Simple Lie algebras of characteristic $p$ with dependent roots

Abstract:

We investigate finite dimensional simple Lie algebras over an algebraically closed field ${\mathbf {F}}$ of characteristic $p \geqslant 7$ having a Cartan subalgebra $H$ whose roots are dependent over ${\mathbf {F}}$. We show that $H$ must be one-dimensional or for some root $\alpha$ relative to $H$ there is a $1$-section ${L^{(\alpha )}}$ such that the core of ${L^{(\alpha )}}$ is a simple Lie algebra of Cartan type $H{(2:\underline m :\Phi )^{(2)}}$ or $W(1:\underline n )$ for some $n > 1$. The results we obtain have applications to studying the local behavior of simple Lie algebras and to classifying simple Lie algebras which have a Cartan subalgebra of dimension less than $p - 2$.