Non-degenerate graded Lie algebras with a degenerate transitive subalgebra

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 $L^{\prime }$ such that $\dim L_1^{\prime }>1,$ then $L$ is an infinite-dimensional Lie algebra.

21 pages