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
sl2​​ 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