The group of units of a connected algebraic monoid†

Abstract

Let S be a connected algebraic monoid with zero and let G denote the group of units of S. In this paper we present some evidence to support the claim that the semigroup S\G can be studied to shed some light on the structure of the algebraic group G. Let T denote a maximal torus of G and let 0=ek <⋅<e 0=1 be a maximal chain in E([Tbar]). For i=1,…,k, let α i denote the number of idempotents of the -class of ei in e i−1[Tbar]. Let W(G) denote the Weyl group of G. Theorem 1 ∣W(G)∣=α1…α k . Theorem 2. The following conditions are equivalent. (1) G is solvable. (2) ∣E([Tbar])∩J∣1 for each -class J of S. (3) S is a semilattice of archimedean semigroups. (4) For all e,f∊E(S) any eigenvalue of ef is either 0 or 1. In particular if E(S) is finite or if S is an orthodox semigroup, then G is solvable. Many other generalizations and related results are also obtained.