Abelianizations of derivation Lie algebras of the free associative algebra and the free Lie algebra

Abstract

Abstract. 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.

Hairy graphs and the unstable homology of Mod(g , s ), Out(F n ) and Aut(F n )

“…Proof. This proof is an adaptation of the proof in [25] to our context. Let G be a generator of C 1 H, that is, a basic hairy graph with one vertex.…”

“…In [25], Morita, Sakasai and Suzuki compute the abelianization of h. In this section, we point out how this follows from the computation of H 1 (H) and the injectivity of the trace map.…”

“…In [25], Morita, Sakasai and Suzuki combined their results with Kontsevich's theorem to prove that the cohomology of mapping class groups vanishes in their virtual cohomological dimension.…”

“…For the associative operad, Kontsevich showed that the homology of hO ∞ is related to the rational homology of the mapping class group [4,17,18]. In Subsection 7.2, we adapt an argument of Morita, Sakasai and Suzuki [25] to compute the first homology group of the hairy O-graph complex in the case O = Assoc. Their argument is part of their result that the abelianization of hO ∞ is precisely equal to the piece determined by Morita [24], and in Theorem 7.9 we show how this also follows from the first homology computation together with injectivity of the trace map.…”

Tropical Curves and Covers and Their Moduli Spaces

“…While this holds true for k = 1 [20,43], Theorem 5.1 shows that it is wrong for k = 2. Morita, Sakasai and Suzuki [44] pointed out that Conjecture 5.2 is implied by a more general conjecture made by Kontsevich around 25 years ago [37].…”