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36
Simple Lie algebras of small characteristic VI. Completion of the classification, in preparation
"... Abstract. Let L be a finitedimensional simple Lie algebra over an algebraically closed field F of characteristic p> 3. We prove in this paper that if for every torus T of maximal dimension in the penvelope of ad L in Der L the centralizer of T in ad L acts triangulably on L, then L is either class ..."
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Abstract. Let L be a finitedimensional simple Lie algebra over an algebraically closed field F of characteristic p> 3. We prove in this paper that if for every torus T of maximal dimension in the penvelope of ad L in Der L the centralizer of T in ad L acts triangulably on L, then L is either classical or of Cartan type. As a consequence we obtain that any finitedimensional simple Lie algebra over an algebraically closed field of characteristic p> 5 is either classical or of Cartan type. This settles the last remaining case of the generalized KostrikinShafarevich conjecture (the case where p = 7). 1. Introduction and
Quantizations of generalizedWitt algebra and of JacobsonWitt algebra in modular case
, 2006
"... Xiuling Wang ..."
STRUCTURES OF G(2) TYPE AND NONINTEGRABLE DISTRIBUTIONS IN CHARACTERISTIC p
"... Structures of G(2) type and nonholonomic distributions in characteristic p by ..."
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Structures of G(2) type and nonholonomic distributions in characteristic p by
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.
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.
Towards classification of simple finite dimensional modular Lie superalgebras in characteristic p
 J. Prime Res. Math
"... Characteristic p is for the time when we retire. 1. ..."
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Characteristic p is for the time when we retire. 1.
SIMPLE LIE SUPERALGEBRAS AND NONINTEGRABLE DISTRIBUTIONS IN CHARACTERISTIC p
, 2006
"... Abstract. Recently, Grozman and Leites returned to the original Cartan’s description of Lie algebras to interpret the Melikyan algebras (for p≤5) and several other littleknown simple Lie algebras over algebraically closed fields for p = 3 as subalgebras of Lie algebras of vector fields preserving n ..."
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Abstract. Recently, Grozman and Leites returned to the original Cartan’s description of Lie algebras to interpret the Melikyan algebras (for p≤5) and several other littleknown simple Lie algebras over algebraically closed fields for p = 3 as subalgebras of Lie algebras of vector fields preserving nonintegrable distributions analogous to (or identical with) those preserved by G(2), O(7), Sp(4) and Sp(10). The description was performed in terms of CartanTanakaShchepochkina prolongs using Shchepochkina’s algorithm and with the help of SuperLie package. Grozman and Leites also found two new series of simple Lie algebras. Here we apply the same method to distributions preserved by one of the two exceptional simple finite dimensional Lie superalgebras over C; for p = 3, we obtain a series of new simple Lie superalgebras and an exceptional one. In memory of Felix Aleksandrovich Berezin F. A. Berezin and supersymmetries are usually associated with physics. However, Lie superalgebras — infinitesimal supersymmetries — appeared in topology at approximately the same time as the word “spin ” appeared in physics and it were these examples that Berezin first had in mind.
GENERATORS OF SIMPLE LIE ALGEBRAS IN ARBITRARY CHARACTERISTICS
"... The motivation for this study goes back to some classical questions concerning generating sets of finite simple groups. It is a wellknown result that finite simple groups are generated by 2 elements (Dixon’s conjecture, see for instance [1, 3, 15]). In [5], Guralnick and Kantor were able to improve ..."
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The motivation for this study goes back to some classical questions concerning generating sets of finite simple groups. It is a wellknown result that finite simple groups are generated by 2 elements (Dixon’s conjecture, see for instance [1, 3, 15]). In [5], Guralnick and Kantor were able to improve this result by showing the following stronger property:
Gradings of nongraded Hamiltonian Lie algebras
"... Abstract. A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2: n; ω2) (of dimension one less than a power of p) from which we construct infinitedimensional thi ..."
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Abstract. A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2: n; ω2) (of dimension one less than a power of p) from which we construct infinitedimensional thin Lie algebras. In the process we provide an explicit identification of H(2: n; ω2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic). 1.
THIN LOOP ALGEBRAS OF ALBERTZASSENHAUS ALGEBRAS
, 2006
"... Abstract. Thin Lie algebras are Lie algebras over a field, graded over the positive integers and satisfying a certain narrowness condition. In particular, all homogeneous components have dimension one or two, and are called diamonds in the latter case. The first diamond is the component of degree on ..."
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Abstract. Thin Lie algebras are Lie algebras over a field, graded over the positive integers and satisfying a certain narrowness condition. In particular, all homogeneous components have dimension one or two, and are called diamonds in the latter case. The first diamond is the component of degree one, and the second diamond can only occur in degrees 3, 5, q or 2q − 1, where q is a power of the characteristic of the underlying field. Here we consider several classes of thin Lie algebras with second diamond in degree q. In particular, we identify the Lie algebras in one of these classes with suitable loop algebras of certain AlbertZassenhaus Lie algebras. We also apply a deformation technique to recover other thin Lie algebras previously produced as loop algebras of certain graded Hamiltonian Lie algebras. A graded Lie algebra