The Lie Algebra Perturbation Lemma

Huebschmann, Johannes

doi:10.1007/978-0-8176-4735-3_8

Cited by 19 publications

(14 citation statements)

References 19 publications

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“…where h σ n = σ −1 h n σ. The idea of symmetrizing the tensor trick homotopy appears in [13,21,18,19] and presumably in many other places, but the author is not aware of any written source where the formal properties of the symmetrized homotopy are worked out in detail. In particular, we believe that the discovery that h Σ is a pseudo-derivation is new, see Theorem 5.4 below.…”

Section: Symmetric Tensor Trickmentioning

confidence: 99%

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Homological perturbation theory for algebras over operads

Berglund¹

2014

Algebr. Geom. Topol.

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Abstract. We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads O. To solve this problem, we introduce thick maps of O-algebras and special thick maps that we call pseudo-derivations that serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory.As an application, we derive explicit formulas for transferring Ω(C)-algebra structures along contractions, where C is any connected cooperad in chain complexes. This specializes to transfer formulas for O∞-algebras for any Koszul operad O, in particular for A∞, C∞, L∞ and G∞-algebras. A key feature is that our formulas are expressed in terms of the compact description of Ω(C)-algebras as coderivation differentials on cofree C-coalgebras. Moreover, we get formulas not only for the transferred structure and a structure on the inclusion, but also for structures on the projection and the homotopy.

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Section: Symmetric Tensor Trickmentioning

confidence: 99%

“…A perturbation lemma for cocommutative coalgebras, yielding transfer of L ∞algebra structures, has been obtained by Huebschmann [18,19] using different methods.…”

Section: Introductionmentioning

confidence: 99%

Homological perturbation theory for algebras over operads

Berglund¹

2014

Algebr. Geom. Topol.

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“…Let G be Lie group with Lie algebra g, P a smooth principal G-bundle, and X a triangulation of the total space of P . Using Theorem 2.16 and the main result of [11] one can transfer the dg Lie algebra structure on Ω • (P, g) to an L ∞ -structure on the complex of g-valued cochains C • (X, g). Let α be connection 1-form on P .…”

Section: Proofmentioning

confidence: 99%

Curved $A_{\infty}$-algebras and Chern classes

Nikolov

Zahariev

2013

Pure and Applied Mathematics Quarterly

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Abstract:We describe two constructions giving rise to curved A ∞ -algebras. The first consists of deforming A ∞ -algebras, while the second involves transferring curved dg structures that are deformations of (ordinary) dg structures along chain contractions. As an application of the second construction, given a vector bundle on a polyhedron X, we exhibit a curved A ∞ -structure on the complex of matrix-valued cochains of sufficiently fine triangulations of X. We use this structure as a motivation to develop a homotopy associative version of Chern-Weil theory.

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“…This transfer of structure began in the case of dg associative algebras with the work of Kadeishvili [14]. The definitive treatment in the Lie case (which is more subtle) is due to Huebschmann [11,12].…”

Section: ∞ -Structure On H(l)mentioning

confidence: 99%

How Dennis and I intersected

Stasheff

2013

Pure and Applied Mathematics Quarterly

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This paper is a slightly modified version of the talk I gave at the Dennisfest. After some reminiscences about how things were in the good old days, including how Dennis' work and mine were somewhat transverse ;-) I will review and update my ancient, unpublished (now arXived) paper with Mike Schlessinger on the deformation theory of rational homotopy types. Then, I will return briefly to another transverse intersection of my work with Dennis'.

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The Lie Algebra Perturbation Lemma

Abstract: We shall establish the following.

Cited by 19 publications

References 19 publications

Homological perturbation theory for algebras over operads

Homological perturbation theory for algebras over operads

Curved $A_{\infty}$-algebras and Chern classes

How Dennis and I intersected

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