The inverse of a real generalized Cartan matrix

Details The inverse of a real generalized Cartan matrix The inverse of a real generalized Cartan matrix

2002-12-16

arxiv.org/abs/math/0212207v1

Mathematics

Representation Theory

6 pages

Scientific paper

The purpose of this note is to give explicit criteria to determine whether a real generalized Cartan matrix is of finite type, affine type or of hyperbolic type by considering the principal minors and the inverse of the matrix. In particular, it will be shown that a real generalized Cartan matrix is of finite type if and only if it is invertible and the inverse is a positive matrix. A real generalized Cartan matrix is of hyperbolic type if and only if it is invertible and the inverse is non-positive.

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