Generic property and conjugacy classes of homogeneous Borel subalgebras of restricted Lie algebras

TL;DR

This paper investigates the structure of homogeneous Borel subalgebras in restricted Lie algebras, proves a conjecture related to regular Cartan subalgebras, and classifies conjugacy classes of Borel subalgebras in Jacobson-Witt algebras for characteristic p>3.

Contribution

It proves a conjecture by Premet linking regular Cartan subalgebras to the generic property and classifies homogeneous Borel subalgebras in simple restricted Lie algebras.

Findings

Validation of Premet's conjecture under the generic property.

Classification of conjugacy classes of homogeneous Borel subalgebras in W(n).

Description of associated solvable subgroups.

Abstract

Let $(\mathfrak{g},[p])$ be a finite-dimensional restricted Lie algebra over an algebraically closed field $\mathbb{K}$ of characteristic $p>0$, and $G$ be the adjoint group of $\mathfrak{g}$. We say that $\mathfrak{g}$ satisfying the {\sl generic property} if $\mathfrak{g}$ admits generic tori introduced in \cite{BFS}. A Borel subalgebra (or Borel for short) of $\mathfrak{g}$ is by definition a maximal solvable subalgebra containing a maximal torus of $\mathfrak{g}$, which is further called generic if additionally containing a generic torus. In this paper, we first settle a conjecture proposed by Premet in \cite{Pr2} on regular Cartan subalgebras of restricted Lie algebras. We prove that the statement in the conjecture for a given $\mathfrak{g}$ is valid if and only if it is the case when $\mathfrak{g}$ satisfies the generic property. We then classify the conjugay classes of homogeneous Borel subalgebras of the restricted simple Lie algebras $\mathfrak{g}=W(n)$ under $G$-conjugation when $p>3$, and present the representatives of these classes. Here $W(n)$ is the so-called Jacobson-Witt algebra, by definition the derivation algebra of the truncated polynomial ring $\mathbb{K}[T_1,\cdots,T_n]\slash (T_1^p,\cdots,T_n^p)$. We also describe the closed connected solvable subgroups of $G$ associated with those representative Borel subalgebras.