Geodesic
Naturally graded Lie algebras of slow growth
Sbornik. Mathematics, Tome 210 (2019) no. 6, pp. 862-909 Cet article a éte moissonné depuis la source Math-Net.Ru

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A pro-nilpotent Lie algebra $\mathfrak g$ is said to be naturally graded if it is isomorphic to its associated graded Lie algebra $\operatorname{gr}\mathfrak g$ with respect to the filtration by the ideals in the lower central series. Finite-dimensional naturally graded Lie algebras are known in sub-Riemannian geometry and geometric control theory, where they are called Carnot algebras. We classify the finite-dimensional and infinite-dimensional naturally graded Lie algebras $\mathfrak g=\bigoplus_{i=1}^{+\infty}\mathfrak g_i$ with the property $$ \dim\mathfrak g_i+\dim\mathfrak g_{i+1}\leqslant3,\qquad i\geqslant1. $$ An arbitrary Lie algebra $\mathfrak g=\bigoplus_{i=1}^{+\infty}\mathfrak g_i$ of this class is generated by the two-dimensional subspace $\mathfrak g_1$, and the corresponding growth function $F_\mathfrak g^\mathrm{gr}(n)$ satisfies the bound $F_\mathfrak g^\mathrm{gr}(n)\leqslant3n/2+1$. Bibliography: 32 titles.
Keywords: graded Lie algebra, Carnot algebra, Kac-Moody algebras, central extension
Mots-clés : automorphism. 
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     author = {D. V. Millionshchikov},
     title = {Naturally graded {Lie} algebras of slow growth},
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     url = {http://geodesic.mathdoc.fr/item/SM_2019_210_6_a4/}
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D. V. Millionshchikov. Naturally graded Lie algebras of slow growth. Sbornik. Mathematics, Tome 210 (2019) no. 6, pp. 862-909. http://geodesic.mathdoc.fr/item/SM_2019_210_6_a4/

[1] A. A. Agrachev, “Topics in sub-Riemannian geometry”, Russian Math. Surveys, 71:6 (2016), 989–1019 | DOI | DOI | MR | Zbl

[2] T. Barron, D. Kerner, M. Tvalavadze, “On varieties of Lie algebras of maximal class”, Canad. J. Math., 67:1 (2015), 55–89 | DOI | MR | Zbl

[3] Y. Benoist, “Une nilvariété non affine”, J. Differential Geom., 41:1 (1995), 21–52 | DOI | MR | Zbl

[4] V. M. Buchstaber, “Polynomial Lie algebras and the Zelmanov–Shalev theorem”, Russian Math. Surveys, 72:6 (2017), 1168–1170 | DOI | DOI | MR | Zbl

[5] A. Caranti, S. Mattarei, M. F. Newman, C. M. Scoppola, “Thin groups of prime-power order and thin Lie algebras”, Quart. J. Math. Oxford Ser. (2), 47:3 (1996), 279–296 | DOI | MR | Zbl

[6] A. Caranti, S. Mattarei, M. F. Newman, “Graded Lie algebras of maximal class”, Trans. Amer. Math. Soc., 349:10 (1997), 4021–4051 | DOI | MR | Zbl

[7] A. Fialovski, “Classification of graded Lie algebras with two generators”, Moscow Univ. Math. Bull., 38:2 (1983), 76–79 | MR | Zbl

[8] D. B. Fuks, Cohomology of infinite-dimensional Lie algebras, Contemp. Soviet Math., Consultants Bureau, New York, 1986, xii+339 pp. | MR | MR | Zbl | Zbl

[9] D. Fuchs, C. Wilmarth, “Laplacian spectrum for the nilpotent Kac–Moody Lie algebras”, Pacific J. Math., 247:2 (2010), 323–334 | DOI | MR | Zbl

[10] L. García Vergnolle, “Sur les algèbres de Lie quasi-filiformes admettant un tore de dérivations”, Manuscripta Math., 124:4 (2007), 489–505 | DOI | MR | Zbl

[11] H. Garland, “Dedekind's $\eta$-function and the cohomology of infinite-dimensional Lie algebras”, Proc. Nat. Acad. Sci. U.S.A., 72:7 (1975), 2493–2495 | DOI | MR | Zbl

