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: Advanced homological algebra 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

Introducing the representation theory of groups and finite dimensional algebras, first studying basic non-commutative ring theory, this book covers the necessary background on elementary homological algebra and representations of groups up to block theory. It further discusses vertices, defect groups, Green and Brauer correspondences and Clifford theory. Whenever possible the statements are presented in a general setting for more general algebras, such as symmetric finite dimensional algebras over a field. Then, abelian and derived categories are introduced in detail and are used to explain stable module categories, as well as derived categories and their main invariants and links between them. Group theoretical applications of these theories are given – such as the structure of blocks of cyclic defect groups – whenever appropriate. Overall, many methods from the representation theory of algebras are introduced. Representation Theory assumes only the most basic knowledge of linear algebra, groups, rings and fields and guides the reader in the use of categorical equivalences in the representation theory of groups and algebras. As the book is based on lectures, it will be accessible to any graduate student in algebra and can be used for self-study as well as for classroom use.

Modern developments in representation theory rely heavily on homological methods. This book for advanced graduate students and researchers introduces these methods from their foundations up and discusses several landmark results that illustrate their power and beauty. Categorical foundations include abelian and derived categories, with an emphasis on localisation, spectra, and purity. The representation theoretic focus is on module categories of Artin algebras, with discussions of the representation theory of finite groups and finite quivers. Also covered are Gorenstein and quasi-hereditary algebras, including Schur algebras, which model polynomial representations of general linear groups, and the Morita theory of derived categories via tilting objects. The final part is devoted to a systematic introduction to the theory of purity for locally finitely presented categories, covering pure-injectives, definable subcategories, and Ziegler spectra. With its clear, detailed exposition of important topics in modern representation theory, many of which were unavailable in one volume until now, it deserves a place in every representation theorist's library.

The Seventh ARTA (“Advances in Representation Theory of Algebras VII”) conference took place at the Instituto de Matemáticas of the Universidad Nacional Autónoma de México, in Mexico City, from September 24–28, 2018, in honor of José Antonio de la Peña's 60th birthday. Papers in this volume cover topics Professor de la Peña worked on, such as covering theory, tame algebras, and the use of quadratic forms in representation theory. Also included are papers on the categorical approach to representations of algebras and relations to Lie theory, Cohen–Macaulay modules, quantum groups and other algebraic structures.

This volume presents an elaborated version of lecture notes for two advanced courses: (Re)Emerging methods in Commutative Algebra and Representation Theory and Building Bridges Between Algebra and Topology, held at the CRM in the spring of 2015. Homological algebra is a rich and ubiquitous area; it is both an active field of research and a widespread toolbox for many mathematicians. Together, these notes introduce recent applications and interactions of homological methods in commutative algebra, representation theory and topology, narrowing the gap between specialists from different areas wishing to acquaint themselves with a rapidly growing field. The covered topics range from a fresh introduction to the growing area of support theory for triangulated categories to the striking consequences of the formulation in the homotopy theory of classical concepts in commutative algebra. Moreover, they also include a higher categories view of Hall algebras and an introduction to the use of idempotent functors in algebra and topology.

Expanding upon the material delivered during the LMS Autumn Algebra School 2020, this volume reflects the fruitful connections between different aspects of representation theory. Each survey article addresses a specific subject from a modern angle, beginning with an exploration of the representation theory of associative algebras, followed by the coverage of important developments in Lie theory in the past two decades, before the final sections introduce the reader to three strikingly different aspects of group theory. Written at a level suitable for graduate students and researchers in related fields, this book provides pure mathematicians with a springboard into the vast and growing literature in each area.