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Morita–Sakasai–Suzuki studied the graded Lie algebra $\mathfrak{h}_{g,1}$ of symplectic derivations, as well as variations $\mathfrak{h}_{g,\ast}$ and $\mathfrak{h}_g$. This is the Lie algebra of those derivations of a free Lie algebra on $H := H_1(\Sigma_{g,1};\mathbb{Q})$ which annihilate the symplectic form $\sum_{i=1}^g [a_i,b_i] \in \Lambda^2(H)$. See for example Symmetry of symplectic derivation Lie algebras of free Lie algebras or Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras by Morita–Sakasai–Suzuki.

Is anything known about the centers of $\mathfrak{h}_{g,1}$, $\mathfrak{h}_{g,*}$, and $\mathfrak{h}_{g}$? Are they trivial?

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