Group of Lie Type - Chevalley Groups

Chevalley Groups

The theory was clarified by the theory of algebraic groups, and the work of Chevalley (1955) on Lie algebras, by means of which the Chevalley group concept was isolated. Chevalley constructed a Chevalley basis (a sort of integral form) for all the complex simple Lie algebras (or rather of their universal enveloping algebras), which can be used to define the corresponding algebraic groups over the integers. In particular, he could take their points with values in any finite field. For the Lie algebras A_n, B_n, C_n, D_n this gave well known classical groups, but his construction also gave groups associated to the exceptional Lie algebras E₆, E₇, E₈, F₄, and G₂. The ones of type G₂ (sometimes called Dickson groups) had already been constructed by Dickson (1905), and the ones of type E₆ by Dickson (1901).