On derivations of free algebras over operads and the generalized divergence

Abstract

For $\mathcal{O}$ a reduced operad, a generalized divergence from the derivations of a free $\mathcal{O}$-algebra to a suitable trace space is constructed. In the case of the Lie operad, this corresponds to Satoh's trace map and, for the associative operad, to the double divergence of Alekseev, Kawazumi, Kuno and Naef. The generalized divergence is shown to be a $1$-cocycle for the usual Lie algebra structure on derivations. These results place the previous constructions into a unified framework; moreover, they are natural with respect to the operad. An important new ingredient is the use of naturality with respect to the category of finite-rank free modules and split monomorphisms over a commutative ring $R$. This allows the notion of torsion for such functors to be exploited. Supposing that the ring $R$ is a PID and that the operad $\mathcal{O}$ is binary, the main result relates the kernel of the generalized divergence to the sub Lie algebra of the Lie algebra of derivations that is generated by the elements of degree one with respect to the grading induced by arity.