| Title:
|
An overview of free nilpotent Lie algebras (English) |
| Author:
|
Benito, Pilar |
| Author:
|
de-la-Concepción, Daniel |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
55 |
| Issue:
|
3 |
| Year:
|
2014 |
| Pages:
|
325-339 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Any nilpotent Lie algebra is a quotient of a free nilpotent Lie algebra of the same nilindex and type. In this paper we review some nice features of the class of free nilpotent Lie algebras. We will focus on the survey of Lie algebras of derivations and groups of automorphisms of this class of algebras. Three research projects on nilpotent Lie algebras will be mentioned. (English) |
| Keyword:
|
Lie algebra |
| Keyword:
|
Levi subalgebra |
| Keyword:
|
nilpotent |
| Keyword:
|
free nilpotent |
| Keyword:
|
derivation |
| Keyword:
|
automorphism |
| Keyword:
|
representation |
| MSC:
|
17B10 |
| MSC:
|
17B30 |
| idZBL:
|
Zbl 06391546 |
| idMR:
|
MR3225613 |
| . |
| Date available:
|
2015-01-19T10:49:49Z |
| Last updated:
|
2016-10-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/143811 |
| . |
| Reference:
|
[1] Ancochea-Bermúdez J.M., Campoamor-Stursberg R., García Vergnolle L.: Classification of Lie algebras with naturally graded quasi-filiform nilradicals.J. Geom. Phys. 61 (2011), no. 11, 2168–2186. Zbl 1275.17023, MR 2827117, 10.1016/j.geomphys.2011.06.015 |
| Reference:
|
[2] Ancochea-Bermúdez J.M., Campoamor-Stursberg R., García Vergnolle L.: Indecomposable Lie algebras with nontrivial Levi decomposition cannot have filiform radical.Int. Math. Forum 1 (2006), no. 7, 309–316. Zbl 1142.17300, MR 2237946 |
| Reference:
|
[3] Auslander L., Scheuneman J.: On certain automorphisms of nilpotent Lie groups.Global Analysis: Proc. Symp. Pure Math. 14 (1970), 9–15. Zbl 0223.22014, MR 0270395 |
| Reference:
|
[4] Del Barco V.J., Ovando G.P.: Free nilpotent Lie algebras admitting ad-invariant metrics.J. Algebra 366 (2012), 205–216. MR 2942650, 10.1016/j.jalgebra.2012.05.016 |
| Reference:
|
[5] Benito P., de-la-Concepción D.: On Levi extensions of nilpotent Lie algebras.Linear Algebra Appl. 439 (2013), no. 5, 1441–1457. Zbl 1281.17014, MR 3067814, 10.1016/j.laa.2013.04.027 |
| Reference:
|
[6] Benito P., de-la-Concepción D.: A note on extensions of nilpotent Lie algebras of Type $2$.arXiv:1307.8419. |
| Reference:
|
[7] Cui R., Wang Y., Deng S.: Solvable Lie algebras with quasifiliforms nilradicals.Comm. Algebra 36 (2008), 4052–4067. MR 2460402, 10.1080/00927870802174629 |
| Reference:
|
[8] Dengyin W., Ge H., Li X.: Solvable extensions of a class of nilpotent linear Lie algebras.Linear Algebra Appl. 437 (2012), 14–25. MR 2917429 |
| Reference:
|
[9] Favre G., Santharoubane L.: Symmetric, invariant, non-degenerate bilinear form on a Lie algebra.J. Algebra 105 (1987), no. 2, 451–464. Zbl 0608.17007, MR 0873679, 10.1016/0021-8693(87)90209-2 |
| Reference:
|
[10] Figueroa-O'Farrill J.M., Stanciu S.: On the structure of symmetric self-dual Lie algebras.J. Math. Phys. 37 (1996), 4121–4134. Zbl 0863.17004, MR 1400838, 10.1063/1.531620 |
| Reference:
|
[11] Gauger M.A.: On the classification of metabelian Lie algebras.Trans. Amer. Math. Soc. 179 (1973), 293–329. Zbl 0267.17015, MR 0325719, 10.1090/S0002-9947-1973-0325719-0 |
| Reference:
|
[12] Gong, Ming-Peng: Classification of nilpotent Lie algebras of dimension $7$ over algebraically closed fields and $\mathbb{R}$.Ph.D. Thesis, Waterloo, Ontario, Canada, 1998. MR 2698220 |
| Reference:
|
[13] Grayson M., Grossman R.: Models for free nilpotent Lie algebras.J. Algebra 35 (1990), 117–191. Zbl 0717.17006, MR 1076084 |
