非可逆矩阵的小极大空间
Small maximal spaces of non-invertible matrices
摘要：
矩阵的向量空间A称为秩临界的，如果任何适当包含A的向量空间具有严格更高的一般秩。给出了A是秩临界的一个充分条件，并应用这个条件证明了某些李代数表示具有秩临界象。这contradition产量，无限多n，一个8维空间的n次n矩阵是最大的奇异（= rank-critical的秩n-1），而-据我所知-最小尺寸的这样一个空间迄今被认为是增长与n。作为另一个应用，证明了对任何半单李代数g，空间ad（g）是End（g）的秩临界子空间.
英文摘要：
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).
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