# A different Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$

In this question $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ is the space of all smooth vector fields on the plane or sphere. A limit cycle for a vector field $X$ is an isolated closed orbit of $X$.

Is there a Lie algebra structure on $\chi^{\infty}(\mathbb{R}^{2})$ or $\chi^{\infty}(S^{2})$ which is not isomorphic to the usual standard structures but satisfy the following:

Two commuting vector fields share on their limit cycles.

I asked this question in MSE but I received no complete answer. In this question in MSE I wrote some motivations for searching such new lie algebra structures:

https://math.stackexchange.com/questions/1276879/certain-lie-algebra-structure-on-chi-infty-mathbbr2-or-chi-inft

• could you add a definition/reference for $\chi^{\infty}(\mathbb{R}^{2})$? – Michael Bächtold May 23 '15 at 8:26
• @MichaelBächtold The space of smooth vector fields on the plane. – Ali Taghavi May 23 '15 at 9:17

I guess you can assume that the bracket $P$ you are looking for is local: $supp(P(X,Y))\subseteq supp(X) \cap supp(Y)$. Then, by the multilinear Peetre theorem, it is given by a bilinear differential operator. In the case of functions, all these algebras have been more or less classified by Kirillov and Lichnerovicz. You surely can carry them over to vector fields. Below are some of the papers on this topic.

• MR0438390 (55 #11304a) Kirillov, A. A. Local Lie algebras. (Russian) Uspehi Mat. Nauk 31 (1976), no. 4(190), 57–76.

• MR0438391 (55 #11304b)
Kirillov, A. A. Letter to the editors: Correction to "Local Lie algebras'' (Uspehi Mat. Nauk 31 (1976), no. 4(190), 57–76). (Russian) Uspehi Mat. Nauk 32 (1977), no. 1, (193), 268.

• MR0380881 (52 #1778) Reviewed Kirillov, A. A. Lie algebra structures that have the property of being local. (Russian) Funkcional. Anal. i Priložen. 9 (1975), no. 2, 75–76.

From the review: Let M be a smooth manifold and consider C∞(M) endowed with the usual topology. A Lie algebra structure (over R) on C∞(M) is said to have the property of being local if the bracket operation is continuous and supp[f1,f2]⊂(suppf1)∩(suppf2) for any f1,f2∈C∞(M). The author proves that any such a structure is locally and generically of the following type: There exists a foliation on M whose leaves are symplectic (or contact) manifolds, the Lie algebra structure on C∞(M) being induced by the Poisson bracket on the leaves. {English translation: Functional Anal. Appl. 9 (1975), no. 2, 158–160.} Reviewed by A. Verona

• MR0881399 (88m:58059) Lichnerowicz, André(F-CDF) Local Lie algebras of Kirillov and geometry of the Jacobi manifolds. Proceedings of the conference commemorating the 1st centennial of the Circolo Matematico di Palermo (Italian) (Palermo, 1984). Rend. Circ. Mat. Palermo (2) Suppl. No. 8 (1985), 193–203.

Introduction: "Consider a real line bundle and the space of its differentiable sections. Introduce on this space a local Lie algebra, the bracket of two sections having a local character. Kirillov gave in 1976 an interesting study of such local Lie algebras. If the bundle is trivial, it is equivalent to study the local Lie algebras on the space of the real-valued functions defined on the base. This "scalar case'' is in fact the more interesting case. Methods and results can be extended in a natural manner to the Kirillov case. "Independently from Kirillov, we introduced in 1975 the notion of general Poisson structure and in 1976–1977 the notion of general Jacobi structure [C. R. Acad. Sci. Paris Sér. A-B 285 (1977), no. 6, A455–A459; MR0455037 (56 #13278); J. Math. Pures Appl. (9) 57 (1978), no. 4, 453–488; MR0524629 (80m:58016)]. Such a structure is a contravariant generalization together of the symplectic or Poisson structures and of the contact structures. To each local Lie algebra is canonically associated such a geometrical structure and conversely. In this talk, we analyze the geometry of these Jacobi structures and the properties of the corresponding Lie algebras of vector fields [op. cit., 1977; op cit., 1978; the author, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 22, 915–920; MR0719276 (84k:58097); ibid. Sér. I Math. 297 (1983), no. 4, 261–266; MR0727184 (84k:17010); the author and F. Guédira , J. Math. Pures Appl. (9) 63 (1984), no. 4, 407–484; MR0789560 (86j:58045)]. The corresponding work has been done in collaboration with Guédira.''

