Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I

Abstract

The automorphism group $\operatorname{Aut}(F_n)$ of the free group $F_n$ acts on a space $A_d(n)$ of Jacobi diagrams of degree $d$ on $n$ oriented arcs. We study the $\operatorname{Aut}(F_n)$-module structure of $A_d(n)$ by using two actions on the associated graded vector space of $A_d(n)$: an action of the general linear group $\operatorname{GL}(n,Z)$ and an action of the graded Lie algebra $\mathrm{gr}(\operatorname{IA}(n))$ of the IA-automorphism group $\operatorname{IA}(n)$ of $F_n$ associated with its lower central series. We extend the action of $\mathrm{gr}(\operatorname{IA}(n))$ to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of $F_n$. By using this action, we study the $\operatorname{Aut}(F_n)$-module structure of $A_d(n)$. We obtain an indecomposable decomposition of $A_d(n)$ as $\operatorname{Aut}(F_n)$-modules for $n\geq 2d$. Moreover, we obtain the radical filtration of $A_d(n)$ for $n\geq 2d$ and the socle of $A_3(n)$.