Operads and triangulation of Loday’s diagram on Leibniz algebras

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.