[Paper Review] Quivers with potentials and their representations I: Mutations

Harm Derksen, Jerzy Weyman+1 more|ArXiv.org|Apr 4, 2007

Algebraic structures and combinatorial models22 references18 citations

TL;DR

This paper introduces a representation-theoretic framework for quivers with potentials (QPs), generalizing Bernstein-Gelfand-Ponomarev reflection functors to arbitrary vertices via a novel mutation procedure. The key contribution is a systematic method to mutate QPs that preserves the Jacobian algebra, resolving technical challenges in removing 2-cycles through a decomposition of potentials into trivial and reduced parts using Theorem 4.6.

ABSTRACT

We study quivers with relations given by non-commutative analogs of Jacobian ideals in the complete path algebra. This framework allows us to give a representation-theoretic interpretation of quiver mutations at arbitrary vertices. This gives a far-reaching generalization of Bernstein-Gelfand-Ponomarev reflection functors. The motivations for this work come from several sources: superpotentials in physics, Calabi-Yau algebras, cluster algebras.

Motivation & Objective

• To extend reflection functors beyond sources and sinks to arbitrary vertices in quiver representations.
• To provide a representation-theoretic interpretation of quiver mutations using noncommutative Jacobian ideals.
• To resolve the technical challenge of removing oriented 2-cycles during mutation while preserving the Jacobian algebra.
• To establish a connection between quivers with potentials and cluster algebras through mutation invariance and decorated representations.
• To generalize existing approaches by avoiding restrictions to hereditary algebras or cluster categories.

Proposed method

• Define potentials as formal linear combinations of cyclic paths in the completed path algebra.
• Introduce the Jacobian ideal as the two-sided ideal generated by noncommutative partial derivatives of the potential.
• Use Theorem 4.6 to decompose any potential into a trivial part (2-cycles) and a reduced part (paths of length ≥3), preserving the Jacobian algebra.
• Construct mutations via a three-step process: create composite arrows, reverse arrows at the mutation vertex, and remove maximal disjoint 2-cycles with corresponding potential modification.
• Define decorated representations and their mutations to extend the framework to cluster algebra contexts.
• Employ duality and topological techniques in D-spaces to prove closure properties of ideals and traces in the completed path algebra.

Experimental results

Research questions

• RQ1 How can quiver mutations be generalized beyond sources and sinks to arbitrary vertices using representation-theoretic tools?
• RQ2 What is the precise mechanism to remove oriented 2-cycles during mutation without altering the Jacobian algebra?
• RQ3 How do potentials and their Jacobian ideals behave under mutation, and what invariants are preserved?
• RQ4 In what way do quivers with potentials unify cluster algebras, Calabi-Yau algebras, and superpotential constructions in physics?
• RQ5 Can decorated representations be systematically mutated to model cluster algebra mutations?

Key findings

• Theorem 4.6 establishes that every potential admits a decomposition into a trivial part (2-cycles) and a reduced part (paths of length ≥3), with isomorphic Jacobian algebras.
• The mutation procedure for QPs is well-defined and preserves the Jacobian algebra, even after removing 2-cycles, by modifying the potential accordingly.
• The mutation of decorated representations is constructed and shown to be compatible with the mutation of QPs, extending reflection functors to arbitrary vertices.
• The Jacobian algebra remains invariant under mutation, establishing a representation-theoretic foundation for cluster algebra mutations.
• Finitely generated left ideals in the completed path algebra are closed, and the trace space respects ideal products, ensuring topological consistency.
• The framework provides a direct, elementary alternative to cluster category approaches, applicable to general quivers without requiring hereditary or acyclic assumptions.