From Poisson algebras to Gerstenhaber algebras
Annales de l'Institut Fourier, Volume 46 (1996) no. 5, pp. 1243-1274.

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.

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.

     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 = {}
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  -
UR  -
UR  -
UR  -
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
%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, Volume 46 (1996) no. 5, pp. 1243-1274. doi : 10.5802/aif.1547.

[A] F. Akman, On some generalizations of Batalin-Vilkovisky algebras, J. Pure Appl. Alg., to appear. | Zbl

[BM] J.V. Beltrán AND J. Monterde, Graded Poisson structures on the algebra of differential forms, Comment. Math. Helvetici, 70 (1995), 383-402. | MR | Zbl

[BMP] P. Bouwknegt, J. Mccarthy and K. Pilch, The W3 algebra : modules, semi-infinite cohomology and BV-algebras, preprint hep-th/9509119. | Zbl

[BP] P. Bouwknegt and K. Pilch, The BV-algebra structure of W3 cohomology, Lect. Notes Phys. 447, G. Aktas, C. Sadioglu, M. Serdaroglu, eds. (1995), pp. 283-291. | MR | Zbl

[B] C. Buttin, Théorie des opérateurs différentiels gradués sur les formes différentielles, Bull. Soc. Math. Fr., 102 (1) (1974), 49-73. | Numdam | MR | Zbl

[CV] A. Cabras and A.M. Vinogradov, Extensions of the Poisson bracket to differential forms and multi-vector fields, J. Geom. Phys., 9 (1992), 75-100. | MR | Zbl

[CNS] L. Corwin, Yu. Neeman and S. Sternberg, Graded Lie algebras in mathematics and physics, Rev. Mod. Phys., 47 (1975), 573-603. | Zbl

[DK1] Yu.L. Daletskii and V.A. Kushnirevitch, Poisson and Nijenhuis brackets for differential forms on non-commutative manifolds, Preprint 698/10/95, Univ. Bielefeld.

[DK2] Yu.L. Daletskii and V.A. Kushnirevitch, Formal differential geometry and Nambu-Takhtajan algebra, Proc. Conf. “Quantum groups and quantum spaces”, Banach Center Publ., to appear.

[DT] Yu.L. Daletsky and B.L. Tsygan, Hamiltonian operators and Hochschild homologies, Funct. Anal. Appl., 19 (4) (1985), 319-321. | Zbl

[dWL] M. De Wilde and P. Lecomte, 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

[D] V.G. Drinfeld, Quantum Groups, Proc. Int. Congress Math. Berkeley, Amer. Math. Soc., (1987), 798-820. | MR | Zbl

[D-VM] M. Dubois-Violette and P. Michor, The Frölicher-Nijenhuis bracket for derivation based non commutative differential forms, J. Pure Appl. Alg., to appear.

[FGV] M. Flato, M. Gerstenhaber and A. Voronov, Cohomology and deformation of Leibniz pairs, Lett. Math. Phys., 34 (1995), 77-90. | MR | Zbl

[FN] A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms. Part I, Indag. Math., 18 (1956), 338-359. | Zbl

[GDT] I.M. Gelfand, Yu.L. Daletskii and B.L. Tsygan, On a variant of noncommutative differential geometry, Soviet Math. Dokl., 40 (2) (1989), 422-426. | Zbl

[G] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. Math., 78 (1963), 267-288. | MR | Zbl

[GS1] M. Gerstenhaber and S.D. Schack, 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

[GS2] M. Gerstenhaber and S.D. Schack, Algebras, bialgebras, quantum groups, and algebraic deformations, Contemporary Math., 134 (1992), 51-92. | MR | Zbl

[Gt] E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. Math. Phys., 159 (1994), 265-285. | MR | Zbl

[KiSV] T. Kimura, J. Stasheff and A.A. Voronov, On operad structures of moduli spaces and string theory, Comm. Math. Phys., 171 (1995), 1-25. | MR | Zbl

[K-S1] Y. Kosmann-Schwarzbach, Jacobian quasi-bialgebras and quasi-Poisson Lie groups, Contemporary Math., 132 (1992), 459-489. | MR | Zbl

[K-S2] Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Applicandae Math., 41 (1995), 153-165. | MR | Zbl

[K-SM] Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Ann. Inst. Henri Poincaré, Phys. Théor., 53 (1) (1990), 35-81. | EuDML | Numdam | MR | Zbl

[Kt] B. Kostant, Graded manifolds, graded Lie theory and prequantization, Lect. Notes Math., 570 (1977), 177-30. | MR | Zbl

[KtS] B. Kostant and S. Sternberg, Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras, Ann. Phys., 176 (1987), 49-113. | MR | Zbl

[K] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in Elie Cartan et les mathématiques d'aujourd'hui, Astérisque, hors série (1985), 257-271. | MR | Zbl

[Ko] J.-L. Koszul, unpublished notes (1990).

[Kr1] I. Krasilshchik, Schouten bracket and canonical algebras, in Global Analysis, Lect. Notes Math., 1334 (1988), 79-110. | MR | Zbl

[Kr2] I. Krasilshchik, Supercanonical algebras and Schouten brackets, Mathematical Notes, 49 (1) (1991), 70-76. | MR | Zbl

[LMS] P. Lecomte, P.W. Michor and H. Schicketanz, The multigraded Nijenhuis-Richardson algebra, its universal property and applications, J. Pure Appl. Alg., 77 (1992), 87-102. | MR | Zbl

[LR] P.B.A. Lecomte et C. Roger, Modules et cohomologie des bigèbres de Lie, C.R. Acad. Sci. Paris, Série I, 310 (1990), 405-410. | MR | Zbl

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

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

[LZ] B.H. Lian and G.J. Zuckerman, New perspectives on the BRST-algebraic structure in string theory, Comm. Math. Phys., 154 (1993), 613-646. | MR | Zbl

[M] P.W. Michor, A generalization of Hamiltonian mechanics, J. Geom. Phys., 2 (2) (1985), 67-82. | MR | Zbl

[N] A. Nijenhuis, The graded Lie algebras of an algebra. Indag. Math., 29 (1967), 475-486. | MR | Zbl

[NR] A. Nijenhuis and R. Richardson, Deformations of Lie algebra structures, J. Math. Mech., 171 (1967), 89-106. | MR | Zbl

[PS] M. Penkava and A. Schwarz, 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

[R] C. Roger, 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] I. Vaisman, Lectures on the Geometry of Poisson Manifolds, Progress in Math. 118, Birkhäuser (1994). | MR | Zbl

[V] A.M. Vinogradov, Unification of Schouten-Nijenhuis and Frölicher-Nijenhuis brackets, cohomology and super-differential operators, Mat. Zametki, 47 (6), 138-140 (1990). | MR | Zbl

[W] E. Witten, A note on the antibracket formalism, Modern Phys. Lett. A, 5 (7) (1990), 487-494. | MR | Zbl

Cited by Sources: