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Transactions of the Moscow Mathematical Society

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Homotopy BV algebras in Poisson geometry

Authors: C. Braun and A. Lazarev
Original publication: Trudy Moskovskogo Matematicheskogo Obshchestva, tom 74 (2013), vypusk 2.
Journal: Trans. Moscow Math. Soc. 2013, 217-227
MSC (2010): Primary 14D15, 16E45, 53D17
Published electronically: April 9, 2014
MathSciNet review: 3235797
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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.

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  • 1. Barannikov S., Kontsevich M., Frobenius manifolds and formality of Lie algebras of polyvector fields, Internat. Math. Res. Notices. 1998. Vol.4. P.201-215. arXiv:alg-geom/9710032, doi:10.1155/S1073792898000166. MR 1609624 (99b:14009)
  • 2. Bruce A.J., On higher Poisson and Koszul-Schouten brackets. arXiv:0910.1992.
  • 3. Chuang J., Lazarev A., $ L$-infinity maps and twistings, Homology Homotopy Appl. 2011. Vol.13, No.2. P.175-195. arXiv:0912.1215, doi:10.4310/HHA.2011.v13.n2.a12. MR 2854334
  • 4. Dotsenko V., Shadrin S., Vallette B., De Rham cohomology and homotopy Frobenius manifolds. arXiv:1203.5077.
  • 5. Fernández M., Ibáñez R., de. León M., The canonical spectral sequences for Poisson manifolds, Israel J. Math. 1998. Vol.106. P.133-155. doi:10.1007/BF02773464. MR 1656861 (99k:58067)
  • 6. Fiorenza D., Manetti M., Formality of Koszul brackets and deformations of holomorphic Poisson manifolds, Homology Homotopy Appl. 2012. Vol.14, No.2. P.63-75. arXiv:1109.4309, doi:10.4310/HHA.2012.v14.n2.a4. MR 3007085
  • 7. Gálvez-Carrillo I., Tonks A., Vallette B., Homotopy Batalin-Vilkovisky algebras, J. Noncommut. Geom. 2012. Vol.6, No.3. P.539-602. arXiv:0907.2246, doi:10.4171/JNCG/99. MR 2956319
  • 8. Katzarkov L., Kontsevich M., and Pantev T., Hodge theoretic aspects of mirror symmetry, From Hodge theory to integrability and TQFT tt*-geometry. Vol.78 of Proc. Sympos. Pure Math. AMS, Providence, RI, 2008. P.87-174. arXiv:0806.0107. MR 2483750 (2009j:14052)
  • 9. Kontsevich M., Deformation quantization of Poisson manifolds, Lett. Math. Phys. 2003. Vol.66, No.3. P.157-216. arXiv:q-alg/9709040v1, doi:10.1023/ MR 2062626 (2005i:53122)
  • 10. Kravchenko O., Deformations of Batalin-Vilkovisky algebras, Poisson geometry (Warsaw, 1998). Vol.51 of Banach Center Publ. Warsaw: Polish Acad. Sci., 2000. P.131-139. arXiv:math/9903191. MR 1764440 (2001e:17036)
  • 11. Khudaverdian H.M., Voronov Th.Th., Higher Poisson brackets and differential forms, Geometric methods in physics. Vol.1079 of AIP Conf. Proc. Melville, New York: Amer. Inst. Phys., 2008. P.203-215. arXiv:0808.3406, doi:10.1063/1.3043861. MR 2757715 (2012e:53163)
  • 12. Loday J.-L., Cyclic homology. Vol.301 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Berlin: Springer-Verlag, 1992. Appendix E by María O.Ronco. MR 1217970 (94a:19004)
  • 13. Merkulov S.A., Formality of canonical symplectic complexes and Frobenius manifolds, Internat. Math. Res. Notices. 1998. No.14. P.727-733. arXiv:math/9805072, doi:10.1155/S1073792898000439. MR 1637093 (99j:58078)
  • 14. Sharygin G. and Talalaev D., On the Lie-formality of Poisson manifolds, J. $ K$-Theory. 2008. Vol.2 (Special issue in memory of Yurii Petrovich Solovyev. Part 1), No.2. P.361-384. arXiv:math/
    0503635, doi:10.1017/is008001011jkt030. MR 2456106 (2009i:17032)
  • 15. Terilla J., Smoothness theorem for differential BV algebras, J. Topol. 2008. Vol.1, No.3. P.693-702. arXiv:0707.1290, doi:10.1112/jtopol/jtn019. MR 2417450 (2009j:17014)
  • 16. Voronov Th., Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra. 2005. Vol.202, No.1-3. P.133-153. arXiv:math/0304038, doi:10.1016/j.jpaa.2005.01.010. MR 2163405 (2006e:17028)
  • 17. Voronov Th., Higher derived brackets for arbitrary derivations, Travaux mathématiques. Fasc. XVI, Trav. Math., XVI. Univ. Luxemb., Luxembourg, 2005. P.163-186. arXiv:math/0412202. MR 2223157 (2007d:17024)

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Additional Information

C. Braun
Affiliation: Centre for Mathematical Sciences, City University London, London, United Kingdom

A. Lazarev
Affiliation: Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom

Keywords: $L_{\infty}$ algebra, BV algebra, Poisson manifold, differential operator
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.
Article copyright: © Copyright 2014 C. Braun, A. Lazarev

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