no. 1
Leibniz cohomology for differentiable manifolds
Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 73-95.

On propose une définition de la cohomologie de Leibniz, HL * , pour les variétés différentiables. Alors HL * devient une version non-commutative de la cohomologie de Gelfand-Fuks. Les calculs de HL * (R n ;R) se réduisent à ceux des champs de vecteurs formels, et peuvent être identifiés avec des invariants de feuilletages.

We propose a definition of Leibniz cohomology, HL * , for differentiable manifolds. Then HL * becomes a non-commutative version of Gelfand-Fuks cohomology. The calculations of HL * (R n ;R) reduce to those of formal vector fields, and can be identified with certain invariants of foliations.

@article{AIF_1998__48_1_73_0,
     author = {Lodder, Jerry M.},
     title = {Leibniz cohomology for differentiable manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {73--95},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
     number = {1},
     year = {1998},
     doi = {10.5802/aif.1611},
     zbl = {0912.17001},
     mrnumber = {99b:17003},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1611/}
}
TY  - JOUR
AU  - Lodder, Jerry M.
TI  - Leibniz cohomology for differentiable manifolds
JO  - Annales de l'Institut Fourier
PY  - 1998
SP  - 73
EP  - 95
VL  - 48
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1611/
DO  - 10.5802/aif.1611
LA  - en
ID  - AIF_1998__48_1_73_0
ER  - 
%0 Journal Article
%A Lodder, Jerry M.
%T Leibniz cohomology for differentiable manifolds
%J Annales de l'Institut Fourier
%D 1998
%P 73-95
%V 48
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1611/
%R 10.5802/aif.1611
%G en
%F AIF_1998__48_1_73_0
Lodder, Jerry M. Leibniz cohomology for differentiable manifolds. Annales de l'Institut Fourier, Tome 48 (1998) no. 1, pp. 73-95. doi : 10.5802/aif.1611. https://aif.centre-mersenne.org/articles/10.5802/aif.1611/

[B] R. Bott, Lectures on Characteristic Classes and Foliations, Lecture Notes in Mathematics, 279 (1972), 1-94. | MR | Zbl

[BS] R. Bott, G. Segal, The Cohomology of the Vector Fields on a Manifold, Topology, 16 (1977), 285-298. | MR | Zbl

[CE] C. Chevalley, S. Eilenberg, Cohomology Theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85-124. | MR | Zbl

[C] A. Connes, Géométrie Non Commutative, Inter Editions, Paris, 1990. | Zbl

[F] D.B. Fuks, Cohomology of Infinite-Dimensional Lie Algebras, Consultants Bureau, 1986 (A.B. Sosinskii translator). | MR | Zbl

[GK] V. Ginzburg, M.M. Kapranov, Koszul Duality for Operads, Duke Jour. Math., 76-1 (1994), 203-272. | MR | Zbl

[Gb] C. Godbillon, Cohomologies d'algèbres de Lie de champs de vecteurs formels, Séminaire Bourbaki, 421 (1972). | Numdam | Zbl

[Gw] T.G. Goodwillie, Relative Algebraic K-Theory and Cyclic Homology, Annals of Math., 124 (1986), 347-402. | MR | Zbl

[H] A. Haefliger, Sur les classes caractéristiques des feuilletages, Séminaire Bourbaki, 412 (1972). | Numdam | Zbl

[HS] G. Hochschild J.-P. Serre, Cohomology of Lie Algebras, Ann. of Math., 57 (1953), 591-603. | MR | Zbl

[K] V. Kač, Vertex Algebras for Beginners, Am. Math. Soc., Univ. Lecture Series, 10 (1996). | Zbl

[K-S] Y. Kosmann-Schwarzbach, From Poisson Algebras to Gerstenhaber Algebra, Ann. Inst. Fourier, Grenoble, 46-5 (1996), 1243-1274. | Numdam | MR | Zbl

[L1] J.-L. Loday, Cyclic Homology, Grund. Math. Wissen. 301, Springer Verlag, 1992. | MR | Zbl

[L2] J.-L. Loday, Une version non commutative des algèbres de Lie: les algèbres de Leibniz, L'Enseignement Math., 39 (1993), 269-293. | MR | Zbl

[L3] J.-L. Loday, Cup product for Leibniz cohomology and dual Leibniz algebras, Math. Scand., 77-2 (1995), 189-196. | MR | Zbl

[L4] J.-L. Loday, La Renaissance des Opérades, Séminaire Bourbaki, 792 (1994-1995). | Numdam | Zbl

[LP] J.-L. Loday, T. Pirashvili, Universal Enveloping Algebras of Leibniz Algebras and (Co)-homology, Math. Annalen, 296 (1993), 139-158. | MR | Zbl

[P] T. Pirashvili, On Leibniz Homology, Ann. Inst. Fourier, Grenoble, 44-2 (1994), 401-411. | Numdam | MR | Zbl

[S] M. Spivak, A Comprehensive Introduction to Differential Geometry, Vol. I, Publish or Perish, Inc. (1979). | Zbl

[W] H. Weyl, The Classical Groups, Princeton University Press, 1946.

Cité par Sources :