Integrable Kondo impurity in one-dimensional q-deformed $t-J$ models

Integrable Kondo impurities in two cases of the one-dimensional q-deformed $t-J$ models are studied by means of the boundary ${\mathbf{Z}}_2$-graded quantum inverse scattering method. The boundary $K$ matrices depending on the local magnetic moments of the impurities are presented as nontrivial realizations of the reflection equation algebras in an impurity Hilbert space. Furthermore, these models are solved by using the algebraic Bethe ansatz method and the Bethe ansatz equations are obtained.