Classification of Certain Regular Subalgebras of $\mathfrak{sl}(n, \mathbb C)$ up to Conjugacy

TL;DR

This paper classifies certain regular upper-triangular subalgebras of rak{sl}(n, \u00a0C) up to conjugacy, focusing on low-dimensional cases and providing general results for higher codimensions using combinatorial methods.

Contribution

It introduces a combinatorial approach to classify regular subalgebras of rak{sl}(n, C) up to conjugacy, including explicit classifications for dimensions 2, codimension 1, and 2.

Findings

Classified 2-dimensional regular subalgebras up to conjugacy.

Classified codimension 1 and 2 regular subalgebras up to conjugacy.

Developed a combinatorial method for higher codimension classification.

Abstract

In this paper we will be classifying some regular upper-triangular subalgebras of $\mathfrak{sl}(n, \mathbb C)$ up to conjugacy by matrices in $SL(n, \mathbb C)$. We do so for dimension 2, codimension 1, and codimension 2 subalgebras. We prove some general results for codimension $k$. The approach we use reduces an abstract classification problem to a combinatorial one, which we solve through a mixture of inductive and computational approaches.