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RESTRICTED SIMPLE LIE ALGEBRAS AND THEIR INFINITESIMAL DEFORMATIONS
, 2007
"... Abstract. In the first two sections, we review the BlockWilsonPremetStrade 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. Rest ..."
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Abstract. In the first two sections, we review the BlockWilsonPremetStrade 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 pLie algebras). Definition 1.1 (Jacobson [JAC37]). A Lie algebra L over F is said to be restricted (or a pLie algebra) if there exits a map (called pmap), [p] : L → L, x ↦ → x [p], which verifies the following conditions: (1) ad(x [p]) = ad(x) [p] for every x ∈ L. (2) (αx)[p] = αpx [p] 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 ∑ adxu(1) ◦ adxu(2) ◦ · · · ◦ adxu(p−1)(x1), r u the summation being over all the maps u: [1, · · · , p − 1] → {0, 1} taking rtimes the value 0. Example. (1) Let A an associative Falgebra. Then the Lie algebra DerFA of Fderivations of A is a restricted Lie algebra with respect to the pmap D ↦ → Dp: = 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 pmap given by the differential of the homomorphism G → G, x ↦ → xp: = x ◦ · · · ◦ x. One can naturally ask when a FLie 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 pmap on L if and only if, for every element x ∈ L, the pth iterate of ad(x) is still an inner derivation. (2) Two such pmaps 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) = α p f(x) for every x ∈ L and α ∈ F.
Simple finite group schemes and their infinitesimal deformations
, 811
"... We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classi ..."
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We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.