This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings admitting a cluster algebra structure. We then present the general definition of a cluster algebra and describe the interplay between cluster variables, coeff...

We develop a new approach to cluster algebras based on the notion of an upper cluster algebra, defined as an intersection of Laurent polynomial rings. Strengthening the Laurent phenomenon established in [6], we show that, under an assumption of “acyclicity”, a cluster algebra coincides with its “upper” counterpart, and is finitely generated; in this case, we also describe its defining ideal, an...

We consider the Ptolemy cluster algebras, which are cluster algebras of finite type A (with non-trivial coefficients) that have been described by Fomin and Zelevinsky using triangulations of a regular polygon. Given any seed Σ in a Ptolemy cluster algebra, we present a formula for the expansion of an arbitrary cluster variable in terms of the cluster variables of the seed Σ. Our formula is give...

We introduce a framework for Z-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a compatibility with the initial exchange matrix, and then one extends this to all cluster variables by mutation. The resulting grading has the property that every (quantu...

We provide a graded and quantum version of the category of rooted cluster algebras introduced by Assem, Dupont and Schiffler and show that every graded quantum cluster algebra of infinite rank can be written as a colimit of graded quantum cluster algebras of finite rank. As an application, for each k we construct a graded quantum infinite Grassmannian admitting a cluster algebra structure, exte...

In [1],the authors defined algebra homomorphisms from the dual RingelHall algebra of certain hereditary abelian categoryA to an appropriate q-polynomial algebra. In the case that A is the representation category of an acyclic quiver, we give an alternative proof by using the cluster multiplication formulas in [9]. Moreover, if the underlying graph of Q is bipartite and the matrix B associated t...

Buan, Marsh and Reiten proved that if a cluster-tilting object T in a cluster category C associated to an acyclic quiver Q satisfies certain conditions with respect to the exchange pairs in C, then the denominator in its reduced form of every cluster variable in the cluster algebra associated to Q has exponents given by the dimension vector of the corresponding module over the endomorphism alge...

We provide an explicit Dynkin diagrammatic description of the c-vectors and the d-vectors (the denominator vectors) of any cluster algebra of finite type with principal coefficients and any initial exchange matrix. We use the surface realization of cluster algebras for types An and Dn, then we apply the folding method to Dn+1 and A2n−1 to obtain types Bn and Cn. Exceptional types are done by di...