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Concept
Cartan matrix
Résumé
In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.
A (symmetrizable) generalized Cartan matrix is a square matrix with integral entries such that
For diagonal entries, .
For non-diagonal entries, .
if and only if
can be written as , where is a diagonal matrix, and is a symmetric matrix.
For example, the Cartan matrix for G2 can be decomposed as such:
The third condition is not independent but is really a consequence of the first and fourth conditions.
We can always choose a D with positive diagonal entries. In that case, if S in the above decomposition is positive definite, then A is said to be a Cartan matrix.
The Cartan matrix of a simple Lie algebra is the matrix whose elements are the scalar products
(sometimes called the Cartan integers) where ri are the simple roots of the algebra. The entries are integral from one of the properties of roots. The first condition follows from the definition, the second from the fact that for is a root which is a linear combination of the simple roots ri and rj with a positive coefficient for rj and so, the coefficient for ri has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let and . Because the simple roots span a Euclidean space, S is positive definite.
Conversely, given a generalized Cartan matrix, one can recover its corresponding Lie algebra. (See Kac–Moody algebra for more details).
An matrix A is decomposable if there exists a nonempty proper subset such that whenever and . A is indecomposable if it is not decomposable.
Let A be an indecomposable generalized Cartan matrix. We say that A is of finite type if all of its principal minors are positive, that A is of affine type if its proper principal minors are positive and A has determinant 0, and that A is of indefinite type otherwise.
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