Operad of formal homogeneous spaces and Bernoulli numbers

Merkulov, Sergei

doi:10.2140/ant.2008.2.407

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Operad of formal homogeneous spaces and Bernoulli numbers

Sergei A. Merkulov

Vol. 2 (2008), No. 4, 407–433

DOI: 10.2140/ant.2008.2.407

Abstract

It is shown that for any morphism, $ϕ : g \to h$ , 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 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.

Keywords

operad, Lie algebra, Bernoulli number

Mathematical Subject Classification 2000

Primary: 18D50

Secondary: 11B68, 55P48

Milestones

Received: 10 October 2007

Revised: 17 January 2008

Accepted: 9 March 2008

Published: 15 June 2008

Authors

Sergei A. Merkulov
Department of Mathematics Stockholm University 10691 Stockholm Sweden

Vol. 2, No. 4, 2008