Manifold structures in regular irreducible algebraic monoids

TL;DR

This paper explores the manifold structures within regular irreducible algebraic monoids over complex numbers, revealing their geometric properties and connections to Grassmann manifolds, and generalizing known dimension results.

Contribution

It demonstrates that Green classes and idempotent spaces in these monoids have natural manifold structures and investigates their interactions with the semigroup structure, extending previous dimension results.

Findings

Green classes have natural manifold structures

Idempotent spaces form manifolds with semigroup interactions

Established relations between these manifolds and Grassmann manifolds

Abstract

In this paper we study regular irreducible algebraic monoids over $\fldc$ equipped with the euclidean topology. It is shown that, in such monoids, the Green classes and the spaces of idempotents in the Green classes all have natural manifold structures. The interactions of these manifold structures and the semigroup structures in these monoids have been investigated. Relations between these manifolds and Grassmann manifolds have been established. A generalisation of a result on the dimension of the manifold of rank $k$ idempotents in the semigroup of linear endomorphisms over $\fldc$ has been proved.