Cluster algebras and cluster categories
This article is a basic survey of cluster algebras, with a focus towards categorification. A cluster algebra is a commutative \(\mathbb{Q}\)-algebra with a set of distinguished generators which are defined recursively from an initial cluster. One source of cluster algebras comes from the category of quivers -- finite, with no loops nor 2-cycles -- and those are described here. The notion of cluster algebra is generalized in a natural way to allow for coefficients, and this generalization is used to provide two geometric examples. Following this, the author turns to categorification. Using the theory of quiver representations and the Caldero-Chapoton formula it is shown how one can define the cluster category for the cluster associated to a Dynkin quiver \(Q\) as a certain orbit category of the bounded derived category of \(\text{mod}kQ,\) \(k\) algebraically closed. Given a cluster, it is shown how to categorify it using cluster-tilting objects. This definition of cluster category is then extended to general acyclic quivers, and it is shown how these categories relate to stable categories of Cohen-Macaulay modules under certain conditions. The work under review does not claim to be an all-encompassing introduction to these topics. As the author points out, there is no discussion of monoidal categorification, quivers with potentials, nor the connection with quantum dilogarithmic identities. However, the author includes citations for these topics.
scientific article; zbMATH DE number 6108068