  The definitions of the main notions related to the quantum inverse scattering
methods are given. The Yang-Baxter equation and reflection equations are
derived as consistency conditions for the factorizable scattering on the whole
line and on the half-line using the Zamolodchikov-Faddeev algebra. Due to the
vertex-IRF model correspondence the face model analogue of the ZF-algebra and
the IRF reflection equation are written down as well as the $Z_2$-graded and
colored algebra forms of the YBE and RE.

High Energy Physics - Theory