Weyl Pair, Current Algebra and Shift Operator

The Abelian current algebra on the lattice is given from a series of the independent Weyl pairs and the shift operator is constructed by this algebra. So the realization of the operators of the braid group is obtained. For $|q|\neq 1$ the shift operator is the product of the theta functions of the generators $w_n$ of the current algebra. For $|q|=1$ it can be expressed by the quantum dilogarithm of $w_n$.