  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.
