Introduction to Lie Algebras and Representation Theory
《Introduction to Lie Algebras and Representation Theory》 · 印刷第 145 页 Chapter VII 的对象约定、第 151 页 Section 26 的基约定、第 159 页 Section 27.2 的有限维表示约定及第 162—163 页 Section 27.4「Passage to an arbitrary field」的构造与命名；保留整格上的除幂转入正特征域的步骤
$L$ is a semisimple Lie algebra over the algebraically closed field $F$ of characteristic $0$, $H$ a CSA, $\Phi$ the root system. […] Fix also a Chevalley basis $\{x_\alpha,\alpha\in\Phi;h_i,1\leq i\leq\ell\}$ of $L$. […] Let $V$ be a finite dimensional $L$-module. To avoid trivialities we assume that $V$ is faithful (in other words, we discard those simple ideals of $L$ which act trivially on $V$). […] Let $\mathbb F_p$ be the prime field of characteristic $p$, $K$ an extension field of $\mathbb F_p$. Let $V$ be a faithful $L$-module; then the weights of $V$ span a lattice between $\Lambda_r$ and $\Lambda$, which we have denoted $\Lambda(V)$. Choose an admissible lattice $M$ in $V$, with stabilizer $L_V=H_V+\coprod_{\alpha\in\Phi}\mathbb Zx_\alpha$ in $L$, $H_V=\{h\in H\mid\lambda(h)\in\mathbb Z\text{ for all }\lambda\in\Lambda(V)\}$ (27.2). Let $V(K)=M\otimes_{\mathbb Z}K$, $L_V(K)=L_V\otimes_{\mathbb Z}K$. […] Since $M$ is stable under the endomorphism $\frac{x_\alpha^t}{t!}$ (i.e., under $\frac{\phi(x_\alpha)^t}{t!}$, if $\phi$ is the representation of $L$ in question), $V(K)$ is stable under the corresponding endomorphism, which we call $x_{\alpha,t}$ (where $x_{\alpha,0}=1$). Notice that for $t<p$, $x_{\alpha,t}$ acts just like $\frac{(x_\alpha\otimes1)^t}{t!}$, whereas for $t\geq p$ this notation no longer has a meaning. At any rate, $x_{\alpha,t}=0$ for large enough $t$, so we may write $\theta_\alpha(1)=\sum_{t=0}^{\infty}x_{\alpha,t}\in\operatorname{End}(V(K))$. Actually, it is clear that $\theta_\alpha(1)$ has determinant $1$, so it belongs to $SL(V(K))$. More generally, we can define automorphisms $\theta_\alpha(c)$ of $V(K)$ by forming $\sum_{t=0}^{\infty}\frac{(Tx_\alpha)^t}{t!}$ and then specializing the indeterminate $T$ to $c\in K$. The group $G_V(K)$ generated by all $\theta_\alpha(c)$ $(\alpha\in\Phi,c\in K)$ is called a Chevalley group of type $\Lambda(V)$, adjoint if $\Lambda(V)=\Lambda_r$, universal if $\Lambda(V)=\Lambda$.