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Cartan matrix
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(Definition)
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Let $R\subset E$ be a reduced root system, with $E$ 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 $$C_{i,j}=\left(\frac{2(\alpha_i,\alpha_j)}{(\alpha_i,\alpha_i)}\right).$$ 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 $E$ .
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"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
There are 2 references to this entry.
This is version 1 of Cartan matrix, born on 2002-12-20.
Object id is 3799, canonical name is CartanMatrix.
Accessed 2750 times total.
Classification:
| AMS MSC: | 17B20 (Nonassociative rings and algebras :: Lie algebras and Lie superalgebras :: Simple, semisimple, reductive ) |
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Pending Errata and Addenda
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