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ON LIE-LIKE COMPLEX FILIFORM LEIBNIZ ALGEBRAS

Part of: Lie algebras and Lie superalgebras General nonassociative rings

Published online by Cambridge University Press:  14 May 2009

B. A. OMIROV  and
I. S. RAKHIMOV
B. A. OMIROV*
Affiliation:
Institute of Mathematics, Uzbekistan Academy of Science, F. Hodjaev str. 29, 100125 Tashkent, Uzbekistan (email: omirovb@mail.ru)
I. S. RAKHIMOV
Affiliation:
Institute for Mathematical Research (INSPEM), Department of Mathematics, Universiti Putra Malaysia, Faculty of Science, 43400 UPM Serdang, Selangor Darul Ehsan, Malaysia (email: isamiddin@science.upm.edu.my, risamiddin@mail.ru)
*
✉For correspondence; e-mail: omirovb@mail.ru
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Abstract

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In this paper we propose an approach to classifying a subclass of filiform Leibniz algebras. This subclass arises from the naturally graded filiform Lie algebras. We reconcile and simplify the structure constants of such a class. In the arbitrary fixed dimension case an effective algorithm to control the behavior of the structure constants under adapted transformations of basis is presented. In one particular case, the precise formulas for less than 10 dimensions are given. We provide a computer program in Maple that can be used in computations as well.

Keywords

Lie algebrafiliform Leibniz algebranatural gradationisomorphism criterion

MSC classification

Secondary: 17A32: Leibniz algebras 17B30: Solvable, nilpotent (super)algebras 17B70: Graded Lie (super)algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 3 , June 2009 , pp. 391 - 404
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

This work was supported by the Science Fund Grant Project 06-01-04-SF0122 MOSTI (Malaysia).

References

[1] Ayupov, Sh. A. and Omirov, B. A., ‘On some classes of nilpotent Leibniz algebras’, Siberian Math. J. (1) 42 (2001), 18–29Google Scholar
[2] Bekbaev, U. D. and Rakhimov, I. S., On classification of finite dimensional complex filiform Leibniz algebras (part 1), http://front.math.ucdavis.edu/, ArXiv:math. RA/01612805, 2006.Google Scholar
[3] Bekbaev, U. D. and Rakhimov, I. S., On classification of finite dimensional complex filiform Leibniz algebras (part 2), http://front.math.ucdavis.edu/, ArXiv:0704.3885v1 [math.RA], 30 Apr 2007.Google Scholar
[4] Gómez, J. R., Jimenéz-Merchán, A. and Khakimdjanov, Y., ‘Low-dimensional filiform Lie algebras’, J. Pure Appl. Algebra 130 (1998), 133–158CrossRefGoogle Scholar
[5] Gómez, J. R. and Omirov, B. A., On classification of complex filiform Leibniz algebras, arXive:math/0612735 v1 [math.R.A.], 23 Dec 2006.Google Scholar
[6] Goze, M. and Khakimjanov, Yu., Nilpotent Lie Algebras, Mathematics and its Applications, 361 (Kluwer Academic Publishers, Dordrecht, 1996), p. 336CrossRefGoogle Scholar
[7] Kosman-Schwarzbach, Y., ‘From Poisson algebras to Gerstenhaber algebras’, Ann. Inst. Fourier (Grenoble) 46 (1996), 1243–1274CrossRefGoogle Scholar
[8] Loday, J.-L., ‘Une version non commutative des algèbres de Lie: les algèbres de Leibniz’, L’Ens. Math. 39 (1993), 269–293Google Scholar
[9] Loday, J.-L. and Pirashvili, T., ‘Universal enveloping algebras of Leibniz algebras and (co)homology’, Math. Ann. 296 (1993), 139–158CrossRefGoogle Scholar
[10] Rakimov, I. S. and Said Husain, S. K., On isomorphism classes and invariants of low-dimensional complex filiform Leibniz algebras (Part 1), http://front.math.ucdavis.edu/, ArXiv:0710.0121 v1.[math RA], 1 Oct 2007.Google Scholar
[11] Rakimov, I. S. and Said Husain, S. K., On isomorphism classes and invariants of low-dimensional complex filiform Leibniz algebras (Part 2), http://front.math.ucdavis.edu/, ArXiv:0806.1803 v1.[math RA], 11 June 2008.Google Scholar
[12] Vergne, M., ‘Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes’, Bull. Soc. Math. France 98 (1970), 81–116CrossRefGoogle Scholar