The Lie algebra perturbation lemma

Details The Lie algebra perturbation lemma The Lie algebra perturbation lemma

2007-08-29

arxiv.org/abs/0708.3977v2

Mathematics

Algebraic Geometry

20 pages; in view of a number of comments of J. Stasheff, the exposition has been improved

Scientific paper

Let g be a differential graded Lie algebra and suppose given a contraction of chain complexes of g onto a general chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S'' to the given Lie algebra g, and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on g onto S'' which is natural in the data. This extends a result established in a joint paper of the author with J. Stashef [Forum math. 14 (2002), 847-868, math.AG/9906036] where only the particular where M is the homology of g has been explored.

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