×

zbMATH — the first resource for mathematics

Exact Gerstenhaber algebras and Lie bialgebroids. (English) Zbl 0837.17014
This work stresses the growing meaning of the theory of Lie algebroids in the sense of J. Pradines [C. R. Acad. Sci., Paris, Sér. A 264, 245-248 (1967; Zbl 0154.21704)] and Lie bialgebroids in the sense of K. Mackenzie and Ping Xu [Duke Math. J. 73, 415-452 (1994)]. The problems considered are important, for example, in string theory which, lately, makes extensive use of these algebraic structures. Lie algebroids appear in many domains of differential geometry. The theory of Poisson manifolds is one of them; it is important from the point of view of some applications to theoretical physics.
This work concerns triangular Lie bialgebroids generalizing both the Lie bialgebroids of Poisson manifolds and the triangular Lie bialgebras. To any such bialgebroid the author assigns two differential Gerstenhaber algebras in duality, one of which is canonically equipped with an operator generating the graded Lie algebra bracket, i.e. with the structure of a Batalin-Vilkovsky algebra.

MSC:
17B70 Graded Lie (super)algebras
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B81 Applications of Lie (super)algebras to physics, etc.
17B66 Lie algebras of vector fields and related (super) algebras
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bhaskara, K. H. and Viswanath, K.: Calculus on Poisson manifolds,Bull. London Math. Soc. 20 (1988), 68-72. · Zbl 0611.58002
[2] Coste, A., Dazord, P., and Weinstein, A.: Groupoïdes symplectiques,Publ. Dép. Math. Univ. Lyon I, 2A (1987). · Zbl 0668.58017
[3] Drinfeld, V. G.: Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equation,Soviet. Math. Dokl. 27(1) (1983), 68-71. (In Russian,Dokl. Akad. Nauk SSSR 268(2) (1983).) · Zbl 0526.58017
[4] Drinfeld, V. G.: Quantum groups,Proc. Internat. Congr. Math. (Berkeley, 1986), Vol. 1, Amer. Math. Soc., Providence, 1987, pp. 798-820.
[5] Gelfand, I. M. and Dorfman, I. Ya.: Hamiltonian operators and the classical Yang-Baxter equation,Funct. Anal. Appl. 16(4) (1982), 241-248. · Zbl 0527.58018
[6] Gerstenhaber, M.: The cohomology structure of an associative ring,Ann. Math. 78 (1963), 267-288. · Zbl 0131.27302
[7] Gerstenhaber, M. and Schack, S. D.: Algebraic cohomology and deformation theory, in M. Hazewinkel and M. Gerstenhaber (eds),Deformation Theory of Algebras and Structures and Applications, Kluwer, Dordrecht, 1988, pp. 11-264. · Zbl 0676.16022
[8] Gerstenhaber, M. and Schack, S. D.: Algebras, bialgebras, quantum groups and algebraic deformations, in M. Gerstenhaber and J. Stasheff (eds),Deformation Theory and Quantum Groups with Applications to Mathematical Physics, Contemporary Mathematics 134, Amer. Math. Soc., Providence, 1992, pp. 51-92. · Zbl 0788.17009
[9] Getzler, E.: Batalin-Vilkovisky algebras and two-dimensional topological field theories,Comm. Math. Phys. 159 (1994), 265-285. · Zbl 0807.17026
[10] Huebschmann, J.: Poisson cohomology and quantization,J. Reine Angew. Math. 408 (1990), 57-113. · Zbl 0699.53037
[11] Karasev, M. V.: Analogues of the objects of Lie group theory for nonlinear Poisson brackets,Math. USSR Izv. 28(3) (1987), 497-527. (In Russian,Izvestyia 50 (1986).) · Zbl 0624.58007
[12] Kosmann-Schwarzbach, Y.: Jacobian quasi-bialgebras and quasi-Poisson Lie groups, in M. Gotay, J. E. Marsden, and V. Moncrief (eds),Mathematical Aspects of Classical Field Theory, Contemporary Mathematics 132, Amer. Math. Soc., Providence, 1992, pp. 459-489. · Zbl 0847.17020
[13] Kosmann-Schwarzbach, Y. and Magri, F.: Poisson-Nijenhuis structures,Ann. Inst. Henri Poincaré A 53(1) (1990), 35-81. · Zbl 0707.58048
[14] Kosmann-Schwarzbach, Y., and Magri, F.: Dualization and deformation of Lie brackets on Poisson manifolds, in J. Janyska and D. Krupka (eds),Differential Geometry and its Applications (Brno, 1989), World Scientific, Singapore, 1990, pp. 79-84. · Zbl 0798.53035
[15] Koszul, J.-L.: Crochet de Schouten-Nijennuis et cohomologie, in’Elie Carton et les mathématiques d’aujourd’hui’, Astérisque, n?hors série, Soc. Math. Fr., 1985, pp. 257-271.
[16] Krasilshchik, I.: Schouten brackets and canonical algebras, inGlobal Analysis III, Lecture Notes Math. 1334, Springer-Verlag, Berlin, 1988, pp. 79-110.
[17] Krasilshchik, I.: Supercanonical algebras and Schouten brackets,Mat. Zametki 49 (1991), 70-76. · Zbl 0723.58020
[18] Lian, B. H. and Zuckerman, G. J.: New perspectives on the BRST-algebraic structure of string theory,Comm. Math. Phys. 154 (1993), 613-646. · Zbl 0780.17029
[19] Mackenzie, K.:Lie Groupoids and Lie Algebroids in Differential Geometry, London Math. Soc. Lect. Notes Series 124, Cambridge University Press, Cambridge, 1987. · Zbl 0683.53029
[20] Mackenzie, K. C. H. and Ping Xu: Lie bialgebroids and Poisson groupoids,Duke Math. J. 73 (1994), 415-452. · Zbl 0844.22005
[21] Magri, F. and Morosi, C.: A geometrical characterization of integrable Hamiltonian systems through the theory of Poisson-Nijenhuis manifolds, Quaderno S 19 (1984), University of Milan.
[22] Palais, R. S.:The Cohomology of Lie Rings, Proc. Symp. Pure Math. 3, Amer. Math. Soc., Providence, 1961, pp. 130-137. · Zbl 0126.03404
[23] Penkava, M. and Schwarz, A.: On some algebraic structures arising in string theory, Preprint hep-tn/912071. · Zbl 0871.17021
[24] Pradines, J.: Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux,C.R. Acad. Sci. Paris, Série A 264 (1967), 245-248.
[25] Roger, C.: Algèbres de Lie graduées et quantification, in P. Donatoet al. (eds),Symplectic Geometry and Mathematical Physics, Progress in Mathematics 99, Birkhäuser, Boston, 1991, pp. 374-421. · Zbl 0748.17028
[26] Vaisman, I.:Lectures on the Geometry of Poisson Manifolds, Progress in Mathematics 118, Birkhäuser, Boston, 1994. · Zbl 0810.53019
[27] Vaisman, I.: Poisson-Nijenhuis structures revisited,Rendiconti Sem. Mat. Torino 52 (1994). · Zbl 0852.58042
[28] Weinstein, A.: Some remarks on dressing transformations,J. Fac. Sci. Univ. Tokyo, IA, Math. 35 (1988), 163-167. · Zbl 0653.58012
[29] Zwiebach, B.: Closed string theory: an introduction, Preprint hep-th/9305026. · Zbl 0856.53057
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.