| Reference:
|
[14] Hall M.: A basis for free Lie rings and higher commutators in free groups.Proc. Amer. Math. Soc. 1 (1950), 575–581. Zbl 0039.26302, MR 0038336, 10.1090/S0002-9939-1950-0038336-7 |
| Reference:
|
[15] Humphreys J.E.: Introduction to Lie algebras and representation theory.vol. 9, Springer, New York, 1972. Zbl 0447.17002, MR 0323842 |
| Reference:
|
[16] Jacobson N.: Lie Algebras.Dover Publications, Inc., New York, 1962. Zbl 0333.17009, MR 0143793 |
| Reference:
|
[17] Kath I., Olbrich M.: Metric Lie algebras with maximal isotropic centre.Math. Z. 246 (2004), no. 1–2, 23–53. Zbl 1046.17003, MR 2031443, 10.1007/s00209-003-0575-2 |
| Reference:
|
[18] Kath I.: Nilpotent metric Lie algebras and small dimension.J. Lie Theory 17 (2007), no. 1, 41–61. MR 2286880 |
| Reference:
|
[19] Lauret J.: Examples of Anosov diffeomorphisms.J. Algebra 262 (2003), no. 1, 201–209. Zbl 1015.37022, MR 1970807, 10.1016/S0021-8693(03)00030-9 |
| Reference:
|
[20] Zhu L.: Solvable quadratic algebras.Science in China: Series A Mathematics 49 (2006), no. 4, 477–493. MR 2250478, 10.1007/s11425-006-0477-y |
| Reference:
|
[21] Mainkar M.G.: Anosov Lie algebras and algebraic units in number fields.Monatsh. Math. 165 (2012), 79–90. Zbl 1259.37020, MR 2886124, 10.1007/s00605-010-0260-6 |
| Reference:
|
[22] Malcev A.I.: On solvable Lie algebras.Izv. Akad. Nauk SSSR Ser. Mat. 9 (1945), 329–352; English transl.: Amer. Math. Soc. Transl. (1) 9 (1962), 228–262; MR 9, 173. MR 0022217 |
| Reference:
|
[23] Medina A., Revoy P.: Algèbres de Lie et produit scalaire invariant (Lie algebras and invariant scalar products).Ann. Sci. École Norm. Sup. (4) 18 (1985), no. 3, 553–561. MR 0826103 |
| Reference:
|
[24] Okubo S.: Gauge theory based upon solvable Lie algebras.J. Phys. A 31 (1998), 7603–7609. Zbl 0951.81015, MR 1652914, 10.1088/0305-4470/31/37/018 |
| Reference:
|
[25] Onishchik A.L., Khakimdzhanov Y.B.: On semidirect sums of Lie algebras.Mat. Zametki 18 (1975), no. 1, 31–40; English transl.: Math. Notes 18 (1976), 600–604. Zbl 0322.17003, MR 0427409 |
| Reference:
|
[26] Onishchick A.L., Vinberg E.B.: Lie Groups and Lie Algebras III.Encyclopaedia of Mathematical Sciences, 41, Springer, 1994. MR 1349140 |
| Reference:
|
[27] Patera J., Zassenhaus H.: The construction of Lie algebras from equidimensional nilpotent algebras.Linear Algebra Appl. 133 (1990), 89–120. MR 1058108 |
| Reference:
|
[28] Payne T.L.: Anosov automorphisms of nilpotent Lie algebras.J. Mod. Dyn. 3 (2009), no. 1, 121–158. Zbl 1188.37031, MR 2481335, 10.3934/jmd.2009.3.121 |
| Reference:
|
[29] Rubin J.L., Winternitz P.: Solvable Lie algebras with Heisenberg ideals.J. Phys. A 26 (1993), no. 5, 1123–1138. Zbl 0773.17004, MR 1211350, 10.1088/0305-4470/26/5/031 |
| Reference:
|
[30] Sato T.: The derivations of the Lie algebras.Tohoku Math. J. 23 (1971), 21–36. Zbl 0253.17012, MR 0288156, 10.2748/tmj/1178242684 |
| Reference:
|
[31] Smale S.: Differentiable dynamical systems.Bull. Amer. Math. Soc. 73 (1967), 747–817. Zbl 0205.54201, MR 0228014, 10.1090/S0002-9904-1967-11798-1 |
| Reference:
|
[32] Šnobl L.: On the structure of maximal solvable extensions and of Levi extensions of nilpotent Lie algebras.J. Phys. A 43 (2010), no. 50, 505202 (17 pages). Zbl 1231.17004, MR 2740380, 10.1088/1751-8113/43/50/505202 |
| Reference:
|
[33] Turkowski P.: Structure of real Lie algebras.Linear Algebra Appl. 171 (1992), 197–212. Zbl 0761.17003, MR 1165454, 10.1016/0024-3795(92)90259-D |
| . |