• MR0864869 (88b:58053) Reviewed Lichnerowicz, A.(F-CDF) Generalized foliations and local Lie algebras of Kirillov. Differential geometry (Santiago de Compostela, 1984), 198–210, Res. Notes in Math., 131, Pitman, Boston, MA, 1985.

Let W be a connected paracompact C∞ manifold, and let N be the class of all real-valued C∞ functions on W. A Lie algebra structure [⋅,⋅] is local if, whenever Ω is open, and u=0 on Ω, then it follows that [u,v]=0 on Ω for all v. A local Lie algebra structure is determined by a Jacobi structure on M, i.e. a pair (Λ,E), such that Λ is a skew-symmetric 2-tensor, E is a vector field, and the identities [Λ,Λ]=2E∧Λ,[E,Λ]=LEΛ=0 hold. (Here [Λ,Λ] is the Schouten bracket of Λ with itself, and LE is Lie differentiation in the direction of E.) Two local Lie algebra structures are equivalent if there is a nowhere vanishing h in N such that multiplication by h establishes an isomorphism between the two structures. This translates into a concept of equivalence of Jacobi structures. An equivalence class of such structures is called a conformal Jacobi structure. To a local Lie algebra structure one can associate a map that produces, for each u in N, a vector field Xu, called the Hamiltonian vector field of u. The class of all these vector fields constitutes a Lie algebra L∗, which is actually an ideal in the Lie algebra L of infinitesimal automorphisms of the conformal Jacobi structure of the given local Lie algebra. Then L∗ defines a smooth field of planes, not necessarily of constant dimension. This field turns out to have the appropriate invariance properties, so as to give rise to a generalized foliation, so that W is partitioned into leaves, possibly of different dimensions. The original Jacobi structure can be restricted to each leaf, giving rise to a transitive local Lie algebra. The structure of the leaves is particularly simple. Odd-dimensional leaves are Pfaffian manifolds. Even-dimensional leaves are locally conformally symplectic. The author then proves several results for the transitive case, showing in particular that, in the odd-dimensional case, there are no differentiably nontrivial deformations of (N,[⋅,⋅]), but such deformations always exist in the even-dimensional globally conformally symplectic case. In the remaining case, i.e. when the Jacobi structure is locally but not globally conformally symplectic, both possibilities can occur. The author makes a detailed study of the Poisson structure on W×R associated with a given Jacobi structure (Λ,E) on W, of distinguished charts associated with Jacobi structures, and of the derivations of the algebras L and L∗.

• MR0848747 (87m:58060) Lichnerowicz, A. Géométrie des algèbres de Lie locales de Kirillov. (French) [Geometry of Kirillov's local Lie algebras] Symplectic geometry and mechanics (Balaruc, 1983), 107–124, Travaux en Cours, Hermann, Paris, 1985.

Let W be a connected, paracompact, differentiable manifold, m=dimW. The author analyses the geometry of Jacobi structures associated with the local Lie algebra structures on N=C∞(W;R). Each Jacobi structure (Λ,E) on W determines two Lie algebras of vector fields: the Lie algebra L of the infinitesimal automorphisms of the associated conformal Jacobi structure, and the ideal L∗ of L of the generalized Hamiltonian vector fields. The author announces geometric characterizations of the transitive local Lie algebras on N in the following cases: (1) m odd; (2) m even, E=0; and (3) m even, (W,Λ,E) is a l.c.s.v. manifold (the author's terminology in previous papers [C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 22, 915–920; MR0719276 (84k:58097); ibid. Sér. I Math. 297 (1983), no. 4, 261–266; MR0727184 (84k:17010)]). The derivations of the Lie algebras associated with a local Lie algebra, as well as some of their deformations, are also studied. Part of the results obtained for the algebra N are extended to the space of cross-sections of a line bundle over W.

