关于Cartan矩阵的对称性

The Symmetry of Cartan Matrix

给出非分裂域上的Ｃａｒｔａｎ矩阵对称的充分必要条件．对域Ｆ上的对称代数Ａ，通过考察其有关单模的自同态代数的维数，确定Ａ的Ｃａｒｔａｎ矩阵是否对称或置换对称（当Ａ为ｑｕａｓｉ－Ｆｒｏｂｅｎｉｕｓ代数）．

Let A be a quasi Frobenius algebra over a field F and C=(C ij ) the Cartan matrix of A. Using endomorphism algebra of simple modules, the author gives a sufficient and necessary condition of C being symmetric or permutation symmetric, according to A as a symmetric algebra or a quasi Frobenius algebra. Specially when F is a splitting field of A, C is symmetric or permutation symmetric.