Letters in Mathematical Physics

, Volume 101, Issue 2, pp 195–222 | Cite as

Open-Closed Homotopy Algebras and Strong Homotopy Leibniz Pairs Through Koszul Operad Theory

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Abstract

Open-closed homotopy algebras (OCHA) and strong homotopy Leibniz pairs (SHLP) were introduced by Kajiura and Stasheff in 2004. In an appendix to their paper, Markl observed that an SHLP is equivalent to an algebra over the minimal model of a certain operad, without showing that the operad is Koszul. In the present paper, we show that both OCHA and SHLP are algebras over the minimal model of the zeroth homology of two versions of the Swiss-cheese operad and prove that these two operads are Koszul. As an application, we show that the OCHA operad is non-formal as a 2-colored operad but is formal as an algebra in the category of 2-collections.

Mathematics Subject Classification

18G55 18D50 

Keywords

Koszul Operads homotopy algebras deformation theory 
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Copyright information

© Springer 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidade Federal do ParanáCuritibaBrazil
  2. 2.Université Paris 13, CNRS, UMR 7539 LAGAVilletaneuseFrance

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