We consider an action of the automorphism group of the free group of rank on the filtered vector space of Jacobi diagrams of degree on oriented arcs. This action induces on the associated graded vector space of , which is identified with the space of open Jacobi diagrams, an action of the general linear group and an action of the graded Lie algebra of the IA-automorphism group of associated with its lower central series. We use these actions on to study the -module structure of . In particular, we consider the case where in detail and give an indecomposable decomposition of . We also construct a polynomial functor of degree from the opposite category of the category of finitely generated free groups to the category of filtered vector spaces, which includes the -module structure of for all .
Nous considérons une action du groupe d’automorphisme s du groupe libre de rang sur l’espace vectoriel filtré des diagrammes de Jacobi de degré sur arcs orientés. Cette action induit sur l’espace vectoriel gradué associé de , qui est identifié à l’espace des diagrammes de Jacobi ouverts, une action du groupe linéaire général et une action de l’algèbre de Lie graduée du groupe d’automorphismes IA de associée à sa série centrale inférieure. Nous utilisons ces actions sur pour étudier la structure de -module de . En particulier, nous considérons en détail le cas où et donnons une décomposition indécomposable de . Nous construisons également un foncteur polynomial de degré de la catégorie opposée de la catégorie des groupes libres finiment engendrés à la catégorie des espaces vectoriels filtrés, qui inclut la structure de -module de pour tout .
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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
@article{AIF_2023__73_4_1489_0, author = {Katada, Mai}, title = {Actions of automorphism groups of free groups on spaces of {Jacobi} diagrams. {I}}, journal = {Annales de l'Institut Fourier}, pages = {1489--1532}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {73}, number = {4}, year = {2023}, doi = {10.5802/aif.3544}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3544/} }
TY - JOUR AU - Katada, Mai TI - Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I JO - Annales de l'Institut Fourier PY - 2023 SP - 1489 EP - 1532 VL - 73 IS - 4 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3544/ DO - 10.5802/aif.3544 LA - en ID - AIF_2023__73_4_1489_0 ER -
%0 Journal Article %A Katada, Mai %T Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I %J Annales de l'Institut Fourier %D 2023 %P 1489-1532 %V 73 %N 4 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3544/ %R 10.5802/aif.3544 %G en %F AIF_2023__73_4_1489_0
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. https://aif.centre-mersenne.org/articles/10.5802/aif.3544/
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