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Simultaneous deformations and Poisson geometry

Part of: General theory of differentiable manifolds Lie algebras and Lie superalgebras Symplectic geometry, contact geometry Pseudogroups, differentiable groupoids and general structures on manifolds

Published online by Cambridge University Press:  04 May 2015

Yaël Frégier  and
Marco Zambon
Yaël Frégier
Affiliation:
UArtois, LML, F-62 300, Lens, France MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Universität Zürich, Winterthurerstr. 190, CH-8057 Zürich, Switzerland email yael.fregier@gmail.com
Marco Zambon
Affiliation:
Universidad Autónoma de Madrid, Spain email marco.zambon@uam.es, marco.zambon@icmat.es ICMAT (CSIC-UAM-UC3M-UCM), Campus de Cantoblanco, 28049 Madrid, Spain Current address: KU Leuven, Department of Mathematics, Celestijnenlaan 200B box 2400, BE-3001 Leuven, Belgium. email marco.zambon@wis.kuleuven.be
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Abstract

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We consider the problem of deforming simultaneously a pair of given structures. We show that such deformations are governed by an $L_{\infty }$-algebra, which we construct explicitly. Our machinery is based on Voronov’s derived bracket construction. In this paper we consider only geometric applications, including deformations of coisotropic submanifolds in Poisson manifolds, of twisted Poisson structures, and of complex structures within generalized complex geometry. These applications cannot be, to our knowledge, obtained by other methods such as operad theory.

Keywords

L∞ -algebradeformationMaurer–Cartan equationPoisson manifoldDirac manifoldgeneralized complex manifold

MSC classification

Primary: 17B70: Graded Lie (super)algebras 53D17: Poisson manifolds; Poisson groupoids and algebroids 58H15: Deformations of structures
Secondary: 53D18: Generalized geometries (à la Hitchin) 58A50: Supermanifolds and graded manifolds

Information

Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 9 , September 2015 , pp. 1763 - 1790
Copyright
© The Authors 2015 

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