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In this thesis, the connection between recently introduced algebraicstructures (tridiagonal algebra, $q$-Onsager algebra, generalized $q-$Onsageralgebras), related representation theory (tridiagonal pair, Leonard pair,orthogonal polynomials), some properties of these algebras and the analysis ofrelated quantum integrable models on the lattice (the $XXZ$ open spin chain atroots of unity) is first reviewed. Then, the main results of the thesis aredescribed: (i) for the class of $q-$Onsager algebras associated with$\widehat{sl_2}$ and ADE type simply-laced affine Lie algebras, higher orderanalogs 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 aredivided polynomials of $q-$Onsager generators) are introduced and some of theirproperties are studied. These two elements together with the two basic elementsof the $q-$Onsager algebra generate a new algebra, which can be understood asan analog of Lusztig's quantum group for the $q-$Onsager algebra. Someperspectives are presented.

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