Abstract
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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.
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Keywords
quiver mutation, presentation, Lie algebra, root system
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Mathematical Subject Classification
Primary: 17B20
Secondary: 17B22, 13F60
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Milestones
Received: 18 July 2024
Revised: 29 April 2025
Accepted: 18 September 2025
Published: 26 November 2025
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| © 2026 The Author(s), under
exclusive license to MSP (Mathematical Sciences Publishers).
Distributed under the Creative Commons
Attribution License 4.0 (CC BY). |
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