We study SU (3)-invariant integrable models solvable by nested algebraic Bethe ansatz. We obtain a determinant representation for the particular case of scalar products of Bethe vectors. This representation can be used for the calculation of form factors and correlation functions of XXX SU (3)-invariant Heisenberg chain.

We study GL(3)-invariant integrable models solvable by the nested algebraic Bethe ansatz. Different formulas are given for the Bethe vectors and the actions of the generators of the Yangian 𝒴(gl3) on the Bethe vectors are considered. These actions are relevant for the calculation of correlation functions and form factors of local operators of the underlying quantum models.

The half-infinite XXZ open spin chain with general integrable boundary conditions is considered within the recently developed 'Onsager's approach'. Inspired by the finite size case, for any type of integrable boundary conditions it is shown that the transfer matrix is simply expressed in terms of the elements of a new type of current algebra recently introduced. In the massive regime −1 < q < 0, level one infinite dimensional representation (q−vertex operators) of the new current algebra are constructed in order to diagonalize the transfer matrix. For diagonal boundary conditions, known results of Jimbo et al. are recovered. For upper (or lower) non-diagonal boundary conditions, a solution is proposed. Vacuum and excited states are formulated within the representation theory of the current algebra using q−bosons, opening the way for the calculation of integral representations of correlation functions for a non-diagonal boundary. Finally, for q generic the long standing question of the hidden non-Abelian symmetry of the Hamiltonian is solved: it is either associated with the q−Onsager algebra (generic non-diagonal case) or the augmented q−Onsager algebra (generic diagonal case). MSC: 81R50; 81R10; 81U15.Keywords: XXZ open spin chain; q−Onsager algebra; q−vertex operators; Thermodynamic limit Potts models [PeAMT, Dav, vGR] are explicit counterexamples of this idea (see also [Ar, AS]).