Related papers: Cluster algebras via cluster categories with infin…
Starting from an arbitrary cluster-tilting object $T$ in a 2-Calabi--Yau category over an algebraically closed field, as in the setting of Keller and Reiten, we define, for each object $L$, a fraction $X(T,L)$ using a formula proposed by…
We review some important results by Gross, Hacking, Keel, and Kontsevich on cluster algebra theory, namely, the column sign-coherence of $C$-matrices and the Laurent positivity, both of which were conjectured by Fomin and Zelevinsky. We…
Derksen-Weyman-Zelevinsky's mutation theory of finite-dimensional representations of quivers with potential is generalized to the framework of infinite-dimensional modules.
In the cluster algebra literature, the notion of a graded cluster algebra has been implicit since the origin of the subject. In this work, we wish to bring this aspect of cluster algebra theory to the foreground and promote its study. We…
Associated to any acyclic cluster algebra is a corresponding triangulated category known as the cluster category. It is known that there is a one-to-one correspondence between cluster variables in the cluster algebra and exceptional…
We investigate two algebra of curves on a topological surface with punctures - the cluster algebra of surfaces defined by Fomin, Shapiro, and Thurston, and the generalized skein algebra constructed by Roger and Yang. By establishing their…
We extend based cluster algebras from the finite rank case to the infinite rank case. By extending (quantum) cluster algebras whose initial seeds are associated with signed words (arising from double Bott--Samelson cells), we recover…
Let $S(\infty)$ denote the infinite symmetric group formed by the finitary permutations of the set of natural numbers; this is a countable group. We introduce its virtual group algebra, a completion of the conventional group algebra…
This is the second paper in a series on enumerative invariants counting self-dual objects in self-dual categories, and is a sequal to (arXiv:2302.00038). Ordinary enumerative invariants in abelian categories can be seen as invariants for…
We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers…
We generalize the theory of integer $C$-, $G$-matrices in cluster algebras to the real case. By a skew-symmetrizing method, we can reduce the problem of skew-symmetrizable patterns to the one of skew-symmetric patterns. In this sense, we…
We show that a certain orbit category considerd by Keller encodes the combinatorics of the $m$-clusters of Fomin and Reading in a fashion similar to the way the cluster category of Buan, Marsh, Reineke, Reiten, and Todorov encodes the…
Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten and Todorov in order to categorify Fomin-Zelevinsky cluster algebras. This survey motivates and outlines the construction of a generalization of cluster categories, and…
We establish certain fundamental properties of $f$-vectors and $F$-matrices for generalized cluster algebras, including the initial and final seed mutation formulas, the compatibility property and the symmetry property. Along the way, we…
We realize Derksen-Weyman-Zelevinsky's mutations of representations as densely-defined regular maps on representation spaces, and study the generic values of Caldero-Chapoton functions with coefficients, giving, for instance, a sufficient…
This paper shows a new phenomenon in higher cluster tilting theory. For each positive integer d, we exhibit a triangulated category C with the following properties. On the one hand, the d-cluster tilting subcategories of C have very simple…
In this note we illustrate by a few examples the general principle: interesting algebras and representations defined over Z_+ come from category theory, and are best understood when their categorical origination has been discovered. We show…
In a categorification of skew-symmetric cluster algebras, each cluster variable corresponds with an indecomposable module over the associated Jacobian algebra. Buan, Marsh and Reiten studied when the denominator vector of each cluster…
This paper gives insight into intriguing connections between two apparently unrelated theories: the theory of skein modules of 3-manifolds and the theory of representations of groups into special linear groups of 2 by 2 matrices. Let R be a…
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 the integers, rationals, or…