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Mutation of signed valued quivers and presentations of simple complex Lie algebras

Joseph Grant, Davide Morigi

TL;DR

This work extends cluster-algebra mutation to signed valued quivers by introducing generalized signed matrices and a mutation rule that respects a Cartan-counterpart. It develops a Serre-like presentation of Lie algebras from these data, and proves mutation equivalence yields isomorphic Lie algebras in mutation Dynkin type, while connecting root systems and companion bases via Parsons’ framework. The authors provide a robust mutation-invariant theory with a detailed A3 example showing an explicit link to $\mathfrak{sl}_4(\mathbb{C})$. Together, these results give new presentations of simple complex Lie algebras arising from signed quiver mutations and illuminate how root-space decompositions behave under mutation.

Abstract

We introduce a signed variant of (valued) quivers and a mutation rule that generalizes the classical Fomin-Zelevinsky mutation of quivers. To any signed valued quiver we associate a matrix that is a signed analogue of the Cartan counterpart appearing in the theory of cluster algebras. From this matrix, we construct a Lie algebra via a "Serre-like" presentation. In the mutation Dynkin case, we define root systems using the signed Cartan counterpart and show compatibility with mutation of roots as defined by Parsons. Using results from Barot-Rivera and Pérez-Rivera, we show that mutation equivalent signed quivers yield isomorphic Lie algebras, giving presentations of simple complex Lie algebras.

Mutation of signed valued quivers and presentations of simple complex Lie algebras

TL;DR

This work extends cluster-algebra mutation to signed valued quivers by introducing generalized signed matrices and a mutation rule that respects a Cartan-counterpart. It develops a Serre-like presentation of Lie algebras from these data, and proves mutation equivalence yields isomorphic Lie algebras in mutation Dynkin type, while connecting root systems and companion bases via Parsons’ framework. The authors provide a robust mutation-invariant theory with a detailed A3 example showing an explicit link to . Together, these results give new presentations of simple complex Lie algebras arising from signed quiver mutations and illuminate how root-space decompositions behave under mutation.

Abstract

We introduce a signed variant of (valued) quivers and a mutation rule that generalizes the classical Fomin-Zelevinsky mutation of quivers. To any signed valued quiver we associate a matrix that is a signed analogue of the Cartan counterpart appearing in the theory of cluster algebras. From this matrix, we construct a Lie algebra via a "Serre-like" presentation. In the mutation Dynkin case, we define root systems using the signed Cartan counterpart and show compatibility with mutation of roots as defined by Parsons. Using results from Barot-Rivera and Pérez-Rivera, we show that mutation equivalent signed quivers yield isomorphic Lie algebras, giving presentations of simple complex Lie algebras.
Paper Structure (15 sections, 49 theorems, 132 equations)

This paper contains 15 sections, 49 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Mutation of signed valued quivers
  3. Generalized skew-symmetrizable matrices
  4. Signed valued quivers
  5. Mutation Dynkin type
  6. Roots
  7. The Cartan counterpart
  8. Root systems
  9. Mutation type $A$
  10. Presentations of Lie algebras
  11. Presentations
  12. Relations
  13. Homomorphisms
  14. Root space decompositions
  15. Example

Key Result

Proposition 2.6

Let $k\in\{1,\ldots, n\}$. Then $\mu_k^{\text{FZ}}(B(1))=(\mu_kB)(1)$, i.e., the following diagram commutes: \xymatrix@C=40pt{B \ar[r]^{\mu_k}\ar[d] & \mu_k(B)\ar[d] \\ B(1)\ar[r]_{\mu_k^{\text{FZ}}} & (\mu_kB)(1)}

Theorems & Definitions (135)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Proposition 2.6
  • proof
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 125 more