New solutions to the slq(2)-invariant Yang-Baxter equations at roots of unity

We find new solutions to the Yang-Baxter equations with the R-matrices possessing slq(2) symmetry at roots of unity, using indecomposable representations. The corresponding quantum one-dimensional chain models, which can be treated as extensions of the XXZ model at roots of unity, are investigated. We consider the case q4=1. The Hamiltonian operators of these models as a rule appear to be non-Hermitian. Taking into account the correspondence between the representations of the quantum algebra slq(2) and the quantum super-algebra ospt(1|2), the presented analysis can be extended to the latter case for the appropriate values of the deformation parameter.