DE

2018 | Book

# Homological Methods, Representation Theory, and Cluster Algebras

Editors: Ibrahim Assem, Sonia Trepode

Publisher:

Book Series : CRM Short Courses

Part of:

insite
SEARCH

This text presents six mini-courses, all devoted to interactions between representation theory of algebras, homological algebra, and the new ever-expanding theory of cluster algebras. The interplay between the topics discussed in this text will continue to grow and this collection of courses stands as a partial testimony to this new development. The courses are useful for any mathematician who would like to learn more about this rapidly developing field; the primary aim is to engage graduate students and young researchers. Prerequisites include knowledge of some noncommutative algebra or homological algebra. Homological algebra has always been considered as one of the main tools in the study of finite-dimensional algebras. The strong relationship with cluster algebras is more recent and has quickly established itself as one of the important highlights of today’s mathematical landscape. This connection has been fruitful to both areas—representation theory provides a categorification of cluster algebras, while the study of cluster algebras provides representation theory with new objects of study.

The six mini-courses comprising this text were delivered March 7–18, 2016 at a CIMPA (Centre International de Mathématiques Pures et Appliquées) research school held at the Universidad Nacional de Mar del Plata, Argentina. This research school was dedicated to the founder of the Argentinian research group in representation theory, M.I. Platzeck.

The courses held were:

Introduction to the representation theory of algebras

Auslander-Reiten theory for algebras of infinite representation type

Cluster algebras arising from surfaces

Cluster tilted algebras

Cluster characters

Introduction to K-theory

Brauer graph algebras and applications to cluster algebras

##### Introduction to the Representation Theory of Finite-Dimensional Algebras: The Functorial Approach
Abstract
These are the notes of a course given at the CIMPA School “Homological Methods, Representation Theory and Cluster Algebras,” Mar del Plata, Argentina, 2016. The aim of this brief course is to give an introduction to the functorial approach to the representation theory of finite-dimensional algebras, developed by Maurice Auslander and Idun Reiten, and is strongly based on the work “A functorial approach to representation theory,” by M. Auslander (Representations of Algebras. Springer, Berlin, 1981) [4].
María Inés Platzeck
##### Auslander–Reiten Theory for Finite-Dimensional Algebras
Abstract
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.
Piotr Malicki
##### Cluster Algebras from Surfaces
Lecture Notes for the CIMPA School Mar del Plata, March 2016
Abstract
Cluster algebras were introduced by Fomin and Zelevinsky [17] in 2002. Their original motivation was coming from canonical bases in Lie Theory. Today cluster algebras are connected to various fields of mathematics, including
• Combinatorics (polyhedra, frieze patterns, green sequences, snake graphs, T-paths, dimer models, triangulations of surfaces)
• Representation theory of finite dimensional algebras (cluster categories, cluster-tilted algebras, preprojective algebras, tilting theory, 2-Calabi–Yau categories, invariant theory)
• Poisson geometry and algebraic geometry (cluster varieties, Grassmannians, stability conditions, scattering diagrams, Poisson structures on $${{\mathrm{SL}}}(n)$$)
• Teichmüller theory (lambda-lengths, Penner coordinates, cluster varieties)
• Knot theory (Chern–Simons invariants, volume conjecture, Legendrian knots)
• Dynamical systems (frieze patterns, pentagram map, integrable systems, T-systems, sine-Gordon Y-systems)
• Mathematical Physics (statistical mechanics, Donaldson–Thomas invariants, quantum dilogarithm identities, BPS particles).
Ralf Schiffler
##### Cluster Characters
Abstract
Shortly after the introduction of cluster algebras in (S. Fomin and A. Zelevinsky (2002). J. Amer. Math. Soc. 15(2), 497–529.) [19], links with an impressively vast number of fields of mathematics were uncovered. Among these is the representation theory of finite-dimensional algebras, whose links to cluster algebras became apparent in, for instance, (R. Marsh, M. Reineke and A. Zelevinsky (2003). Trans. Amer. Math. Soc. 355(10), 4171–4186.) [39], (A.B Buan, R. Marsh, M. Reineke, I. Reiten and G. Todorov (2006). Adv. Math. 204(2), 572–618.) [8], (P. Caldero and F. Chapoton (2006). Comment. Math. Helv. 81(3), 595–616.) [10].
Pierre-Guy Plamondon
##### A Course on Cluster Tilted Algebras
Abstract
These notes are an expanded version of a mini-course given in the CIMPA School ‘Homological Methods, Representation Theory and Cluster Algebras’, held from the 7th to the 18th of March 2016 in Mar del Plata (Argentina). The aim of the course was to introduce the participants to cluster tilted algebras and their applications in the representation theory of algebras.
Ibrahim Assem
##### Brauer Graph Algebras
A Survey on Brauer Graph Algebras, Associated Gentle Algebras and Their Connections to Cluster Theory
Abstract
This survey starts with a motivation of the study of Brauer graph algebras by relating them to special biserial algebras. The definition of Brauer graph algebras is given in great detail with many examples to illustrate the concepts. An interpretation of Brauer graphs as decorated ribbon graphs is included. A section on gentle algebras and their associated ribbon graph, trivial extensions of gentle algebras, admissible cuts of Brauer graph algebras and a first connection of Brauer graph algebras with Jacobian algebras associated to triangulations of marked oriented surfaces follows. The interpretation of flips of diagonals in triangulations of marked oriented surfaces as derived equivalences of Brauer graph algebras and the comparison of derived equivalences of Brauer graph algebras with derived equivalences of frozen Jacobian algebras is the topic of the next section. In the last section, after defining Green’s walk around the Brauer graph, a complete description of the Auslander Reiten quiver of a Brauer graph algebra is given.
Sibylle Schroll
Title
Homological Methods, Representation Theory, and Cluster Algebras
Editors
Ibrahim Assem
Sonia Trepode