Infinitesimal deformations of restricted simple Lie algebras II

Abstract

Communicated by E.M. Friedlander MSC: Primary: 17B50 secondary: 17B20 17B56 a b s t r a c t We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan-type: the Contact and the Hamiltonian Lie algebras.

Deformations of the Restricted Melikian Lie Algebra

“…Namely, the restricted Melikian algebra M can be realized as a subalgebra of the restricted contact algebra K(5) of rank 5 (see [MEL80]) in such a way that the negative elements of M coincide with the negative elements of K(5). This allows to reduce the above result to the analogous result for the contact algebras (see [VIV2]). …”

“…Rudakov ([RUD71]) proved that the simple Lie algebras of classical type are rigid, in analogy of what happens in characteristic zero. On the other hand, the author computed the infinitesimal deformations of the four families (Witt-Jacobson, Special, Hamiltonian, Contact) of restricted simple algebras of Cartan-type (see [VIV1] and [VIV2]), showing that these are never rigid.…”

Simple finite group schemes and their infinitesimal deformations

“…The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in [33] (building upon [30], [31] and [32]) the infinitesimal deformations of the simple finite group schemes associated to simple and restricted Lie algebras of Cartan type.…”

Restricted Infinitesimal Deformations of Restricted Simple Lie Algebras

“…We prove these theorems using the computation of the second ordinary cohomology group H 2 (g, g) that we performed in [VIV1], [VIV2] and [VIV3], together with the 6-term exact sequence of Hochschild (see section 2.1) that relates the ordinary and restricted cohomology.…”