  Cecotti and Vafa introduced the tt*-equation (topological-antitopological
fusion equation), whose solutions describe massive deformations of
supersymmetric conformal field theories. We describe some solutions of the
tt*-equation constructed from the SU(2)$_k$-fusion algebra. The idea of the
construction is due to Cecotti and Vafa, but we give a precise mathematical
formulation and a description of the "holomorphic data" corresponding to the
solutions by using the DPW method. Furthermore, we give a relation between the
solutions and the representations of SU(2). As a special case, we consider the
solutions corresponding to the supersymmetric A$_k$-minimal model.
