presented the celebrated example of a non-linearizable action of $\mathfrak{sl}(2,\mathbb{R})$ on $\mathbb{R}^{3}$ , constructed via a perturbation involving the radial vector field with flat coefficients. This example has been foundational in the literature, inspiring the construction of other examples with profound geometric implications, such as Weinstein’s non-stable Poisson structure example [19].

When the semisimple Lie algebras are of compact type, the linearization of the action can be achieved by combining the local integration of the Lie group action with Bochner’s theorem, leading to the linearization of the associated Lie algebra action [6].

Linearization techniques also play a significant role in Hamiltonian systems. When a Hamiltonian system arises from a symplectic action of a compact Lie group fixing a point, the equivariant version of Darboux’s theorem ([18], [3]) ensures that the group action can be linearized in Darboux coordinates near the fixed point. It is worth exploring if similar results apply beyond the compact case.

For complete integrable systems, an associated abelian symplectic action emerges. When the integrable system in local coordinates has a “linear part” linked to a Cartan subalgebra, this leads to non-degenerate singularities [4]. As shown in [5], [4], [10], [11] and [12], complete integrable systems near non-degenerate singular points are equivalent to their linear models. Consequently, the Hamiltonian system itself is equivalent to the linear one. This result provides normal forms for integrable systems near singular non-degenerate points and, specifically, ensures the simultaneous linearization of Hamiltonian vector fields near a common zero.

The next challenge involves Hamiltonian systems with a semisimple linear part, as proposed by Eliasson in [5]. In the formal or analytic setting, results by Guillemin and Sternberg [8] and Kushnirenko [9] demonstrate that such systems are equivalent to the linear model when the symplectic form is disregarded. In this note, we establish that not only can the Hamiltonian vector fields be linearized, but they can also be linearized in Darboux coordinates.

Following Guillemin and Sternberg’s approach, we prove that if a symplectic Lie algebra action of semisimple type fixes a point, there exist analytic Darboux coordinates in which the analytic vector fields generating the Lie algebra action are linear. This result also extends to complex analytic Lie algebra actions on complex analytic manifolds. Additionally, we construct an example of a Hamiltonian system with a semisimple linear part that is not $C^{\infty}$ -linearizable.

Organization of this article: In Section 2, we prove that linearizable actions on symplectic manifolds can be locally linearized in Darboux coordinates. In Section 3, we apply this to show that any real analytic symplectic action of a semisimple Lie algebra can be linearized in real analytic Darboux coordinates in a neighborhood of a fixed point. Furthermore, this result extends to analytic complex manifolds and complex analytic actions of semisimple Lie algebras. In Section 4, we present a counterexample proving that the linearization result does not hold in general for smooth Hamiltonian actions of semisimple Lie algebras.

Acknowledgements: I would like to thank Häkan Eliasson, Ghani Zeghib, David Martínez-Torres and Marco Castrillón for their valuable feedback on an earlier version of this note.

2 Linearizable actions in Darboux coordinates

Let $\mathfrak{g}$ be a Lie algebra and let $\rho:\mathfrak{g}\longrightarrow L_{analytic}$ stand for a representation of $\mathfrak{g}$ in the algebra of real (or complex) analytic vector fields on a real (or complex) analytic manifold $M$ .

We say that $p\in M$ is a fixed point for $\rho$ if the vector fields in $\rho(\mathfrak{g})$ vanish at $p$ . We say that $\rho$ can be linearized in a neighborhood of a fixed point if there exist local coordinates in a neighbourhood of $p$ such that the vector fields in the image of $\rho$ can be simultaneously linearized (i.e, $\rho$ is equivalent to a linear representation).

Assume that the Lie algebra action is (analytically/smoothly) linearizable and assume that $M$ is endowed with a symplectic structure (smooth, analytic). We first prove that it is then symplectically linearizable.

Theorem 2.1.

Let $\mathfrak{g}$ be a Lie algebra and let $(M,\omega)$ be a (real or complex) analytic symplectic manifold. Let $\rho$ be a representation by analytic symplectic vector fields. Let $p$ be a fixed point for $\rho$ and assume that $\rho$ can be linearized. Then there exist local analytic coordinates $(x_{1},y_{1},\dots,x_{n},y_{n})$ in a neighborhood of $p$ for $\rho$ such that $\rho$ is a linear representation and $\omega$ can be written as,

\omega=\sum_{i=1}^{n}dx_{i}\wedge dy_{i}.

Proof.

