Abstract. Let L be a finite-dimensional 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 p-envelope 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 finite-dimensional 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 Kostrikin-Shafarevich conjecture (the case where p = 7). 1. Introduction and

Xiuling Wang

Structures of G(2) type and nonholonomic distributions in characteristic p by

Abstract. We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan-type: the Contact and the Hamiltonian Lie algebras. 1.

Abstract. In the first two sections, we review the Block-Wilson-Premet-Strade 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 p-Lie algebras). Definition 1.1 (Jacobson [JAC37]). A Lie algebra L over F is said to be restricted (or a p-Lie algebra) if there exits a map (called p-map), [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 r-times the value 0. Example. (1) Let A an associative F-algebra. Then the Lie algebra DerFA of F-derivations of A is a restricted Lie algebra with respect to the p-map 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 p-map given by the differential of the homomorphism G → G, x ↦ → xp: = x ◦ · · · ◦ x. One can naturally ask when a F-Lie 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 p-map on L if and only if, for every element x ∈ L, the p-th iterate of ad(x) is still an inner derivation. (2) Two such p-maps 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.

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 little-known 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 Cartan-Tanaka-Shchepochkina 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.

Characteristic p is for the time when we retire. 1.

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 infinite-dimensional 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.

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 Albert-Zassenhaus 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

Abstract. We compute the infinitesimal deformations of the restricted Melikian Lie algebra in characteristic 5. 1.