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Differential Batalin-Vilkovisky algebras arising from twilled Lie-Rinehart algebras

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Twilled L(ie-)R(inehart)-algebras generalize, in the Lie-Rinehart context, complex structures on smooth manifolds. An almost complex manifold determines an "almost twilled pre-LR algebra", which is a true twilled LR-algebra iff the almost complex structure is integrable. We characterize twilled LR structures in terms of certain associated differential (bi)graded Lie and G(erstenhaber)-algebras; in particular the G-algebra arising from an almost complex structure is a (strict) d(ifferential) G-algebra iff the almost complex structure is integrable. Such G-algebras, endowed with a generator turning them into a B(atalin-)V(ilkovisky)-algebra, occur on the B-side of the mirror conjecture. We generalize a result of Koszul to those dG-algebras which arise from twilled LR-algebras. A special case thereof explains the relationship between holomorphic volume forms and exact generators for the corresponding dG-algebra and thus yields in particular a conceptual proof of the Tian-Todorov lemma. We give a differential homological algebra interpretation for twilled LR-algebras and by means of it we elucidate the notion of a generator in terms of homological duality for differential graded LR-algebras.
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  • Université des Sciences et Technologies de Lille, UFR de Mathématiques, F-59 655 Villeneuve d'Ascq Cedex, France
  • [1] S. Barannikov and M. Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, alg-geom/9710032, Internat. Math. Res. Notices 4 (1998), 201-215.
  • [2] I. A. Batalin and G. S. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev. D 28 (1983) 2567-2582.
  • [3] I. A. Batalin and G. S. Vilkovisky, Closure of the gauge algebra, generalized Lie equations and Feynman rules, Nucl. Phys. B 234 (1984), 106-124.
  • [4] I. A. Batalin and G. S. Vilkovisky, Existence theorem for gauge algebra, Jour. Math. Phys. 26 (1985), 172-184.
  • [5] F. A. Bogomolov, Hamiltonian Kähler varieties, Sov. Math. Dokl. 19 (1978), 1462-1465; translated from: Dokl. Akad. Nauk SSSR 243 (1978), 1101-1104.
  • [6] F. A. Bogomolov, Kähler manifolds with trivial canonical class, preprint, Institut des Hautes Etudes Scientifiques 1981, pp. 1-32.
  • [7] A. Cannas de Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Mathematics Lecture Notes, Volume 10, Amer. Math. Soc. 1999.
  • [8] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85-124.
  • [9] A. Frölicher, Relations between the cohomology groups of Dolbeault and topological invariants, Proc. Nat. Acad. Sci. USA 41 (1955), 641-644.
  • [10] A. Frölicher and A. Nijenhuis, Theory of vector-valued differential forms, Part I: Derivations in the graded ring of differential forms, Proc. Kon. Ned. Akad. Wet. Amsterdam 59 (1956), 338-359.
  • [11] A. Frölicher and A. Nijenhuis, Some new cohomological invariants for complex manifolds, I, Proc. Kon. Ned. Akad. Wet. Amsterdam 59 (1956), 540-564.
  • [12] M. Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math. 78 (1963), 267-288.
  • [13] M. Gerstenhaber and S. D. Schack, Algebras, bialgebras, quantum groups and algebraic deformations, in: Deformation theory and quantum groups with applications to mathematical physics, M. Gerstenhaber and J. Stasheff (eds.), Cont. Math. 134, American Mathematical Society, Providence, (1992), 51-92.
  • [14] E. Getzler, Batalin-Vilkovisky algebras and two-dimensional topological field theories, Comm. in Math. Phys. 195 (1994), 265-285.
  • [15] W. M. Goldman and J. J. Millson, The homotopy invariance of the Kuranishi space, Illinois J. of Math. 34 (1990), 337-367.
  • [16] J. Herz, Pseudo-algèbres de Lie, C. R. Acad. Sci. Paris 236 (1953), 1935-1937.
  • [17] J. Huebschmann, Poisson cohomology and quantization, J. für die Reine und Angew. Math. 408 (1990), 57-113.
  • [18] J.. Huebschmann, Duality for Lie-Rinehart algebras and the modular class, J. für die Reine und Angew. Math. 510 (1999), 103-159.
  • [19] J. Huebschmann, Lie-Rinehart algebras, Gerstenhaber algebras, and Batalin- Vilkovisky algebras, Annales de l'Institut Fourier 48 (1998), 425-440.
