Homotopy BV algebras in Poisson geometry
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- by C. Braun and A. Lazarev
- Trans. Moscow Math. Soc. 2013, 217-227
- DOI: https://doi.org/10.1090/S0077-1554-2014-00216-8
- Published electronically: April 9, 2014
Abstract:
We define and study the degeneration property for $\mathrm {BV}_\infty$ algebras and show that it implies that the underlying $L_{\infty }$ algebras are homotopy abelian. The proof is based on a generalisation of the well-known identity $\Delta (e^{\xi })=e^{\xi }\Big (\Delta (\xi )+\frac {1}{2}[\xi ,\xi ]\Big )$ which holds in all BV algebras. As an application we show that the higher Koszul brackets on the cohomology of a manifold supplied with a generalised Poisson structure all vanish.References
- Sergey Barannikov and Maxim Kontsevich, Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices 4 (1998), 201–215. MR 1609624, DOI 10.1155/S1073792898000166
- Bruce A. J., On higher Poisson and Koszul–Schouten brackets. arXiv:0910.1992.
- Joseph Chuang and Andrey Lazarev, $L$-infinity maps and twistings, Homology Homotopy Appl. 13 (2011), no. 2, 175–195. MR 2854334, DOI 10.4310/HHA.2011.v13.n2.a12
- Dotsenko V., Shadrin S., Vallette B., De Rham cohomology and homotopy Frobenius manifolds. arXiv:1203.5077.
- Marisa Fernández, Raúl Ibáñez, and Manuel de León, The canonical spectral sequences for Poisson manifolds, Israel J. Math. 106 (1998), 133–155. MR 1656861, DOI 10.1007/BF02773464
- Domenico Fiorenza and Marco Manetti, Formality of Koszul brackets and deformations of holomorphic Poisson manifolds, Homology Homotopy Appl. 14 (2012), no. 2, 63–75. MR 3007085, DOI 10.4310/HHA.2012.v14.n2.a4
- Imma Gálvez-Carrillo, Andrew Tonks, and Bruno Vallette, Homotopy Batalin-Vilkovisky algebras, J. Noncommut. Geom. 6 (2012), no. 3, 539–602. MR 2956319, DOI 10.4171/JNCG/99
- L. Katzarkov, M. Kontsevich, and T. Pantev, Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt*-geometry, Proc. Sympos. Pure Math., vol. 78, Amer. Math. Soc., Providence, RI, 2008, pp. 87–174. MR 2483750, DOI 10.1090/pspum/078/2483750
- Maxim Kontsevich, Deformation quantization of Poisson manifolds, Lett. Math. Phys. 66 (2003), no. 3, 157–216. MR 2062626, DOI 10.1023/B:MATH.0000027508.00421.bf
- Olga Kravchenko, Deformations of Batalin-Vilkovisky algebras, Poisson geometry (Warsaw, 1998) Banach Center Publ., vol. 51, Polish Acad. Sci. Inst. Math., Warsaw, 2000, pp. 131–139. MR 1764440
- H. M. Khudaverdian and Th. Th. Voronov, Higher Poisson brackets and differential forms, Geometric methods in physics, AIP Conf. Proc., vol. 1079, Amer. Inst. Phys., Melville, NY, 2008, pp. 203–215. MR 2757715
- Jean-Louis Loday, Cyclic homology, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301, Springer-Verlag, Berlin, 1992. Appendix E by María O. Ronco. MR 1217970, DOI 10.1007/978-3-662-21739-9
- S. A. Merkulov, Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices 14 (1998), 727–733. MR 1637093, DOI 10.1155/S1073792898000439
- G. Sharygin and D. Talalaev, On the Lie-formality of Poisson manifolds, J. K-Theory 2 (2008), no. 2, Special issue in memory of Yurii Petrovich Solovyev., 361–384. MR 2456106, DOI 10.1017/is008001011jkt030
- John Terilla, Smoothness theorem for differential BV algebras, J. Topol. 1 (2008), no. 3, 693–702. MR 2417450, DOI 10.1112/jtopol/jtn019
- Theodore Voronov, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra 202 (2005), no. 1-3, 133–153. MR 2163405, DOI 10.1016/j.jpaa.2005.01.010
- Theodore Th. Voronov, Higher derived brackets for arbitrary derivations, Travaux mathématiques. Fasc. XVI, Trav. Math., vol. 16, Univ. Luxemb., Luxembourg, 2005, pp. 163–186. MR 2223157
Bibliographic Information
- C. Braun
- Affiliation: Centre for Mathematical Sciences, City University London, London, United Kingdom
- Email: Christopher.Braun.1@city.ac.uk
- A. Lazarev
- Affiliation: Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom
- Email: a.lazarev@lancaster.ac.uk
- Published electronically: April 9, 2014
- Additional Notes: This work was partially supported by EPSRC grants EP/J00877X/1 and EP/J008451/1.
The authors would like to thank the Isaac Newton Institute for their hospitality during this work
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- © Copyright 2014 C. Braun, A. Lazarev
- Journal: Trans. Moscow Math. Soc. 2013, 217-227
- MSC (2010): Primary 14D15, 16E45, 53D17
- DOI: https://doi.org/10.1090/S0077-1554-2014-00216-8
- MathSciNet review: 3235797