Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I
Annales de l'Institut Fourier, Volume 73 (2023) no. 4, pp. 1489-1532.

We consider an action of the automorphism group Aut(F n ) of the free group F n 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,) and an action of the graded Lie algebra of the IA-automorphism group of F n associated with its lower central series. We use these actions on B d (n) to study the Aut(F n )-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(F n )-module structure of A d (n) for all n0.

Nous considérons une action du groupe d’automorphisme s Aut(F n ) du groupe libre F n 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,) et une action de l’algèbre de Lie graduée du groupe d’automorphismes IA de F n associée à sa série centrale inférieure. Nous utilisons ces actions sur B d (n) pour étudier la structure de Aut(F n )-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(F n )-module de A d (n) pour tout n0.

Published online:
DOI: 10.5802/aif.3544
Classification: 20F12, 20F28, 57K16
Keywords: Jacobi diagrams, Automorphism groups of free groups, General linear groups, IA-automorphism groups of free groups.
Mot clés : Diagrammes de Jacobi, groupes d’automorphisme des groupes libres, groupes linéaires généraux, groupes d’automorphisme IA des groupes libres.
Katada, Mai 1

1 Department of Mathematics Kyoto University Kyoto 606-8502 (Japan)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Actions of automorphism groups of free groups on spaces of {Jacobi} diagrams. {I}},
     journal = {Annales de l'Institut Fourier},
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Katada, Mai. Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I. Annales de l'Institut Fourier, Volume 73 (2023) no. 4, pp. 1489-1532. doi : 10.5802/aif.3544.

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