Integrable models and star structures

We consider the representations of Hopf algebras involved in some physical models, namely, factorizable S-matrix models (FSM's), one-dimensional quantum spin chains (QSC's) and statistical vertex models (SVM's). These physical representations have definite hermiticity assignments and lead to star structures on the corresponding Hopf algebras. It turns out that for FSM's and the quantum mechanical time-evolution of QSC's the corresponding stars are compatible with the Hopf structures. However, in the case of statistical models the resulting star structure is not a Hopf one but what we call a twisted star. Real representations of a twisted star Hopf algebra do not close under the usual tensor product of representations. We briefly comment on the relation of these results with the Wick rotation.