On Lie algebras of type F₄ and Chevalley groups F₄(K), E₆(K), and ²E₆(K) for fields K of characteristic two

Abstract

In this article, we give an elementary and self-contained approach to construct the Lie algebras of type $F_4(K)$ over an arbitrary field K of characteristic two. The Lie algebras are represented as subalgebras of $\text{End}(A_K)$, where A_K is a 27-dimensional vector space over K. The Lie algebras of type $E_6(K)$ for fields K of characteristic two have been constructed by the authors, using the notion of M sets. Here we follow the same notion to give an easy and effective construction of the corresponding Chevalley groups $E_6(K),F_4(K)$, and ${}^2{E}_6(K)$. It is remarkable to mention that most of the available literature on Chevalley groups does not deal with fields of characteristic two. Hence, this work aims to contribute in this regard.

Keywords:

• Chevalley groups
• Lie algebra

2000 Mathematics subject classification:

• 17A75
• 17A45

Acknowledgements

The first author would like to thank Kuwait Foundation for the Advancement of Sciences and PAAET for supporting project Nos. P115-16-SM, BE-15-13 respectively. The second author would like to thank H. J. Schaeefer, C. Hering and A. Alazemi for their remarks on this article.