Non-degenerate graded Lie algebras with a degenerate transitive subalgebra

TL;DR

This paper investigates the properties of modular graded Lie algebras, focusing on the conditions under which non-degenerate algebras containing certain degenerate subalgebras become infinite-dimensional.

Contribution

It characterizes when non-degenerate graded Lie algebras with degenerate transitive subalgebras are infinite-dimensional in characteristic p>2.

Findings

Non-degenerate Lie algebras with degenerate subalgebras and im L'_1>1 are infinite-dimensional.

The analysis extends Weisfeiler's work on degeneration of modular graded Lie algebras.

Abstract

The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler, is analyzed. Transitive irreducible graded Lie algebras $L=\sum_{i\in \mathbb Z}L_i,$ over an algebraically closed field of characteristic $p>2,$ with classical reductive component $L_0$ are considered. We show that if a non-degenerate Lie algebra $L$ contains a transitive degenerate subalgebra $L'$ such that $\dim L'_1>1,$ then $L$ is an infinite-dimensional Lie algebra.