The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two classifications is different: roughly speaking, Kac-Moody algebras are associated with (symmetrizable) Cartan matrices, while cluster algebras correspond to s...

We continue the study of quivers with potentials and their representations initiated in the first paper of the series. Here we develop some applications of this theory to cluster algebras. As shown in the “Cluster algebras IV” paper, the cluster algebra structure is to a large extent controlled by a family of integer vectors called g-vectors, and a family of integer polynomials called F -polyno...

A family of quantum cluster algebras is introduced and studied. In general, these algebras are new, but sub-classes have been studied previously by other authors. The algebras are indexed by double partitions or double flag varieties. Equivalently, they are indexed by broken lines L. By grouping together neighboring mutations into quantum line mutations we can mutate from the cluster algebra of...

Abstract Let $Q$ be a finite acyclic valued quiver. We give the cluster multiplication formulas in quantum algebra of with arbitrary coefficients, by applying certain quotients derived Hall subalgebras $Q$. These can viewed as version theorem classical proved Caldero–Keller for type, Hubery affine and Xiao–Xu quivers.

We initiate the study of cluster algebras in Feynman integrals dimensional regularization. provide evidence that four-point with one off-shell leg are described by a $C_{2}$ algebra, and we find adjacency relations restrict allowed function space. By embedding inside $A_3$ identify these adjacencies extended Steinmann for six-particle massless scattering. The algebra connection restricts functi...

The Goulden–Jackson cluster method is a powerful tool for counting words by occurrences of prescribed subwords, and was adapted Elizalde Noy permutations consecutive patterns. In this paper, we lift the to Malvenuto–Reutenauer algebra. Upon applying standard homomorphisms, our result specializes both as well q-analogue which keeps track inversion number statistic. We construct additional homomo...

Let Λ be an artin algebra and M n-cluster tilting subcategory of mod Λ. We show that has additive generator if only the n-almost split sequences form a basis for relations Grothendieck group every effaceable functor M→Ab finite length. As consequence we type then

first we show that the cosets of a fuzzy ideal μ in a bck-algebrax form another bck-algebra  x/μ (called the fuzzy quotient bck-algebra of x by μ). also we show thatx/μ is a fuzzy partition of x and we prove several some isomorphism theorems. moreover we prove that if the associated fuzzy similarity relation of a fuzzy partition p of a commutative bck-algebra iscompatible, then p is a fuzzy quo...