CLASSIFICATION AND LIE STRUCTURES OF GIVEN SOLVABLE LEIBNIZ ALGEBRAS WITH NIL RADICALS
Keywords:
Derivation; Leibniz algebra; nilpotency; nilradical; solvabilityAbstract
This work is devoted to the classification of solvable Leibniz algebras with an abelian nilradical. We consider (k–1)-dimensional extension of
k-dimensional abelian algebras and classify all (2k–1)-dimensional solvable Leibniz algebras with an abelian nilradical of dimension k.
References
[1] Adashev, J. Q., Ladra, M., Omirov, B. A. (2017). Solvable Leibniz algebras with naturally graded non-Lie p- filiform nilradicals. Commun. Algebra 45(10):4329–4347. DOI: 10.1080/00927872.2016.1263311.
[2] Ancochea, J. M., Campoamor-Stursberg, R., Garc´ıa, L. (2006). Indecomposable Lie algebras with nontrivial Levi decomposition cannot have filiform radical. Int. Math. Forum. 1(5–8):309–316.
[3] Barnes, D. W. (2012). On Levi’s theorem for Leibniz algebras. Bull. Aust. Math. Soc. 86(2):184–185. DOI: 10.1017/S0004972711002954.
[4] Bosko-Dunbar, L., Dunbar, J. D., Hird, J. T., Stagg, K. (2015). Solvable Leibniz algebras with Heisenberg nilradical. Commun. Algebra 43(6):272–2281.
[5] Bloh, A. (1965). On a generalization of the concept of Lie algebra. Dokl. Akad. Nauk. SSSR. 165:471–473.
[6] Can~ete, E. M., Khudoyberdiyev, A. Kh. (2013). The classification of 4-dimensional Leibniz algebras. Linear Algebra Appl. 439(1):273–288. DOI: 10.1016/j.laa.2013.02.035.
[7] Casas, J. M., Ladra, M., Omirov, B. A., Karimjanov, I. A. (2013). Classification of solvable Leibniz algebras with null-filiform nilradical. Linear Multilinear Algebra 61(6):758–774. DOI: 10.1080/03081087.2012.703194.
[8] Casas, J. M., Insua, M. A., Ladra, M., Ladra, S. (2012). An algorithm for the classification of 3-dimensional complex Leibniz algebras. Linear Algebra Appl. 436(9):3747–3756. DOI: 10.1016/j.laa.2011.11.039.
[9] Jacobson, N. (1962). Lie Algebras. New York: Interscience Publishers, Wiley.
[10] Khudoyberdiyev, A. Kh., Rakhimov, I. S., Said Husain, S. K. (2014). On classification of 5-dimensional solv- able Leibniz algebras. Linear Algebra Appl. 457:428–454. DOI: 10.1016/j.laa.2014.05.034.
[11] Khudoyberdiyev, A. Kh., Ladra, M., Omirov, B. A. (2014). On solvable Leibniz algebras whose nilradical is a direct sum of null-fililiform algebras. Linear Multilinear Algebra 62(9):1220–1239. DOI: 10.1080/ 03081087.2013.816305.
[12] Loday, J.-L. (1993). Une version non commutative des alg`ebres de Lie: les alg`ebres de Leibniz. Enseign. Math. 39(3–4):269–293.
[13] Mubarakzjanov, G. M. (1963). On solvable Lie algebras (Russian). Izv. Vysˇs. Uˇcehn. Zaved. Matematika. 32(1):114–123.
[14] Ndogmo, J. C., Winternitz, P. (1994). Solvable Lie algebras with abelian nilradicals. J. Phys. A Math. Gen.
27(2):405–423. DOI: 10.1088/0305-4470/27/2/024.
[15] Shabanskaya, A. (2018). Solvable extensions of the naturally graded quasi-filiform Leibniz algebra of second type L2. Commun. Algebra 46(11):5006–5031.
[16] Shabanskaya, A. (2017). Solvable extensions of naturally graded quasi-filiform Leibniz algebras of second
type L1 and L3. Commun. Algebra 45(10):4492–4520.
[17] ˇSnobl, L., Winternitz, P. (2005). A class of solvable Lie algebras and their Casimir invariants. J. Phys. A Math. Gen. 38(12):2687–2700.
