Graded Lie algebras of maximal class of type $n$
Sandro Mattarei, Simone Ugolini
TL;DR
This work advances the classification program for graded Lie algebras of maximal class of type $n$ (with $n>1$) by determining the possible lengths of the first constituent under the hypothesis that the characteristic $p$ of the base field is large relative to $n$ and $p$ is odd. The authors develop a polynomial method that translates a sequence of Lie algebra relations into vanishing conditions on coefficients of a polynomial $(x-1)^{k}g(x)$, yielding sharp restrictions on the first-constituent length $\ell$, which in turn controls the metabelian quotients of $L$. They introduce and analyze an exceptional family $\mathcal{E}$ of algebras, constructed via a divided-power ring and graded derivations, whose first constituent lengths fill in the remaining admissible values predicted by the main polynomial criterion. The full argument combines polynomial bounds, detailed constituent analysis, and intricate Lie-bracket computations to exclude spurious possibilities and obtain a clear dichotomy: either $\ell=2q$ (with $q$ a power of $p$) or $\ell$ lies in a $p$-dependent interval near $q$, with explicit parity restrictions. The paper also provides explicit constructions and generating functions for the exceptional algebras, contributing concrete examples to the classification landscape and linking the type-$n$ theory to the type-$1$ uncovered subalgebras and coclass-inspired phenomena.
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.