This talk, based on [14], is a report on recent work by B. Leclerc on a new type of categorification for cluster algebras. Cluster algebras were invented by Fomin and Zelevinsky [8] at the beginning of this decade. Since then, a major effort has gone into their categorification (cf. for example [15] [1] [2] [3] [10]). Namely, in many cases, it was proved that for a given cluster algebra A, ther...

We introduce a Poisson variety compatible with a cluster algebra structure and a compatible toric action on this variety. We study Poisson and topological properties of the union of generic orbits of this toric action. In particular, we compute the number of connected components of the union of generic toric orbits for cluster algebras over real numbers. As a corollary we compute the number of ...

let a be a banach algebra and e be a banach a-bimodule then s = a  e, the l1-direct sum of a and e becomes a module extension banach algebra when equipped with the algebras product (a; x):(a′; x′) = (aa′; a:x′ + x:a′). in this paper, we investigate △-amenability for these banach algebras and we show that for discrete inverse semigroup s with the set of idempotents es, the module extension bana...

In this paper, we propose a novel method for comparing the shape of similar objects. From the viewpoint of linear algebra, we turn this identifiable region detection problem into a low-rank submatrices searching process, and solve it with biclustering. Comparing with traditional cluster analysis, our method looks for structural information on both object index and local shape dimensions, which ...

We express explicitly the Aomoto trilogarithm by classical trilogarithms and investigate the algebraic-geometric structures staying behind: different realizations of the weight three motivic complexes. Applying these results we describe the motivic structure of the Grassmannian tetralogarithm function and determine the structure of the motivic Lie coalgebra in degrees ≤ 4. Using this we give an...

Despite being one of the most common approach in unsupervised data analysis, a very small literature exists on the formalization of clustering algorithms. This paper proposes a semiring-based methodology, named Feature-Cluster Algebra, which is applied to abstract the representation of a labeled tree structure representing a hierarchical categorical clustering algorithm, named CCTree. The eleme...

The exchange graph of a quiver is the graph of mutation-equivalent quivers whose edges correspond to mutations. The exchange graph admits a natural acyclic orientation called the oriented exchange graph. Building on work of Iyama, Reiten, Thomas, and Todorov, we show that this directed graph is a semidistributive lattice by using the isomorphism to the lattice of functorially finite torsion cla...

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin–Zelevinsky cluster algebra. In this model, the tilting obj...

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...

Type A, or Ptolemy cluster algebras are a prototypical example of finite type cluster algebras, as introduced by Fomin and Zelevinsky. Their combinatorics is that of triangulations of a polygon. Lam and Pylyavskyy have introduced a generalization of cluster algebras where the exchange polynomials are not necessarily binomial, called Laurent phenomenon algebras. It is an interesting and hard que...