Potentials for some tensor algebras

Abstract

This paper generalizes former works of Derksen, Weyman and Zelevinsky about quivers with potentials. We consider semisimple finite-dimensional algebras $E$ over a field $F$, such that $E \otimes_{F} E^{op}$ is semisimple. We assume that $E$ contains a certain type of $F$-basis which is a generalization of a multiplicative basis. We study potentials belonging to the algebra of formal power series, with coefficients in the tensor algebra over $E$, of any finite-dimensional $E$-$E$-bimodule on which $F$ acts centrally. In this case, we introduce a cyclic derivative and to each potential we associate a Jacobian ideal. Finally, we develop a mutation theory of potentials, which in the case that the bimodule is $Z$-free, it behaves as the quiver case; but allows us to obtain realizations of a certain class of skew-symmetrizable integer matrices.