C. Braun, A. Lazarev, “Homotopy BV algebras in Poisson geometry”, Tr. Mosk. Mat. Obs., 74, no. 2, MCCME, M., 2013, 265–277; Trans. Moscow Math. Soc., 74 (2013), 217

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Trudy Moskovskogo Matematicheskogo Obshchestva, 2013, Volume 74, Issue 2, Pages 265–277 (Mi mmo548)

This article is cited in 16 scientific papers (total in 16 papers)

Homotopy BV algebras in Poisson geometry

C. Braun^a, A. Lazarev^b

^a Centre for Mathematical Sciences, City University London, London, UK
^b Departament of Mathematics and Statistics, Lancaster University, Lancaster, UK

Full-text PDF (289 kB) Citations (16)

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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 $\Delta(e^\xi)=e^\xi\left(\Delta(\xi)+\frac12[\xi,\xi]\right)$ 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: 17 entries.

Key words and phrases: $L_\infty$ algebra, BV algebra, Poisson manifold, differential operator.

Received: 15.05.2013

English version:
Transactions of the Moscow Mathematical Society, 2013, Volume 74, Pages 217–227
DOI: https://doi.org/10.1090/s0077-1554-2014-00216-8

Bibliographic databases:

Document Type: Article

UDC: 512.66

MSC: 14D15, 16E45, 53D17

Language: English

Citation: C. Braun, A. Lazarev, “Homotopy BV algebras in Poisson geometry”, Tr. Mosk. Mat. Obs., 74, no. 2, MCCME, M., 2013, 265–277; Trans. Moscow Math. Soc., 74 (2013), 217–227

Citation in format AMSBIB

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Linking options:

https://www.mathnet.ru/eng/mmo548

https://www.mathnet.ru/eng/mmo/v74/i2/p265

This publication is cited in the following 16 articles:

Joana Cirici, Scott O. Wilson, “Homotopy BV-algebras in Hermitian geometry”, Journal of Geometry and Physics, 2024, 105275
Kai Behrend, Matt Peddie, Ping Xu, “Quantization of ($-1$)-Shifted Derived Poisson Manifolds”, Commun. Math. Phys., 402:3 (2023), 2301
Jonathan P Pridham, “Quantisation of derived Lagrangians”, Geom. Topol., 26:6 (2022), 2405
Lawrence R., Ranade N., Sullivan D., “Quantitative Towers in Finite Difference Calculus Approximating the Continuum”, Q. J. Math., 72:1-2 (2021), 515–545
Gwilliam O., Williams B.R., “Higher Kac-Moody Algebras and Symmetries of Holomorphic Field Theories”, Adv. Theor. Math. Phys., 25:1 (2021), 129–239
Voronov A.A., “Quantizing Deformation Theory II”, Pure Appl. Math. Q., 16:1, 3, SI (2020), 125–152
Ch. Braun, J. Maunder, “Minimal models of quantum homotopy Lie algebras via the BV-formalism”, J. Math. Phys., 59:6 (2018), 063512
O. Gwilliam, R. Haugseng, “Linear Batalin–Vilkovisky quantization as a functor of $\infty$-categories”, Sel. Math.-New Ser., 24:2 (2018), 1247–1313
N. Kowalzig, “When Ext is a Batalin–Vilkovisky algebra”, J. Noncommutative Geom., 12:3 (2018), 1080–1130
M. Markl, A. A. Voronov, “The MV formalism for IBL$_\infty$- and BV$_\infty$-algebras”, Lett. Math. Phys., 107:8 (2017), 1515–1543
D. Bashkirov, A. A. Voronov, “The BV formalism for $L_\infty$-algebras”, J. Homotopy Relat. Struct., 12:2 (2017), 305–327
A. J. Bruce, A. G. Tortorella, “Kirillov structures up to homotopy”, Differ. Geom. Appl., 48 (2016), 72–86
Kowalzig N., “Batalin-Vilkovisky Algebra Structures on (Co)Tor and Poisson Bialgebroids”, J. Pure Appl. Algebr., 219:9 (2015), 3781–3822
Iacono D., “Deformations and Obstructions of Pairs (X, D)”, Int. Math. Res. Notices, 2015, no. 19, 9660–9695
Kowalzig N., “Gerstenhaber and Batalin-Vilkovisky Structures on Modules Over Operads”, Int. Math. Res. Notices, 2015, no. 22, 11694–11744
Vitagliano L., “Representations of Homotopy Lie-Rinehart Algebras”, Math. Proc. Camb. Philos. Soc., 158:1 (2015), 155–191

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Trudy Moskovskogo Matematicheskogo Obshchestva

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