Restricted one-dimensional central extensions of the restricted filiform Lie algebras [formula omitted

Abstract

Consider the filiform Lie algebra m0 with nonzero Lie brackets [e1,ei]=ei+1 for 1<i<p, where the characteristic of the field F is p>0. We show that there is a family m0λ(p) of restricted Lie algebra structures parameterized by elements λ∈Fp. We explicitly describe both the ordinary and restricted 1-cohomology spaces and show that for p≥3 these spaces are equal. We also describe the ordinary and restricted 2-cohomology spaces and interpret our results in the context of one-dimensional central extensions.