The Spectral Problem and Algebras Associated with Extended Dynkin Graphs

TL;DR

This paper solves the Spectral Problem for all star-shaped simply laced extended Dynkin graphs, providing explicit spectral descriptions for these algebraic structures.

Contribution

It extends previous solutions to include all star-shaped simply laced extended Dynkin graphs, broadening the understanding of spectral properties in these algebraic contexts.

Findings

Explicit spectral descriptions for all star-shaped simply laced extended Dynkin graphs.

Generalization of previous solutions to a wider class of graphs.

Enhanced understanding of algebraic structures associated with these graphs.

Abstract

The Spectral Problem is to describe possible spectra $\sigma (A_j)$ for an irreducible $n$-tuple of Hermitian operators s.t. $A_1+...+A_n$ is a scalar operator. In case when $m_j= | \sigma (A_j)|$ are finite and a rooted tree ${\rm T}_{m_1,..., m_n}$ with $n$ branches of lengths $m_1, ..., m_n$ is a Dynkin graph the explicit answer to the Spectral Problem was given recently by S. A. Kruglyak, S. V. Popovych, and Yu. S. Samo\v\i{}lenko. In present work the solution of the Spectral Problem for all star-shaped simply laced extended Dynkin graphs, i.e. when $(m_1, ..., m_n) \in \{(2,2,2,2), (3,3,3), (4,4,2)$, $(6,3,2)\}$ is presented.