Bundle structure of the Green classes in $M_{n}(\mathbb{K})$

Abstract

It is shown using the semigroup structure of $M_{n}(\mathbb{K})$ that the Green classes of the multiplicative semigroup $M_{n}(\mathbb{K})$ of linear endomorphisms of an n-dimensional vector space V over $\mathbb{K}$ (=ℝ, or ℂ) are the total spaces of certain fibre bundles having various Grassmann manifolds as the base spaces and the maps x↦R(x), x↦N(x) as the projection maps. In the case of a -class the fibre space is a certain Stiefel manifold and in the case of - and -classes the fibre space is a general linear group. The bundle structures are established by constructing representative coordinate bundles. It is also shown that the bundles having x↦R(x) as the projection map are equivalent to the bundles having x↦N(x) as the projection map.