Structure of symplectic invariant Lie subalgebras of symplectic derivation Lie algebras

Abstract

ABSTRACT. We study the structure of the symplectic invariant part h Sp g,1 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 Sp g,1 , 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.

The Goldman–Turaev Lie bialgebra in genus zero and the Kashiwara–Vergne problem

“…Morita, Sakasai and Suzuki [32] determined the structure of M k for k = 6. For k ≤ 5, see the references in [32].…”

“…Here, the group structure of the target is given by the BCH series. The canonical map gr g(Σ) ∼ = |A| → sDer(A) induces an isomorphism gr L + (Σ) ∼ = h g,1 , and the associated graded map of (33) coincides with the classical one (32). Let f be a framing on Σ g,1 and denote by I f g,1 the subgroup of I g,1 consisting of elements preserving f .…”

An Abelian Quotient of the Symplectic Derivation Lie Algebra of the Free Lie Algebra

“…Remark 3.2. Since h 1,1 (12) Sp = 0 as mentioned in [20], we have H 1 (h + 1,1 ) Sp 12 = 0. Therefore our bound of genus for the non-triviality of H 1 (h + g,1 ) Sp 12 is best possible.…”

Morita’s trace maps on the group of homology cobordisms

“…. , k + 2} into (k/2 + 1) ordered pairs, we can apply to h k the linear map We will use Table 1, which is borrowed to [MSS15]. For instance, the 1-st, 3-rd, 5-th and 8-th rows of this table read…”

“…Indeed, using some algebraic results of Morita, Suzuki and the second-named author [MSS15,MSS16], we derive from the 1-cocycle Ab θ a non-trivial group homomorphism I k : H → Q.…”