Nilpotent varieties of some finite dimensional restricted Lie algebras

In the late 1980s, A. Premet conjectured that the variety of nilpotent elements of any finite dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$ is irreducible. This conjecture remains open, but it is known to hold for a large class of simple restricted Lie algebras, e.g. for Lie algebras of connected algebraic groups, and for Cartan series $W, S$ and $H$. In this thesis we start by proving that Premet's conjecture can be reduced to the semisimple case. The proof is straightforward. However, the reduction of the semisimple case to the simple case is very non-trivial in prime characteristic as semisimple Lie algebras are not always direct sums of simple ideals. Then we consider some semisimple restricted Lie algebras. Under the assumption that $p>2$, we prove that Premet's conjecture holds for the semisimple restricted Lie algebra whose socle involves the special linear Lie algebra $\mathfrak{sl}_2$ tensored by the truncated polynomial ring $k[X]/(X^{p})$. Then we extend this example to the semisimple restricted Lie algebra whose socle involves $S\otimes \mathcal{O}(m; \underline{1})$, where $S$ is any simple restricted Lie algebra such that $\text{ad} S=\text{Der} S$ and its nilpotent variety $\mathcal{N}(S)$ is irreducible, and $\mathcal{O}(m; \underline{1})=k[X_1, \dots, X_m]/(X_1^p, \dots, X_m^p)$ is the truncated polynomial ring in $m\geq 2$ variables. In the final chapter we assume that $p>3$. We confirm Premet's conjecture for the minimal $p$-envelope $W(1; n)_{p}$ of the Zassenhaus algebra $W(1; n)$ for all $n\geq 2$. This is the main result of the research paper [3] (see thesis) which was published in the Journal of Algebra and Its Applications.