Coxeter变换的周期性及其Euler形式的非负性
On the periodicity of Coxeter transformations and the non-negativity of their Euler forms
摘要：
我们表明，分段遗传代数，Coxeter变换的周期性意味着欧拉形式的非负性。与以前的假设相反，分段遗传的条件不能被忽略，即使是三角代数，如偏序集的关联代数所证明的。我们还给出了一个简单、直接的证明，证明了对任意一个主对角线上为2的方阵A，特别是对应于广义Cartan矩阵的Coxeter变换，定义的某些反射积可以表示为$-A_{+}^{-1} A_{-}^t$，其中A_{+}，A_{-}与A的上、下三角部分密切相关。
英文摘要：
We show that for piecewise hereditary algebras, the periodicity of the Coxeter transformation implies the non-negativity of the Euler form. Contrary to previous assumptions, the condition of piecewise heredity cannot be omitted, even for triangular algebras, as demonstrated by incidence algebras of posets. We also give a simple, direct proof, that certain products of reflections, defined for any square matrix A with 2 on its main diagonal, and in particular the Coxeter transformation corresponding to a generalized Cartan matrix, can be expressed as $-A_{+}^{-1} A_{-}^t$, where A_{+}, A_{-} are closely associated with the upper and lower triangular parts of A.
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