Let $\rho$ be an analytic symplectic action of a Lie algebra on a manifold $M$ , with a fixed point $p\in M$ . Choose analytic coordinates $(x_{1},y_{1},\dots,x_{n},y_{n})$ centered at $p$ in which the action $\rho$ is linear. Let $\omega_{1}$ denote the symplectic form in these coordinates. Although $\rho$ is now linear, $\omega_{1}$ need not be of Darboux type.

We denote by $\omega_{0}$ the constant (degree-zero) term in the Taylor expansion of $\omega_{1}$ at the origin. Since $\omega_{1}$ is preserved by $\rho$ and $\rho$ is linear, it follows that $\omega_{0}$ is preserved by the linearized action $\rho^{(1)}=\rho$ . In particular, $\omega_{0}$ is a constant symplectic form invariant under the action. Our goal is to construct a local analytic diffeomorphism $\phi$ , fixing the origin, such that $\phi^{*}(\omega_{1})=\omega_{0}$ and $\phi$ commutes with $\rho$ . That is, we seek an equivariant analytic Darboux theorem for $\omega_{1}$ , linearizing the form while preserving the linear action $\rho$ .

To this end, we apply the path method [16] for analytic symplectic structures. By the Poincaré lemma, there exists an analytic 1-form $\alpha$ such that

\omega_{1}=\omega_{0}+d\alpha.

Define a path of symplectic forms:

\omega_{t}=t\omega_{1}+(1-t)\omega_{0},\quad t\in[0,1].

Each $\omega_{t}$ is an analytic symplectic form in a neighbourhood of the origin. Moreover, the action $\rho$ preserves both $\omega_{0}$ and $\omega_{1}$ , hence it preserves the entire path $\omega_{t}$ .

We now define the time-dependent analytic vector field $X_{t}$ by Moser’s equation:

i_{X_{t}}\omega_{t}=-\alpha.

(2.1)

In order to ensure that $X_{t}$ is invariant under $\rho$ , it suffices to construct $\alpha$ invariant under $\rho$ . For this purpose, we apply the standard homotopy operator used in the proof of the Poincaré lemma, adapted to our equivariant setting.

Let $R=\sum x_{i}\partial_{x_{i}}+y_{i}\partial_{y_{i}}$ be the radial vector field, and $h_{t}$ the homothety $x\mapsto tx$ . Then, we define

\alpha:=\int_{0}^{1}\frac{1}{t}h_{t}^{*}(i_{R}\beta)\,dt,\quad\text{where }% \beta=\omega_{1}-\omega_{0}.

Because $\beta$ is $\rho$ -invariant and $\rho$ commutes with $R$ , it follows that $\alpha$ is also $\rho$ -invariant. Thus, the vector field $X_{t}$ is invariant under $\rho$ .

Let $\phi_{t}$ denote the flow of $X_{t}$ , satisfying the differential equation

\frac{\partial\phi_{t}}{\partial t}(q)=X_{t}(\phi_{t}(q)),\quad\phi_{0}=% \mathrm{id}.

(2.2)

Since $X_{t}$ is $\rho$ -invariant, the flow $\phi_{t}$ commutes with the action $\rho$ . Moreover, because $\alpha$ vanishes at the origin, so does $X_{t}$ , ensuring that each $\phi_{t}$ fixes the origin.

By construction, $\phi_{t}^{*}(\omega_{t})=\omega_{0}$ , and in particular $\phi_{1}^{*}(\omega_{1})=\omega_{0}$ . The diffeomorphism $\phi:=\phi_{1}$ is then the desired equivariant analytic transformation taking $\omega_{1}$ to $\omega_{0}$ while preserving the linear action $\rho$ .

This completes the proof. ∎

Remark 2.2.

The theorem above is stated in the analytic category; however, if the linearization is assumed to hold in the smooth category, the symplectic diffeomorphism obtained from the proof is also smooth.

3 The case of analytic semisimple Lie algebra actions

Guillemin and Sternberg provided in [8] a complete characterization of analytically linearizable actions. They demonstrated that a necessary and sufficient condition for the representation

{\rho}:g\to L_{\text{analytic}}

to be locally analytically linearizable is the existence of an analytic vector field $X$ , defined in a neighborhood of $p$ , vanishing at $p$ , with the identity matrix as its Jacobian at $p$ , and commuting with all the vector fields in $\tilde{g}$ .

This condition was elegantly recast in cohomological terms in [8]. They proved that the first cohomology group $H^{1}(g,V^{*})$ acts as an obstruction to analytic linearization. For semisimple $g$ , $H^{1}(g,V^{*})$ vanishes for all representation spaces $V$ , ensuring the possibility of analytic linearization. On the other hand, for non-semisimple $g$ , one can construct a representation space $V$ such that $H^{1}(g,V^{*})\neq 0$ , which precludes analytic linearization.

