Restricted one-dimensional central extensions of the restricted filiform Lie algebras m0λ(p)

Consider the filiform Lie algebra m(0) with nonzero Lie brackets [e(1), e(i)] = e(i+1) for 1 < i < p, where the characteristic of the field F is p > 0. We show that there is a family m(0)(lambda)(p) of restricted Lie algebra structures parameterized by elements lambda is an element of F-P. 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. (C) 2018 Elsevier Inc. All rights reserved.