The higher order $q$-Dolan-Grady relations and quantum integrable systems
Abstract
In this thesis, the connection between recently introduced algebraic structures (tridiagonal algebra, -Onsager algebra, generalized 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 open spin chain at roots of unity) is first reviewed. Then, the main results of the thesis are described: (i) for the class of Onsager algebras associated with 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 spin chain at roots of unity, new elements (that are divided polynomials of Onsager generators) are introduced and some of their properties are studied. These two elements together with the two basic elements of the Onsager algebra generate a new algebra, which can be understood as an analog of Lusztig's quantum group for the Onsager algebra. Some perspectives are presented.
Cite
@article{arxiv.1601.06253,
title = {The higher order $q$-Dolan-Grady relations and quantum integrable systems},
author = {Thi-Thao Vu},
journal= {arXiv preprint arXiv:1601.06253},
year = {2016}
}
Comments
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:1312.5897