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 hat E ⊗F Eop 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.