关于广义Cartan矩阵的注记

Notes on Generahsed Cartan Matrices

本文证明了有限型、仿射型及严格双曲型的广义Ｃａｒｔａｎ矩阵中每一元素的代数余子式均大于０，而双曲型广义Ｃａｒｔａｎ矩阵中每一元素的代数余子式都是非负的。还证明了双曲型广义Ｃａｒｔａｎ矩阵的行列式小于０。对一类所谓的超双曲型广义Ｃａｒｔａｎ矩阵，给出了其分类定理。

In this paper,we investigate some properties of generalised Cartanmatrices. First we prove that the cofactors of generalised Cartan matrices of finitetype。 affine type or strictly hyperbolic type are positive and the cofactors of generalisedCartan matrices of hyperbolic type are non一negative。Then we prove that the determinantsof generalised Cartan matrices of hyperbolic type are negative。Finally, we give the classifi-cation theorem for the generalised Cartan matrices of Super一hyperbolic type。