Symplectic Jacobi Diagrams and the Lie Algebra of Homology Cylinders

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.