Small maximal spaces of non-invertible matrices

Abstract

A vector space A of matrices is called rank-critical if any vector space that properly contains A has a strictly higher generic rank. I present a sufficient condition for A to be rank-critical, and apply this condition to prove that certain Lie algebra representations have rank-critical images. This contruction yields, for infinitely many n, an 8-dimensional space of n-times-n matrices that is maximal singular (= rank-critical of rank n-1), while--to the best of my knowledge--the minimal dimension of such a space was hitherto believed to grow with n. As another application, I prove that for any semisimple Lie algebra g, the space ad(g) is a rank-critical subspace of End(g).