A Course on Cluster Tilted Algebras

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Methods and results from the representation theory of quivers and finite dimensional algebras have led to many interactions with other areas of mathematics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further develop such interactions and to stimulate progress in the representation theory of algebras.

Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47). Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … [↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu. [1]: #xref-corresp-1-1

This article is based on a course given at the CIMPA School “Homological Methods, Representation Theory and Cluster Algebras,” held in March 2016 in Mar del Plata. The aim of the course, consisting of four lectures, was to provide a brief introduction to the notion of an almost split sequence and its use in the representation theory of finite-dimensional algebras. The first two sections are reduced, and the next three sections are extended in comparison with the above-mentioned course.

Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton–Okounkov bodies, PBW- filtrations and cluster algebras come to push the subject forward.

Commutative algebra is a vast subject, with connections to many different areas of mathematics, and beyond. The focus of this workshop was on three areas, all concerned with resolutions in various forms. One is the resolution of singularities of algebraic varieties, which remains a vibrant topic of research. The second is the theory of noncommutative resolution of singularities. Introduced two decades ago, this subject has witnessed remarkable growth developing connections to algebraic geometry, commutative algebra, cluster algebras, and the representation theory of algebras, both commutative and noncommutative, among others. The third intended meaning of the world “resolution” is as in free resolutions of algebras and modules in commutative algebra. There is another sense in which the title is appropriate: recently three long standing open problems in commutative algebra have been resolved. This workshop brought together experts and early career researchers in these various fields, to facilitate exchange of ideas and to explore potential collaborations.

This work presents a survey (with some proofs) of the recent developments in the study of Coxeter transformations and their applications to representation theory of associative algebras. Let Q be a quiver and C = C(Q) the associated Coxeter matrix. We consider results on the eigenvalues of C: multiplicities, distribution, bounds, relation with the symmetries of Q... We consider applications to the study of the indecomposable modules over the hereditary algebra k[Q] (k a field): position of modules, growth of the Auslander-Reiten translates... Other applications concern growth of components of the representation-quiver of finite dimensional algebras, towers of algebras...

We give an example of a cluster-tilted algebra $$\Lambda $$ with quiver Q, such that the associated cluster algebra $$\mathcal {A}(Q)$$ has a denominator vector which is not the dimension vector of any indecomposable $$\Lambda $$ -module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra $$\Lambda $$ , we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid $$\Lambda $$ -modules. In order to do this it is necessary, and of independent interest, to first classify the indecomposable rigid $$\Lambda $$ -modules in this case.

Affine.m—Mathematica package for computations in representation theory of finite-dimensional and affine Lie algebras

Since its inception in the early seventies, the study of Enveloping Algebras has undergone a significant and continuous evolution and moreover has inspired a wide variety of developments in many areas of mathematics including Ring Theory, Differential Operators, Invariant Theory, Quantum Groups and Hecke Algebras. As indicated above, one of the main goals behind this meeting was to bring together a group of participants with a wide range of interests in and around the geometric and the combinatorial side of the representation theory of Lie groups and algebras. We strongly believe that such an approach to representation theory, in particular interaction between geometry and representation theory, will open up new avenues of thought and lead to progress in a number of areas. This diversity was well reflected in the expertize represented by the conference participants, as well as in the wide range of topics covered. They may broadly be summarized under the following three headings: (1) The study of the structure and representation theory through a) Slodowy slices and the Joseph ideal, b) Equivalence of categories of representations and certain geometric counterparts, c) Hall algebras, Quotient schemes and canonical bases, d) Richardson elements and birationality questions, e) free Lie algebras and current algebras, f) Dirac cohomology, h) Gelfand-Zeitlin theory (2) Combinatorial Aspects of Lie Systems particularly through a) Saturation problems and Buildings, b) Affine and double Affine Hecke algebras, Cherednik algebras, c) Solvable lattice models (3) Geometric Structures including a) Representation theoretic methods in enumerative geometry, b) Multiplicity free symplectic reduction, c) Monodromy actions of braid groups, d) Associated varieties for Lie super algebras, e) affine Grassmannians, f) Schubert varieties and Demazure modules The single most important development reported in this conference seems to us to be the generalization of the categorical Satake isomorphism to quantum groups at roots of unity, which allows to bring in a whole lot of new geometry into representation theory. It seems likely that it will lead to the proofs of some open conjectures of Lusztig. We were particularly happy to have quite a few very strong younger participants and some highly promising postdocs. We tried to accomodate young postdocs, as much as feasible, to present their work by devoting one afternoon to their (shorter) talks.