• MR0789560 (86j:58045) Reviewed Guedira, Fouzia; Lichnerowicz, André Géométrie des algèbres de Lie locales de Kirillov. (French) [Geometry of Kirillov's local Lie algebras] J. Math. Pures Appl. (9) 63 (1984), no. 4, 407–484.

Let W be a connected paracompact manifold and let N denote the set of all smooth functions on W. A local Lie algebra on W is a Lie algebra (N,[,]) such that supp[u,v]⊂(suppu)∩(suppv) for all u,v∈N. Restating a result by A. A. Kirillov , the authors first deduce that the local Lie algebras coincide with the so-called Jacobi manifolds. (A Jacobi structure on W is a pair consisting of a contravariant antisymmetric tensor field Λ of degree 2 and a vector field E with the property that the Lie derivative of Λ with respect to E vanishes and the Schouten bracket of Λ with itself is 2E∧Λ.) In analogy with the classical case, every u∈N determines a generalized Hamiltonian vector field σ(u) on W and the set σ(N)=L∗ is an ideal in the algebra L of all infinitesimal automorphisms of (N,[,]). The following theorem completes two previous results by Lichnerowicz [C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 22, 915–920; MR0719276 (84k:58097); ibid. Sér. I Math. 297 (1983), no. 4, 261–266; MR0727184 (84k:17010)]. If (N,[,]) is a local transitive Lie algebra on an even-dimensional manifold such that the corresponding Jacobi manifold is locally conformally true symplectic, then (N,[,]) is isomorphic to L∗ by means of σ and coincides with the derived ideal. An explicit formula for the derivations of L∗ is also deduced and the cohomology space H2diff(N;N) is described. Every Jacobi structure on a manifold W induces a Poisson structure on W×R. The authors first study some distinguished local charts on the Poisson manifold W×R. This enables them to deduce a detailed description of the derivations in (N,[,]). In conclusion, the authors outline how their results can be extended to the local Lie algebras defined on the section of a line bundle.

• MR0772089 (85k:17015) Reviewed Lichnerowicz, André(F-CDF); Rubio, Ramon(F-CDF) Algèbres de Lie locales de Kirillov transitives et déformations. (French. English summary) [Transitive Kirillov local Lie algebras and deformations] C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 15, 761–765.

Authors' summary: "If K→W is a nontrivial line bundle, Γ(K) the space of its sections, we study the local Lie algebras (Γ(K),[,]) transitive on W and the corresponding geometric structures of W. If W is odd-dimensional, (Γ(K),[,]) is rigid. If W is even-dimensional, the nature of deformations is connected with the study of the bundle K2→W; if it is symplectically trivial, there are always nontrivial infinitesimal deformations.''

• MR0727184 (84k:17010)
Lichnerowicz, André Dérivations d'algèbres de Lie attachées à une algèbre de Lie locale de Kirillov. (French. English summary) [Derivations of the Lie algebras associated with a Kirillov local Lie algebra] C. R. Acad. Sci. Paris Sér. I Math. 297 (1983), no. 4, 261–266.

Author's summary: "Let (Γ(K),[ ,]) be a local Lie algebra on the space Γ(K) of sections of a line bundle K→W. There is on K a canonical Poisson structure, homogeneous of degree -1; conversely such a structure determines a local algebra on Γ(K). We introduce distinguished charts on W. Let L be the Lie algebra of the infinitesimal automorphisms of the associated conformal Jacobi structure, L∗ the ideal of L given by the `Hamiltonian' vector fields. We determine the derivations of L∗ and L, and study the nonlocal derivations of (Γ(K),[ ,]).'' Citations

• MR0719276 (84k:58097) Reviewed Lichnerowicz, André(F-CDF) Sur les algèbres de Lie locales de Kirillov-Shiga. (French. English summary) [Local Lie algebras in the sense of Kirillov-Shiga] C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 22, 915–920.
• Prof. Michor Thank you very much for your very interesting and helpful materials. – Ali Taghavi Jun 17 '15 at 15:18
• Do you think that the dynamical condition in my question(about limit cycles) automatically implies Peetre (support) condition? – Ali Taghavi Jun 20 '15 at 17:55
• You seem to want to have the usual integral curves. Thus you have the same action on functions. In fact I do not believe that any other Lie bracket exists satisfying your condition. – Peter Michor Jun 20 '15 at 20:16