Infinitesimal deformations of restricted simple Lie algebras I

Abstract

We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan-type: the WittJacobson and the Special Lie algebras.

Infinitesimal deformations of restricted simple Lie algebras II

“…This action becomes homogeneous with respect to the weight decomposition of K (n) ≥0 if we give the weight −2 n to the generator of F λ−σ . As remarked in [11,Sec. 2.1], in this situation we have that…”

“…The strategy of the proof is exactly the same as that of the proposition [11,Prop. 3.10], that is to compute, step-by-step as d increases, the truncated invariant cohomology groups…”

Restricted Infinitesimal Deformations of Restricted Simple Lie Algebras

“…Since the restricted infinitesimal deformations of a restricted Lie algebra g correspond to the infinitesimal deformations of the associated finite group scheme G of height one, the above results give the infinitesimal deformations of some simple finite group schemes (see [VIV4] and [VIV5] for more details). In order to complete the picture, it would be very interesting to extend the above computations to the minimal p-envelope of all the 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.…”

Deformations of the Restricted Melikian Lie Algebra

“…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.…”