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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.
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.
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.
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.
INFINITESIMAL DEFORMATIONS OF SYMMETRIC SIMPLE MODULAR LIE ALGEBRAS AND LIE SUPERALGEBRAS
, 807
"... Abstract. Over algebraically closed fields of positive characteristic, infinitesimal deformations of simple finite dimensional symmetric (the ones that with every root have its opposite of the same multiplicity) Lie algebras and Lie superalgebras are described for small ranks. The results are obtain ..."
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Abstract. Over algebraically closed fields of positive characteristic, infinitesimal deformations of simple finite dimensional symmetric (the ones that with every root have its opposite of the same multiplicity) Lie algebras and Lie superalgebras are described for small ranks. The results are obtained by means of the Mathematica based code SuperLie. The infinitesimal deformation given by any odd cocycle is integrable. The moduli of the deformations form, in general, a supervariety. Not each even cocycle is integrable; but for those that are integrable, the global deforms (the results of deformations) are linear with respect to the parameter. In characteristic 2, the simple 3dimensional Lie algebra admits a parametric family of nonisomorphic simple deforms. Some of Shen’s ”variations of G(2) theme ” are interpreted as two global deforms corresponding to the several of the 20 infinitesimal deforms first found by Chebochko; we give their explicit form. 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.
DEFORMATIONS OF SIMPLE RESTRICTED LIE ALGEBRAS I
, 2006
"... Abstract. We compute the infinitesimal deformations of two families of simple restricted modular Lie algebras of Cartantype: the WittJacobson and the Special Lie algebras. 1. ..."
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Abstract. We compute the infinitesimal deformations of two families of simple restricted modular Lie algebras of Cartantype: the WittJacobson and the Special Lie algebras. 1.