Operad of formal homogeneous spaces and Bernoulli numbers

Abstract

It is shown that for any morphism, $\varphi :\mathfrak{g}\to \mathfrak{h}$, of Lie algebras the vector space underlying the Lie algebra $\mathfrak{h}$ is canonically a $\mathfrak{g}$-homogeneous formal manifold with the action of $\mathfrak{g}$ being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from a study of the 2-coloured operad of formal homogeneous spaces whose minimal resolution gives a new conceptual explanation of both Ziv Ran’s Jacobi–Bernoulli complex and Fiorenza–Manetti’s $L_{\infty }$-algebra structure on the mapping cone of a morphism of two Lie algebras. All these constructions are iteratively extended to the case of a morphism of arbitrary $L_{\infty }$-algebras.