Michael Barot

Cluster Algebras of Finite Type and Positive Symmetrizable Matrices

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 skew-symmetrizable matrices. We study an interplay between the two classes of matrices, in particular, establishing a new criterion for deciding whether a given skew-symmetrizable matrix gives rise to a cluster algebra of finite type.

The cluster category of a canonical algebra

We study the cluster category of a canonical algebra A in terms of the hereditary category of coherent sheaves over the corresponding weighted projective line X. As an application we determine the automorphism group of the cluster category and show that the cluster-tilting objects form a cluster structure in the sense of Buan, Iyama, Reiten and Scott. The tilting graph of the sheaf category always coincides with the tilting or exchange graph of the cluster category. We show that this graph is connected if the Euler characteristic of X is non-negative, or equivalently, if A is of tame (domestic or tubular) representation type.

One-point extensions and derived equivalence

Abstract Work of the first author with de la Pena [M. Barot, J.A. de la Pena, Proc. Amer. Math. Soc. 127 (1999) 647–655], concerned with the class of algebras derived equivalent to a tubular algebra, raised the question whether a derived equivalence between two algebras can be extended to one-point extensions. The present paper yields a positive answer.

Reflection group presentations arising from cluster algebras

We give a presentation of a finite crystallographic reflection group in terms of an arbitrary seed in the corresponding cluster algebra of finite type and interpret the presentation in terms of companion bases in the associated root system.

Tubular cluster algebras II: Exponential growth

Among the mutation finite cluster algebras the tubular ones are a particularly interesting class. We show that all tubular (simply laced) cluster algebras are of exponential growth by two different methods: first by studying the automorphism group of the corresponding cluster category and second by giving explicit sequences of mutations.

Cluster Tilted Algebras with a Cyclically Oriented Quiver

In association with a finite dimensional algebra A of global dimension two, we consider the endomorphism algebra of A, viewed as an object in the triangulated hull of the orbit category of the bounded derived category, in the sense of Amiot. We characterize the algebras A of global dimension two such that its endomorphism algebra is isomorphic to a cluster-tilted algebra with a cyclically oriented quiver. Furthermore, in the case that the cluster tilted algebra with a cyclically oriented quiver is of Dynkin or extended Dynkin type then A is derived equivalent to a hereditary algebra of the same type.

Estimating the size of a union of random subsets of fixed cardinality

Jose Antonio de la Pena got his Ph.D. from UNAM, Mexico in 1983. He made a postdoctoral stay at the University of Zurich, Switzerland from 1984 to 1986. Since then he has a research position at the Instituto de Matematicas, UNAM. His main research area is the representation theory of algebras but he has also done some work in combinatorics. At this moment, he is Director of the Instituto de Matematicas, UNAM.

Derived Tubularity: a Computational Approach

Let A be a finite dimensional algebra over an algebraically closed field. We denote by mod A the category of finite dimensional left A-modules and by D b (A) the derived category of mod A (see for example [17]] for definitions). We say that two algebras A and B are derived equivalent if their derived categories D b (A) and D b (B) are equivalent as triangulated categories.

Indecomposables and Dimensions

Some of the deepest results in the theory of representations of algebras are presented within this chapter. All of them deal with the representation type, in other words, with the amount of non-isomorphic indecomposable modules with a fixed dimension vector. Several results will only be indicated without proof.

The Auslander-Reiten Theory

In this chapter the insight into the structure of module category is deepened. One section is devoted to the example of the Kronecker algebra. A series of notions and results named after Auslander and Reiten is presented, all of them constitute the Auslander-Reiten theory: The “Auslander-Reiten translate”, which associates to every indecomposable non-projective module an indecomposable non-injective module and the the “Auslander Reiten sequences”, which are short exact sequences and in a certain sense minimal among the non-split sequences. The Theorem of Auslander-Reiten concerns the structure of module categories and yield a remarkable deep insight. Since it is valid in full generality it can be considered as the crown jewel of representation theory. It makes it possible to calculate combinatorially, via a process called “knitting” studied in the next chapter, important information about certain parts—and in some cases all of it—of the module category over some algebra.