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Some Special Classes of Cartan Matrices

Published online by Cambridge University Press:  20 November 2018

A. P. Ogg
A. P. Ogg*
Affiliation:
Max-Planck-Institut für Mathematik, Bonn, West Germany
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Let A = (Aij)l≦ij≦l be a Cartan matrix, i.e., Aii = 2 for all i and Aij is an integer ≦ 0 for i ≠ j, with Aij = 0 if Aji = 0. The size l of A is called its rank, for Lie-theoretic reasons, and may be larger than its matrix rank. We associate to A its Dynkin diagram, with vertices 1, 2, … , l, with AijAji lines joining i to j, and with an arrow pointing from i to j if Aij/Aji < 1, i.e., pointing toward the shorter root (see below). The Cartan matrix A is indecomposable if its diagram is connected, and symmetrizable if there exist positive rational numbers q1 … , ql with

qiAij = qjAji for all i and j.

Symmetrizability is automatic if the diagram contains no cycle.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 5 , 01 October 1984 , pp. 800 - 819
Copyright
Copyright © Canadian Mathematical Society 1984

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