On montre que l’on peut construire un crochet de Poisson pair à partir d’un crochet de Gerstenhaber, à l’aide d’une dérivation impaire de carré nul, dans la catégorie des algèbres de Loday (algèbres munies d’un crochet non antisymétrique, généralisant les crochets de Lie, appelées jusqu’à présent dans la littérature, algèbres de Leibniz). Ces “crochets dérivés” donnent des crochets de Lie sur certains quotients, et sur certaines sous-algèbres abéliennes. On peut expliquer ainsi l’origine du crochet de Lie sur l’espace des formes différentielles co-exactes sur une variété de Poisson. Nous étudions les crochets dérivés sur l’espace des cochaînes sur une algèbre associative ou de Lie. Enfin nous relions les résultats précédents à diverses généralisations de la notion d’algèbre de Batalin-Vilkovisky.
Constructing an even Poisson algebra from a Gerstenhaber algebra by means of an odd derivation of square 0 is shown to be possible in the category of Loday algebras (algebras with a non-skew-symmetric bracket, generalizing the Lie algebras, heretofore called Leibniz algebras in the literature). Such “derived brackets” give rise to Lie brackets on certain quotient spaces, and also on certain Abelian subalgebras. The construction of these derived brackets explains the origin of the Lie bracket on the space of co-exact differential forms on a Poisson manifold. We further examine the derived brackets on the space of cochains of an associative or Lie algebra. Finally, we relate the previous result to various generalizations of the notion of BV-algebra.
@article{AIF_1996__46_5_1243_0, author = {Kosmann-Schwarzbach, Yvette}, title = {From {Poisson} algebras to {Gerstenhaber} algebras}, journal = {Annales de l'Institut Fourier}, pages = {1243--1274}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {46}, number = {5}, year = {1996}, doi = {10.5802/aif.1547}, zbl = {0858.17027}, mrnumber = {98b:17032}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1547/} }
TY - JOUR AU - Kosmann-Schwarzbach, Yvette TI - From Poisson algebras to Gerstenhaber algebras JO - Annales de l'Institut Fourier PY - 1996 SP - 1243 EP - 1274 VL - 46 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1547/ DO - 10.5802/aif.1547 LA - en ID - AIF_1996__46_5_1243_0 ER -
%0 Journal Article %A Kosmann-Schwarzbach, Yvette %T From Poisson algebras to Gerstenhaber algebras %J Annales de l'Institut Fourier %D 1996 %P 1243-1274 %V 46 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1547/ %R 10.5802/aif.1547 %G en %F AIF_1996__46_5_1243_0
Kosmann-Schwarzbach, Yvette. From Poisson algebras to Gerstenhaber algebras. Annales de l'Institut Fourier, Tome 46 (1996) no. 5, pp. 1243-1274. doi : 10.5802/aif.1547. https://aif.centre-mersenne.org/articles/10.5802/aif.1547/
[A] On some generalizations of Batalin-Vilkovisky algebras, J. Pure Appl. Alg., to appear. | Zbl
,[BM] Graded Poisson structures on the algebra of differential forms, Comment. Math. Helvetici, 70 (1995), 383-402. | MR | Zbl
AND ,[BMP] The W3 algebra : modules, semi-infinite cohomology and BV-algebras, preprint hep-th/9509119. | Zbl
, and ,[BP] The BV-algebra structure of W3 cohomology, Lect. Notes Phys. 447, G. Aktas, C. Sadioglu, M. Serdaroglu, eds. (1995), pp. 283-291. | MR | Zbl
and ,[B] Théorie des opérateurs différentiels gradués sur les formes différentielles, Bull. Soc. Math. Fr., 102 (1) (1974), 49-73. | Numdam | MR | Zbl
,[CV] Extensions of the Poisson bracket to differential forms and multi-vector fields, J. Geom. Phys., 9 (1992), 75-100. | MR | Zbl
and ,[CNS] Graded Lie algebras in mathematics and physics, Rev. Mod. Phys., 47 (1975), 573-603. | Zbl
, and ,[DK1] Poisson and Nijenhuis brackets for differential forms on non-commutative manifolds, Preprint 698/10/95, Univ. Bielefeld.
and ,[DK2] Formal differential geometry and Nambu-Takhtajan algebra, Proc. Conf. “Quantum groups and quantum spaces”, Banach Center Publ., to appear.
and ,[DT] Hamiltonian operators and Hochschild homologies, Funct. Anal. Appl., 19 (4) (1985), 319-321. | Zbl
and ,[dWL] Formal deformations of the Poisson-Lie algebra of a symplectic manifold and star-products, in Deformation Theory of Algebras and Structures and Applications, M. Gerstenhaber and M. Hazewinkel, eds., 897-960, Kluwer, 1988. | MR | Zbl
and ,[D] Quantum Groups, Proc. Int. Congress Math. Berkeley, Amer. Math. Soc., (1987), 798-820. | MR | Zbl
,[D-VM] The Frölicher-Nijenhuis bracket for derivation based non commutative differential forms, J. Pure Appl. Alg., to appear.
