Equivalences of higher derived brackets

Cattaneo, Alberto S.; Schätz, Florian

doi:10.1016/j.jpaa.2008.03.013

Cited by 25 publications

(84 citation statements)

References 9 publications

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“…The first one is an immediate corollary of Theorem 1.1 and Theorem 3.3. The second one should be confronted with the results of [6]. …”

Section: ) Finally We Have a Commutative Diagrammentioning

confidence: 99%

“…Higher derived brackets have been applied in the study of coisotropic deformations, cf. [32,6,12]. Let X be a differentiable manifold, and let T X be the tangent bundle: we denote by V * X = Γ( * T X) the Gerstenhaber algebra of polyvector fields on X, equipped with the Schouten-…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…When X is the total space of a vector bundle on Z (and Z is embedded as the zero section) the above sequence admits a natural splitting, whose desuspension sends s −1 N * Z|X onto the abelian Lie subalgebra of s −1 V * X consisting of vertical polyvector fields constant along the fibers, cf. for instance [6,32]: in general one reduces to this situation via the choice of an embedding of N Z onto a tubular neighborhood of Z in X. For a coisotropic Z the higher derived brackets…”

Section: Examples and Applicationsmentioning

confidence: 99%

“…[24,34,35,4]. Another list of examples from deformation theory [12,13,32,6] can be better understood in light of our Theorem 1.1: in fact homotopy fibers naturally occur when considering semitrivial deformation problems (cf. [29,9]).…”

Section: Introduction Let I : (L D [· ·]) → (M D [· ·])mentioning

confidence: 99%

“…[29,9]). Consider for instance the case of deformations of a coisotropic submanifold of a Poisson manifold, where the usual approach [32,6,12], using higher derived brackets, can be paralleled with the more recent one in [2], using homotopy fibers, cf. Example 6.6.…”

Section: Introduction Let I : (L D [· ·]) → (M D [· ·])mentioning

confidence: 99%

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Nonabelian higher derived brackets

Bandiera

2015

Journal of Pure and Applied Algebra

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Abstract. Let M be a graded Lie algebra, together with graded Lie subalgebras L and A such that as a graded space M is the direct sum of L and A, and A is abelian. Let D be a degree one derivation of M squaring to zero and sending L into itself, then Voronov's construction of higher derived brackets associates to D a L∞ structure on A[−1]. It is known, and it follows from the results of this paper, that the resulting L∞ algebra is a weak model for the homotopy fiber of the inclusion of differential graded Lie algebras i :. We prove this fact using homotopical transfer of L∞ structures, in this way we also extend Voronov's construction when the assumption A abelian is dropped: the resulting formulas involve Bernoulli numbers. In the last section we consider some example and some further application. Introductionbe the inclusion of a differential graded Lie subalgebra, recall that its homotopy fiber is the differential graded Lie algebra (dgla) In this paper we find explicit formulas under the additional assumption that A ⊂ M is a graded Lie subalgebra of M , then the L ∞ structure on A[−1] is given (after décalage) by the family of degree one symmetric brackets Φ(D) i :where S i is the symmetric group, ε(σ) = ε(σ; a 1 , . . . , a i ) is the Koszul sign and the B j are the Bernoulli numbers. ThewhereOur first main result is that with these definitions Date: September 9, 2013.

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“…The first one is an immediate corollary of Theorem 1.1 and Theorem 3.3. The second one should be confronted with the results of [6]. …”

Section: ) Finally We Have a Commutative Diagrammentioning

confidence: 99%

Section: Examples and Applicationsmentioning

confidence: 99%

Section: Examples and Applicationsmentioning

confidence: 99%

Section: Introduction Let I : (L D [· ·]) → (M D [· ·])mentioning

confidence: 99%

Section: Introduction Let I : (L D [· ·]) → (M D [· ·])mentioning

confidence: 99%

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Nonabelian higher derived brackets

Bandiera

2015

Journal of Pure and Applied Algebra

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BFV-Complex and Higher Homotopy Structures

Schätz

2008

Commun. Math. Phys.

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We present a connection between the BFV-complex (appreviation for Batalin-Fradkin-Vilkovisky complex) and the so-called strong homotopy Lie algebroid associated to a coisotropic submanifold of a Poisson manifold. We prove that the latter structure can be derived from the BFV-complex by means of homotopy transfer along contractions. Consequently the BFV-complex and the strong homotopy Lie algebroid structure are L∞ quasi-isomorphic and control the same formal deformation problem.However there is a gap between the non-formal information encoded in the BFV-complex and in the strong homotopy Lie algebroid respectively. We prove that there is a one-to-one correspondence between coisotropic submanifolds given by graphs of sections and equivalence classes of normalized Maurer-Cartan elemens of the BFV-complex. This does not hold if one uses the strong homotopy Lie algebroid instead.

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Formality and Kontsevich–Duflo type theorems for Lie pairs

Liao

Stiénon

2019

Advances in Mathematics

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Equivalences of higher derived brackets

Cited by 25 publications

References 9 publications

Nonabelian higher derived brackets

Nonabelian higher derived brackets

BFV-Complex and Higher Homotopy Structures

Formality and Kontsevich–Duflo type theorems for Lie pairs

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