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2004 | 2 | 5 | 811-825
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Lie algebraic characterization of manifolds

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Results on characterization of manifolds in terms of certain Lie algebras growing on them, especially Lie algebras of differential operators, are reviewed and extended. In particular, we prove that a smooth (real-analytic, Stein) manifold is characterized by the corresponding Lie algebra of linear differential operators, i.e. isomorphisms of such Lie algebras are induced by the appropriate class of diffeomorphisms of the underlying manifolds.
Opis fizyczny
  • Polish Academy of Sciences
  • University of Luxembourg
  • [1] K. Abe: “Pursell-Shanks type theorem for orbit spaces and G-manifolds”, Publ. Res. Inst. Math. Sci., Vol. 18, (1982), pp. 265–282.
  • [2] I. Amemiya: “Lie algebra of vector fields and complex structure”, J. Math. Soc. Japan, Vol. 27, (1975), pp. 545–549.
  • [3] C.J. Atkin and J. Grabowski: “Homomorphisms of the Lie algebras associated with a symplectic manifolds”, Compos. Math., Vol. 76, (1990), pp. 315–348.
  • [4] F. Boniver, S. Hansoul, P. Mathonet and N. Poncin: “Equivariant symbol calculus for differential operators acting on forms”, Lett. Math. Phys., Vol. 62, (2002), pp. 219–232.
  • [5] A. Campillo, J. Grabowski and G. Müller G: “Derivation algebras of toric varieties”, Comp. Math., Vol. 116, (1999), pp. 119–132.
  • [6] P. Cohen, Yu Manin and D. Zagier: “Automorphic pseudodifferential operators, Algebraic Aspects of Integrable Systems”, Progr. Nonlinear Differential Equations Appl., Vol. 26, (1997), pp. 17–47.
  • [7] M. De Wilde and P. Lecomte: “Some Characterizations of Differential Operators on Vector Bundles”, E.B. Christoffel, P. Butzer and F. Feher (Eds.), Brikhäuser Verlag, Basel, 1981, pp. 543–549.
  • [8] M. De Wilde and P. Lecomte: “Cohomology of the Lie Algebra of Smooth Vector Fields of a Manifold, associated to the Lie Derivative of Smooth Forms”, J. Math. pures et appl., Vol. 62, (1983), pp. 197–214.
  • [9] C. Duval and V. Ovsienko: “Space of second order linear differential operators as a module over the Lie algebra of vector fields”, Adv. in Math., Vol. 132(2), (1997), pp. 316–333.
  • [10] M. Flato and A. Lichnerowicz: “Cohomologie des représentations definies par la derivation de Lie et à valeurs dans les formes, de l'algèbre de Lie des champs de vecteurs d'une variété différentiable. Premiers espaces de cohomologie. Applications”, C. R. Acad. Sci., Sér. A, Vol. 291, (1980), pp. 331–335.
  • [11] H. Gargoubi and V. Ovsienko: “Space of linear differential operators on the real line as a module over the Lie algebra of vector fields”, Internat. Math. Res. Notices, Vol. 5, (1996), pp. 235–251.
  • [12] K. Grabowska and J. Grabowski: “The Lie algebra of a Lie algebroid”, In: J. Kubarski et al. (Eds.): Lie Algebroids and Related Topics in Differential Geometry, Vol. 54, Banach Center Publications, Warszawa, 2001, pp. 43–50.
  • [13] I. Gel'fand and A. Kolmogoroff: “On rings of continuous functions on topological spaces”, C. R. (Dokl.) Acad. Sci. URSS, Vol. 22, (1939), pp. 11–15.
  • [14] J. Grabowski: “Isomorphisms and ideals of the Lie algebras of vector fields”, Invent. math., Vol. 50, (1978), pp. 13–33.
  • [15] J. Grabowski: “Lie algebras of vector fields and generalized foliations”, Publ. Matem., Vol. 37, (1993), pp. 359–367.
  • [16] J. Grabowski: “Isomorphisms of Poisson and Jacobi brackets”, In: J. Grabowski and P. Urbański (Eds.): Poisson Geometry, Vol. 51, Banach Center Publications, Warszawa, 2000, pp. 79–85.
  • [17] J. Grabowski: “Isomorphisms of algebras of smooth functions revisited”, Archiv Math., to appear, electronic version at
  • [18] J. Grabowski and N. Poncin: “Automorphisms of quantum and classical Poisson algebras”, Comp. Math. London Math. Soc., Vol. 140, (2004), pp. 511–527 v1
  • [19] H. Grauert: “On Levi's problem and the embedding of real analytic manifolds”, Ann. Math., Vol. 68, (1958), pp. 460–472.
  • [20] H. Hauser and G. Müller: “Affine varieties and Lie algebras of vector fields”, Manusc. Math., Vol. 80, (1993), pp. 309–337.
  • [21] A. Koriyama: “On Lie algebras of vector fields with invariant submanifolds”, Nagoya Math. J., Vol. 55, (1974), pp. 91–110.
  • [22] A. Koriyama, Y. Maeda and H. Omori: “On Lie algebras of vector fields”, Trans. Amer. Math. Soc., Vol. 226, (1977), pp. 89–117.
  • [23] A. Kriegl and P.W. Michor: “The Convenient Setting of Global Analysis”, Math. Surv. Monog., Vol. 53, American Mathematical Society, (1997).
  • [24] P. Lecomte: “On some sequence of graded Lie algebras associated to manifolds”, Ann. Glob. Anal. Geom., Vol. 12, (1994), pp. 183–192.
  • [25] P. Lecomte, P. Mathonet and E. Tousset: “Comparison of some modules of the Lie algebra of vector fields”, Indag. Math., Vol. 7(4), (1996), pp 461–471.
  • [26] P. Lecomte and V. Ovsienko: “Projectively equivariant symbol calculus”, Lett. Math. Phys., Vol. 49, (1999), pp. 173–196.
  • [27] J. Mrčun: On isomorphisms of algebras of smooth functions, electronic version at
  • [28] H. Omori: “Infinite dimensional Lie transformation groups”, Lect. Notes in Math., Vol. 427, (1976).
  • [29] N. Poncin: “Cohomologie de l'algèbre de Lie des opérateurs différentiels sur une variété, à coefficients dans les fonctions”, C.R.A.S. Paris, Vol. 328, Serie I, (1999), pp. 789–794.
  • [30] N. Poncin: “Equivariant Operators between some Modules of the Lie Algebra of Vector Fields”, Comm. in Alg., Vol. 32(7), (2004), pp. 2559–2572.
  • [31] N. Poncin: Equivariant Operators between some Modules of the Lie Algebra of Vector Fields, preprint, Centre Universitaire de Luxembourg, (
  • [32] M.E. Shanks and L.E. Pursell: “The Lie algebra of a smooth manifolds”, Proc. Amer. Math. Soc., Vol. 5, (1954), pp. 468–472.
  • [33] T. Siebert: “Lie algebras of derivations and affine differential geometr over fields of characteristic 0”, Mat. Ann., Vol. 305, (1996), pp. 271–286.
  • [34] S.M. Skryabin: The regular Lie rings of derivations of commutative rings, preprint WINITI 4403-W87 (1987).
  • [35] H. Whitney: “Differentiable manifolds”, Ann. Math., Vol. 37, (1936), pp. 645–680.
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