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This textbook introduces the representation theory of algebras by
focusing on two of its most important aspects: the Auslander-Reiten
theory and the study of the radical of a module category. It starts
by introducing and describing several characterisations of the
radical of a module category, then presents the central concepts of
irreducible morphisms and almost split sequences, before providing
the definition of the Auslander-Reiten quiver, which encodes much
of the information on the module category. It then turns to the
study of endomorphism algebras, leading on one hand to the
definition of the Auslander algebra and on the other to tilting
theory. The book ends with selected properties of
representation-finite algebras, which are now the best understood
class of algebras. Intended for graduate students in representation
theory, this book is also of interest to any mathematician wanting
to learn the fundamentals of this rapidly growing field. A graduate
course in non-commutative or homological algebra, which is standard
in most universities, is a prerequisite for readers of this book.
The Seventh ARTA ('Advances in Representation Theory of Algebras
VII') conference took place at the Instituto de Matematicas of the
Universidad Nacional Autonoma de Mexico, in Mexico City, from
September 24-28, 2018, in honor of Jose Antonio de la Pena's 60th
birthday. Papers in this volume cover topics Professor de la Pena
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 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 Mathematiques Pures
et Appliquees) 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
This textbook introduces the representation theory of algebras by
focusing on two of its most important aspects: the Auslander-Reiten
theory and the study of the radical of a module category. It starts
by introducing and describing several characterisations of the
radical of a module category, then presents the central concepts of
irreducible morphisms and almost split sequences, before providing
the definition of the Auslander-Reiten quiver, which encodes much
of the information on the module category. It then turns to the
study of endomorphism algebras, leading on one hand to the
definition of the Auslander algebra and on the other to tilting
theory. The book ends with selected properties of
representation-finite algebras, which are now the best understood
class of algebras. Intended for graduate students in representation
theory, this book is also of interest to any mathematician wanting
to learn the fundamentals of this rapidly growing field. A graduate
course in non-commutative or homological algebra, which is standard
in most universities, is a prerequisite for readers of this book.
This first part of a two-volume set offers a modern account of the
representation theory of finite dimensional associative algebras
over an algebraically closed field. The authors present this topic
from the perspective of linear representations of finite-oriented
graphs (quivers) and homological algebra. The self-contained
treatment constitutes an elementary, up-to-date introduction to the
subject using, on the one hand, quiver-theoretical techniques and,
on the other, tilting theory and integral quadratic forms. Key
features include many illustrative examples, plus a large number of
end-of-chapter exercises. The detailed proofs make this work
suitable both for courses and seminars, and for self-study. The
volume will be of great interest to graduate students beginning
research in the representation theory of algebras and to
mathematicians from other fields.
This first part of a two-volume set offers a modern account of the
representation theory of finite dimensional associative algebras
over an algebraically closed field. The authors present this topic
from the perspective of linear representations of finite-oriented
graphs (quivers) and homological algebra. The self-contained
treatment constitutes an elementary, up-to-date introduction to the
subject using, on the one hand, quiver-theoretical techniques and,
on the other, tilting theory and integral quadratic forms. Key
features include many illustrative examples, plus a large number of
end-of-chapter exercises. The detailed proofs make this work
suitable both for courses and seminars, and for self-study. The
volume will be of great interest to graduate students beginning
research in the representation theory of algebras and to
mathematicians from other fields.
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