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

Thi-Thao Vu

Published 2016-01-23Version 1

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 considered. The main results of the thesis are: (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; (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 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.