On the derivation algebra of the free Lie algebra and trace maps

Enomoto, Naoya¹ ; Satoh, Takao²

¹Department of Mathematics, Graduate School of Science, Kyoto University, Kitashirakawaoiwake-cho, Sakyo-ku, Kyoto city 606-8502, Japan
²Department of Mathematics, Faculty of Science Division II, Tokyo University of Science, 1-3 Kagurazaka, Shinjyuku-ku, Tokyo 162-8601, Japan

Algebraic and Geometric Topology, Tome 11 (2011) no. 5, pp. 2861-2901

Résumé

We mainly study the derivation algebra of the free Lie algebra and the Chen Lie algebra generated by the abelianization H of a free group, and trace maps. To begin with, we give the irreducible decomposition of the derivation algebra as a GL(n,Q)–module via the Schur–Weyl duality and some tensor product theorems for GL(n,Q). Using them, we calculate the irreducible decomposition of the images of the Johnson homomorphisms of the automorphism group of a free group and a free metabelian group.

Next, we consider some applications of trace maps: Morita’s trace map and the trace map for the exterior product of H. First, we determine the abelianization of the derivation algebra of the Chen Lie algebra as a Lie algebra, and show that the abelianization is given by the degree one part and Morita’s trace maps. Second, we consider twisted cohomology groups of the automorphism group of a free nilpotent group. In particular, we show that the trace map for the exterior product of H defines a nontrivial twisted second cohomology class of it.

Détail

DOI : 10.2140/agt.2011.11.2861

Classification : 17B40, 20C15, 20F28
Keywords: derivation, free Lie algebra, Chen Lie algebra, trace map, Johnson homomorphism, automorphism group, free nilpotent group

Bibliographie

[1] S Andreadakis, On the automorphisms of free groups and free nilpotent groups, Proc. London Math. Soc. $(3)$ 15 (1965) 239

[2] S Andreadakis, Generators for $\mathrm{Aut} G,$ $G$ free nilpotent, Arch. Math. $($Basel$)$ 42 (1984) 296

[3] S Bachmuth, Induced automorphisms of free groups and free metabelian groups, Trans. Amer. Math. Soc. 122 (1966) 1

[4] J S Birman, Braids, links, and mapping class groups, Annals of Math. Studies 82, Princeton Univ. Press (1974)

[5] N Bourbaki, Lie groups and Lie algebras. Chapters 1–3, Elements of Math., Springer (1989)

[6] R M Bryant, C K Gupta, Automorphism groups of free nilpotent groups, Arch. Math. $($Basel$)$ 52 (1989) 313

[7] K T Chen, Integration in free groups, Ann. of Math. $(2)$ 54 (1951) 147

[8] F R Cohen, J Pakianathan, On automorphism groups of free groups and their nilpotent quotients, preprint

[9] F R Cohen, J Pakianathan, On subgroups of the automorphism group of a free group and associated graded Lie algebras, preprint

[10] H S M Coxeter, W O J Moser, Generators and relations for discrete groups, Ergebnisse der Math. und ihrer Grenzgebiete 14, Springer (1972)

[11] B Farb, Automorphisms of $F_n$ which act trivially on homology, in preparation

[12] A M Garsia, Combinatorics of the free Lie algebra and the symmetric group, from: "Analysis, et cetera" (editors P H Rabinowitz, E Zehnder), Academic Press (1990) 309

[13] A V Goryaga, Generating elements of the group of automorphisms of a free nilpotent group, Algebra i Logika 15 (1976) 458, 488

[14] R Hain, Infinitesimal presentations of the Torelli groups, J. Amer. Math. Soc. 10 (1997) 597

[15] D Johnson, An abelian quotient of the mapping class group $\mathcal{I}_{g}$, Math. Ann. 249 (1980) 225

[16] D Johnson, The structure of the Torelli group—III: The abelianization of $\mathscr T$, Topology 24 (1985) 127

[17] M D Kassabov, On the automorphism tower of free nilpotent groups, PhD thesis, Yale University (2003)

[18] N Kawazumi, Cohomological aspects of Magnus expansions

[19] A A Kljačko, Lie elements in a tensor algebra, Sibirsk. Mat. Ž. 15 (1974) 1296, 1430

[20] K Koike, On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters, Adv. Math. 74 (1989) 57

[21] M Kontsevich, Formal (non)commutative symplectic geometry, from: "The Gel$'`$fand Mathematical Seminars, 1990–1992" (editors L Corwin, I Gel′‘fand, J Lepowsky), Birkhäuser (1993) 173

[22] M Kontsevich, Feynman diagrams and low-dimensional topology, from: "First European Congress of Mathematics, Vol. II (Paris, 1992)" (editors A Joseph, F Mignot, F Murat, B Prum, R Rentschler), Progr. Math. 120, Birkhäuser (1994) 97

[23] W Kraśkiewicz, J Weyman, Algebra of coinvariants and the action of a Coxeter element, Bayreuth. Math. Schr. (2001) 265

[24] S Krstić, J Mccool, The non-finite presentability of $\mathrm{IA}(F_3)$ and $\mathrm{GL}_2(\mathbf{Z}[t,t^{-1}])$, Invent. Math. 129 (1997) 595

[25] W Lin, Presentations of the automorphism groups of free nilpotent groups, Comm. Algebra 24 (1996) 3403

[26] W Magnus, Über $n$–dimensionale Gittertransformationen, Acta Math. 64 (1935) 353

[27] S Morita, Families of Jacobian manifolds and characteristic classes of surface bundles. I, Ann. Inst. Fourier (Grenoble) 39 (1989) 777

[28] S Morita, Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993) 699

[29] S Morita, The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993) 197

[30] S Morita, Structure of the mapping class groups of surfaces: a survey and a prospect, from: "Proceedings of the Kirbyfest (Berkeley, CA, 1998)" (editors J Hass, M Scharlemann), Geom. Topol. Monogr. 2 (1999) 349

[31] S Morita, Cohomological structure of the mapping class group and beyond, from: "Problems on mapping class groups and related topics" (editor B Farb), Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 329

[32] J Nielsen, Die Isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924) 169

[33] A Pettet, The Johnson homomorphism and the second cohomology of $\mathrm{IA}_n$, Algebr. Geom. Topol. 5 (2005) 725

[34] C Reutenauer, Free Lie algebras, London Math. Soc. Monogr. New Series 7, Oxford Science Publ., The Clarendon Press, Oxford Univ. Press (1993)

[35] T Satoh, New obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group, J. London Math. Soc. $(2)$ 74 (2006) 341

[36] T Satoh, Twisted first homology groups of the automorphism group of a free group, J. Pure Appl. Algebra 204 (2006) 334

[37] T Satoh, The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group, Trans. Amer. Math. Soc. 361 (2009) 2085

[38] T Satoh, On the lower central series of the IA-automorphism group of a free group, preprint (2009)

[39] T Satoh, On the fourth Johnson homomorphism of the automorphism group of a free group, J. Algebra 323 (2010) 3182

[40] T Satoh, A reduction of the target of the Johnson homomorphisms of the automorphism group of a free group, Trans. Amer. Math. Soc. 363 (2011) 1631

[41] E Witt, Treue Darstellung Liescher Ringe, J. Reine Angew. Math. 177 (1937) 152