This result establishes the semisimple case as a natural candidate for analytic linearization.

Guillemin and Sternberg [8] and Kushnirenko [9] proved the following.

Theorem 3.1 (Guillemin-Sternberg, Kushnirenko).

The representation $\rho:\mathfrak{g}\longrightarrow L_{analytic}$ with $\mathfrak{g}$ semisimple is locally equivalent, via an analytic diffeomorphism, to a linear representation of $\mathfrak{g}$ in a neighbourhood of a fixed point for $\rho$ .

As an application of theorem 2.1: When the representation is done by Hamiltonian vector fields (locally symplectic), the analytic diffeomorphism that gives the equivalence of the initial representation to the linear representation can be chosen to take the initial symplectic form to the Darboux one. Namely,

Corollary 3.2.

Let $\mathfrak{g}$ be a semisimple Lie algebra and let $(M,\omega)$ be a (real or complex) analytic symplectic manifold. Let $\rho:\mathfrak{g}\longrightarrow L_{analytic}$ be a representation by analytic symplectic vector fields. Then there exist local analytic coordinates $(x_{1},y_{1},\dots,x_{n},y_{n})$ in a neighbourhood of a fixed point $p$ for $\rho$ such that $\rho$ is a linear representation and $\omega$ can be written as,

\omega=\sum_{i=1}^{n}dx_{i}\wedge dy_{i}.

4 Non-linearizable semisimple smooth actions

References

[1] S. Bochner, Compact groups of differentiable transformations. Ann. of Math. (2) 46, (1945). 372–381.
[2] G. Cairns and E. Ghys, The local linearization problem for smooth $SL(n)$ -actions, Enseign. Math. (2) 43 (1997), no. 1-2, 133-171.
[3] M. Chaperon, Quelques outils de la théorie des actions différentiables Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), 259–275, Astérisque, 107-108, Soc. Math. France, Paris, 1983.
[4] L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals, Ph.D. Thesis (1984).
[5] L.H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case, Comment. Math. Helv. 65 (1990), no. 1, 4–35.
[6] R. L. Fernandes and Ph. Monnier, Linearization of Poisson brackets, Lett. Math. Phys. 69, (2004) 89-114.
[7] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1990.
[8] V. Guillemin and S. Sternberg, Remarks on a paper of Hermann, Trans. Amer. Math. Soc. 130 (1968) 110–116.
[9] A.G. Kushnirenko,Linear-equivalent action of a semisimple Lie group in the neighbourhood of a stationary point, Functional Anal. Appll, 1, (1967) 89-90.
[10] E. Miranda, On symplectic linearization of singular Lagrangian foliations, Ph.D. thesis, Universitat de Barcelona, ISBN: 9788469412374, 2003.
[11] E. Miranda, Integrable systems and group actions. Cent. Eur. J. Math. 12 (2014), no. 2, 240–270.
[12] E. Miranda and N.T. Zung, Equivariant normal forms for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. Ecole Norm. Sup., 37, no. 6, 2004, pp. 819–839.
[13] E. Miranda and N. T. Zung, A note on equivariant normal forms of Poisson structures, Math. Research Notes, 2006, vol 13-6, 1001–1012.
[14] E. Miranda, Some rigidity results for Symplectic and Poisson group actions, XV International Workshop on Geometry and Physics, 177–183, Publ. R. Soc. Mat. Esp., 11, R. Soc. Mat. Esp., Madrid, 2007.
[15] E. Miranda, P. Monnier and N.T. Zung, Rigidity of Hamiltonian actions on Poisson manifolds. Adv. Math. 229, No. 2, 1136-1179 (2012).
[16] J. Moser, On the volume elements on a manifold, Trans. Am. Math. Soc. 120, 286-294 (1965).
[17] R. Palais, Equivalence of nearby differentiable actions of a compact group. Bull. Amer. Math. Soc. 67 1961 362–364.
[18] A. Weinstein, Lectures on symplectic manifolds. Regional Conference Series in Mathematics, No.29. American Mathematical Society, Providence, R.I., 1977.
[19] A. Weinstein, The local structure of Poisson manifolds., J. Differential Geom. 18 (1983), no. 3, 523–557.

