$H$-based Quivers with potentials and their representations

We generalize Derksen-Weyman-Zelevinsky's theory of quivers with potentials (QPs) to an $H$-based setting by considering quivers with exactly one loop at each vertex, asking the loops to be nilpotent and so attaching a truncated polynomial ring $H_i$ to each vertex. The algebra is then defined by taking the quotient of the complete path algebra by relations arising from analogs of the Jacobian ideals of a given potential. We develop the mutation theory for such $H$-based QPs and their decorated representations in general position. As an application, we consider generalized cluster algebras introduced by Chekhov-Shapiro. For those algebras corresponding to $H$-based quivers $(Q,\mathbf{d})$ that have mutation degree $d_k\leq 2$ at each vertex $k$ and admit nondegenerate potentials $S$ making $(Q,\mathbf{d},S)$ locally free, we provide a representation-theoretic interpretation of $\mathbf{g}$-vectors and $F$-polynomials. When the exchange matrix $B(Q)$ has full rank, we further construct generic character for upper generalized cluster algebras.