Introduction to SH Lie algebras for physicists

Lada, Tom; Stasheff, Jim

doi:10.1007/bf00671791

Cited by 508 publications

(660 citation statements)

References 31 publications

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“…We only emphasize that an L ∞ morphism of DGLAs has a linear part that is a morphism of complexes and therefore it induces a morphism in cohomology. For the detailed descriptions of such structures we refer to [LS93,LM95,Ma02,Fu03,Kon03,Get04,Ma04,FiMa07,Ia08]. The homotopy fibre of a morphism of DGLA χ : L → M is the DGLA…”

Section: Review Of Logarithmic Differentialsmentioning

confidence: 99%

Deformations and Obstructions of Pairs (X,D)

Iacono

2014

Int Math Res Notices

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Abstract. We study infinitesimal deformations of pairs (X, D) with X smooth projective variety and D a smooth or a normal crossing divisor, defined over an algebraically closed field of characteristic 0. Using the differential graded Lie algebras theory and the Cartan homotopy construction, we are able to prove in a completely algebraic way the unobstructedness of the deformations of the pair (X, D) in many cases, e.g., whenever (X, D) is a log Calabi-Yau pair, in the case of a smooth divisor D in a Calabi Yau variety X and when D is a smooth divisor in | − mKX|, for some positive integer m.

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Section: Review Of Logarithmic Differentialsmentioning

confidence: 99%

Deformations and Obstructions of Pairs (X,D)

Iacono

2014

Int Math Res Notices

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“…Here C n are graded symmetric multilinear operations satisfying special quadratic relations generating L ∞ -like structure [15], [16], such that…”

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confidence: 99%

Perturbed beta–gamma systems and complex geometry

Zeitlin

2008

Nuclear Physics B

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We consider the equations, arising as the conformal invariance conditions of the perturbed curved beta-gamma system. These equations have the physical meaning of Einstein equations with a B-field and a dilaton on a hermitian manifold, where the B-field 2-form is imaginary and proportional to the canonical form associated with hermitian metric. We show that they decompose into linear and bilinear equations and lead to the vanishing of the first Chern class of the manifold where the system is defined. We discuss the relation of these equations to the generalized Maurer-Cartan structures related to BRST operator. Finally we describe the relations of the generalized Maurer-Cartan bilinear operation and the Courant/Dorfman brackets.

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“…We will work in the category of L ∞ [1] algebras and L ∞ [1] morphisms between them: this is isomorphic to the usual category of L ∞ algebras from [27,14] via the so called décalage isomorphisms, cf. for instance [9] (we recall that in this way L ∞ algebra structures on a graded space V correspond to L ∞ [1] algebra structures on the desuspension s −1 V ).…”

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

confidence: 99%

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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Introduction to SH Lie algebras for physicists

Cited by 508 publications

References 31 publications

Deformations and Obstructions of Pairs (X,D)

Deformations and Obstructions of Pairs (X,D)

Perturbed beta–gamma systems and complex geometry

Nonabelian higher derived brackets

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