Mutation of representations and nearly Morita equivalence

TL;DR

This paper demonstrates a construction of a quasi-inverse functor to the mutation functor between Jacobian algebras of quivers with potential, avoiding the use of the Axiom of Choice.

Contribution

It provides an explicit quasi-inverse to the mutation functor, removing the need for the Axiom of Choice in establishing near Morita equivalence.

Findings

Constructed a quasi-inverse functor without Axiom of Choice.

Confirmed the near Morita equivalence of Jacobian algebras under mutation.

Simplified the proof of equivalence by explicit construction.

Abstract

Buan-Iyama-Reiten-Smith proved, based on Derksen-Weyman-Zelevinsky work, that the Jacobian algebra of two quivers with potential related by a QP-mutation are nearly Morita equivalent. They proved, using Axiom of Choice, that the natural functor $\mu_k$ is an equivalence by showing that $\mu_k$ is full, faithfull and dense. In this note we provide a quasi-inverse $\mu_k^-$ to $\mu_k$ without Axiom of Choice.