Universal geometric cluster algebras

TL;DR

This paper introduces the concept of universal geometric cluster algebras over exchange matrices, connecting polyhedral geometry, mutation fans, and basis constructions to extend the theory of cluster algebras.

Contribution

It defines universal geometric coefficients for cluster algebras, relates them to mutation fans and g-vectors, and constructs these coefficients in specific types like rank 2 and finite type.

Findings

Universal geometric coefficients constructed in rank 2 and finite type.

Mutation fan F_B linked to g-vectors and universal coefficients.

Polyhedral geometry is central to understanding universal geometric cluster algebras.

Abstract

We consider, for each exchange matrix B, a category of geometric cluster algebras over B and coefficient specializations between the cluster algebras. The category also depends on an underlying ring R, usually the integers, rationals, or reals. We broaden the definition of geometric cluster algebras slightly over the usual definition and adjust the definition of coefficient specializations accordingly. If the broader category admits a universal object, the universal object is called the cluster algebra over B with universal geometric coefficients, or the universal geometric cluster algebra over B. Constructing universal coefficients is equivalent to finding an R-basis for B (a "mutation-linear" analog of the usual linear-algebraic notion of a basis). Polyhedral geometry plays a key role, through the mutation fan F_B, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between F_B and g-vectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.