The spectral problem and algebras associated with extended Dynkin graphs. I

TL;DR

This paper explores the connection between *-representations of algebras linked to extended Dynkin graphs and the spectral problem of Hermitian operators, providing explicit parameters and an algorithm for constructing representations.

Contribution

It offers a detailed description of parameters for *-representations of algebras associated with extended Dynkin graphs and introduces an algorithm for their construction.

Findings

Explicit parameter descriptions for *-representations

Algorithm for constructing *-representations

Connection established between spectral problem and algebra representations

Abstract

There is a connection between *-representations of algebras associated with graphs and the problem about the spectrum of a sum of Hermitian operators (spectral problem). For algebras associated with extended Dynkin graphs we give an explicit description of the parameters for which there are $*$-representations and an algorithm for constructing these representations.