Naive Lie Theory
《Naive Lie Theory》 · 印刷第 124 页 Section 7.4「Root data」，7.4.1 的四元组定义、反射公式及两条公理
7.4.1. A root datum is a quadruple $\Psi=(X,R,X^\vee,R^\vee)$, where (a) $X$ and $X^\vee$ are free abelian groups of finite rank, in duality by a pairing $X\times X^\vee\to\mathbb Z$, denoted by $\langle\, ,\,\rangle$; (b) $R$ and $R^\vee$ are finite subsets of $X$ and $X^\vee$, and we are given a bijection $\alpha\mapsto\alpha^\vee$ of $R$ onto $R^\vee$. For $\alpha\in R$ we define endomorphisms $s_\alpha$ and $s_\alpha^\vee$ of $X$ and $X^\vee$ by $s_\alpha(x)=x-\langle x,\alpha^\vee\rangle\alpha$, $s_\alpha^\vee(y)=y-\langle\alpha,y\rangle\alpha^\vee$ $(x\in X,\ y\in X^\vee)$. The following axioms are imposed. (RD 1) If $\alpha\in R$ then $\langle\alpha,\alpha^\vee\rangle=2$; (RD 2) If $\alpha\in R$ then $s_\alpha R=R$, $s_\alpha^\vee(R^\vee)=R^\vee$.