Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I (Q6169424)

If \(F_{n}\) is a free group of finite rank \(n\geq 1\), then \(\Aut(F_{n})\) acts on the filtered vector space \(A_{d}(n)\) of Jacobi diagrams of degree \(d\) on \(n\) oriented arcs. This action induces on the associated graded vector space of \(A_{d}(n)\), which is identified with the space \(B_{d}(n)\) of open Jacobi diagrams, an action of \(\mathrm{GL}(n,\mathbb{Z})\) and an action of the graded Lie algebra of the IA-automorphism group of \(F_{n}\) associated with its lower central series. In the paper under review, the author uses these actions on \(B_{d}(n)\) to study the \(\Aut(F_{n})\)-module structure of \(A_{d}(n)\). In particular he gives an indecomposable decomposition of \(A_{2}(n)\) and studies the case \(d=2\) in great detail. Furthermore, the author constructs a polynomial functor \(A_{d}\) 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 \(\Aut(F_{n})\)-module structure of \(A_{d}(n)\).