Cartan matrix

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In mathematics, the term Cartan matrix has two meanings. Both of these are named after the French mathematician Élie Cartan. In an example of Stigler's law of eponymy, Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan.

[edit] Lie algebras

A generalized Cartan matrix is a square matrix A = (aij) with integer entries such that

  1. For diagonal entries, aii = 2.
  2. For non-diagonal entries, a_{ij} \leq 0.
  3. aij = 0 if and only if aji = 0
  4. A can be written as DS, where D is a diagonal matrix, and S is a symmetric matrix.

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

a_{ij}={2 (r_i,r_j)\over (r_i,r_i)}

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 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 ri and rj with a positive coefficient for ri and so, the coefficient for ri has to be nonnegative. The third is true because orthogonality is a symmetric relation. And lastly, let D_{ij}={\delta_{ij}\over (r_i,r_i)} and Sij = 2(ri,rj). Because the simple roots span a Euclidean space, S is positive definite.

[edit] Representations of finite-dimensional algebras

In modular representation theory, and more generally in the theory of representations of finite-dimensional algebras A that are not semisimple, a Cartan matrix is defined by considering a (finite) set of principal indecomposable modules and writing composition series for them in terms of projective modules, yielding a matrix of integers counting the number of occurrences of a projective module.

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