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

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.Comment: PhD thesis, November 2014; 136 pages; Some basic material of Chapter 1,2 taken from other works (Terwilliger and coauthors, arXiv:math/0406555, ...; Baseilhac and co-authors arXiv:0906.1482, ...). Main results described in Chapter 3, published in arXiv:1312.3433, arXiv:1312.589