Tensor algebras and decorated representations

TL;DR

This paper extends the theory of quivers with potentials to decorated representations using tensor algebras, proving mutation involution and a near Morita equivalence between related Jacobian algebras.

Contribution

It generalizes the existing framework to decorated representations and establishes key properties like mutation involution and Morita equivalence.

Findings

Mutation of decorated representations is an involution.

Existence of nearly Morita equivalence between related Jacobian algebras.

Extension of quiver potential theory to decorated representations.

Abstract

In arXiv:1506.05880 we gave a generalization of the theory of quivers with potentials introduced by Derksen-Weyman-Zelevinsky, via completed tensor algebras over $S$-bimodules where $S$ is a finite dimensional basic semisimple algebra. In this paper we show how to extend this construction to the level of decorated representations and we prove that mutation of decorated representations is an involution. Moreover, we prove that there exists a nearly Morita equivalence between the Jacobian algebras which are related via mutation. This generalizes the construction given by Buan-Iyama-Reiten-Smith.