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

Katada, Mai

doi:10.5802/aif.3544

Katada, Mai ¹

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

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

Abstract (Original language)
Résumé

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 $n \geq 0$ .

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 $n \geq 0$ .

Article information
Cite this article

Received: 2021-09-21
Revised: 2021-12-02
Accepted: 2022-02-17
Published online: 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.
Mots-clés : Diagrammes de Jacobi, groupes d’automorphisme des groupes libres, groupes linéaires généraux, groupes d’automorphisme IA des groupes libres.

Author's affiliations:

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

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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References
Cited by

[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 | MR | DOI | 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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