Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I
Abstract
We consider an action of the automorphism group Aut(Fn) of the free group Fn of rank n on the filtered vector space Ad(n) of Jacobi diagrams of degree d on n oriented arcs. This action induces on the associated graded vector space of Ad(n), which is identified with the space Bd(n) of open Jacobi diagrams, an action of the general linear group GL(n,Z) and an action of the graded Lie algebra of the IA-automorphism group of Fn associated with its lower central series. We use these actions on Bd(n) to study the Aut(Fn)-module structure of Ad(n). In particular, we consider the case where d=2 in detail and give an indecomposable decomposition of A2(n). We also construct a polynomial functor Ad of degree 2d from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces, which includes the Aut(Fn)-module structure of Ad(n) for all n≥ 0.