Cohomology of Restricted Filiform Lie Algebras $\mathfrak{m}_2^λ(p)$

Tyler J. Evans, Alice Fialowski

Published 2019-01-20Version 1

Consider the $p$-dimensional filiform Lie algebra $\mathfrak{m}_2(p)$ over a field $\mathbb{F}$ of prime characteristic $p$ with nonzero Lie brackets $[e_1,e_i]=e_{i+1}$ for $1<i<p$ and $[e_2,e_i]=e_{i+2}$ for $2<i<p-1$. We show that there is a family $\mathfrak{m}_2^{\lambda}(p)$ of restricted Lie algebra structures parameterized by elements $\lambda \in \mathbb{F}^p$. We explicitly describe bases for the the ordinary and restricted 1- and 2-cohomology spaces, and give formulas for the bracket and $[p]$-operations in the corresponding restricted one-dimensional central extensions.