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Braces and Poisson additivity

Part of: Categories with structure Homological algebra

Published online by Cambridge University Press:  18 July 2018

Pavel Safronov
Pavel Safronov*
Affiliation:
Max-Planck-Institut für Mathematik, Bonn, Germany email psafronov@mpim-bonn.mpg.de
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Abstract

We relate the brace construction introduced by Calaque and Willwacher to an additivity functor. That is, we construct a functor from brace algebras associated to an operad ${\mathcal{O}}$ to associative algebras in the category of homotopy ${\mathcal{O}}$-algebras. As an example, we identify the category of $\mathbb{P}_{n+1}$-algebras with the category of associative algebras in $\mathbb{P}_{n}$-algebras. We also show that under this identification there is an equivalence of two definitions of derived coisotropic structures in the literature.

Keywords

operadsKoszul dualityshifted Poisson structures

MSC classification

Primary: 18D50: Operads
Secondary: 18G55: Homotopical algebra
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 8 , August 2018 , pp. 1698 - 1745
Copyright
© The Author 2018 

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Footnotes

1

Current address: Intitut für Mathematik, Winterthurerstrasse 190, 8057 Zürich, Switzerland

References

Berger, C. and Moerdijk, I., Resolution of coloured operads and rectification of homotopy algebras , in Categories in algebra, geometry and mathematical physics, Contemporary Mathematics, vol. 431 (American Mathematical Society, Providence, RI, 2007), 31–58.Google Scholar
Calaque, D., Pantev, T., Toën, B., Vaquié, M. and Vezzosi, G., Shifted Poisson structures and deformation quantization , J. Topol. 10 (2017), 483–584.Google Scholar
Calaque, D. and Willwacher, T., Triviality of the higher formality theorem , Proc. Amer. Math. Soc. 143 (2015), 5181–5193.Google Scholar
Cartier, P., A primer of Hopf algebras , in Frontiers in number theory, physics, and geometry II (Springer, Berlin, 2007), 537–615.Google Scholar
Costello, K. and Gwilliam, O., Factorization algebras in quantum field theory, Vol. 2, (2016), http://people.mpim-bonn.mpg.de/gwilliam/vol2may8.pdf.Google Scholar
Dolgushev, V. and Willwacher, T., Operadic twisting—with an application to Deligne’s conjecture , J. Pure Appl. Algebra 219 (2015), 1349–1428.Google Scholar
Gerstenhaber, M. and Voronov, A., Homotopy G-algebras and moduli space operad , Int. Math. Res. Not. IMRN 3 (1995), 141–153.Google Scholar
Gwilliam, O. and Haugseng, R., Linear Batalin–Vilkovisky quantization as a functor of ∞-categories , Selecta Math. 24 (2018), 1247–1313.Google Scholar
Hinich, V., Homological algebra of homotopy algebras , Comm. Algebra 25 (1997), 3291–3323.Google Scholar
Hirsh, J. and Millès, J., Curved Koszul duality theory , Math. Ann. 354 (2012), 1465–1520.Google Scholar
Kapranov, M., Rozansky–Witten invariants via Atiyah classes , Compositio Math. 115 (1999), 71–113.Google Scholar
Loday, J.-L. and Vallette, B., Algebraic operads, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 346 (Springer, Heidelberg, 2012), xxiv+634.Google Scholar
Lurie, J., Derived algebraic geometry $X$ : formal moduli problems.http://math.harvard.edu/∼lurie/papers/DAG-X.pdf, (2011).Google Scholar
Lurie, J., Higher algebra, http://math.harvard.edu/∼lurie/papers/HA.pdf, (2017).Google Scholar
Melani, V. and Safronov, P., Derived coisotropic structures I: affine case, Preprint (2016),arXiv:1608.01482 [math.AG].Google Scholar
Oudom, J.-M. and Guin, D., On the Lie enveloping algebra of a pre-Lie algebra , J. K-Theory 2 (2008), 147–167.Google Scholar
Pavlov, D. and Scholbach, J., Admissibility and rectification of colored symmetric operads, Preprint (2014), arXiv:1410.5675 [math.AT].Google Scholar
Positselski, L., Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence , Mem. Amer. Math. Soc. 212 (2011), vi + 133pp.Google Scholar
Safronov, P., Poisson reduction as a coisotropic intersection , Higher Structures 1 (2017), 87–121.Google Scholar
Tamarkin, D., Deformation complex of a $d$ -algebra is a $(d+1)$ -algebra, Preprint (2000),arXiv:math/0010072.Google Scholar
Tamarkin, D., Quantization of Lie bialgebras via the formality of the operad of little disks , Geom. Funct. Anal. 17 (2007), 537–604.Google Scholar
Toën, B., Operations on derived moduli spaces of branes, Preprint (2013), arXiv:1307.0405 [math.AG].Google Scholar
Vallette, B., Homotopy theory of homotopy algebras, Preprint (2014), arXiv:1411.5533 [math.AT].Google Scholar