Sommaire:

The analysis of the XXZ open spin chain within the framework of an Onsager's approach will be presented. Starting from the finite size model, the thermodynamic limit will be considered. The transfer matrix - expressed in terms of \(O_q(\widehat{sl}_2)\) \(q-\)Onsager algebra's currents - is diagonalized in the massive regime using level one infinite dimensional representations (\(q-\)vertex operators) of \(U_q(\widehat{sl}_2)\) for (i) diagonal boundary conditions, in which case results of Jimbo et al. are recovered; (ii) triangular boundary conditions. In this latter case, integral representations for correlation functions and form factors are derived. As a byproduct, a summation formula for the expectation values \(<\sigma_a>\), \(a=x,y,z\) is obtained for the first time as well as identities among \(n-\)fold multiple integrals of theta functions. In this approach, the long-standing question of the hidden non-Abelian symmetry of the model is solved. Finally, open problems such that the root of unity case will be briefly presented. Work done in collaboration with S. Belliard (arXiv:1211.6304v2), T. Kojima (arXiv:1309.7785 + in prep.).

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