On Graded Lie Algebras of Characteristic Three With Classical Reductive Null Component

Abstract

We consider finite-dimensional irreducible transitive graded Lie algebras $L={\sum }_{i=-q}^r{L}_i$ over algebraically closed fields of characteristic three. We assume that the null component $L_0$ is classical and reductive. The adjoint representation of $L$ on itself induces a representation of the commutator subalgebra $L_0^{\prime }$ of the null component on the minus-one component $L_{-1}.$ We show that if the depth $q$ of $L$ is greater than one, then this representation must be restricted.

46 pages; correct typos; change argument; one hypothesis of result changed: height of Lie algebra now required to be at least two