Gabriel's theorem

《Introduction to Representation Theory》 · 印刷第 8 页 Section 2.2 的基域约定、第 160 页 Section 6.4 的图与内积约定、第 163 页 Definition 6.5.1 及第 164 页 Theorem 6.5.2 完整陈述

Let $k$ be a field. Unless stated otherwise, we will always assume that $k$ is algebraically closed, i.e., any nonconstant polynomial with coefficients in $k$ has a root in $k$. […] From now on, let $\Gamma$ be a fixed graph of type $A_n,D_n,E_6,E_7,E_8$. We denote the adjacency matrix of $\Gamma$ by $R_\Gamma$. Definition 6.4.1 (Cartan matrix). We define the Cartan matrix of $\Gamma$ as $A_\Gamma=2\operatorname{Id}-R_\Gamma$. On the lattice $\mathbb Z^n$ (or the space $\mathbb R^n$) we then define an inner product $B(x,y)=x^TA_\Gamma y$ corresponding to the graph $\Gamma$. […] Definition 6.5.1. Let $Q$ be a quiver with any labeling $1,\ldots,n$ of the vertices. Let $V=(V_1,\ldots,V_n)$ be a representation of $Q$. We then call $d(V)=(\dim V_1,\ldots,\dim V_n)$ the dimension vector of this representation. […] Theorem 6.5.2 (Gabriel's theorem). Let $Q$ be a quiver of type $A_n,D_n,E_6,E_7,E_8$. Then $Q$ has finitely many indecomposable representations. Namely, the dimension vector of any indecomposable representation is a positive root (with respect to $B_\Gamma$) and for any positive root $\alpha$ there is exactly one indecomposable representation with dimension vector $\alpha$.