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DEFORMATIONS OF RESTRICTED SIMPLE LIE ALGEBRAS II
, 2007
"... Abstract. We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartantype: the Contact and the Hamiltonian Lie algebras. 1. ..."
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Abstract. We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartantype: the Contact and the Hamiltonian Lie algebras. 1.
Classification of finite dimensional simple Lie algebras in prime characteristics, arxiv:math.RA/0601380 v2
, 2006
"... Abstract. We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of positive characteristic and announce that the classification of all finite dimensional simple Lie algebras over an algebraically closed field of characteristic p> 3 is n ..."
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Abstract. We give a comprehensive survey of the theory of finite dimensional Lie algebras over an algebraically closed field of positive characteristic and announce that the classification of all finite dimensional simple Lie algebras over an algebraically closed field of characteristic p> 3 is now complete. Any such Lie algebra is up to isomorphism either classical or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5. Unless otherwise specified, all Lie algebras in this survey are assumed to be finite dimensional. In the first two sections, we review some basics of modular Lie theory including absolute toral rank, generalized Winter exponentials, sandwich elements, and standard filtrations. In Section 3, we give a systematic description of all known simple Lie algebras of characteristic p> 3 with emphasis on graded and filtered Cartan type Lie algebras. We also discuss the Melikian algebras of characteristic 5 and their analogues in characteristic 3 and 2. Our main result (Theorem 7) is stated in Section 4 which also contains formulations of several important theorems frequently used in the course of classifying simple Lie algebras. The main principles of our proof of Theorem 7, with emphasis on the rank two case, are outlined in Section 5. 1. The beginnings The theory of Lie algebras over a field F of characteristic p> 0 was initiated by Jacobson, Witt and Zassenhaus. In [J 37], Jacobson has investigated purely inseparable field extensions E/F of the form E = F(c1,..., cn) where c p i ∈ F for all i ≤ n. Although such field extensions do not possess nontrivial Fautomorphisms, Jacobson has developed for them a version of Galois Theory. The rôle of Galois automorphisms in his theory was played by Fderivations. The set DerF E of all Fderivations of E carries the following three structures: • a natural structure of a vector space over E, • a natural pstructure given by the pth power map D ↦ → Dp, • a Lie algebra structure given by the commutator product. Let F denote the set of all subfields of E containing F and L the set of all Esubspaces of DerF E stable under the pth power map and Lie bracket in DerF E. Both sets F and L are partially ordered by inclusion. Given a subset X in DerF E
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.
Block Degeneracy and Cartan Invariants for Graded Lie Algebras of Cartan Type
 J. Algebra
, 1993
"... Let L be a finitedimensional Lie algebra over an algebraically closed field F of characteristic p ≥ 5. An element x ∈ L, x = 0, is an absolute zero divisor if (ad x) 2 = 0 [K1]. (In more recent terminology x is sometimes referred to as a sandwich element [Z].) In the early 60’s Kostrikin [K2] show ..."
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Let L be a finitedimensional Lie algebra over an algebraically closed field F of characteristic p ≥ 5. An element x ∈ L, x = 0, is an absolute zero divisor if (ad x) 2 = 0 [K1]. (In more recent terminology x is sometimes referred to as a sandwich element [Z].) In the early 60’s Kostrikin [K2] showed that these elements play a fundamental role in the structure theory of simple modular
DEFORMATIONS OF W1(n) ⊗ A AND MODULAR SEMISIMPLE LIE ALGEBRAS WITH A SOLVABLE MAXIMAL SUBALGEBRA
, 2003
"... Abstract. In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided pow ..."
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Abstract. In one of his last papers, Boris Weisfeiler proved that if modular semisimple Lie algebra possesses a solvable maximal subalgebra which defines in it a long filtration, then associated graded algebra is isomorphic to one constructed from the Zassenhaus algebra tensored with the divided powers algebra. We completely determine such class of algebras, calculating in process lowdimensional cohomology groups of Zassenhaus algebra tensored with any associative commutative algebra.
RESTRICTED INFINITESIMAL DEFORMATIONS OF RESTRICTED SIMPLE LIE ALGEBRAS
, 705
"... Abstract. We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic p ≥ 5. 1. ..."
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Abstract. We compute the restricted infinitesimal deformations of the restricted simple Lie algebras over an algebraically closed field of characteristic p ≥ 5. 1.
Simple finite group schemes and their infinitesimal deformations, preprint available at arXiv:0811.2668
"... Abstract. 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 simple objects have been classified. We review this ..."
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Abstract. 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 simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes. 1.
DEFORMATIONS OF SIMPLE FINITE GROUP SCHEMES
, 705
"... Abstract. Simple finite group schemes over an algebraically closed field of positive characteristic p ̸ = 2, 3 have been classified. We consider the problem of determining their infinitesimal deformations. In particular, we compute the infinitesimal deformations of the simple finite group schemes of ..."
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Abstract. Simple finite group schemes over an algebraically closed field of positive characteristic p ̸ = 2, 3 have been classified. We consider the problem of determining their infinitesimal deformations. In particular, we compute the infinitesimal deformations of the simple finite group schemes of height one corresponding to the restricted simple Lie algebras. 1.
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.