Potentials for some tensor algebras

TL;DR

This paper extends the theory of quivers with potentials to more general tensor algebras over semisimple algebras, introducing cyclic derivatives and mutation theory applicable to a broader class of algebraic structures.

Contribution

It generalizes the concept of potentials and mutations from quivers to tensor algebras over semisimple algebras, broadening the scope of algebraic and combinatorial applications.

Findings

Developed a cyclic derivative for potentials in tensor algebras.

Established a mutation theory for these potentials.

Realized skew-symmetrizable matrices through this framework.

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.