Vertex Operators and Matrix Elements of \(U_q(su(2)_k)\) via Bosonization

We construct bosonized vertex operators (VOs) and conjugate vertex operators (CVOs) of \(U_q(su(2)_k)\) for arbitrary level \(k\) and representation \(j\leq k/2\). Both are obtained directly as two solutions of the defining condition of vertex operators - namely that they intertwine \(U_q(su(2)_k)\) modules. We construct the screening charge and present a formula for the n-point function. Specializing to \(j=1/2\) we construct all VOs and CVOs explicitly. The existence of the CVO allows us to place the calculation of the two-point function on the same footing as \(k=1\); that is, it is obtained without screening currents and involves only a single integral from the CVO. This integral is evaluated and the resulting function is shown to obey the q-KZ equation and to reduce simply to both the expected \(k=1\) and \(q=1\) limits.