The Spectral Problem and Algebras Associated with Extended Dynkin Graphs

Stanislav Popovych

paper

The Spectral Problem and Algebras Associated with Extended Dynkin Graphs

arXiv:0904.0968

Abstract

The Spectral Problem is to describe possible spectra $σ (A_{j})$ for an irreducible $n$ -tuple of Hermitian operators s.t. $A_{1} + ... + A_{n}$ is a scalar operator. In case when $m_{j} = ∣ σ (A_{j}) ∣$ are finite and a rooted tree $T_{m_{1}, ..., m_{n}}$ 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\vılenko. 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.