2. Connected Reductive Groups and Their Lie Algebras

Abstract

The geometrical objects considered are defined over an algebraically closed field k of characteristic p. In this chapter, we first introduce some notation which will be used throughout this book. We then discuss some properties about algebraic groups and their Lie algebras related to the characteristic p. These results will be used to give an explicit bound on p for which the main result of [Lus87] applies. For any prime r, we choose once for all an algebraic closure $\overline{\mathbb{F}}_r$ of the finite field $\mathbb{F}_r = \mathbb{Z}/r\mathbb{Z}$. Then we denote by $\mathbb{F}_{r^n}$ the unique extension of degree n > 0 of $\mathbb{F}_r$ in $\overline{\mathbb{F}}_r$.