Cohomology of Restricted Filiform Lie Algebras mλ₂(p)

Abstract

For the p-dimensional filiform Lie algebra m₂(p) over a field F of prime characteristic p≥5 with nonzero Lie brackets [e₁,eᵢ]=eᵢ₊₁ for 1 < i < p and [e₂,eᵢ]=eᵢ₊₂ for 2 < i < p − 1, we show that there is a family mλ₂(p) of restricted Lie algebra structures parameterized by elements λ ∈ 𝔽ᵖ. We explicitly describe bases for the ordinary and restricted 1- and 2-cohomology spaces with trivial coefficients, and give formulas for the bracket and [p]-operations in the corresponding restricted one-dimensional central extensions.