### Abstract

We study the structure of the symplectic invariant part h_{g,1} ^{Sp} of the Lie algebra h_{g,1} consisting of symplectic derivations of the free Lie algebra generated by the rational homology group of a closed oriented surface Σ_{g} 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 h_{g,1}. Second we clarify the relations among h_{g,1} and the other two related Lie algebras h_{g,⁎} and h_{g} which correspond to the cases of a closed surface Σ_{g} with and without base point ⁎∈Σ_{g}. 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 h_{g,1} ^{Sp}, we reveal a considerable difference between two particular submodules of it, one is the Sp-invariant part of a certain ideal j_{g,1} and the other is that of the Johnson image. Finally we combine these results to determine the structure of h_{g,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.

Original language | English |
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Pages (from-to) | 291-334 |

Number of pages | 44 |

Journal | Advances in Mathematics |

Volume | 282 |

DOIs | |

Publication status | Published - 10 Sep 2015 |

### Keywords

- Derivation Lie algebra
- Free Lie algebra
- Johnson homomorphism
- Symplectic representation
- Young diagram

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## Cite this

*Advances in Mathematics*,

*282*, 291-334. https://doi.org/10.1016/j.aim.2015.06.017