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LIE ALGEBRAS


A generalized Cartan matrix is a Square Matrix A = (a_{ij}) with Integral entries such that

# For diagonal entries, a_{ii} = 2.
# For non-diagonal entries, a_{ij} \leq 0 .
# a_{ij} = 0 if and only if a_{ji} = 0
# 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 Product s

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

where r_i are the Simple Roots of the algebra. The entries are integral from one of the properties of Root s. The first condition follows from the definition, the second from the fact that for i
eq 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 Root s 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 D_{ij}={\delta_{ij}\over (r_i,r_i)} and S_{ij}=2(r_i,r_j). Because the simple roots span a Euclidean Space , S is positive definite.


REPRESENTATIONS OF FINITE-DIMENSIONAL ALGEBRAS


In Modular Representation Theory , and more generally in the theory of representations of Finite-dimensional Algebra s ''A'' that are ''not'' Semisimple , a Cartan matrix is defined by considering a (finite) set of Principal Indecomposable Module s and writing Composition Series for them in terms of Projective Module s, yielding a matrix of integers counting the number of occurrences of a projective module.