Nilpotent algebraic monoids (Q1911323)

Irreducible linear algebraic monoids \(M\) defined over an algebraically closed field are studied. \(M\) is called nilpotent if its group \(G\) of units is a nilpotent group. A characterization of such monoids is obtained in case \(M\) is regular. In particular, nilpotency is then equivalent to the fact that the set \(E(M)\) of idempotents of \(M\) is finite and the minimum two-sided ideal \(\text{ker}(M)\) of \(M\) is a nilpotent group. We note that arbitrary linear semigroups which are nilpotent in the sense of Malcev are studied in a forthcoming paper by the reviewer [``Nilpotent semigroups of matrices'', Math. Proc. Camb. Philos. Soc. (to appear)].