The half-infinite XXZ chain in Onsager's approach

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 {\it 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).