**Journal of Prime Research in Mathematics**

Vol. 1 (2007), Issue 1, pp. 111 – 119

ISSN: 1817-3462 (Online) 1818-5495 (Print)

ISSN: 1817-3462 (Online) 1818-5495 (Print)

**Matrix lie rings that contains a one-dimentional lie algebra of semi-simple matrices**

**Evgenii L. Bashkirov
**Belorussian State University of Informatics and Radioelectronics, P. Brovki st. 6, Minsk 220013, Belarus.

\(^{1}\)Corresponding Author: bashkirov57@mail.ru

Copyright © 2007 Evgenii L. Bashkirov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Published:**December, 2007.

### Abstract

Let \(k\) be a field and \(\overline{k}\) an algebraic closure of \(k\). Suppose that \(k\)

contains more than five elements if char \(k \neq 2\). Let \(h\) be a one-dimensional subalgebra of the Lie \(k-\)algebra \(sl_{2}\overline{k}\) consisting of semi-simple matrices. In this paper, it is proved that if g is a subring of the Lie ring \(sl_{2}\overline{k}\) containing h, then g is either solvable or there exists a quaternion algebra A over a subfield \(F\) of \(\overline{k}\) such that \(F ⊇ k\) and g is isomorphic to the Lie \(F-\)algebra of all elements in A that are skew-symmetric with respect to a symplectic type involution defined on A.

contains more than five elements if char \(k \neq 2\). Let \(h\) be a one-dimensional subalgebra of the Lie \(k-\)algebra \(sl_{2}\overline{k}\) consisting of semi-simple matrices. In this paper, it is proved that if g is a subring of the Lie ring \(sl_{2}\overline{k}\) containing h, then g is either solvable or there exists a quaternion algebra A over a subfield \(F\) of \(\overline{k}\) such that \(F ⊇ k\) and g is isomorphic to the Lie \(F-\)algebra of all elements in A that are skew-symmetric with respect to a symplectic type involution defined on A.

#### Keywords:

Lie rings, Lie algebras, Semi-simple matrices.