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

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.

24 pages. The title is changed. In the revised version, we limit to classify the conjugacy classes of homogeneous Borel subalgebras

References in corpus (3)

• Localization of modules for a semisimple Lie algebra in prime characteristic
• On the smoothness of normalisers and the subalgebra structure of modular Lie algebras
• Borel subalgebras of the Witt algebra $W_1$

Cited by in corpus (2)

• Toral stabilizers of restricted Lie algebras
• Borel subalgebras of Cartan Type Lie Algebras