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Indecomposable Representations of Graphs and Algebras

Vlastimil DlabClaus Michael Ringel

出版	American Mathematical Soc., 1976
主題	Mathematics / Graphic Methods
ISBN	08218187329780821818732
URL	http://books.google.com.hk/books?id=r_DTCQAAQBAJ&hl=&source=gbs_api
EBook	SAMPLE

註釋

I.N. Bernstein, I.M. Gelfand and V.A. Ponomarev have recently shown that the bijection, first observed by P. Gabriel, between the indecomposable representations of graphs ("quivers") with a positive definite quadratic form and the positive roots of this form can be proved directly. Appropriate functors produce all indecomposable representations from the simple ones in the same way as the canonical generators of the Weyl group produce all positive roots from the simple ones. This method is extended in two directions. In order to deal with all Dynkin diagrams rather than with those having single edges only, we consider valued graphs ("species"). In addition, we consider valued graphs with positive semi-definite quadratic form, i.e. extended Dynkin diagrams. The main result of the paper describes all indecomposable representations up to the homogeneous ones, of a valued graph with positive semi-definite quadratic form. These indecomposable representations are of two types: those of discrete dimension type, and those of continuous dimension type.