On Maximal Subalgebras

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.