有限维结合代数的Coxeter矩阵

Abstract

Coxeter矩阵理论在李理论,有限维结合代数的表示理论等学科起着重要作用.由Gabriel定理,代数闭域上基的,连通的有限维结合代数A同构于一个由连通有限箭图Q确定的路代数的商代数.本文先证明了当Q中无有向圈时,对顶点集适当排序后,A的整体维数有限,进而A的Cartan矩阵在整数环上可逆.然后利用A的Cartan矩阵和对称双线性型定义了A的基本反射,并利用数学归纳法证明了在Q无有向圈的条件下,A的Coxeter矩阵可分解为基本反射的乘积.

Coxeter matrices of finite dimensional associative algebras play an important role in many topics,such as Lie theory and the representation theory of associative algebras.From the famous Gabriel's theorem,a basic connected finite dimensional associative algebra A over an algebraically closed field can be looked as a quotient of a path algebra decided by a connected finite quiver Q.This pater first prove that A has finite global dimension,if Q has no oriented cycles.Then the concept of fundamental reflection of A is introduced.At last,this pater prove that the Coxeter matrix of A has a composition of the fundamental reflections with the condition that Q has no oriented cycles.