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root lattice

根格

本站译名 表示论与李理论 义项编号TS-1335

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Introduction to Lie Algebras and Representation Theory 封面

《Introduction to Lie Algebras and Representation Theory》 · 第 67 页 · Section 13「Abstract theory of weights」的根系约定及 Section 13.1「Weights」(省略中间的习题引用,保留权格记号及配对公式以明确环境)

Let $\Phi$ be a root system in a euclidean space $E$, with Weyl group $\mathcal W$. […] Let $\Lambda$ be the set of all $\lambda\in E$ for which $\langle\lambda,\alpha\rangle\in\mathbb Z$ $(\alpha\in\Phi)$, and call its elements weights. Since $\langle\lambda,\alpha\rangle=\frac{2(\lambda,\alpha)}{(\alpha,\alpha)}$ depends linearly on $\lambda$, $\Lambda$ is a subgroup of $E$ including $\Phi$. […] Denote by $\Lambda_r$ the root lattice ($=$ subgroup of $\Lambda$ generated by $\Phi$).

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