Additive functions on trees

Piroska Lakatos¹

¹ Institute of Mathematics and Informatics University of Debrecen P.O. Box 12 4010 Debrecen, Hungary

Colloquium Mathematicum, Tome 89 (2001) no. 1, pp. 135-145.

Résumé

The motivation for considering positive additive functions on trees was a characterization of extended Dynkin graphs (see I. Reiten [R]) and applications of additive functions in representation theory (see H. Lenzing and I. Reiten [LR] and T. Hübner [H]).We consider graphs equipped with integer-valued functions, i.e. valued graphs (see also [DR]). Methods are given for constructing additive functions on valued trees (in particular on Euclidean graphs) and for characterizing their structure. We introduce the concept of almost additive functions, which are additive on each vertex of a graph except one (called the exceptional vertex). On (valued) trees (with fixed exceptional vertex) the almost additive functions are unique up to rational multiples. For valued trees a necessary and sufficient condition is given for the existence of positive almost additive functions.

Détail

DOI : 10.4064/cm89-1-10

Keywords: motivation considering positive additive functions trees characterization extended dynkin graphs see reiten applications additive functions representation theory see lenzing reiten bner consider graphs equipped integer valued functions valued graphs see methods given constructing additive functions valued trees particular euclidean graphs characterizing their structure introduce concept almost additive functions which additive each vertex graph except called exceptional vertex valued trees fixed exceptional vertex almost additive functions unique rational multiples valued trees necessary sufficient condition given existence positive almost additive functions