Abstract
We consider a Lie algebraL over an arbitrary field that is decomposable into the sumL=A+B of an almost Abelian subalgebraA and a subalgebraB finite-dimensional over its center. We prove that this algebra is almost solvable, i.e., it contains a solvable ideal of finite codimension. In particular, the sum of the Abelian and almost Abelian Lie algebras is an almost solvable Lie algebra.
Similar content being viewed by others
References
N. Ito, “Ueber das Produkt von zwei abelschen Gruppen,”Math. Z.,62, No. 4, 400–401 (1955).
B. Kolman, “Semi-modular Lie algebras”,J. Sci. Hiroshima Univ., Ser. A1,29, 149–163 (1965).
Kourovka Notebook (Unsolved Problems in Group Theory) [in Russian], 11th ed., Institute of Mathematics, Academy of Sciences of the USSR, Siberian Division, Novosibirsk (1990).
O. H. Kegel, “On the solvability of some factorized linear groups,”Ill. J. Math.,9, No. 3, 535–547 (1965).
Yu. Bahturin, and O. H. Kegel, “Universal sums of Abelian subalgebras,”Commun. Algebra,23, 2975–2990 (1995).
N. S. Chernikov,Groups Factorizable into a Product of Commutative Subgroups [in Russian], Naukova Dumka, Kiev (1987).
I. Stewart, “Lie algebras generated by finite-dimensional ideals”,Res. Notes Math.,2 (1972).
A. P. Petravchuk, “On infinite-dimensional Lie algebras with solvable ideals of finite codimension”, in:Algebraic Proceedings, [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1996), pp. 158–167.
A. I. Kostrikin, “A criterion of solvability of a finite-dimensional Lie algebra”,Vestn. Mosk. Univ., Ser. Mat. Mekh., No. 2, 5–8 (1982).
A. P. Petravchuk, “On the sum of two Lie algebras with finite-dimensional commutants”,Ukr. Mat. Zh.,47, No. 8, 1089–1096 (1995).
Additional information
Kiev University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 5, pp. 636–644, May, 1999.
Rights and permissions
About this article
Cite this article
Petravchuk, A.P. On the sum of an almost abelian Lie algebra and a Lie algebra finite-dimensional over its center. Ukr Math J 51, 707–715 (1999). https://doi.org/10.1007/BF02591706
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02591706