Cartan matrix

In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. In fact, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.

A generalized Cartan matrix is a square matrix ${\displaystyle A=(a_{ij})}$ with integral entries such that

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 r_i 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 ${\displaystyle i\neq j,r_{j}-{2(r_{i},r_{j}) \over (r_{i},r_{i})}r_{i}}$ is a root which is a linear combination of the simple roots r_i and r_j with a positive coefficient for r_j and so, the coefficient for r_i has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let ${\displaystyle D_{ij}={\delta _{ij} \over (r_{i},r_{i})}}$ and ${\displaystyle S_{ij}=2(r_{i},r_{j})}$. Because the simple roots span a Euclidean space, S is positive definite.

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