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

TL;DR

This paper demonstrates that Lie-infinity morphisms induce Leibniz-infinity morphisms between derived Leibniz algebras associated with Maurer-Cartan elements, providing an explicit construction and connecting to deformation quantization.

Contribution

It explicitly constructs Leibniz-infinity morphisms induced by Lie-infinity morphisms between DGLAs, extending the understanding of derived brackets and their functorial properties.

Findings

Leibniz-infinity morphism construction from Lie-infinity morphisms.

Relation between derived Leibniz algebras via these morphisms.

Application to formula in deformation quantization.

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.