  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.
