  We find new solutions to the Yang-Baxter equations with the $R$-matrices
possessing $sl_q(2)$ 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 $q^4=1$. 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 $sl_q(2)$ and
the quantum super-algebra $osp_t(1|2)$, the presented analysis can be extended
to the latter case for the appropriate values of the deformation parameter.
