Related papers: Tilting theory and cluster algebras
This is a brief and informal introduction to cluster algebras. It roughly follows the historical path of their discovery, made jointly with A.Zelevinsky. Total positivity serves as the main motivation.
Let D be the cluster category of Dynkin type A_{\infty}. This paper provides a bijection between torsion theories in D and certain configurations of arcs connecting non-neighbouring integers.
We give the first example of a non-trivial cluster tilting module in a local finite dimensional algebra. To do this, we give an explicit calculation of the corresponding higher Auslander algebra by quiver and relations using the GAP-package…
The goal of this paper is to consider some relations between varieties of representations of groups and varieties of associative algebras. The main emphasis is put on the varieties of representations of groups induced by the varieties of…
This is a companion article to my papers on Kazhdan-Lusztig polynomials and character formulae for the Lie superalgebras gl(m|n) (much revised!) and q(n). The goal is to develop the general theory of tilting modules for Lie superalgebras,…
In this paper, we will place clusters in type $\tilde{\mathbb{A}}$ (equivalently triangluations of an annulus) into infinite families parametrized by winding numbers of certain arcs in the corresponding triangulation. We will count how many…
We introduce a relative tilting theory in abelian categories and show that this work offers a unified framework of different previous notions of tilting, ranging from Auslander-Solberg relative tilting modules on Artin algebras to…
We compute the Hochschild cohomology groups of the cluster-tilted algebras of finite representation type.
To every minimal model of a complete local isolated cDV singularity Donovan--Wemyss associate a finite dimensional symmetric algebra known as the contraction algebra. We construct the first known standard derived equivalences between these…
These are lecture notes for a course on the theory of Clifford algebras, with special emphasis on their wide range of applications in mathematics and physics. Clifford algebra is introduced both through a conventional tensor algebra…
This is an expanded version of the notes of our lectures given at the conference "Current Developments in Mathematics 2003" held at Harvard University on November 21--22, 2003. We present an overview of the main definitions, results and…
In the 1980s, Harada introduced a new class of algebras now called Harada algebras. Harada algebras provides us with a rich source of Auslander's 1-Gorenstein algebras. In this paper, we have two main results about Harada algebras. The…
Clustering is one of the most common tasks of Machine Learning. In this paper we examine how ideas from topology can be used to improve clustering techniques.
Koszul algebras have arisen in many contexts; algebraic geometry, combinatorics, Lie algebras, non-commutative geometry and topology. The aim of this paper and several sequel papers is to show that for any finite dimensional algebra there…
This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out…
Tilting theory in cluster categories of hereditary algebras has been developed in [BMRRT] and [BMR]. These results are generalized to cluster categories of hereditary abelian categories. Furthermore, for any tilting object $T$ in a…
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…
Let $\mathcal{C}$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster tilting object. Under some constructibility assumptions on $\mathcal{C}$ which are satisfied for instance by cluster categories, by generalized cluster…
First, we give a new example of silting-discrete algebras. Second, one explores when the algebra of triangular matrices over a finite dimensional algebra is $\tau$-tilting finite. In particular, we classify algebras over which triangular…
We initiate a study of the dependence on the choice of ground ring on the question of whether a cluster algebra is equal to its upper cluster algebra. A condition for when there is equality of the cluster algebra and upper cluster algebra…