Algebra & Number Theory

Operad of formal homogeneous spaces and Bernoulli numbers

Sergei Merkulov

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It is shown that for any morphism, ϕ:gh, 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.

Article information

Algebra Number Theory, Volume 2, Number 4 (2008), 407-433.

Received: 10 October 2007
Revised: 17 January 2008
Accepted: 9 March 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 18D50: Operads [See also 55P48]
Secondary: 11B68: Bernoulli and Euler numbers and polynomials 55P48: Loop space machines, operads [See also 18D50]

operad Lie algebra Bernoulli number


Merkulov, Sergei. Operad of formal homogeneous spaces and Bernoulli numbers. Algebra Number Theory 2 (2008), no. 4, 407--433. doi:10.2140/ant.2008.2.407.

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