ON NATURALLY-GRADED 2-FILIFORM LIE ALGEBRAS OF LENGTH N-1
Keywords:
Lie algebras, n-dimensional, 2-filiform, natural graduaded, maximal length.Abstract
It is known from the classical theory of Lie algebras that any finite-dimensional Lie algebra over a field of characteristic zero can be decomposed into a semidirect sum of its maximal solvable ideal and its semisimple subalgebra. Similarly, finite-dimensional Leibniz algebras also decompose into a semidirect sum of their maximal solvable ideal and a semisimple Lie algebra. The study of solvable algebras with nilradicals of special types is related to various models in physics. Thus, similar to the case of Lie algebras, studying Leibniz algebras with given nilradicals is an important problem.
It should be noted that nilpotent Lie algebras are a special type of solvable algebras. Since describing nilpotent Lie algebras is an enormous task, their study must be conducted with additional constraints. In particular, one of the main constraints when studying nilpotent algebras is the limitation on the nilpotency index. It is important to emphasize that the maximum nil-index for a Lie algebra coincides with the dimension of the algebra itself, and such algebras are called filiform algebras.
Although Leibniz filiform algebras in the class of nilpotent algebras have relatively simple restrictions, they possess a complex structure, which is conveniently studied by imposing a gradation condition. The effectiveness of the maximal grading lies in the fact that it provides the most precise information about the structural constants in the multiplication table of the algebra.
"Currently, the structural theory of Lie algebras holds an important place in the study of other branches of group theory. Numerous articles have been published on these algebras, and their structural theory has been studied.
As is known from the classical theory of finite-dimensional Lie algebras, the study of finite-dimensional Lie algebras is reduced to the study of nilpotent Lie algebras. The class of Lie algebras with maximal nilindex is an important part of the class of nilpotent Lie algebras.
References
1. Ancochea-Bermudez J.M., Campoamor-Strucberg R., Garcia-Vergnolle L. Classification of Lie algebras with naturally graded quasi-filiform nilradicals. // J. Geom. and Phys., 2011, Vol. 61, P. 2168-2186.
2. Aleksandrov I. Algebraic Structures. - M.: 1964 - 356 pages.
3. Bordemann M., Gomez J.R., Khakimjanov Yu., Navarro R.M. Some deformations of nilpotent Lie superalgebras. // J. Geom. and Phys., 2007, 〖Vol.57,P.1391-1403.〗_
4. Carles R. Sur la structure des algebres de Lie rigides. // Ann. Inst. Fourier (Grenoble), 1984, Vol. 8, P. 155-158.
5. Dixmier J., Lister W.G. Derivations of nilpotent Lie algebras. // Proc. Amer. Math. Soc., 1957, Vol. 8, P. 155-158.
6. Gomez J.R., Goze M., Khakimdjanov Yu. On the k-abelian filiform Lie algebras. // Comm. in Algebra, 1997, Vol. 25, No. 2, P. 431-450.
7. Gomez J.R., Jimenez-Merchen A. Naturally graded quasi-filiform Lie algebras. // J. of Algebra, 2002, Vol. 256, P. 211-228.
8. Goze M., Khakimdjanov Yu. Nilpotent Lie Algebras. // Kluwer Academic Publishers, Dordrecht, 1996, Vol. 361, 336 pages.
9. J.R. Gomez et al. / Linear Algebra and its Applications, 335 (2001), P. 119-135.
10. Khakimdjanov Yu.B. Characteristically nilpotent Lie algebras. // Math. USSR Sbornik, 1991, Vol. 70, No. 1, P. 65-78.
11. Weimar-Woods E., Contractions, generalized Inonu-Wigner contractions and deformations of finite-dimensional Lie algebras. // Rev. Math. Phys., 2000, Vol. 12, No. 11, P. 1505-1529.
12. Aleksandrov I. Group and Algebraic Structures. Kavkaz IX - M., 1986, 174 pages.
13. Additional References:
14. Leger G., Togo S. Characteristically nilpotent Lie algebras. // Duke Math. J., 1959, Vol. 26, P. 340-341.
15. Kac V.G. Classification of simple Lie superalgebras. // Funktional Anal. Appl., 1975, Vol. 9, P. 263-265.
16. Navarro R.M. Superalgebras of Nilpotent Lie Algebras: Ph.D. Thesis - University of Sevilla, 2001, 197 pages.
17. Maevov A.E. On the Representations of Lie Algebras. // Izv. Akad. Nauk. USSR, 1945, No. 9, P. 329-356.
