An integrable structure related with tridiagonal algebras

TL;DR

This paper explores the algebraic structures underlying integrable models, revealing new commuting operators, their relation to quadratic algebras, and implications for superintegrability in quantum systems.

Contribution

It introduces a new family of commuting operators from tridiagonal algebras, extending the Dolan-Grady construction and connecting to reflection equations and superintegrable models.

Findings

New family of mutually commuting operators derived from tridiagonal algebras

Connection established between tridiagonal algebras and quadratic reflection equations

Identification of superintegrability in related quantum models

Abstract

The standard generators of tridiagonal algebras, recently introduced by Terwilliger, are shown to generate a new (in)finite family of mutually commuting operators which extends the Dolan-Grady construction. The involution property relies on the tridiagonal algebraic structure associated with a deformation parameter $q$. Representations are shown to be generated from a class of quadratic algebras, namely the reflection equations. The spectral problem is briefly discussed. Finally, related massive quantum integrable models are shown to be superintegrable.