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

2024/03/21 by Grant, Joseph, Morigi, Davide
#13F60 #FOS: Mathematics #Primary: 17B20 #Representation Theory (math.RT) #Rings and Algebras (math.RA) #Secondary: 17B22

paper · doi:10.48550/arxiv.2403.14595

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.

Related

• Signed quivers, symmetric quivers, and root systems
• Quivers with potentials and their representations I: Mutations
• Quivers for silting mutation
• Reduction Techniques to Identify Connected Components of Mutation Quivers
• Subquivers of mutation-acyclic quivers are mutation-acyclic