A semigroup approach to linear algebraic groups. II: Roots
type: Item
instance of: scholarly article
title: A semigroup approach to linear algebraic groups. II: Roots
zbMATH Open document ID: 0578.20056
DOI: 10.1016/0022-4049(86)90142-8
published in: Journal of Pure and Applied Algebra
publication date: 1986-01-01T00:00:00Z
review text: 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.
reviewed by: Takayuki Tamura
Mathematics Subject Classification ID: 20M10
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 3925124
zbMATH Keywords: algebraic monoid
zbMATH Keywords: group of units
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: roots
zbMATH Keywords: Borel subgroup
zbMATH Keywords: maximal torus
zbMATH Keywords: idempotents
author: Mohan S. Putcha
MaRDI profile type: Publication
full work available at URL: 0022-4049(86)90142-8
OpenAlex ID: W2015175180
cites work: Q4744461
cites work: Corrigendum: On linear algebraic semigroups. III
cites work: Green's relations on a connected algebraic monoid†
cites work: Connected Algebraic Monoids
cites work: The J-class structure of connected algebraic monoids
cites work: Linear algebraic semigroups
cites work: A semigroup approach to linear algebraic groups
cites work: Reductive groups and regular semigroups
cites work: Idempotent cross-sections of J-classes
cites work: Determinant functions on algebraic monoids
cites work: Linear algebraic monoids.
cites work: Reductive monoids are von Neumann regular
cites work: Classification of Semisimple Rank One Monoids
cites work: Linear algebraic groups
cites work: The Unit Groups of Affine Algebraic Monoids
description: scientific article; zbMATH DE number 3925124