Algebraic monoids whose nonunits are products of idempotents

Putcha, Mohan S.

doi:10.1090/S0002-9939-1988-0938640-8

Algebraic monoids whose nonunits are products of idempotents
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by Mohan S. Putcha PDF

Proc. Amer. Math. Soc. 103 (1988), 38-40 Request permission

Abstract:

Let $M$ be a connected regular linear algebraic monoid with zero and group of units $G$. Suppose $G$ is nearly simple, i.e. the center of $G$ is one dimensional and the derived group $G’$ is a simple algebraic group. Then it is shown that $S = M\backslash G$ is an idempotent generated semigroup. If $M$ has a unique nonzero minimal ideal, the converse is also proved. It follows that if ${G_0}$ is any simple algebraic group defined over an algebraically closed field $K$ and if $\Phi :{G_0} \to GL(n,K)$ is any representation of ${G_0}$, then the nonunits of the monoid $M(\Phi ) = \overline {K\Phi ({G_0})}$ form an idempotent generated semigroup.

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