Vol. 2, No. 4, 2008

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Operad of formal homogeneous spaces and Bernoulli numbers

Sergei A. Merkulov

Vol. 2 (2008), No. 4, 407–433
Abstract

It is shown that for any morphism, ϕ : 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 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-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-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