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.

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

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.

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 3-dimensional Lie algebra admits a parametric family of non-isomorphic 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.