  Within the framework of the discrete Wess-Zumino-Novikov-Witten theory we
analyze the structure of vertex operators on a lattice. In particular, the
lattice analogues of operator product expansions and braid relations are
discussed. As the main physical application, a rigorous construction for the
discrete counterpart g_n of the group valued field g(x) is provided. We study
several automorphisms of the lattice algebras including discretizations of the
evolution in the WZNW model. Our analysis is based on the theory of modular
Hopf algebras and its formulation in terms of universal elements. Algebras of
vertex operators and their structure constants are obtained for the deformed
universal enveloping algebras U_q(G). Throughout the whole paper, the abelian
WZNW model is used as a simple example to illustrate the steps of our
construction.
