Symplectic Jacobi diagrams and the Lie algebra of homology cylinders

TL;DR

This paper establishes a deep connection between the algebraic structure of homology cylinders over surfaces and symplectic Jacobi diagrams, using an extension of the LMO invariant to describe their graded Lie algebra.

Contribution

It introduces a functorial extension of the LMO invariant that links the Y-filtration on homology cylinders to symplectic Jacobi diagrams, providing new insights into their algebraic structure.

Findings

The graded Lie algebra of homology cylinders is isomorphic to symplectic Jacobi diagrams.

The embedding of the Torelli group into homology cylinders preserves the lower central series.

The Lie algebra map is injective in degree two, with higher degree injectivity discussed.

Abstract

Let S be a compact connected oriented surface, whose boundary is connected or empty. A homology cylinder over the surface S is a cobordism between S and itself, homologically equivalent to the cylinder over S. The Y-filtration on the monoid of homology cylinders over S is defined by clasper surgery. Using a functorial extension of the Le-Murakami-Ohtsuki invariant, we show that the graded Lie algebra associated to the Y-filtration is isomorphic to the Lie algebra of ``symplectic Jacobi diagrams.'' This Lie algebra consists of the primitive elements of a certain Hopf algebra whose multiplication is a diagrammatic analogue of the Moyal-Weyl product. The mapping cylinder construction embeds the Torelli group into the monoid of homology cylinders, sending the lower central series to the Y-filtration. We give a combinatorial description of the graded Lie algebra map induced by this embedding, by connecting Hain's infinitesimal presentation of the Torelli group to the Lie algebra of symplectic Jacobi diagrams. This Lie algebra map is shown to be injective in degree two, and the question of the injectivity in higher degrees is discussed.