Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 印刷第 42 页 Section 9.1 的欧氏空间及配对记号约定；第 55 页 Section 11 的根系约定与 Section 11.1「Cartan matrix of Φ」的定义段
Throughout this chapter we are concerned with a fixed euclidean space $E$, i.e., a finite dimensional vector space over $\mathbb R$ endowed with a positive definite symmetric bilinear form $(\alpha,\beta)$. […] Since the number $2(\beta,\alpha)/(\alpha,\alpha)$ occurs frequently, we abbreviate it by $\langle\beta,\alpha\rangle$. […] In this section $\Phi$ denotes a root system of rank $\ell$, $\mathcal W$ its Weyl group, $\Delta$ a base of $\Phi$. […] Fix an ordering $(\alpha_1,\ldots,\alpha_\ell)$ of the simple roots. The matrix $(\langle\alpha_i,\alpha_j\rangle)$ is then called the Cartan matrix of $\Phi$. Its entries are called Cartan integers.