Algebraic monoids with group kernels
type: Item
instance of: scholarly article
title: Algebraic monoids with group kernels
zbMATH Open document ID: 0855.20052
DOI: 10.1007/BF02574107
published in: Semigroup Forum
publication date: 1997-01-27T00:00:00Z
full work available at URL: https://eudml.org/doc/135457
review text: Let \(M\) be an algebraic monoid, that is \(M\) be both an affine variety over an algebraically closed field \(K\) and a monoid for which the operation of multiplication \(M\times M\to M\) is an affine variety morphism. An algebraic monoid \(M\) is irreducible if it is so as an affine variety. \(M\) is regular if \(a\in aMa\) for all \(a\in M\). An algebraic monoid is called nilpotent [solvable, unipotent, reductive] if it is irreducible and its unit group is nilpotent [solvable, unipotent, reductive]. Let \(M^c\) denote the irreducible component of \(1\in M\). For an idempotent \(f\in M\) by \(M_f\) is denoted \(\{x\in M\mid xf=fx=f\}^c\) and by \(G_f\) is denoted the unit group of \(M_f\). By \(R(G)\) \([R_u(G)]\) is denoted the solvable [unipotent] radical of \(G\). Let \(M\) be an irreducible closed submonoid of the algebraic monoid \(M_n(K)\) of all \(n\times n\) matrices over the field \(K\) with \(G(M)=G\) and \(f\) be a central idempotent of \(M\). Then (i) \(R(G_f)=R(G)_f=(R(G)\cap G_f)^c\); (ii) \(R_u(G_f)=R_u(G)_f=((R(G))_f)_u\); (iii) if \(M\) is regular, \(\dim_u(G)=\dim(R_u(fG))\). So, the map \(R_u(G)\to R_u(fG)\) given by \(u\mapsto fu\), is a finite morphism, and if moreover, \(\text{char }K=0\), it is an isomorphism of algebraic groups. Using this theorem the author proves several corollaries, in particular, the Putcha-Renner theorem.
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 883913
zbMATH Keywords: algebraic monoids
zbMATH Keywords: affine varieties
zbMATH Keywords: unit groups
zbMATH Keywords: irreducible closed submonoids
zbMATH Keywords: central idempotents
zbMATH Keywords: algebraic groups
zbMATH Keywords: Putcha-Renner theorem
author: Huang Wenxue
MaRDI profile type: Publication
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OpenAlex ID: W1987317990
description: scientific article; zbMATH DE number 883913