[12] J. R. Gómez, A. Jimenéz-Merchán, J. Reyes, “Maximum length filiform Lie algebras”, Extracta Math., 16:3 (2001), 405–421 (Spanish) | MR | Zbl

[13] J. R. Gómez, A. Jiménez-Merchán, “Naturally graded quasi-filiform Lie algebras”, J. Algebra, 256:1 (2002), 211–228 | DOI | MR | Zbl

[14] M. Gromov, “Carnot–Carathéodory spaces seen from within”, Sub-Riemannian geometry, Progr. Math., 144, Birkhäuser, Basel, 1996, 79–323 | MR | Zbl

[15] V. G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, Cambridge, 1990, xxii+400 pp. | DOI | MR | MR | Zbl | Zbl

[16] V. G. Kac, “Some problems on infinite dimensional Lie algebras and their representations”, Lie algebras and related topics (New Brunswick, NJ, 1981), Lecture Notes in Math., 933, Springer, Berlin–New York, 1982, 117–126 | DOI | MR | Zbl

[17] G. R. Krause, T. H. Lenagan, Growth of algebras and Gelfand–Kirillov dimension, Grad. Stud. Math., 22, Rev. ed., Amer. Math. Soc., Providence, RI, 2000, x+212 pp. | MR | Zbl

[18] S. Kumar, “Geometry of Schubert cells and cohomology of Kac–Moody Lie algebras”, J. Differential Geom., 20:2 (1984), 389–431 | DOI | MR | Zbl

[19] J. Lepowsky, “Generalized Verma modules, loop spaces cohomology and MacDonald-type identities”, Ann. Sci. École Norm. Sup. (4), 12:2 (1979), 169–234 | DOI | MR | Zbl

[20] J. Lepowsky, S. Milne, “Lie algebraic approaches to classical partition identities”, Adv. in Math., 29:1 (1978), 15–59 | DOI | MR | Zbl

[21] O. Mathieu, “Classification of simple graded Lie algebras of finite growth”, Invent. Math., 108 (1990), 455–519 | DOI | MR | Zbl

[22] D. V. Millionshchikov, “Filiform $\mathbb N$-graded Lie algebras”, Russian Math. Surveys, 57:2 (2002), 422–424 | DOI | DOI | MR | Zbl

[23] D. V. Millionshchikov, “Graded filiform Lie algebras and symplectic nilmanifolds”, Geometry, topology, and mathematical physics, Amer. Math. Soc. Transl. Ser. 2, 212, Adv. Math. Sci., 55, Amer. Math. Soc., Providence, RI, 2004, 259–279 | DOI | MR | Zbl

[24] D. V. Millionshchikov, “Characteristic Lie algebras of the sinh-Gordon and Tzitzeica equations”, Russian Math. Surveys, 72:6 (2017), 1174–1176 | DOI | DOI | MR | Zbl

[25] J. Milnor, “On fundamental groups of complete affinely flat manifolds”, Adv. in Math., 25:2 (1977), 178–187 | DOI | MR | Zbl

[26] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Math. Surveys Monogr., 91, Amer. Math. Soc., Providence, RI, 2002, xx+259 pp. | MR | Zbl

[27] V. V. Morozov, “Klassifikatsiya nilpotentnykh algebr Li shestogo poryadka”, Izv. vuzov. Matem., 1958, no. 4, 161–171 | MR | Zbl

[28] V. Petrogradsky, “Nil Lie $p$-algebras of slow growth”, Comm. Algebra, 45:7 (2017), 2912–2941 | DOI | MR | Zbl

[29] A. V. Rozhkov, “Lower central series of a group of tree automorphisms”, Math. Notes, 60:2 (1996), 165–174 | DOI | DOI | MR | Zbl

[30] A. Shalev, E. I. Zelmanov, “Narrow Lie algebras: a coclass theory and a characterization of the Witt algebra”, J. Algebra, 189:2 (1997), 294–331 | DOI | MR | Zbl

[31] A. Shalev, E. I. Zelmanov, “Narrow algebras and groups”, J. Math. Sci. (N. Y.), 93:6 (1999), 951–963 | DOI | MR | Zbl

[32] M. Vergne, “Cohomologie des algèbres de Lie nilpotentes. Application à l'étude de la variété des algèbres de Lie nilpotentes”, Bull. Soc. Math. France, 98 (1970), 81–116 | DOI | MR | Zbl