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...

Abstract We investigate the role of gentle algebras in higher homological algebra. In first part paper, we show that if module category a algebra Λ contains d -cluster tilting subcategory for some d ≥ 2 {d\geq 2} , then is radical square zero Nakayama This gives complete cla...

Abstract In the present paper, we first give a characterization for Bongartz completion in $\tau $-tilting theory via $c$-vectors. Motivated by this characterization, definition of cluster algebras using Then prove existence and uniqueness algebras. We also that admits certain commutativity. two applications As application, full subquiver exchange quiver (or known as oriented graph) algebra $\m...

We describe all Poisson brackets compatible with the natural cluster algebra structure in the open Schubert cell of the Grassmannian Gk(n) and show that any such bracket endows Gk(n) with a structure of a Poisson homogeneous space with respect to the natural action of SLn equipped with an R-matrix Poisson-Lie structure. The corresponding R-matrices belong to the simplest class in the Belavin-Dr...

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...

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...

A bstract We uncover a connection between two seemingly separate subjects in integrable models: the representation theory of affine Temperley-Lieb algebra, and algebraic structure solutions to Bethe equations XXZ spin chain. study solution analytically by computational geometry, find that space encodes rich information about algebra. Using these connections, we compute partition function comple...

In this article, we prove that for a finite quiver Q the equivalence class of potential up to formal change variables complete path algebra $$\widehat{{{\mathbb {C}}}Q}$$ , is determined by its Jacobi together with in 0-th Hochschild homology represented assuming dimensional. This noncommutative analogue famous theorem Mather and Yau on isolated hypersurface singularities. We also right suffici...

Cluster computing is presently a major research area, mostly for high performance computing. The work herein presented refers to the application of cluster computing in a small scale where a virtual machine is composed by a small number of off-the-shelf personal computers connected by a low cost network. A methodology to determine the optimal number of processors to be used in a computation is ...