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Operads and triangulation of Loday’s diagram on Leibniz algebras

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Abstract

We factor the classical functors \({ As}\mathop {\longrightarrow }\limits ^{-} { Lie}\) and \({ Dias}\mathop {\longrightarrow }\limits ^{-}{ Leib}\) through the categories \({ Pre}\hbox {-}{} { Lie}\) and \({ Pre}\hbox {-}{} { Leib}\) of two new types of algebras. Thanks to Koszul duality for binary quadratic operads, we deduce two more categories of algebras \({ Perm}\) and \({ Ricod}\) giving rise to other factorizations. This yields a triangulation of Loday’s commutative diagram of functors on Leibniz algebras and associated operads. As an application, we define a notion of extended Leibniz algebras.

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Acknowledgments

I am grateful to the Institut de Recherche Mathématique Avancée of Strasbourg (Université Louis Pasteur and CNRS) for their support and hospitality during jobless years. Also, I would like to warmly thank Professors Jim Stasheff, Jean-Louis Loday and Benoit Fresse for their helpful comments and suggestions. Special thoughts to SylviKA Ndjekorboa in souvenir of that Automn 2001’ when a wall fell down.

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  1. Département de Mathématiques, Faculté des Sciences Exactes et Appliquées, Université de N’Djaména, B.P.: 1027, N’Djaména, Chad

    Allahtan Victor Gnedbaye

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  1. Allahtan Victor Gnedbaye
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Correspondence to Allahtan Victor Gnedbaye.

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Gnedbaye, A.V. Operads and triangulation of Loday’s diagram on Leibniz algebras. Afr. Mat. 28, 109–118 (2017). https://doi.org/10.1007/s13370-016-0431-2

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  • DOI: https://doi.org/10.1007/s13370-016-0431-2

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