Classification of Semisimple Algebraic Monoids

Let A' be a semisimple algebraic monoid with unit group G. Associated with E is its polyhedral root system (X, 0, C), where X = X(T) is the character group of the maximal torus T c G, $ c X(T) is the set of roots, and C = X(T) is the character monoid of T c E (Zariski closure). The correspondence £-»(A\ O, C) is a complete and discriminating invariant of the semisimple monoid £, 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. Consider the classification problem for semisimple, algebraic monoids over the algebraically closed field k. What sort of problem is this? First the definition: A semisimple, algebraic monoid is an irreducible, affine, algebraic variety E, defined over k together with an associative morphism m: E X E -* E and a two-sided unit 1 e E for m. We assume further that E has a 0, the unit group G (which is always linear, algebraic and dense in E), is reductive (e.g. G\n(k)), dim ZG = 1 and is is a normal variety. The problem then, presents us with two familiar objects. Let Tc G be a maximal torus. Then Z = T c E (Zariski closure) is an affine, torus embedding and G c E is a reductive, algebraic group. Torus embeddings have been introduced by Demazure in [6] in his study of Cremona groups, and are classified numerically using rational, polyhedral cones [14]. On the other hand, reductive groups have been studied, at least in principle, since the nineteenth century; their classification in the modern sense being achieved largely by Chevalley [4]. That numerical classification uses the now familiar root systems [11, Chapter 3] of Killing that were introduced by him [15] in his penetrating formulation of the classification (E. Cartan's!) of semisimple Lie algebras. Thus, in the classification of semisimple monoids we are compelled to consider the root system (X, $) = (X(T), ®(T)) of G, and the polyhedral cone C = X(Z) c X(T) of T c Z. The two objects are canonically related by the Weyl group action on X, which leaves C stable. Received by the editors November 16, 1984. 1980 Mathematics Subject Classification. Primary 14M99; Secondary 20M99.