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

Abstract

The automorphism group Aut(Fn) of the free group Fn acts on a space A d (n) of Jacobi diagrams of degree d on n oriented arcs. We study the Aut(Fn)-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 GL(n, Z) and an action of the graded Lie algebra gr(IA(n)) of the IA-automorphism group IA(n) of Fn associated with its lower central series. We extend the action of gr(IA(n)) to an action of the associated graded Lie algebra of the Andreadakis filtration of the endomorphism monoid of Fn. By using this action, we study the Aut(Fn)-module structure of A d (n). We obtain a direct decomposition of A d (n) as Aut(Fn)-modules for general d, which is indecomposable for d = 3, 4. Moreover, we obtain the radical of A d (n) for general d and the socle of A 3 (n). Contents 1. Introduction 1 2. Preliminaries 6 3. Andreadakis filtration E * (n) of End(F n ) 11 4. Action of gr(E * (n)) on B d (n) 17 5. Contraction map 26 6. Correspondence between the map βr d,k and the map γ r d,k 30 7. The GL(V n )-module structure of B d (n) 33 8. The Aut(F n )-module structure of A d (n) 40 9. The functor A d 50 Appendix A. Presentation of the category A L 51 References 54

On the functors associated with beaded open Jacobi diagrams

“…She also gives a complete description of the functor A 1 (0, −) and the more complicated case of the functor A 2 (0, −). In [Katb,Theorem 10.1], she gives a direct decomposition of the functor A d (0, −) for d ≥ 1 (see also Proposition 6.16 for another proof) and obtains in [Katb,Proposition 10.2] that this is an indecomposable decomposition.…”

“…By [Ves18, Theorem 4.2], Ext 1 F (gr;K) (F, S 2d • a) = 0, for F a polynomial functor so S 2d • a is an injective object in the category of polynomial functors on gr, so S 2d • a # is a projective object in the category of polynomial functors on gr op . This allows us to give another proof of [Katb,Theorem 10.1]. Proposition 6.16.…”

“…In particular, we have A 1 (0, −) ≃ S 2 • a # . In Section 6.7 we give another proof, based on [PV18], of [Katb,Theorem 10.1], giving a direct sum decomposition of the functor A d (0, −) in the category of functors on gr op . One of the main ingredients of this paper is the use of the equivalence of categories given by Powell in [Powa]:…”

Outer functors and a general operadic framework

“…The original motivating work, [PV18], shows how outer functors appear naturally in connection with the study of the higher Hochschild homology of a wedge of circles, building upon ideas of Turchin and Willwacher [TW19]. Recently, Katada [Kat21a,Kat21b] has exhibited another important and, at first sight, surprising context in which they arise. Her work studies functors on gr op constructed from Jacobi diagrams, based on constructions of Habiro and Massuyeau [HM21].…”