An overview of free nilpotent Lie algebras

Pilar Benito; Daniel de-la-Concepción

Commentationes Mathematicae Universitatis Carolinae (2014)

  • Volume: 55, Issue: 3, page 325-339
  • ISSN: 0010-2628

Abstract

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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.

How to cite

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Benito, Pilar, and de-la-Concepción, Daniel. "An overview of free nilpotent Lie algebras." Commentationes Mathematicae Universitatis Carolinae 55.3 (2014): 325-339. <http://eudml.org/doc/261867>.

@article{Benito2014,
abstract = {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.},
author = {Benito, Pilar, de-la-Concepción, Daniel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Lie algebra; Levi subalgebra; nilpotent; free nilpotent; derivation; automorphism; representation; Lie algebra; Levi subalgebra; nilpotent; free nilpotent; derivation; automorphism; representation},
language = {eng},
number = {3},
pages = {325-339},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {An overview of free nilpotent Lie algebras},
url = {http://eudml.org/doc/261867},
volume = {55},
year = {2014},
}

TY - JOUR
AU - Benito, Pilar
AU - de-la-Concepción, Daniel
TI - An overview of free nilpotent Lie algebras
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2014
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 55
IS - 3
SP - 325
EP - 339
AB - 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.
LA - eng
KW - Lie algebra; Levi subalgebra; nilpotent; free nilpotent; derivation; automorphism; representation; Lie algebra; Levi subalgebra; nilpotent; free nilpotent; derivation; automorphism; representation
UR - http://eudml.org/doc/261867
ER -

References

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  1. Ancochea-Bermúdez J.M., Campoamor-Stursberg R., García Vergnolle L., 10.1016/j.geomphys.2011.06.015, J. Geom. Phys. 61 (2011), no. 11, 2168–2186. Zbl1275.17023MR2827117DOI10.1016/j.geomphys.2011.06.015
  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. Zbl1142.17300MR2237946
  3. Auslander L., Scheuneman J., On certain automorphisms of nilpotent Lie groups, Global Analysis: Proc. Symp. Pure Math. 14 (1970), 9–15. Zbl0223.22014MR0270395
  4. Del Barco V.J., Ovando G.P., 10.1016/j.jalgebra.2012.05.016, J. Algebra 366 (2012), 205–216. MR2942650DOI10.1016/j.jalgebra.2012.05.016
  5. Benito P., de-la-Concepción D., 10.1016/j.laa.2013.04.027, Linear Algebra Appl. 439 (2013), no. 5, 1441–1457. Zbl1281.17014MR3067814DOI10.1016/j.laa.2013.04.027
  6. Benito P., de-la-Concepción D., A note on extensions of nilpotent Lie algebras of Type 2 , arXiv:1307.8419. 
  7. Cui R., Wang Y., Deng S., 10.1080/00927870802174629, Comm. Algebra 36 (2008), 4052–4067. MR2460402DOI10.1080/00927870802174629
  8. Dengyin W., Ge H., Li X., Solvable extensions of a class of nilpotent linear Lie algebras, Linear Algebra Appl. 437 (2012), 14–25. MR2917429
  9. Favre G., Santharoubane L., 10.1016/0021-8693(87)90209-2, J. Algebra 105 (1987), no. 2, 451–464. Zbl0608.17007MR0873679DOI10.1016/0021-8693(87)90209-2
  10. Figueroa-O'Farrill J.M., Stanciu S., 10.1063/1.531620, J. Math. Phys. 37 (1996), 4121–4134. Zbl0863.17004MR1400838DOI10.1063/1.531620
  11. Gauger M.A., 10.1090/S0002-9947-1973-0325719-0, Trans. Amer. Math. Soc. 179 (1973), 293–329. Zbl0267.17015MR0325719DOI10.1090/S0002-9947-1973-0325719-0
  12. Gong, Ming-Peng, Classification of nilpotent Lie algebras of dimension 7 over algebraically closed fields and ℝ , Ph.D. Thesis, Waterloo, Ontario, Canada, 1998. MR2698220
  13. Grayson M., Grossman R., Models for free nilpotent Lie algebras, J. Algebra 35 (1990), 117–191. Zbl0717.17006MR1076084
  14. Hall M., 10.1090/S0002-9939-1950-0038336-7, Proc. Amer. Math. Soc. 1 (1950), 575–581. Zbl0039.26302MR0038336DOI10.1090/S0002-9939-1950-0038336-7
  15. Humphreys J.E., Introduction to Lie algebras and representation theory, vol. 9, Springer, New York, 1972. Zbl0447.17002MR0323842
  16. Jacobson N., Lie Algebras, Dover Publications, Inc., New York, 1962. Zbl0333.17009MR0143793
  17. Kath I., Olbrich M., 10.1007/s00209-003-0575-2, Math. Z. 246 (2004), no. 1–2, 23–53. Zbl1046.17003MR2031443DOI10.1007/s00209-003-0575-2
  18. Kath I., Nilpotent metric Lie algebras and small dimension, J. Lie Theory 17 (2007), no. 1, 41–61. MR2286880
  19. Lauret J., 10.1016/S0021-8693(03)00030-9, J. Algebra 262 (2003), no. 1, 201–209. Zbl1015.37022MR1970807DOI10.1016/S0021-8693(03)00030-9
  20. Zhu L., 10.1007/s11425-006-0477-y, Science in China: Series A Mathematics 49 (2006), no. 4, 477–493. MR2250478DOI10.1007/s11425-006-0477-y
  21. Mainkar M.G., 10.1007/s00605-010-0260-6, Monatsh. Math. 165 (2012), 79–90. Zbl1259.37020MR2886124DOI10.1007/s00605-010-0260-6
  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. MR0022217
  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. MR0826103
  24. Okubo S., 10.1088/0305-4470/31/37/018, J. Phys. A 31 (1998), 7603–7609. Zbl0951.81015MR1652914DOI10.1088/0305-4470/31/37/018
  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. Zbl0322.17003MR0427409
  26. Onishchick A.L., Vinberg E.B., Lie Groups and Lie Algebras III, Encyclopaedia of Mathematical Sciences, 41, Springer, 1994. MR1349140
  27. Patera J., Zassenhaus H., The construction of Lie algebras from equidimensional nilpotent algebras, Linear Algebra Appl. 133 (1990), 89–120. MR1058108
  28. Payne T.L., 10.3934/jmd.2009.3.121, J. Mod. Dyn. 3 (2009), no. 1, 121–158. Zbl1188.37031MR2481335DOI10.3934/jmd.2009.3.121
  29. Rubin J.L., Winternitz P., 10.1088/0305-4470/26/5/031, J. Phys. A 26 (1993), no. 5, 1123–1138. Zbl0773.17004MR1211350DOI10.1088/0305-4470/26/5/031
  30. Sato T., 10.2748/tmj/1178242684, Tohoku Math. J. 23 (1971), 21–36. Zbl0253.17012MR0288156DOI10.2748/tmj/1178242684
  31. Smale S., 10.1090/S0002-9904-1967-11798-1, Bull. Amer. Math. Soc. 73 (1967), 747–817. Zbl0205.54201MR0228014DOI10.1090/S0002-9904-1967-11798-1
  32. Šnobl L., 10.1088/1751-8113/43/50/505202, J. Phys. A 43 (2010), no. 50, 505202 (17 pages). Zbl1231.17004MR2740380DOI10.1088/1751-8113/43/50/505202
  33. Turkowski P., 10.1016/0024-3795(92)90259-D, Linear Algebra Appl. 171 (1992), 197–212. Zbl0761.17003MR1165454DOI10.1016/0024-3795(92)90259-D

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