Transferring algebra structures up to homology equivalence

Johansson, Leif; Lambe, Larry A.

doi:10.7146/math.scand.a-14337

Cited by 21 publications

(29 citation statements)

References 23 publications

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“…In our setting, µ 0 is the (background) curvature, and we now indicate briefly how such a curved A ∞ -algebra structure arises. Following [29,30,36], we first work on the level of the chain bundles and define λ m inductively, for m 2, by…”

Section: Curved a ∞ -Algebrasmentioning

confidence: 99%

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Calderbank¹,

Diemer²

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

102

260

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Abstract. We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential "cup product" on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A∞-algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of helicity raising and lowering for conformally invariant field equations.

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Section: Curved a ∞ -Algebrasmentioning

confidence: 99%

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Calderbank¹,

Diemer²

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

102

260

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“…Recall (see, for example, [14] The nature and rôle of these side conditions is explained in the following proposition, proved in [26]. The difference between ordinary homotopy diagrams and strongly homotopy ones is measured by the homology of the nerve of the category S. From this point of view, diagram (18) related to moves (M1) and (M3) was very complicated.…”

Section: Structure Of Strongly Homotopy Diagramsmentioning

confidence: 99%

Homotopy algebras are homotopy algebras

Markl¹

2004

Forum Mathematicum

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“…Our main references will be [Gug72, LS87, GLS91], papers which contain the results needed for the present application. There are other ways to solve this question, see [Kel01] and [JL01] for instance, however perturbation lemma and its application nicknamed tensor trick have a key advantage for us: they provide explicit formulas for which we can checked that the new higher intersection products still satisfy basic properties such as the projection formula.…”

Section: Perturbation Theory and Intersection Theory For Cycle Complexesmentioning

confidence: 99%

Cycle modules and the intersection A ∞-algebra

Ivorra

2013

manuscripta math.

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Abstract. -In his paper Chow Group with Coefficients, M. Rost has developed a generalization of the classical Chow groups based on Milnor K-theory and an axiomatic generalization of it called cycles modules. For a cycle module M with a ring structure and a smooth scheme X of finite type over a field, we show that Rost's cycle complex with coefficients C * (X , M ) has a structure of an A ∞ -algebra. In the case of Milnor K -theory it provides an homotopy model for the classical intersection theory of cycles. To construct the A ∞ -algebra structure we first construct such an A ∞ -algebra structure on a homotopy invariant version of Rost's complex, this step relies on geometry, and then we apply homological perturbation theory. Résumé. -Dans son article Chow Group with Coefficients, M. Rost a généralisé les groupes deChow classiqueà partir de la K-théorie de Milnor et une généralisation axiomatique de cette dernière: les modules de cycles.Étant donné un module de cycle M muni d'un produit et un schéma lisse X de type fini sur un corps, nous montrons dans ce travail que le complexe de cycle C * (X , M )à coefficients dans M introduit par M. Rost possède une structure naturelle de A ∞ -algèbre. Dans le cas particulier de la K-théorie de Milnor cette structure relève la théorie classique de l'intersection des cycles algébriques au niveau des complexes. La construction présente deux facettes, une facette géométrique reposant sur des espaces de déformation au cône normal ad hoc et une facette homologique reposant sur le lemme de perturbation.

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Transferring algebra structures up to homology equivalence

Abstract: Given a strong deformation retract M of an algebra A, there are several apparently distinct ways ([9]

Cited by 21 publications

References 23 publications

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Homotopy algebras are homotopy algebras

Cycle modules and the intersection A ∞-algebra

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