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

Katada, Mai

doi:10.5802/aif.3544

Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I
[Actions des groupes d’automorphismes des groupes libres sur les espaces de diagrammes de Jacobi. I]

Katada, Mai ¹

¹ Department of Mathematics Kyoto University Kyoto 606-8502 (Japan)

Annales de l'Institut Fourier, Tome 73 (2023) no. 4, pp. 1489-1532.

Résumé
Abstract

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 $GL (n, ℤ)$ 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 $2 d$ 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 $n \geq 0$ .

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 $GL (n, ℤ)$ 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 $2 d$ 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 $n \geq 0$ .

Reçu le : 2021-09-21
Révisé le : 2021-12-02
Accepté le : 2022-02-17
Publié le : 2023-07-07

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.

Affiliations des auteurs :

Katada, Mai ¹

¹ Department of Mathematics Kyoto University Kyoto 606-8502 (Japan)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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},
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Katada, Mai. Actions of automorphism groups of free groups on spaces of Jacobi diagrams. I. Annales de l'Institut Fourier, Tome 73 (2023) no. 4, pp. 1489-1532. doi : 10.5802/aif.3544. https://aif.centre-mersenne.org/articles/10.5802/aif.3544/

Bibliographie
Cité par

[1] Andreadakis, S. On the automorphisms of free groups and free nilpotent groups, Proc. London Math. Soc. (3), Volume 15 (1965), pp. 239-268 | DOI | MR | Zbl

[2] Bar-Natan, Dror On the Vassiliev knot invariants, Topology, Volume 34 (1995) no. 2, pp. 423-472 | DOI | MR | Zbl

[3] Bar-Natan, Dror Vassiliev homotopy string link invariants, J. Knot Theory Ramifications, Volume 4 (1995) no. 1, pp. 13-32 | DOI | MR | Zbl

[4] Bar-Natan, Dror Some computations related to Vassiliev invariants (1996) (preprint)

[5] Conant, Jim; Kassabov, Martin Hopf algebras and invariants of the Johnson cokernel, Algebr. Geom. Topol., Volume 16 (2016) no. 4, pp. 2325-2363 | DOI | MR | Zbl

[6] Eilenberg, Samuel; Mac Lane, Saunders On the groups $H (Π, n)$ . II. Methods of computation, Ann. of Math. (2), Volume 60 (1954), pp. 49-139 | DOI | MR | Zbl

[7] Fulton, William Young tableaux, London Mathematical Society Student Texts, 35, Cambridge University Press, Cambridge, 1997, x+260 pages (With applications to representation theory and geometry) | MR | Zbl

[8] Fulton, William; Harris, Joe Representation theory. A first course, Graduate Texts in Mathematics, 129, Springer-Verlag, New York, 1991, xvi+551 pages (Readings in Mathematics) | DOI | MR | Zbl

[9] Habegger, Nathan; Lin, Xiao-Song The classification of links up to link-homotopy, J. Amer. Math. Soc., Volume 3 (1990) no. 2, pp. 389-419 | DOI | MR | Zbl

[10] Habiro, Kazuo Bottom tangles and universal invariants, Algebr. Geom. Topol., Volume 6 (2006), pp. 1113-1214 | DOI | MR | Zbl

[11] Habiro, Kazuo; Massuyeau, Gwénaël Generalized Johnson homomorphisms for extended N-series, J. Algebra, Volume 510 (2018), pp. 205-258 | DOI | MR | Zbl

[12] Habiro, Kazuo; Massuyeau, Gwénaël The Kontsevich integral for bottom tangles in handlebodies, Quantum Topol., Volume 12 (2021) no. 4, pp. 593-703 | DOI | MR | Zbl

[13] Hartl, Manfred; Pirashvili, Teimuraz; Vespa, Christine Polynomial functors from algebras over a set-operad and nonlinear Mackey functors, Int. Math. Res. Not. IMRN (2015) no. 6, pp. 1461-1554 | DOI | MR | Zbl

[14] Kassel, Christian Quantum groups, Graduate Texts in Mathematics, 155, Springer-Verlag, New York, 1995, xii+531 pages | DOI | MR | Zbl

[15] Katada, Mai Actions of automorphism groups of free groups on spaces of Jacobi diagrams. II., Journal of the Institute of Mathematics of Jussieu (2022), p. 1â69 | DOI

[16] Kawazumi, Nariya Cohomological aspects of Magnus expansions (2005) (https://arxiv.org/abs/math/0505497)

[17] Kielak, Dawid Low-dimensional free and linear representations of $Out (F_{3})$ , J. Group Theory, Volume 18 (2015) no. 6, pp. 913-949 | DOI | MR | Zbl

[18] Kontsevich, Maxim Vassiliev’s knot invariants, I. M. Gelʼfand Seminar (Adv. Soviet Math.), Volume 16, Amer. Math. Soc., Providence, RI, 1993, pp. 137-150 | MR | Zbl

[19] Magnus, Wilhelm Über $n$ -dimensionale Gittertransformationen, Acta Math., Volume 64 (1935) no. 1, pp. 353-367 | DOI | MR | Zbl

[20] Magnus, Wilhelm; Karrass, Abraham; Solitar, Donald Combinatorial group theory. Presentations of groups in terms of generators and relations, Dover Publications, Inc., New York, 1976, xii+444 pages | MR | Zbl

[21] Nielsen, Jakob Die Gruppe der dreidimensionalen Gittertransformationen, Det Kgl. Danske Videnskabernes Selskab., Mathematisk-fysiske Meddelelser, Volume 5 (1924) no. 12, pp. 1-29

[22] Nielsen, Jakob Die Isomorphismengruppe der freien Gruppen, Math. Ann., Volume 91 (1924) no. 3-4, pp. 169-209 | DOI | MR

[23] Ohtsuki, Tomotada Quantum invariants. A study of knots, 3-manifolds, and their sets, Series on Knots and Everything, 29, World Scientific Publishing Co., Inc., River Edge, NJ, 2002, xiv+489 pages | MR | Zbl

[24] Powell, Geoffrey; Vespa, Christine Higher Hochschild homology and exponential functors (2018) (https://arxiv.org/abs/1802.07574)

[25] Turchin, Victor; Willwacher, Thomas Hochschild–Pirashvili homology on suspensions and representations of $Out (F_{n})$ , Ann. Sci. Éc. Norm. Supér. (4), Volume 52 (2019) no. 3, pp. 761-795 | DOI | MR | Zbl

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