  We determine the abelianizations of the following three kinds of graded Lie
algebras in certain stable ranges: derivations of the free associative algebra,
derivations of the free Lie algebra and symplectic derivations of the free
associative algebra. In each case, we consider both the whole derivation Lie
algebra and its ideal consisting of derivations with positive degrees. As an
application of the last case, and by making use of a theorem of Kontsevich, we
obtain a new proof of the vanishing theorem of Harer concerning the top
rational cohomology group of the mapping class group with respect to its
virtual cohomological dimension.

Algebraic Topology;Algebraic Geometry;Geometric Topology;Quantum Algebra