Graded Lie algebras of maximal class of type $p$

TL;DR

This paper extends the classification of infinite-dimensional graded Lie algebras of maximal class over fields of positive characteristic p, generalizing previous results from characteristic 2 to arbitrary primes.

Contribution

It provides an extended classification of graded Lie algebras of maximal class for any prime characteristic p, building upon prior work in characteristic 2.

Findings

Classification extended to arbitrary prime p

Key steps in the proof of the classification

New structural properties identified

Abstract

The algebras of the title are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, over a field of positive characteristic $p$, that are generated by an element of degree $1$ and an element of degree $p$, and satisfy $[L_i,L_1]=L_{i+1}$ for $i\ge p$. In case $p=2$ such algebras were classified by Caranti and Vaughan-Lee in 2003. We announce an extension of that classification to arbitrary prime characteristic, and prove several major steps in its proof.