Algebraic monoids with group kernels
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.
scientific article; zbMATH DE number 883913
scientific article