Nilpotent variety of a reductive monoid.
An irreducible linear algebraic monoid \(M\) with zero is referred to as reductive if its unit group is so; \(M\) is semisimple if it is reductive with \(1\)-dimensional center. The author studies the nilpotent vatiety \(M_{\text{nil}}:=\{a\in M\mid a^n=0\text{ for some }n\}\). In general this variety can be reducible and has a completely different structure from the unipotent variety of the unit group \(G\) of \(M\). When \(M\) is \(\mathcal J\)-irreducible or \(\mathcal J\)-coirreducible, the author obtains a precise description of the irreducible components of \(M_{\text{nil}}\) via the Renner monoid of \(M\) and the Weyl group of \(G\). In particular, the author finds necessary and sufficient conditions for the nilpotent variety to be irreducible.
scientific article; zbMATH DE number 5281979
scientific article