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Journal of Lie Theory 16 (2006), No. 1, 019--037
Copyright Heldermann Verlag 2006

On the Poisson Bracket on the Free Lie Algebra in two Generators

Leila Schneps
Equipe Analyse Algébrique, Université de Paris 6, 175 rue du Chevaleret, 75013 Paris, France

We prove a combinatorial formula for the Poisson bracket of two elements of the free Lie algebra on two generators, which has a particularly nice cocycle form when the two elements are Lie monomials containing only one y. By relating this cocycle form with the period polynomials introduced by Eichler-Shimura and Zagier, we completely describe and classify a set of fundamental relations in Ihara's stable derivation algebra, generalizing the first few cases of these relations which he had observed and computed by hand.

Keywords: Poisson bracket, graded Lie algebras, modular forms.

MSC: 17B63, 17B70, 11F11

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