Tensor algebras and decorated representations

In arXiv:1506.05880 we gave a generalization of the theory of quivers with potentials introduced by Derksen-Weyman-Zelevinsky, via completed tensor algebras over $S$-bimodules where $S$ is a finite dimensional basic semisimple algebra. In this paper we show how to extend this construction to the level of decorated representations and we prove that mutation of decorated representations is an involution. Moreover, we prove that there exists a nearly Morita equivalence between the Jacobian algebras which are related via mutation. This generalizes the construction given by Buan-Iyama-Reiten-Smith.