Locally-scalar representations of graphs in the category of Hilbert spaces

TL;DR

This paper develops a theory of graph representations in Hilbert spaces with local scalarity constraints, establishing a classification analogous to Gabriel's theorem that connects finite representability to Dynkin graphs.

Contribution

It introduces a new framework for graph representations in Hilbert spaces and proves a key theorem linking finite representability to Dynkin graphs.

Findings

Finite representability in Hilbert spaces characterized by Dynkin graphs

Established a Gabriel-type theorem for Hilbert space representations

Extended classical graph representation theory to infinite-dimensional settings

Abstract

In this paper authors consider representations of graphs in Hilbert spaces applying a restriction of local scalarity on them. It enables to obtain a theory, similar to the classical theory of representations of graphs in vector spaces. In particular, it is obtained a theorem analogous to the well-known Gabriel theorem: a connected finite graph (wood) is finitely representable in the category of Hilbert spaces if and only if it is a Dynkin graph.