Orthoscalar representations of quivers on the category of Hilberts spaces. II

TL;DR

This paper extends the theory of orthoscalar (locally scalar) representations of quivers in Hilbert spaces, classifying certain indecomposable representations related to extended Dynkin graphs and exploring their connection to broader representation categories.

Contribution

It establishes a link between indecomposable orthoscalar representations and all representations, classifies regular orthoscalar representations for extended Dynkin graphs, and advances the understanding of quiver representations in Hilbert spaces.

Findings

Classified regular orthoscalar representations for extended Dynkin graphs.

Connected indecomposable orthoscalar representations to the larger category of all representations.

Extended the Gabriel theorem analogue to Hilbert space representations.

Abstract

As it is known, finitely presented quivers correspond to Dynkin graphs (Gabriel, 1972) and tame quivers -- to extended Dynkin graphs (Donovan and Freislich, Nazarova, 1973). In the article "Locally scalar reresentations of graphs in the category of Hilbers spaces" (Func. Anal. and Appl., 2005) authors showed the way to tranfer these results to Hilbert spaces, constructed Coxeter functors and proved an analogue of Gabriel theorem fol locally scalar (orthoscalar in the sequel) representations (up to the unitary equivalence). The category of orthoscalar representations of a quiver can be considered as a subcategory in the category of all representations (over a field $\mathbb C$). In the present paper we study the connection between indecomposable orthoscalar representations in the subcategory and in the category of all representations. For the quivers, corresponded to extended Dynkin graphs, orthoscalar representations which cannot be obtained from the simplest by Coxeter functors (regular representations) are classified.