Potentials and Jacobian algebras for tensor algebras of bimodules

TL;DR

This paper develops a framework for potentials, mutations, and Jacobian algebras in tensor algebras of bimodules over symmetric algebras, extending classical quiver theories to more general settings with applications to cluster algebras.

Contribution

It introduces a generalized theory of potentials and Jacobian algebras for tensor algebras of bimodules, including graded and non simply-laced cases, and constructs associated Ginzburg dg-algebras.

Findings

Unified framework for potentials in tensor bimodule algebras

Extension to graded and non simply-laced contexts

Construction of generalized Ginzburg dg-algebras

Abstract

We introduce and study potentials, mutations and Jacobian algebras in the framework of tensor algebras associated with symmetrizable dualizing pairs of bimodules on a symmetric algebra over any commutative ground ring. The graded context is also considered by starting from graded bimodules, and the classical non simply-laced context of modulated quivers with potentials is a particular case. The study of potentials in this framework is related to symmetrically separable algebras, and we have two kinds of potentials: the symmetric and the non symmetric ones. When the Casimir ideal of the symmetric algebra coincides with its center, all potentials appear as symmetric potentials and their manipulation mimics the simply laced study of quivers with potentials. This useful information suggests that, for applications to cluster algebras theory and related fields, one may restrict a further study of modulated quivers with potentials to the setting where the ground symmetric algebra is separable over a field. Associated with this work is a generalized construction of Ginzburg dg-algebras and cluster categories associated with graded modulated quivers with potentials.