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

## TL;DR

This paper computes the cohomology of a family of restricted filiform Lie algebras over fields of prime characteristic, providing explicit bases and formulas for their algebraic structures and extensions.

## Contribution

It explicitly describes the cohomology and algebraic structures of a new family of restricted filiform Lie algebras parameterized by elements of the field.

## Key findings

- Explicit bases for ordinary and restricted cohomology spaces.
- Formulas for brackets and p-operations in central extensions.
- Classification of restricted Lie algebra structures for the given family.

## Abstract

For the $p$-dimensional filiform Lie algebra ${\mathfrak m}_2(p)$ over a field ${\mathbb F}$ of prime characteristic $p\ge 5$ 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 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.