Homological perturbation theory for algebras over operads

By Alexander Berglund. 2009

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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 .C/‐algebra structures along contractions, where C is any connected cooperad in chain complexes. This specializes to transfer formulas for O1 ‐algebras for any Koszul operad O , in particular for A1 ‐, C1 ‐, L1 ‐ and G1 ‐algebras. A key feature is that our formulas are expressed in terms of the compact description of .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. 18D50, 55P48

Subfields: Algebra and Number Theory, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/59fbbe39be83087ee88dce337d9e7fb2456eb6f6

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In Algebra and Number Theory

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Citations in the corpus

Pre-Lie deformation theory (2015, methodology): “(1) A somewhat indirect proof of the latter relation is also contained in [Ber14, Proof of Lemma 7.”