Journal of Prime Research in Mathematics

# 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

### 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.

#### Keywords:

Lie rings, Lie algebras, Semi-simple matrices.