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''X'' != weight lattice P, rather: Q (root lattice) \subseteq X \subseteq P with strict inclusions in general! |
m →Definition: dual lattice --> dual; wikilink was to unrelated topic, and lattice is not really appropriate as the modules in question don't come embedded in vector spaces |
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:<math>(X^\ast, \Phi, X_\ast, \Phi^\vee)</math>,
where
* <math>X^\ast</math> and <math>X_\ast</math> are free abelian groups of finite [[Rank (linear algebra)|rank]] together with a [[perfect pairing]] between them with values in <math>\mathbb{Z}</math> which we denote by ( , ) (in other words, each is identified with the
* <math>\Phi</math> is a finite subset of <math>X^\ast</math> and <math>\Phi^\vee</math> is a finite subset of <math>X_\ast</math> and there is a bijection from <math>\Phi</math> onto <math>\Phi^\vee</math>, denoted by <math>\alpha\mapsto\alpha^\vee</math>.
* For each <math>\alpha</math>, <math>(\alpha, \alpha^\vee)=2</math>.
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