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 $$\mathbb {Z},\,\mathbb {Q}$$ , or $$\mathbb {R}$$ . 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 geometric 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 $${\mathcal {F}}_B$$ , which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between $${\mathcal {F}}_B$$ and $$\mathbf{g}$$ -vectors. We construct universal geometric coefficients in rank $$2$$ and in finite type and discuss the construction in affine type.