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Homological perturbation theory for algebras over operads
Stockholm University, Faculty of Science, Department of Mathematics.
2014 (English)In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 14, no 5, p. 2511-2548Article in journal (Refereed) Published
Abstract [en]

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

Place, publisher, year, edition, pages
2014. Vol. 14, no 5, p. 2511-2548
Keywords [en]
operads, strong homotopy algebras
National Category
Algebra and Logic Geometry
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-111889DOI: 10.2140/agt.2014.14.2511ISI: 000350399400001OAI: oai:DiVA.org:su-111889DiVA, id: diva2:777018
Available from: 2015-01-08 Created: 2015-01-08 Last updated: 2017-12-05Bibliographically approved

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