Varieties of tori and Cartan subalgebras of restricted Lie algebras

This paper investigates varieties of tori and Cartan subalgebras of a finite-dimensional restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically closed field $k$ of positive characteristic $p$. We begin by showing that schemes of tori may be used as a tool to retrieve results by A. Premet on regular Cartan subalgebras. Moreover, they give rise to principal fibre bundles, whose structure groups coincide with the Weyl groups in case $\mathfrak{g}= \operatorname{Lie}(\mathcal{G})$ is the Lie algebra of a smooth group $\mathcal{G}$. For solvable Lie algebras, varieties of tori are full affine spaces, while simple Lie algebras of classical or Cartan type cannot have varieties of this type. In the final sections the quasi-projective variety of Cartan subalgebras of minimal dimension ${\rm rk}(\mathfrak{g})$ is shown to be irreducible of dimension $\dim_k\mathfrak{g}-{\rm rk}(\mathfrak{g})$, with Premet's regular Cartan subalgebras belonging to the regular locus.