@MISC{Kay99semi-affinecoxeter-dynkin, author = {John M C Kay}, title = {SEMI-AFFINE COXETER-DYNKIN GRAPHS AND G ⊆ SU2(C)}, year = {1999} }

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Abstract

Abstract. The semi-affine Coxeter-Dynkin graph is introduced, generalizing both the affine and the finite types. Semi-affine graphs. It is profitable to treat the so-called Coxeter-Dynkin diagrams as graphs. A classification of finite graphs with an adjacency matrix having 2 as the largest eigenvalue is made in a paper of John Smith [JHS]. It is in a combinatorial context and no reference is made to Coxeter-Dynkin diagrams there. This maximal eigenvalue property is a defining property of the affine diagrams. What is introduced in this note is a more weakly constrained graph, and we examine its eigenvalues and interpret the rational functions which arise in terms of my correspondence [M1,K]. Since these semi-affine graphs do not have symmetrizable matrices, this appears to imply a connection with singularities rather than Lie algebras. Here we shall deal only with those of type A, D, and E. Undirected edges are treated as a pair of edges directed in opposing directions as in [FM,M1,M2]. By so doing, we can introduce the semi-affine graph which may be defined in terms of a graph of finite type with an additional edge (two for A-type) directed toward the affine node; equivalently it may be