Cluster algebras and preprojective algebras : the non simply-laced case

TL;DR

This paper extends the theory of cluster algebras to non simply-laced Lie groups, showing that their cluster structures are projections of simply-laced cases and proving linear independence of cluster monomials.

Contribution

It generalizes known results to non simply-laced cases, establishing a link between cluster structures in different Lie group types.

Findings

Cluster structures in non simply-laced cases are projections of simply-laced cases.

Cluster monomials are linearly independent in the non simply-laced case.

Provides a framework for understanding cluster algebras across different Lie group types.

Abstract

We generalize to the non simply-laced case results of Gei\ss, Leclerc and Schr\"oer about the cluster structure of the coordinate ring of the maximal unipotent subgroups of simple Lie groups. In this way, cluster structures in the non simply-laced case can be seen as projections of cluster structures in the simply-laced case. This allows us to prove that cluster monomials are linearly independent in the non simply-laced case.