A semigroup approach to linear algebraic groups. II: Roots
The author has studied finding connections between the structures of the algebraic monoid and its group of units [Part I, J. Algebra 80, 164-185 (1983; Zbl 0519.20043)]. This paper is a continuation of his work. The main topics in this paper are parabolic subgroups and roots of the group of units. A closed subgroup of the group G of units containing a Borel subgroup is called parabolic. Let B be a Borel subgroup containing a maximal torus T, \(E(\bar T)\) the finite set of all ''diagonal'' idempotents, \(\Lambda\) the set of elements e of \(E(\bar T)\) satisfying \(ae=eae\) for all \(a\in B\). The main theorem in this paper shows that for a parabolic subgroup P of G containing B, there is a chain \(\Gamma\) \(\subseteq \Lambda\) such that P equals the set of \(a\in G\) satisfying \(as=eae\) for all \(a\in \Gamma\). The author also finds interesting connections between the roots of the group of units and some structure of the monoid through the determinants.
scientific article; zbMATH DE number 3925124