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Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. (English) Zbl 1137.17012
The author performs a painstaking task of comparison of numerous existing classifications of low-dimensional nilpotent Lie algebras, point out errors in them and put the classification on algorithmic ground: for any given nilpotent Lie algebra of dimension not greater than 6, there is a procedure allowing to identify it with an algebra from the classification list. Moreover, the procedure is implemented in GAP. The major ingredients of the proof are the Skjelbred-Sund method of constructing nilpotent Lie algebras as central extensions [{\it T. Skjelbred and T. Sund}, C. R. Acad. Sci., Paris, Sér. A 286, 241--242 (1978; Zbl 0375.17006)], and identification of Lie algebras by method of Gröbner bases due to the author.

MSC:
17B30Solvable, nilpotent Lie (super)algebras
17-08Computational methods
Software:
Magma; GAP
WorldCat.org
Full Text: DOI
References:
[1] Beck, R. E.; Kolman, B.: Construction of nilpotent Lie algebras over arbitrary fields. Proceedings of the 1981 ACM symposium on symbolic and algebraic computation, 169-174 (1981) · Zbl 0554.17003
[2] Bosma, W.; Cannon, J.; Playoust, C.: The magma algebra system. I. the user language. J. symbolic comput. 24, No. 3 -- 4, 235-265 (1997) · Zbl 0898.68039
[3] M. Costantini, W.A. de Graaf, C. Schneider, liealgdb, a database of Lie algebras, a GAP4 package, 2006, in preparation
[4] Cox, D.; Little, J.; O’shea, D.: Ideals, varieties and algorithms: an introduction to computational algebraic geometry and commutative algebra. (1992) · Zbl 0756.13017
[5] Eick, B.; O’brien, E. A.: Enumerating p-groups. J. austral. Math. soc. Ser. A 67, No. 2, 191-205 (1999) · Zbl 0979.20021
[6] Group, The Gap: GAP --- groups, algorithms, and programming, version 4.4. (2004)
[7] M.-P. Gong, Classification of nilpotent Lie algebras of dimension 7, PhD thesis, University of Waterloo, Waterloo, Canada, 1998
[8] De Graaf, W. A.: Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2. (2005)
[9] De Graaf, W. A.: Classification of solvable Lie algebras. Experiment. math. 14, No. 1, 15-25 (2005) · Zbl 1173.17300
[10] Morozov, V. V.: Classification of nilpotent Lie algebras of sixth order. Izv. vysš. Učebn. zaved. Mat. 1958, No. 4 (5), 161-171 (1958) · Zbl 0198.05501
[11] Newman, M. F.; O’brien, E. A.; Vaughan-Lee, M. R.: Groups and nilpotent Lie rings whose order is the sixth power of a prime. J. algebra 278, No. 1, 383-401 (2004) · Zbl 1072.20022
[12] Nielsen, O. A.: Unitary representations and coadjoint orbits of low-dimensional nilpotent Lie groups. Queen’s papers in pure and appl. Math. 63 (1983)
[13] O’brien, E. A.: The p-group generation algorithm. J. symbolic comput. 9, No. 5 -- 6, 677-698 (1990)
[14] O’brien, E. A.; Vaughan-Lee, M. R.: The groups with order p7 for odd prime p. J. algebra 292, No. 1, 243-258 (2005)
[15] Schneider, C.: A computer-based approach to the classification of nilpotent Lie algebras. Experiment. math. 14, No. 2, 153-160 (2005) · Zbl 1093.17004
[16] Skjelbred, Tor; Sund, Terje: Sur la classification des algèbres de Lie nilpotentes. C. R. Acad. sci. Paris sér. A -- B 286, No. 5 (1978) · Zbl 0375.17006