Restricted Simple Lie Algebras and Their Infinitesimal Deformations

In the first two sections, we review the Block-Wilson-PremetStrade classification of restricted simple Lie algebras. In the third section, we compute their infinitesimal deformations. In the last section, we indicate some possible generalizations by formulating some open problems. 1. Restricted Lie algebras We fix a field F of characteristic p > 0 and we denote with Fp the prime field with p elements. All the Lie algebras that we will consider are of finite dimension over F . We are interested in particular class of Lie algebras, called restricted (or p-Lie algebras). Definition 1.1 (Jacobson [JAC37]). A Lie algebra L over F is said to be restricted (or a p-Lie algebra) if there exits a map (called p-map), [p] : L → L, x 7→ x, which verifies the following conditions: (1) ad(x) = ad(x) for every x ∈ L. (2) (αx) = αx for every x ∈ L and every α ∈ F . (3) (x0 + x1) [p] = x [p] 0 + x [p] 1 + ∑p−1 i=1 si(x0, x1) for every x, y ∈ L, where the element si(x0, x1) ∈ L is defined by si(x0, x1) = − 1 r ∑ u adxu(1) ◦ adxu(2) ◦ · · · ◦ adxu(p−1)(x1), the summation being over all the maps u : [1, · · · , p − 1] → {0, 1} taking r-times the value 0. Example. (1) Let A an associative F -algebra. Then the Lie algebra DerFA of F -derivations of A is a restricted Lie algebra with respect to the p-map D 7→ D := D ◦ · · · ◦D. (2) Let G a group scheme over F . Then the Lie algebra Lie(G) associated to G is a restricted Lie algebra with respect to the p-map given by the differential of the homomorphism G → G, x 7→ x := x ◦ · · · ◦ x. One can naturally ask when a F -Lie algebra can acquire the structure of a restricted Lie algebra and how many such structures there can be. The following criterion of Jacobson answers to that question. Proposition 1.2 (Jacobson). Let L be a Lie algebra over F . Then (1) It is possible to define a p-map on L if and only if, for every element x ∈ L, the p-th iterate of ad(x) is still an inner derivation. (2) Two such p-maps differ by a semilinear map from L to the center Z(L) of L, that is a map f : L → Z(L) such that f(αx) = αf(x) for every x ∈ L and α ∈ F . 1991 Mathematics Subject Classification. Primary 17B50; Secondary 17B20, 17B56.