Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras

Abstract

We study the structure of the symplectic invariant part hg,1Sp of the Lie algebra hg,1 consisting of symplectic derivations of the free Lie algebra generated by the rational homology group of a closed oriented surface Σg of genus g.First we describe the orthogonal direct sum decomposition of this space which is induced by the canonical metric on it and compute it explicitly up to degree 20. In this framework, we give a general constraint which is imposed on the Sp-invariant component of the bracket of two elements in hg,1. Second we clarify the relations among hg,1 and the other two related Lie algebras hg,⁎ and hg which correspond to the cases of a closed surface Σg with and without base point ⁎∈Σg. In particular, based on a theorem of Labute, we formulate a method of determining these differences and describe them explicitly up to degree 20. Third, by giving a general method of constructing elements of hg,1Sp, we reveal a considerable difference between two particular submodules of it, one is the Sp-invariant part of a certain ideal jg,1 and the other is that of the Johnson image.Finally we combine these results to determine the structure of hg,1 completely up to degree 6 including the unstable cases where the genus 1 case has an independent meaning. In particular, we see a glimpse of the Galois obstructions explicitly from our point of view.