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

Abstract

We consider finite-dimensional irreducible transitive graded Lie algebras $L = \sum_{i=-q}^rL_i$ 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 $L_0'$ 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.