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Then g/FI ∼ = pslm+1 is simple. The simple Lie algebras obtained by the process of reducing a Chevalley basis modulo p (and factoring out the center if necessary) are called the classical simple Lie algebras. 2. GENERAL INFORMATION ABOUT THE CLASSICAL LIE ALGEBRAS 33 under the general rubric of classical Lie algebras. A classical reductive Lie algebra modulo its center is a classical Lie algebra. Assumption (a) of the Recognition Theorem asserts that g0 is classical reductive. 4) e[p] α = 0 for α ∈ Φ, and [p] h i = hi for i = 1, .

SIMPLE LIE ALGEBRAS AND ALGEBRAIC GROUPS Given a linear algebraic F-group H, we denote by X∗ (H) the set of all rational homomorphisms from the one-dimensional torus F× to H. 17) λ(t)|gj = tj idgj ∀ t ∈ F× , ∀ j ∈ Z . Clearly, λ(F× ) is a one-dimensional torus of G. Since any torus of G is contained in a maximal torus and all maximal tori of G are conjugate, we can assume in the rest of the proof that λ(F× ) ⊆ T . In other words, we can assume that λ belongs to X∗ (T ), the lattice of rational cocharacters of T .

Xpm . The Jacobson-Witt Lie algebra is the derivation algebra W (m; 1) := Der(F[x1 , . . , xm ]/ xp1 , . . , xpm ). In the special case that m = 1, the resulting Lie algebra W (1; 1) is called the p-dimensional Witt algebra or simply the Witt algebra. The JacobsonWitt algebras may be viewed as “thickenings” of it by addition of variables. By identifying cosets with their representatives, we may assume that the elements xa := xa11 · · · xamm with a = (a1 , . . , am ) and 0 ≤ ai < p for all i determine a basis for F[x1 , .

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