On Maximal Subalgebras

TL;DR

This paper classifies all maximal subalgebras of certain algebraic structures over an algebraically closed field, providing the first such classification for algebras of dimension greater than one.

Contribution

It offers the first classification of maximal subalgebras in higher-dimensional algebras, specifically for one-dimensional domains and certain two-dimensional cases.

Findings

Classified all maximal subalgebras of one-dimensional finitely generated domains.

Classified all maximal subalgebras of extbf{k}[t, t^{-1}, y] in dimension two.

Provided examples of maximal subalgebras not containing a coordinate.

Abstract

Let $\textbf{k}$ be an algebraically closed field. We classify all maximal $\textbf{k}$-subalgebras of any one-dimensional finitely generated $\textbf{k}$-domain. In dimension two, we classify all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, t^{-1}, y]$. To the authors' knowledge, this is the first such classification result for an algebra of dimension $> 1$. In the course of this study, we classify also all maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that contain a coordinate. Furthermore, we give examples of maximal $\textbf{k}$-subalgebras of $\textbf{k}[t, y]$ that do not contain a coordinate.