Classification of semisimple algebraic monoids

Abstract:

Let $X$ be a semisimple algebraic monoid with unit group $G$. Associated with $E$ is its polyhedral root system $(X,\Phi ,C)$, where $X = X(T)$ is the character group of the maximal torus $T \subseteq G$, $\Phi \subseteq X(T)$ is the set of roots, and $C = X(\overline T )$ is the character monoid of $\overline T \subseteq E$ (Zariski closure). The correspondence $E \to (X,\Phi ,C)$ is a complete and discriminating invariant of the semisimple monoid $E$, extending the well-known classification of semisimple groups. In establishing this result, monoids with preassigned root data are first constructed from linear representations of $G$. That done, we then show that any other semisimple monoid must be isomorphic to one of those constructed. To do this we devise an extension principle based on a monoid analogue of the big cell construction of algebraic group theory. This, ultimately, yields the desired conclusions.