× S. Bochner, Compact groups of differentiable transformations. Ann. of Math. (2) 46, (1945). 372–381.
× G. Cairns and E. Ghys, The local linearization problem for smooth $SL(n)$ -actions, Enseign. Math. (2) 43 (1997), no. 1-2, 133-171.
× M. Chaperon, Quelques outils de la théorie des actions différentiables Third Schnepfenried geometry conference, Vol. 1 (Schnepfenried, 1982), 259–275, Astérisque, 107-108, Soc. Math. France, Paris, 1983.
× L. H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals, Ph.D. Thesis (1984).
× L.H. Eliasson, Normal forms for Hamiltonian systems with Poisson commuting integrals—elliptic case, Comment. Math. Helv. 65 (1990), no. 1, 4–35.
× R. L. Fernandes and Ph. Monnier, Linearization of Poisson brackets, Lett. Math. Phys. 69, (2004) 89-114.
× V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, 1990.
× V. Guillemin and S. Sternberg, Remarks on a paper of Hermann, Trans. Amer. Math. Soc. 130 (1968) 110–116.
× A.G. Kushnirenko,Linear-equivalent action of a semisimple Lie group in the neighbourhood of a stationary point, Functional Anal. Appll, 1, (1967) 89-90.
× E. Miranda, On symplectic linearization of singular Lagrangian foliations, Ph.D. thesis, Universitat de Barcelona, ISBN: 9788469412374, 2003.
× E. Miranda, Integrable systems and group actions. Cent. Eur. J. Math. 12 (2014), no. 2, 240–270.
× E. Miranda and N.T. Zung, Equivariant normal forms for nondegenerate singular orbits of integrable Hamiltonian systems, Ann. Sci. Ecole Norm. Sup., 37, no. 6, 2004, pp. 819–839.
× E. Miranda and N. T. Zung, A note on equivariant normal forms of Poisson structures, Math. Research Notes, 2006, vol 13-6, 1001–1012.
× E. Miranda, Some rigidity results for Symplectic and Poisson group actions, XV International Workshop on Geometry and Physics, 177–183, Publ. R. Soc. Mat. Esp., 11, R. Soc. Mat. Esp., Madrid, 2007.
× E. Miranda, P. Monnier and N.T. Zung, Rigidity of Hamiltonian actions on Poisson manifolds. Adv. Math. 229, No. 2, 1136-1179 (2012).
× J. Moser, On the volume elements on a manifold, Trans. Am. Math. Soc. 120, 286-294 (1965).
× R. Palais, Equivalence of nearby differentiable actions of a compact group. Bull. Amer. Math. Soc. 67 1961 362–364.
× A. Weinstein, Lectures on symplectic manifolds. Regional Conference Series in Mathematics, No.29. American Mathematical Society, Providence, R.I., 1977.
× A. Weinstein, The local structure of Poisson manifolds., J. Differential Geom. 18 (1983), no. 3, 523–557.

$\displaystyle\widetilde{X}$	$\displaystyle=\rho(X)+f\,R\,,$	(4.2)
$\displaystyle\widetilde{Y}$	$\displaystyle=\rho(Y)+g\,R\,,$
$\displaystyle\widetilde{Z}$	$\displaystyle=\rho(Z)\,,$

	$\displaystyle\hat{\rho}(X)$	$\displaystyle=\rho(X)+\frac{xz}{r^{2}}g(r^{2}-z^{2})R,$
	$\displaystyle\hat{\rho}(Y)$	$\displaystyle=\rho(Y)-\frac{yz}{r^{2}}g(r^{2}-z^{2})R,$
	$\displaystyle\hat{\rho}(Z)$	$\displaystyle=\rho(Z)+g(r^{2}-z^{2})R,$

	$\displaystyle\hat{\rho}(X)$	$\displaystyle=\rho(X)+\frac{xz}{r^{2}}g(r^{2}-z^{2})R,$
	$\displaystyle\hat{\rho}(Y)$	$\displaystyle=\rho(Y)-\frac{yz}{r^{2}}g(r^{2}-z^{2})R,$
	$\displaystyle\hat{\rho}(Z)$	$\displaystyle=\rho(Z)+g(r^{2}-z^{2})R,$

On symplectic linearizable actions

Abstact.

1 Introduction

2 Linearizable actions in Darboux coordinates

Theorem 2.1.

Proof.

Remark 2.2.

3 The case of analytic semisimple Lie algebra actions

Theorem 3.1 (Guillemin-Sternberg, Kushnirenko).

Corollary 3.2.

4 Non-linearizable semisimple smooth actions

4.1 The counterexample of Cairns and Ghys

4.2 The counterexample of Guillemin and Sternberg

4.3 A Hamiltonian counterexample

Proposition 4.1.

Proof.

Remark 4.2.

4.4 The case of semisimple Lie algebras of compact type

Acknowledgments

References

Contents