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Quivers with potentials and their representations I: Mutations

Harm Derksen, Jerzy Weyman, Andrei Zelevinsky

TL;DR

This paper develops a comprehensive framework for quivers with potentials (QPs) and their representations by working in the completed path algebra and using Jacobian ideals to define Jacobian algebras. It introduces mutations at arbitrary vertices, proves a Splitting Theorem to separate trivial and reduced parts, and shows mutation is an involutive operation on right-equivalence classes of reduced QPs and their decorated representations. The work establishes mutation-invariants (finite-dimensionality, rigidity), defines nondegenerate QPs and discusses when they exist, and connects Jacobian algebras to cluster-tilted algebras in Dynkin-type settings. In addition to numerous structural results, the paper provides rich examples (including three-vertex and band representations) and opens several directions, including potential cluster-category analogues for non-acyclic QPs.

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.

Quivers with potentials and their representations I: Mutations

TL;DR

This paper develops a comprehensive framework for quivers with potentials (QPs) and their representations by working in the completed path algebra and using Jacobian ideals to define Jacobian algebras. It introduces mutations at arbitrary vertices, proves a Splitting Theorem to separate trivial and reduced parts, and shows mutation is an involutive operation on right-equivalence classes of reduced QPs and their decorated representations. The work establishes mutation-invariants (finite-dimensionality, rigidity), defines nondegenerate QPs and discusses when they exist, and connects Jacobian algebras to cluster-tilted algebras in Dynkin-type settings. In addition to numerous structural results, the paper provides rich examples (including three-vertex and band representations) and opens several directions, including potential cluster-category analogues for non-acyclic QPs.

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.

Paper Structure

This paper contains 13 sections, 61 theorems, 240 equations, 1 table.

Table of Contents

  1. Introduction
  2. Quivers and path algebras
  3. Potentials and their Jacobian ideals
  4. Quivers with potentials
  5. Mutations of quivers with potentials
  6. Some mutation invariants
  7. Nondegenerate QPs
  8. Rigid QPs
  9. Relation to cluster-tilted algebras
  10. Decorated representations and their mutations
  11. Some three-vertex examples
  12. Some open problems
  13. Appendix. Proof of Lemma \ref{['lem:tr-IJ']}

Key Result

Proposition 2.4

Any pair $(\varphi^{(1)}, \varphi^{(2)})$ of $R$-bimodule homomorphisms $\varphi^{(1)}: A \to A'$ and $\varphi^{(2)}: A \to {\mathfrak m}(A')^2$ gives rise to a unique homomorphism of topological algebras $\varphi: R\langle\langle A\rangle \rangle \to R\langle\langle A'\rangle \rangle$ such that $\v

Theorems & Definitions (134)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • Definition 3.4
  • ...and 124 more