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 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 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 B d (n) to study the Aut(Fn)-module structure of A d (n). In particular, we consider the case where d = 2 in detail and give an indecomposable decomposition of A 2 (n). We also construct 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(Fn)-module structure of A d (n) for all n 0.Résumé. -Nous considérons une action du groupe d'automorphisme s Aut(Fn) du groupe libre Fn de rang n sur l'espace vectoriel filtré A d (n) des diagrammes de Jacobi de degré d sur n arcs orientés. Cette action induit sur l'espace vectoriel gradué associé de A d (n), qui est identifié à l'espace B d (n) des diagrammes de Jacobi ouverts, une action du groupe linéaire général GL(n, Z) et une action de l'algèbre de Lie graduée du groupe d'automorphismes IA de Fn associée à sa série centrale inférieure. Nous utilisons ces actions sur B d (n) pour étudier la structure de Aut(Fn)-module de A d (n). En particulier, nous considérons en détail le cas où d = 2 et donnons une décomposition indécomposable de A 2 (n). Nous construisons également un foncteur polynomial A d de degré 2d de la catégorie opposée de la catégorie des groupes libres finiment engendrés à la catégorie des espaces vectoriels filtrés, qui inclut la structure de Aut(Fn)-module de A d (n) pour tout n 0.

Actions of Automorphism Groups of Free Groups on Spaces of Jacobi Diagrams. Ii

“…Self duality of A ′′ 2 (3) In this section, we will prove that A ′′ 2 (3) is a self-dual Aut(F 3 )-module, which appeared in our previous paper [16,Remark 7.13] as a conjecture.…”

“…Here we recall the vector spaces of Jacobi diagrams and an action of Aut(F n ) on the spaces of Jacobi diagrams, which was studied in [16,14].…”

“…In this paper, we will study the homology of IA n with coefficients in the polynomial Aut(F n )-module A 2 (n) of Jacobi diagrams of degree 2 on ncomponent oriented arcs. The Aut(F n )-module structure on the spaces A d (n) of Jacobi diagrams of degree d was introduced and studied in [16,14]. (See Section 2.2 for details.…”

“…We have an action of the handlebody group H n on the space B(n) of n-component bottom tangles [13]. The action of Aut(F n ) on A d (n) can be interpreted as a restriction of this action of H n on B(n) via the surjective group homomorphism from H n to Aut(F n ) that is induced by the action of H n on the fundamental group of the handlebody [16]. Therefore, the homology of Aut(F n ) with coefficients in A d (n) can be regarded as an approximation of the homology of H n with coefficients in B(n).…”

“…It follows from the work of Kielak [18] that for n = 3, their module U I 3 gives the first example of such Out(F 3 )-modules of the smallest dimension 7. In [16], we proved that the Aut(F n )-action on A d (n) induces an action of the outer automorphism group Out(F n ) of F n . The dimension of A 2,1 (3) and…”