  We extend the notion of space shifts introduced by L. D. Faddeev and A. Yu.
Volkov (Phys. Lett. B 315 (1993)) for certain quantum light cone lattice
equations of sine-Gordon type at root of unity. As a result we obtain a
compatibility equation for the roots of central elements within the algebra of
observables (also called current algebra). The equation which is obtained by
exponentiating these roots is exactly the evolution equation for the "classical
background" as described in V. Bazhanov, A. Bobenko, N. Reshetikhin (Comm.
Math. Phys. 175 (1996)).
  As an application for the introduced constructions, a one to one
correspondence between a special case of the quantum light cone lattice
equations of sine-Gordon type and free massive fermions on a lattice as
constructed in Destri and de Vega (Nucl. Phys. B 290 (1987)) is derived.