  • [20] J. Huebschmann, Extensions of Lie-Rinehart algebras and the Chern-Weil construction, in: Festschrift in honor of J. Stasheff's 60th birthday, Cont. Math. 227 (1999), 145-176, Amer. Math. Soc., Providence R. I.
  • [21] J. Huebschmann, Twilled Lie-Rinehart algebras and differential Batalin-Vilkovisky algebras, math.DG/9811069.
  • [22] J. Huebschmann, The modular class and master equation for Lie-Rinehart bialgebras, in preparation.
  • [23] J. Huebschmann and J. D. Stasheff, Formal solution of the master equation via HPT and deformation theory, math.AG/9906036.
  • [24] K. Kodaira and D. C. Spencer, On deformations of complex analytic structures. I. II., Ann. of Math. 67 (1958), 328-401, 403-466.
  • [25] Y. Kosmann-Schwarzbach, Exact Gerstenhaber algebras and Lie bialgebroids, Acta Applicandae Mathematicae 41 (1995), 153-165.
  • [26] Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Annales de l'Institut Fourier 46 (1996), 1243-1274.
  • [27] Y. Kosmann-Schwarzbach, The Lie bialgebroid of a Poisson-Nijenhuis manifold, Letters in Math. Physics 38 (1996), 421-428.
  • [28] Y. Kosmann-Schwarzbach and F. Magri, Poisson-Lie groups and complete integrability. I. Drinfeld bigebras, dual extensions and their canonical representations, Annales Inst. H. Poincaré Série A (Physique théorique) 49 (1988), 433-460.
  • [29] J. L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie, in: E. Cartan et les Mathématiciens d'aujourd'hui, Lyon, 25-29 Juin, 1984, Astérisque, hors-série, (1985), 251-271.
  • [30] B. H. Lian and G. J. Zuckerman, New perspectives on the BRST-algebraic structure of string theory, Comm. in Math. Phys. 154 (1993), 613-646.
  • [31] J.-H. Lu and A. Weinstein, Poisson Lie groups, dressing transformations, and Bruhat decompositions, J. of Diff. Geom. 31 (1990), 501-526.
  • [32] K. Mackenzie, Double Lie algebroids and the double of a Lie bialgebroid, preprint 1998; math.DG/9808081.
  • [33] K. C. Mackenzie and P. Xu, Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), 415-452.
  • [34] S. Mac Lane, Homology, Die Grundlehren der mathematischen Wissenschaften No. 114, Springer, Berlin, 1963.
  • [35] S. Majid, Matched pairs of Lie groups associated to solutions of the Yang-Baxter equation, Pac. J. of Math. 141 (1990), 311-332.
  • [36] Yu. I. Manin, Three constructions of Frobenius manifolds, Atiyah-Festschrift (to appear), math.QA/9801006.
  • [37] T. Mokri, Matched pairs of Lie algebroids Glasgow Math. J. 39 (1997), 167-181.
  • [38] R. S. Palais, The cohomology of Lie rings, Proc. Symp. Pure Math. III (1961), 130-137.
  • [39] G. Rinehart, Differential forms for general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195-222.
  • [40] V. Schechtman, Remarks on formal deformations and Batalin-Vilkovisky algebras, math. AG/9802006.
  • [41] P. Stachura, Double Lie algebras and Manin triples, q-alg/9712040.
  • [42] J. D. Stasheff, Deformation theory and the Batalin-Vilkovisky master equation, in: Deformation Theory and Symplectic Geometry, Proceedings of the Ascona meeting, June 1996, D. Sternheimer, J. Rawnsley, S. Gutt (eds.), Mathematical Physics Studies Vol. 20, Kluwer Academic Publishers, Dordrecht/Boston/London, 1997, 271-284.
  • [43] G. Tian, A note on Kaehler manifolds with $c_1=0$, preprint.
  • [44] A. N. Todorov, The Weil-Petersson geometry of the moduli space of su(n) (n ≥ 3) (Calabi-Yau) manifolds, I., Comm. Math. Phys. 126 (1989), 325-346.
  • [45] E. Witten, Mirror manifolds and topological field theory, in: Essays on mirror manifolds, S. T. Yau (ed.), International Press Co. Hong Kong, 1992, 230-310.
  • [46] P. Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545-560.
  • [47] S. Zakrzewski, Poisson structures on the Poincaré groups, Comm. Math. Phys. 185 (1997), 285-311.
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