PlanetMath: weight lattice

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weight lattice

(Definition)

The weight lattice $\Lambda_W$ of a root system $R\subset E$ is the lattice

$\displaystyle \Lambda_W=\left\{ e\in E \left\vert \frac{(e,\alpha)}{(\alpha,\alpha)}\in\mathbb{Z} \text{ for all } r\in R \right. \right\} .$

Weights which lie in the weight lattice are called integral. If $R\subset\mathfrak{h}$ is the root system of a semi-simple Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$ , then $\Lambda_W$ is exactly the set of weights appearing in finite dimensional representations of $\mathfrak{g}$ .

"weight lattice" is owned by bwebste.

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Also defines:

integral weight

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Cross-references: representations, finite dimensional, Cartan subalgebra, semi-simple Lie algebra, weights, lattice, root system
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This is version 6 of weight lattice, born on 2002-12-05, modified 2007-06-14.
Object id is 3660, canonical name is WeightLattice.
Accessed 2966 times total.

Classification:

17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive )

Pending Errata and Addenda

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