Cluster Algebras From Surfaces and Extended Affine Weyl Groups

Abstract

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space V , and with every triangulation a basis in V , such that any mutation of a cluster (i.e., a flip of a triangulation) transforms the corresponding bases into each other by partial reflections. Furthermore, every triangulation gives rise to an extended affine Weyl group of type A, which is invariant under flips. The construction is also extended to exceptional skew-symmetric mutation-finite cluster algebras of types E.

Mutation-finite quivers with real weights

“…This implies that most non-acyclic mutation classes of punctured surfaces or orbifolds do not possess admissible realisations by reflections. Geometric realisations by reflections of all mutation classes of quivers originating from unpuctured surfaces were constructed in [5]. There is strong evidence for the following conjecture.…”

“…Geometric realisations by reflections of all mutation classes of quivers originating from unpuctured surfaces were constructed in [5]. There is strong evidence for the following conjecture.…”