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

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.