Czechoslovak Mathematical Journal, online first, 13 pp.

Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center

Ren Bin, Zhu Lin Sheng

Received May 23, 2016.   First published August 14, 2017.

Ren Bin, Department of Mathematics, University of Science and Technology of Suzhou, 1 Kerui Road, SND, 215009 Suzhou, Jiangsu, China, e-mail: renbinsz@163.com, Zhu Lin Sheng, Department of Mathematics, Huaiyin Normal University, 111 Changjiang W Road, 223300 Huaiyin, Huaian, Jiangsu, China

Abstract: A Lie algebra $L$ is called 2-step nilpotent if $L$ is not abelian and $[L, L]$ lies in the center of $L$. 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.

Keywords: related set; basis; derivation

Classification (MSC 2010): 17B05, 17B30

DOI: 10.21136/CMJ.2017.0253-16

Full text available as PDF.


References:
[1] J. M. Ancochea-Bermudez, M. Goze: Classification des algèbres de Lie filiformes de dimension 8. Arch. Math. 50 (1988), 511-525. (In French.) DOI 10.1007/BF01193621 | MR 0948265 | Zbl 0628.17005
[2] J. M. Ancochea-Bermudez, M. Goze: Classification des algèbres de Lie nilpotentes complexes de dimension 7. Arch. Math. 52 (1989), 175-185. (In French.) DOI 10.1007/BF01191272 | MR 0985602 | Zbl 0672.17005
[3] L. Y. Galitski, D. A. Timashev: On classification of metabelian Lie algebras. J. Lie Theory 9 (1999), 125-156. MR 1680007 | Zbl 0923.17015
[4] M. A. Gauger: On the classification of metabelian Lie algebras. Trans. Am. Math. Soc. 179 (1973), 293-329. DOI 10.2307/1996506 | MR 0325719 | Zbl 0267.17015
[5] M.-P. Gong: Classification of Nilpotent Lie Algebras of Dimension 7 (over Algebraically Closed Fields and R). Ph.D. Thesis, University of Waterloo, Waterloo (1998). MR 2698220
[6] M. Goze, Y. Khakimdjanov: Nilpotent Lie Algebras. Mathematics and Its Applications 361, Kluwer Academic Publishers, Dordrecht (1996). DOI 10.1007/978-94-017-2432-6 | MR 1383588 | Zbl 0845.17012
[7] M. Goze, E. Remm: $k$-step nilpotent Lie algebras. Georgian Math. J. 22 (2015), 219-234. DOI 10.1515/gmj-2015-0022 | MR 3353570 | Zbl 06458841
[8] B. Khuhirun, K. C. Misra, E. Stitzinger: On nilpotent Lie algebras of small breadth. J. Algebra 444 (2015), 328-338. DOI 10.1016/j.jalgebra.2015.07.036 | MR 3406181 | Zbl 1358.17013
[9] G. Leger, E. Luks: On derivations and holomorphs of nilpotent Lie algebras. Nagoya Math. J. 44 (1971), 39-50. DOI 10.1017/s0027763000014525 | MR 0297828 | Zbl 0264.17003
[10] E. Remm: Breadth and characteristic sequence of nilpotent Lie algebras. Commun. Algebra 45 (2017), 2956-2966. DOI 10.1080/00927872.2016.1233238 | MR 3594570
[11] B. Ren, D. J. Meng: Some completable 2-step nilpotent Lie algebras I. Linear Algebra Appl. 338 (2001), 77-98. MR 1860314 | Zbl 0992.17005
[12] B. Ren, L. S. Zhou: Classification of 2-step nilpotent Lie algebras of dimension 8 with 2-dimensional center. Commun. Algebra 39 (2011), 2068-2081. DOI 10.1080/00927872.2010.483342 | MR 2813164 | Zbl 1290.17004
[13] P. Revoy: Algèbres de Lie métabéliennes. Ann. Fac. Sci. Toulouse, V. Ser., Math. 2 (1980), 93-100. (In French.) DOI 10.5802/afst.547 | MR 0595192 | Zbl 0447.17007
[14] L. J. Santharoubane: Kac-Moody Lie algebras and the classification of nilpotent Lie algebras of maximal rank. Can. J. Math. 34 (1982), 1215-1239. DOI 10.4153/CJM-1982-084-5 | MR 0678665 | Zbl 0495.17011
[15] C. Seeley: 7-dimensional nilpotent Lie algebras. Trans. Am. Math. Soc. 335 (1993), 479-496. DOI 10.2307/2154390 | MR 1068933 | Zbl 0770.17003
[16] K. A. Umlauf: Über die Zusammensetzung der endlichen continuierlichen Transformationsgruppen insbesondere der Gruppen vom Range Null. Ph.D. Thesis, University of Leipzig, Leipzig (1891). (In German.)
[17] Z. Yan, S. Deng: The classification of two step nilpotent complex Lie algebras of dimension 8. Czech. Math. J. 63 (2013), 847-863. DOI 10.1007/s10587-013-0057-6 | MR 3125659 | Zbl 1291.17013


Access to the full text of journal articles on this site is restricted to the subscribers of Myris Trade. To activate your access, please contact Myris Trade at myris.cz.
Subscribers of Springer need to access the articles on their site, which is http://link.springer.com/journal/10587.


[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]