# Borel subalgebras of Cartan Type Lie Algebras

## TL;DR

This paper classifies certain subalgebras of the Jacobson-Witt algebra over fields with positive characteristic, focusing on trigonalizable subalgebras related to Borel subalgebras, and determines their conjugation classes and properties.

## Contribution

It introduces a classification of trigonalizable subalgebras related to Borel subalgebras in Jacobson-Witt algebras, extending previous work on homogeneous Borel subalgebras.

## Key findings

- Conjugation classes of these subalgebras are determined.
- Properties such as filtration and dimension are analyzed.
- Relations to previously studied homogeneous Borel subalgebras are established.

## Abstract

Let $ W(n) $ be Jacobson-Witt algebra over algebraic closed field $ \mathbb{K} $ with positive characteristic $ p>2. $ It is difficult to classify all Borel subalgebras of $ W(n) $ or non-classical restricted simple Lie algebras. The present paper and \cite{S7} study two kinds of subalgebras which are easily to understand and highly related to Borel subalgebras. In \cite{S7}, the last author investigates a class of special Borel subalgebras of $W(n)$ which is called homogeneous Borel subalgebras. The present paper focuses on subalgebras of $ W(n) $ which are related to Borel subalgebras such that firstly, they could be trigonalizable; and secondly, they essentially belong to the ones investigated in \cite{S7}. In this paper, the conjugation classes of these subalgebras and representative for each class will be determined. Then some properties such as filtration and dimension will be investigated.