The higher order $q$-Dolan-Grady relations and quantum integrable systems

TL;DR

This thesis explores advanced algebraic structures related to quantum integrable systems, proving higher order relations and introducing new algebraic elements for the $XXZ$ spin chain at roots of unity, advancing understanding of quantum algebraic symmetries.

Contribution

It introduces and proves higher order $q$-Dolan-Grady relations for specific $q$-Onsager algebras and constructs a new algebraic framework analogous to Lusztig's quantum group.

Findings

Higher order relations for $q$-Onsager algebras are conjectured and proven.

New algebraic elements are constructed for the $XXZ$ spin chain at roots of unity.

A new algebra generalizing Lusztig's quantum group is proposed.

Abstract

In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, $q$-Onsager algebra, generalized $q-$Onsager algebras), related representation theory (tridiagonal pair, Leonard pair, orthogonal polynomials), some properties of these algebras and the analysis of related quantum integrable models on the lattice (the $XXZ$ open spin chain at roots of unity) is first reviewed. Then, the main results of the thesis are described: (i) for the class of $q-$Onsager algebras associated with $\widehat{sl_2}$ and ADE type simply-laced affine Lie algebras, higher order analogs of Lusztig's relations are conjectured and proven in various cases, (ii) for the open $XXZ$ spin chain at roots of unity, new elements (that are divided polynomials of $q-$Onsager generators) are introduced and some of their properties are studied. These two elements together with the two basic elements of the $q-$Onsager algebra generate a new algebra, which can be understood as an analog of Lusztig's quantum group for the $q-$Onsager algebra. Some perspectives are presented.