and ,[FGV] Cohomology and deformation of Leibniz pairs, Lett. Math. Phys., 34 (1995), 77-90. | MR | Zbl
, and ,[FN] Theory of vector-valued differential forms. Part I, Indag. Math., 18 (1956), 338-359. | Zbl
and ,[GDT] On a variant of noncommutative differential geometry, Soviet Math. Dokl., 40 (2) (1989), 422-426. | Zbl
, and ,[G] The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267-288. | MR | Zbl
,[GS1] Algebraic cohomology and deformation theory, in Deformation Theory of Algebras and Structures and Applications, M. Gerstenhaber and M. Hazewinkel, eds., ASI C247, 11-264, Kluwer, 1988. | MR | Zbl
and ,[GS2] Algebras, bialgebras, quantum groups, and algebraic deformations, Contemporary Math., 134 (1992), 51-92. | MR | Zbl
and ,[Gt] Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys., 159 (1994), 265-285. | MR | Zbl
,[KiSV] On operad structures of moduli spaces and string theory, Comm. Math. Phys., 171 (1995), 1-25. | MR | Zbl
, and ,[K-S1] Jacobian quasi-bialgebras and quasi-Poisson Lie groups, Contemporary Math., 132 (1992), 459-489. | MR | Zbl
,[K-S2] Exact Gerstenhaber algebras and Lie bialgebroids, Acta Applicandae Math., 41 (1995), 153-165. | MR | Zbl
,[K-SM] Poisson-Nijenhuis structures, Ann. Inst. Henri Poincaré, Phys. Théor., 53 (1) (1990), 35-81. | EuDML | Numdam | MR | Zbl
and ,[Kt] Graded manifolds, graded Lie theory and prequantization, Lect. Notes Math., 570 (1977), 177-30. | MR | Zbl
,[KtS] Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Phys., 176 (1987), 49-113. | MR | Zbl
and ,[K] Crochet de Schouten-Nijenhuis et cohomologie, in Elie Cartan et les mathématiques d'aujourd'hui, Astérisque, hors série (1985), 257-271. | MR | Zbl
,[Ko]
, unpublished notes (1990).[Kr1] Schouten bracket and canonical algebras, in Global Analysis, Lect. Notes Math., 1334 (1988), 79-110. | MR | Zbl
,[Kr2] Supercanonical algebras and Schouten brackets, Mathematical Notes, 49 (1) (1991), 70-76. | MR | Zbl
,[LMS] The multigraded Nijenhuis-Richardson algebra, its universal property and applications, J. Pure Appl. Alg., 77 (1992), 87-102. | MR | Zbl
, and ,[LR] Modules et cohomologie des bigèbres de Lie, C.R. Acad. Sci. Paris, Série I, 310 (1990), 405-410. | MR | Zbl
et ,[L1] Cyclic homology, Grund. Math. Wiss. 301, Springer-Verlag, 1992. | MR | Zbl
,[L2] Une version non commutative des algèbres de Lie: les algèbres de Leibniz, L'Enseignement Mathématique, 39 (1993), 269-293. | MR | Zbl
,[LZ] New perspectives on the BRST-algebraic structure in string theory, Comm. Math. Phys., 154 (1993), 613-646. | MR | Zbl
and ,[M] A generalization of Hamiltonian mechanics, J. Geom. Phys., 2 (2) (1985), 67-82. | MR | Zbl
,[N] The graded Lie algebras of an algebra. Indag. Math., 29 (1967), 475-486. | MR | Zbl
,[NR] Deformations of Lie algebra structures, J. Math. Mech., 171 (1967), 89-106. | MR | Zbl
and ,[PS] On some algebraic structures arising in string theory, in Perspectives in Mathematical Physics, vol. 3, R. Penner and S.T. Yau, eds., International Press, 1994. | MR | Zbl
and ,[R] Algèbres de Lie graduées et quantification, in Symplectic Geometry and Mathematical Physics, P. Donato et al., eds., Progress in Math. 99, Birkhäuser, (1991), pp. 374-421. | MR | Zbl
,[Va] Lectures on the Geometry of Poisson Manifolds, Progress in Math. 118, Birkhäuser (1994). | MR | Zbl
,[V] Unification of Schouten-Nijenhuis and Frölicher-Nijenhuis brackets, cohomology and super-differential operators, Mat. Zametki, 47 (6), 138-140 (1990). | MR | Zbl
,[W] A note on the antibracket formalism, Modern Phys. Lett. A, 5 (7) (1990), 487-494. | MR | Zbl
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