Operad of formal homogeneous spaces and Bernoulli numbers

TL;DR

This paper introduces a new operadic framework for formal homogeneous spaces, revealing how Bernoulli numbers influence the nonlinear actions of Lie algebra morphisms, and extends these ideas to L-infinity algebras.

Contribution

It develops a 2-colored operad approach to formal homogeneous spaces, providing a conceptual explanation for Bernoulli number appearances in Lie algebra structures and extending to L-infinity algebras.

Findings

Canonical g-homogeneous formal manifolds from Lie algebra morphisms

Operadic interpretation of Bernoulli numbers in Lie algebra actions

Extension of constructions to L-infinity algebra morphisms

Abstract

It is shown that for any morphism, i: g --> h, of Lie algebras the vector space underlying the Lie algebra h is canonically a g-homogeneous formal manifold with the action of g being highly nonlinear and twisted by Bernoulli numbers. This fact is obtained from the study of a 2-coloured operad of formal homogeneous spaces and its minimal resolution, and is used to give a new conceptual explanation of both Ziv Ran's Jacobi-Bernoulli complex and Fiorenza-Manetti's L-infinity 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-infinity algebras.