On the Hamiltonian integrability of the bi-Yang-Baxter σ -model

The bi-yang-baxter σ-model is a certain two-parameter deformation of the principal chiral model on a real lie group g for which the left and right g-symmetries of the latter are both replaced by poisson-lie symmetries. it was introduced by c. klimčík who also recently showed it admits a lax pair, thereby proving it is integrable at the lagrangian level. by working in the hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset σ-model on g × g/g diag, we show that it also admits a lax matrix whose poisson bracket is of the standard r/s-form characterised by a twist function which we determine. a number of results immediately follow from this, including the identification of certain complex poisson commuting kac-moody currents as well as an explicit description of the q-deformed symmetries of the model. moreover, the model is also shown to fit naturally in the general scheme recently developed for constructing integrable deformations of σ-models. finally, we show that although the poisson bracket of the lax matrix still takes the r/s-form after fixing the g diag gauge symmetry, it is no longer characterised by a twist function.