… cluster algebra with generic coefficients (At, OAt ) is the Cox ring (homogeneous coordinate
ring… In particular, our ⋆-upper cluster algebra agrees with the usual upper cluster algebra (A, …

… cluster algebra structure on the Cox ring of a smooth complex variety Z, depending on a base
cluster structure on the ring … ) cluster variety, and prove that its Cox ring is a graded upper …

… algebraic definition [14] of a cluster complex for a cluster algebra of finite type. Although
inspired by cluster algebra theory, this paper does not rely on any of its results (cf. Remark 3.13). …

… on canonical bases of cluster algebras. We interpret a cluster algebra as the ring of global
functions … We describe a canonical basis of a cluster algebra determined by tropical counts of …

… cluster algebra with the Cox ring of Xφ. This is a slight simplification of the discussion: see the
main text for precise statements. The Cox ring … of factorial cluster algebras remarked on, eg…

… ring A2,n as a quotient of the polynomial ring in Plücker variables C[pij : 1 ≤ i < j ≤ n] by the
Plücker ideal I2,n. It was shown in [23] that A2,n is a cluster algebra in which the cluster … Cox …

… cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables.
These algebras … This paper is a step towards understanding the notion of cluster superalgebra…

… A Mori dream space is a variety whose Cox ring is finitely generated as k-algebra. The … the
property of having finitely generated Cox ring. Notice that Cox(X) finitely generated implies the …

… class of the cluster algebra and the upper cluster algebra. It … try to identify cluster algebra
structures on a given ring. So, … the Cox ring of a certain partial compactification of the cluster …

… of years before the discovery of cluster algebras. In retrospect, it can be seen as the first
step towards introducing the (upper) cluster algebra structure in the coordinate ring C[Gu,v]. (…