Classification of Semisimple Algebraic Monoids

Abstract

Abstract.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.

Smooth Projective Symmetric Varieties With Picard Number One

“…It is easily showed that every complete symmetric variety is covered by simple open symmetric G-subvarieties whose closed orbits are complete. We generalize the results of Timashev (see [16]) and Renner (see [12]) on the equivariant embeddings of reductive groups. The idea of the proof is the following one.…”

“…Observe that G acts on End C (V (µ i )) ⊂ End C (V ) and the action coincides with the one over V ⊗ V * through the canonical isomorphism End C (V ) ∼ = V ⊗ V * . Also Renner (see [12]) considers the case where G/H is isomorphic to a reductive groupĠ, but he studies the affine (normal) equivariant embeddings ofĠ with a G-fixed point. He classifies the smooth ones in the case where the center ofĠ has dimension 1 and the derived group ofĠ is simple.…”

Conjugacy Classes in Algebraic Monoids

“…We will further asssume that the group of units G of M is reductive. Then by [6,10], M is unit regular, i.e. M = E(M)G. Here E(M) is the idempotent set of M. We fix a maximal torus T of G. If Y Ç E(T), then we let CrG(Y) = {aE G\ae = eae for all e E Y}, ClG(Y) = {a E G\ea = eae for all e € T}.…”

“…This means [1,3] that the unipotent radical of G is trivial. Then by [6,10], M is unit regular, i.e. M = E(M)G where E -E(M) = {e E M\e2 = e}.…”

Representation Theory of Reductive Normal Algebraic Monoids

“…Thus it seems reasonable in trying to extend representation theory from reductive groups to reductive monoids to look first at the case when M is normal. Second, L. Renner [14] has obtained a classification theorem for such monoids under the additional assumptions that the center Z(M ) is 1-dimensional and that M has a zero element. Renner calls such algebraic monoids semisimple and proves that they are classified by data of the form (X(T ), Φ, X(T )), where (X(T ), Φ) is the usual root data which classifies the reductive group G and X(T ) is the character group of the closure (in M ) of the maximal torus T .…”