Graded Lie algebras of maximal class of type $p$

TL;DR

This work extends the classification of modular graded Lie algebras of maximal class to type $p$, showing that such algebras over a field of characteristic $p>0$ are either subalgebras of uncovered type $1$ algebras, translates thereof, or members of a countable exceptional family ${\mathcal E}$ (empty when $p=2$). The authors develop a framework of constituents and translates, then leverage a polynomial-constraint approach, using Lucas’ theorem, to bound the possible first constituent length $\ell$ to the cases $\ell=2q$ or $\ell=q+p$ (with $q$ a power of $p$), while isolating exceptional instances. An explicit construction of the exceptional algebras via a ring of divided powers underpins ${\mathcal E}$, completing the three-part classification stated in the introduction. The results connect to earlier work in characteristic two and general odd characteristics, providing a unified view of type $p$ algebras and enriching the theory of coclass for modular Lie algebras.

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.

Theorems & Definitions (29)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 5
• Proposition 6: Ugolini
• proof
• Remark 7
• Lemma 8
• proof