Lex E. Renner

Classification of Semisimple Algebraic Monoids

Let X be a semisimple algebraic monoid with unit group G. Associated with E is its polyhedral root system ( X , @, 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 E + ( X , @, 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. Thls, 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 morphlsm 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. Gl,(k)), dim ZG = 1and E is a normal variety. The problem then, presents us with two familiar objects. Let T c 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 [ l l , Chapter 31 of Killing that were introduced by him [15]in his penetrating formulation of the classification (E. Cartans!) 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.

Reductive monoids are Von Neumann regular

Let k be an algebraically closed field. An a@e algebraic monoid E is an affme algebraic variety, defined over k, together with an associative morphism m: E X E + E and a two-sided unit 1 E E for m. In categorical terminology, an afftne algebraic monoid is a representable functor from the category of afftne varieties to the category of monoids. Algebraic group theory is concerned mainly with reductive groups and their representations whereas the theory of monoids is concerned largely with von Neumann regular monoids. The main result of this paper provides a fundamental link between these two objects.

The canonical compactification of a finite group of Lie type

Let G be a finite group of Lie type. We construct a finite monoid M having G as the group of units. M has properties analogous to the canonical compactification of a reductive group. The complex representation theory of M yields Harish-Chandras philosophy of cuspidal representations of G. The main purpose of this paper is to determine the irreducible modular representations of M. We then show that all the irreducible modular representations of G come (via the 1942 work of Clifford) from the one-dimensional representations of the maximal subgroups of M. This yields a semigroup approach to the modular representation theory of G, via the full rank factorizations of the «sandwich matrices» of M. We then determine the irreducible modular representations of any finite monoid of Lie type

Completely regular algebraic monoids

Abstract By previous results of Putcha and the author, an irreducible algebraic monoid M is regular if and only if the Zariski closure R(G) ⊆ M is completely regular, where R(G) is the solvable radical of G. Thus, the classification problem leads initially to extreme cases; reductive monoids and completely regular monoids with solvable unit groups. In this paper we classify normal, completely regular (NCR) monoids with solvable unit group. It turns out that each NCR monoid M is determined by its unit group G = TU and the closure Z of T in M. For the converse, we find the exact conditions on a diagram T =Z↩T↪G for which there exists an NCR monoid M with Z= T ⊂M .

Algebraic Semigroups Are Strongly π-Regular

Let S be an algebraic semigroup (not necessarily linear) defined over a field F. We show that there exists a positive integer n such that x n belongs to a subgroup of S(F) for any x∈S(F). In particular, the semigroup S(F) is strongly π-regular.

Observability in invariant theory

We consider actions G ×X → X of the affine, algebraic group G on the affine, algebraic variety X. We say that G × X → X is observable in codimension one if for any height-one, G-invariant, prime ideal p ⊂ k[X], pG = (0). Many familiar actions are observable in codimension one. We characterize such actions geometrically and indicate how they fit into the general framework of invariant theory. We look at what happens if we impose further restrictions, such as G being reductive or X being factorial. We indicate how Grosshans subgroups are involved.

Algebras with finitely many orbits

Abstract Any algebra of finite representation type has a finite number of two-sided ideals. But there are stronger finiteness conditions that should be considered here. We consider finite-dimensional K -algebras that have only a finite number of left (respectively, principal left) ideals, up to conjugacy. We then characterize K -algebras A whose Jacobson radical satisfies J ( A ) 2 =0, and with finitely many classes of principal left ideals. Finally, we consider basic algebras with J ( A ) 2 =0. Here we characterize such algebras with finitely many classes of left ideals.

Modular representations of finite monoids of Lie type

Abstract Let M be a finite monoid of Lie type of characteristic p . In this paper we compute the number of irreducible modular representations of M in characteristic p . To do this we combine the theory of semigroup representations, of Munn-Ponizovskii, with Richens theory of modular representations of finite groups of Lie type. Each of these representations is determined by a certain triple ( I , J , χ ) where lϵ 2 s is a subset of the simple roots, Jϵ U (M) is J -class and χ:P 1 → F q ∗ is a character.

Canonical Cellular Decomposition for 𝒥-Irreducible Monoids

A reductive monoid M is called 𝒥-irreducible if M∖{0} has exactly one minimal G × G-orbit. There is a canonical cellular decomposition for such monoids. These cells are defined in terms of idempotents, B × B-orbits, and other natural monoid notions. But they can also be obtained by the method of one-parameter subgroups. This decomposition leads to a number of important combinatorial and topologial properties of the monoid of B × B-orbits of M. In case M∖{0} is rationally smooth these cells are closely related to affine spaces. They can be used to calculate the Betti numbers of a certain projective variety.

The ring of regular functions of an algebraic monoid

Let M be an irreducible normal algebraic monoid with unit group G. It is known that G admits a Rosenlicht decomposition, G=G_antG_aff, where G_ant is the maximal anti-affine subgroup of G, and G_aff the maximal normal connected affine subgroup of G. In this paper we show that this decomposition extends to a decomposition M=G_antM_aff, where M_aff is the affine submonoid M_aff=\bar{G_aff}. We then use this decomposition to calculate