PlanetMath: Cartan matrix

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Cartan matrix

(Definition)

Let $R\subset E$ be a reduced root system, with a Euclidean vector space, with inner product $(\cdot,\cdot)$ , and let $\Pi=\{\alpha_1,\cdots,\alpha_n\}$ be a base of this root system. Then the Cartan matrix of the root system is the matrix

$\displaystyle C_{i,j}=\left(\frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}\right).$

The Cartan matrix uniquely determines the root system, and is unique up to simultaneous permutation of the rows and columns. It is also the basis change matrix from the basis of fundamental weights to the basis of simple roots in

"Cartan matrix" is owned by bwebste.

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Cross-references: simple roots, weights, basis, basis change, columns, rows, permutation, matrix, root system, base, inner product, Euclidean vector space, reduced root system
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This is version 1 of Cartan matrix, born on 2002-12-20.
Object id is 3799, canonical name is CartanMatrix.
Accessed 2281 times total.

Classification:

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

Pending Errata and Addenda

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