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.