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 Z, Q, or 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 FB, which we suspect to be an important object beyond its role in constructing universal geometric coefficients. We make the connection between FB and g-vectors. We construct universal geometric coefficients in rank 2 and in finite type and discuss the construction in affine type.