We describe a connection between the subjects of cluster algebras, polynomial identity algebras, and discriminants. For this, we define the notion of root of unity quantum cluster algebras and prove that they are polynomial identity algebras. Inside each such algebra we construct a (large) canonical central subalgebra, which can be viewed as a far reaching generalization of the central subalgebras of big quantum groups constructed by De Concini, Kac, and Procesi and used in representation theory. Each such central subalgebra is proved to be isomorphic to the underlying classical cluster algebra of geometric type. When the root of unity quantum cluster algebra is free over its central subalgebra, we prove that the discriminant of the pair is a product of powers of the frozen variables times an integer. An extension of this result is also proved for the discriminants of all subalgebras generated by the cluster variables of nerves in the exchange graph. These results can be used for the effective computation of discriminants. As an application we obtain an explicit formula for the discriminant of the integral form over of each quantum unipotent cell of De Concini, Kac, and Procesi for arbitrary symmetrizable Kac–Moody algebras, where is a root of unity.

Almost split sequences and Auslander-Reiten quivers (species) have turned out to be an extremely useful tool in the representation theory of artinian algebras and integral representations of complete orders. Whereas the theory is well developed in the artinian situations, it is far behind for integral representations. The main reason is that the powerful tool of universal coverings and covering techniques — as developed by Riedtmann [14], Bongartz — Gabriel [4] — has not yet found its integral analogue. The main difficulty is that it is unclear to me, when a representation quiver — in the sense of C.Riedtmann [14] — is the Auslander-Reiten quiver of an order of finite lattice type; in particular, one obstacle is the multiplication with the parameter of the ground ring. A little progress has been made in this direction by Wiedemann. Apart from my general philosophy, that most of the results in the representation theory of artinian algebras have more or less obvious analoga in integral representation theory, there is a more solid connection between integral representations and some categories of torsionfree modules over artinian algebras as observed by C.M.Ringel and the author [15], such that in some instances integral representation theory might also influence the representation theory of artinian algebras — e.g. that some torsionfree categories of artinian modules have almost split sequences was — before its general treatment by M.Auslander and S.O.Smalo [1] — in secial cases — influenced by the results in integral representation theory — noted by the author [17] and independently by R.Bautista and R.Martinez [3].

This is a report on advances in the representation theory of finite dimensional algebras in the years 1984 – 1990. During these years, the German research council (DFG) has sponsered a Forschungsschwerpunkt devoted to the representation theory of finite groups and finite dimensional algebras; it started in 1984 and will be finished by 1991.KeywordsEndomorphism RingIndecomposable ModuleFinite Dimensional AlgebraHall AlgebraHereditary AlgebraThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Equivalences between cluster categories

This paper is a report of my lecture at the International Conference on the Representation Theory of Algebras which preceded the Euroconference on Infinite Length Modules. While it is true that the aim of the research is the homological algebra of finitely presented modules, a major point of the lecture was the role that infinitely generated modules could play in the investigation. The methods of idempotent modules and the theory of support varieties for infinite dimensional modules have already had a significant impact on group representation theory. It seems certain that there will be a lot more to follow. I am honored by the invitation to include the report in the conference proceedings, and I would like to thank the organizers of the conference and the workshop for the stimulating experience.KeywordsExact SequenceDirect SummandFull SubcategoryProjective ModuleEndomorphism RingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.