Graded Lie algebras of maximal class of type $n$

TL;DR

This paper investigates the structure and classification of infinite-dimensional graded Lie algebras of a specific type, extending previous classifications for type 2 to arbitrary types over large fields.

Contribution

It establishes the foundational classification framework for algebras of arbitrary type n, focusing on the first constituent length as a key invariant.

Findings

Classifies possible first constituent lengths for type n algebras

Extends classification results from type 2 to arbitrary n

Provides a detailed description of algebraic structures based on constituent length

Abstract

Let $n>1$ be an integer. The algebras of the title, which we abbreviate as algebras of type $n$, are infinite-dimensional graded Lie algebras $L= \bigoplus_{i=1}^{\infty}L_i$, which are generated by an element of degree $1$ and an element of degree $n$, and satisfy $[L_i,L_1]=L_{i+1}$ for $i\ge n$. Algebras of type $2$ were classified by Caranti and Vaughan-Lee in 2000 over any field of odd characteristic. In this paper we lay the foundations for a classification of algebras of arbitrary type $n$, over fields of sufficiently large characteristic relative to $n$. Our main result describes precisely all possibilities for the first constituent length of an algebra of type $n$, which is a numerical invariant closely related to the dimension of its largest metabelian quotient.