Classification of semisimple rank one monoids

Abstract:

Consider the classification problem for irreducible, normal, algebraic monoids with unit group $G$. We obtain complete results for the groups $\operatorname {Sl}_2(K) \times {K^\ast }$, $\operatorname {Gl}_2(K)$ and $\operatorname {PGl}_2(K) \times {K^\ast }$. If $G$ is one of these groups let $\mathcal {E}(G)$ denote the set of isomorphy types of normal, algebraic monoids with zero element and unit group $G$. Our main result establishes a canonical one-to-one correspondence $\mathcal {E}(G) \cong {{\mathbf {Q}}^ + }$, where ${{\mathbf {Q}}^ + }$ is the set of positive rational numbers. The classification is achieved in two steps. First, we construct a class of monoids from linear representations of $G$. That done, we show that any other $E$ must already be one of those constructed. To do this, we devise an extension principle analogous to the big cell construction of algebraic group theory. This yields a birational comparison morphism $\varphi :{E_r} \to E$, for some $r \in {{\mathbf {Q}}^ + }$, which is ultimately an isomorphism because the monoid ${E_r} \in \mathcal {E}(G)$ is regular. The relatively insignificant classification problem for normal monoids with group $G$ and no zero element is also solved. For each $G$ there is only one such $E$ with $G \subsetneqq E$.