Locally
finite Lie algebras with complemented one-dimensional
subalgebras

A.P. Petravchuk

It is proved that all one-dimensional subalgebras of a locally finite-dimensional Lie algebra $L$ with countable basis over a perfect field of characteristic $\not= 2$ are complemented iff $L$ can be isomorphically embedded into a Cartesian sum of 3-dimensional simple Lie algebras of type~$A_{1}$. It is also shown that all one-dimensional subalgebras of a locally finite-dimensional Lie algebra $L$ over an arbitrary field of characteristic $p=2$ are complemented in $L$ iff $L$ is solvable and can be isomorphically embedded into a Cartesian sum of 2-dimensional non-abelian Lie algebras.

1. Ph. Hall Complemented groups // Journ. London Math. Soc. – 1937 V. 12 p. 201–204

2. N.V. Chernikova Groups with complemented subgroups // Mat. sbornik – 1956 V. 3 p. 273–292 in Russian

3. A.P. Petravchuk Lie algebras with complemented one-dimensional subalgebras // Visnyk Kyiv univ. Mat. i mech – 1999 in Ukrainian, to appear

4. K.H. Hofmann Hyperplane subalgebras of real algebras // Geometriae Dedicata – 1990 V. 36 № 2-3 p. 207–224

5. D. Poguntke The theorem of Lie and hyperplane subalgebras in Lie algebras // Geometriae Dedicata – 1992 V. 43 № 1 p. 83–91

6. Yu.M. Gorchakov Primitively factorized groups // Uchenye zapiski Permsk. univ. – 1960 V. 17 p. 15–31 in Russian

7. N.Jacobson Lie algebras ,addr Moscow , Nauka – 1964 translated into Russian

	Locally finite Lie algebras with complemented one-dimensional subalgebras
Author	A.P. Petravchuk Kyiv University
Abstract	It is proved that all one-dimensional subalgebras of a locally finite-dimensional Lie algebra $L$ with countable basis over a perfect field of characteristic $\not= 2$ are complemented iff $L$ can be isomorphically embedded into a Cartesian sum of 3-dimensional simple Lie algebras of type~$A_{1}$. It is also shown that all one-dimensional subalgebras of a locally finite-dimensional Lie algebra $L$ over an arbitrary field of characteristic $p=2$ are complemented in $L$ iff $L$ is solvable and can be isomorphically embedded into a Cartesian sum of 2-dimensional non-abelian Lie algebras.
Keywords	one-dimensional subalgebras, complemented subalgebras, Cartesian sum, solvable Lie algebras, non-abelian Lie algebras
DOI	doi:10.30970/ms.11.2.135-140
Reference	1. Ph. Hall Complemented groups // Journ. London Math. Soc. – 1937 V. 12 p. 201–204 2. N.V. Chernikova Groups with complemented subgroups // Mat. sbornik – 1956 V. 3 p. 273–292 in Russian 3. A.P. Petravchuk Lie algebras with complemented one-dimensional subalgebras // Visnyk Kyiv univ. Mat. i mech – 1999 in Ukrainian, to appear 4. K.H. Hofmann Hyperplane subalgebras of real algebras // Geometriae Dedicata – 1990 V. 36 № 2-3 p. 207–224 5. D. Poguntke The theorem of Lie and hyperplane subalgebras in Lie algebras // Geometriae Dedicata – 1992 V. 43 № 1 p. 83–91 6. Yu.M. Gorchakov Primitively factorized groups // Uchenye zapiski Permsk. univ. – 1960 V. 17 p. 15–31 in Russian 7. N.Jacobson Lie algebras ,addr Moscow , Nauka – 1964 translated into Russian
Pages	135-140
Volume	11
Issue	2
Year	1999
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Locally finite Lie algebras with complemented one-dimensional subalgebras