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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.
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.
Classification of simple finite dimensional modular Lie superalgebras with indecomposable Cartan matrix
, 2007
"... Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical hypotheses. Either these Lie superalgebras are simple or the quotients of their derived algebras modulo center are simple. Twelve new exceptional simp ..."
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Cited by 4 (2 self)
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Finite dimensional modular Lie superalgebras over algebraically closed fields with indecomposable Cartan matrices are classified under some technical hypotheses. Either these Lie superalgebras are simple or the quotients of their derived algebras modulo center are simple. Twelve new exceptional simple modular Lie superalgebras are discovered.
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
New simple modular Lie superalgebras as generalized prolongations, Funktsional. Anal. i Prilozhen
 English transl.: Funct. Anal. Appl
"... Abstract. Over algebraically closed fields of characteristic p> 2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exce ..."
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Abstract. Over algebraically closed fields of characteristic p> 2, prolongations of the simple finite dimensional Lie algebras and Lie superalgebras with Cartan matrix are studied for certain simplest gradings of these algebras. Several new simple Lie superalgebras are discovered, serial and exceptional, including superBrown and superMelikyan superalgebras. Simple Lie superalgebras with Cartan matrix of rank 2 are classified. 1.
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.
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.
On the Structure of Cohomology of Hamiltonian pAlgebras
, 2004
"... Abstract. We demonstrate advantages of nonstandard grading for computing cohomology of restricted Hamiltonian and Poisson algebras. These algebras contain the inner grading element in the properly defined symmetric grading compatible with the symplectic structure. Using modulo p analog of the theor ..."
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Abstract. We demonstrate advantages of nonstandard grading for computing cohomology of restricted Hamiltonian and Poisson algebras. These algebras contain the inner grading element in the properly defined symmetric grading compatible with the symplectic structure. Using modulo p analog of the theorem on the structure of cohomology of Lie algebra with inner grading element, we show that all nontrivial cohomology classes are located in the grades which are the multiples of the characteristic p. Besides, this grading implies another symmetries in the structure of cohomology. These symmetries are based on the Poincaré duality and symmetry with respect to transpositions of conjugate variables of the symplectic space. Some results obtained by computer program utilizing these peculiarities in the cohomology structure are presented. 1