Infinis morphismes de Leibniz pour les crochets d\'eriv\'es

Abstract

The derived bracket of a Maurer-Cartan element in a differential graded Lie algebra (DGLA) is well-known to define a differential graded Leibniz algebra. It is also well-known that a Lie infinity morphism between DGLAs maps a Maurer-Cartan element to a Maurer-Cartan element. Given a Lie-infinity morphism, a Maurer-element and its image, we show that both derived differential graded Leibniz algebras are related by a Leibniz-infinity morphism, and we construct it explicitely. As an application, we recover a well-known formula of Dominique Manchon about the commutator of the star-product. Keywords: Leibniz algebras, Lie-infinity algebras, formality and quantization.