Finite Dimensional Algebras

Author : Yurj A. Drozd,Vladimir V. Kirichenko
Publisher : Springer Science & Business Media
Page : 260 pages
File Size : 44,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642762444

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Finite Dimensional Algebras by Yurj A. Drozd,Vladimir V. Kirichenko Pdf

This English edition has an additional chapter "Elements of Homological Al gebra". Homological methods appear to be effective in many problems in the theory of algebras; we hope their inclusion makes this book more complete and self-contained as a textbook. We have also taken this occasion to correct several inaccuracies and errors in the original Russian edition. We should like to express our gratitude to V. Dlab who has not only metic ulously translated the text, but has also contributed by writing an Appendix devoted to a new important class of algebras, viz. quasi-hereditary algebras. Finally, we are indebted to the publishers, Springer-Verlag, for enabling this book to reach such a wide audience in the world of mathematical community. Kiev, February 1993 Yu.A. Drozd V.V. Kirichenko Preface The theory of finite dimensional algebras is one of the oldest branches of modern algebra. Its origin is linked to the work of Hamilton who discovered the famous algebra of quaternions, and Cayley who developed matrix theory. Later finite dimensional algebras were studied by a large number of mathematicians including B. Peirce, C.S. Peirce, Clifford, ·Weierstrass, Dedekind, Jordan and Frobenius. At the end of the last century T. Molien and E. Cartan described the semisimple algebras over the complex and real fields and paved the first steps towards the study of non-semi simple algebras.

Author : Nathanial Patrick Brown,Narutaka Ozawa
Publisher : American Mathematical Soc.
Page : 530 pages
File Size : 48,6 Mb
Release : 2008
Category : C*-algebras
ISBN : 9780821843819

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$\textrm {C}^*$-Algebras and Finite-Dimensional Approximations by Nathanial Patrick Brown,Narutaka Ozawa Pdf

$\textrm{C}*$-approximation theory has provided the foundation for many of the most important conceptual breakthroughs and applications of operator algebras. This book systematically studies (most of) the numerous types of approximation properties that have been important in recent years: nuclearity, exactness, quasidiagonality, local reflexivity, and others. Moreover, it contains user-friendly proofs, insofar as that is possible, of many fundamental results that were previously quite hard to extract from the literature. Indeed, perhaps the most important novelty of the first ten chapters is an earnest attempt to explain some fundamental, but difficult and technical, results as painlessly as possible. The latter half of the book presents related topics and applications--written with researchers and advanced, well-trained students in mind. The authors have tried to meet the needs both of students wishing to learn the basics of an important area of research as well as researchers who desire a fairly comprehensive reference for the theory and applications of $\textrm{C}*$-approximation theory.

Author : Nathan Jacobson
Publisher : Springer Science & Business Media
Page : 290 pages
File Size : 43,9 Mb
Release : 2009-12-09
Category : Mathematics
ISBN : 9783642024290

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Finite-Dimensional Division Algebras over Fields by Nathan Jacobson Pdf

Here, the eminent algebraist, Nathan Jacobsen, concentrates on those algebras that have an involution. Although they appear in many contexts, these algebras first arose in the study of the so-called "multiplication algebras of Riemann matrices". Of particular interest are the Jordan algebras determined by such algebras, and thus their structure is discussed in detail. Two important concepts also dealt with are the universal enveloping algebras and the reduced norm. However, the largest part of the book is the fifth chapter, which focuses on involutorial simple algebras of finite dimension over a field.

Author : Peter Gabriel,Andrei V. Roiter
Publisher : Springer Science & Business Media
Page : 188 pages
File Size : 47,9 Mb
Release : 1997-09-12
Category : Mathematics
ISBN : 3540629904

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Representations of Finite-Dimensional Algebras by Peter Gabriel,Andrei V. Roiter Pdf

From the reviews: "... [Gabriel and Roiter] are pioneers in this subject and they have included proofs for statements which in their opinions are elementary, those which will help further understanding and those which are scarcely available elsewhere. They attempt to take us up to the point where we can find our way in the original literature. ..." --The Mathematical Gazette

Author : V. Dlab,Leonard Scott
Publisher : Springer Science & Business Media
Page : 392 pages
File Size : 55,9 Mb
Release : 2013-04-18
Category : Mathematics
ISBN : 9789401715560

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Finite Dimensional Algebras and Related Topics by V. Dlab,Leonard Scott Pdf

Based on invited lectures at the 1992 Canadian Algebra Seminar, this volume represents an up-to-date and unique report on finite-dimensional algebras as a subject with many serious interactions with other mathematical disciplines, including algebraic groups and Lie theory, automorphic forms, sheaf theory, finite groups, and homological algebra. It will interest mathematicians and graduate students in these and related subjects as an introduction to research in an area of increasing relevance and importance.

Author : Peter Gabriel,Andrei V. Roiter
Publisher : Springer Science & Business Media
Page : 190 pages
File Size : 51,6 Mb
Release : 1992-10-08
Category : Mathematics
ISBN : 3540537325

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Representations of Finite-Dimensional Algebras by Peter Gabriel,Andrei V. Roiter Pdf

From the reviews: "... [Gabriel and Roiter] are pioneers in this subject and they have included proofs for statements which in their opinions are elementary, those which will help further understanding and those which are scarcely available elsewhere. They attempt to take us up to the point where we can find our way in the original literature. ..." --The Mathematical Gazette

Author : Bangming Deng
Publisher : American Mathematical Soc.
Page : 790 pages
File Size : 53,8 Mb
Release : 2008
Category : Finite fields (Algebra).
ISBN : 9780821841860

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Finite Dimensional Algebras and Quantum Groups by Bangming Deng Pdf

"The interplay between finite dimensional algebras and Lie theory dates back many years. In more recent times, these interrelations have become even more strikingly apparent. This text combines, for the first time in book form, the theories of finite dimensional algebras and quantum groups. More precisely, it investigates the Ringel-Hall algebra realization for the positive part of a quantum enveloping algebra associated with a symmetrizable Cartan matrix and it looks closely at the Beilinson-Lusztig-MacPherson realization for the entire quantum $\mathfrak{gl}_n$. The book begins with the two realizations of generalized Cartan matrices, namely, the graph realization and the root datum realization. From there, it develops the representation theory of quivers with automorphisms and the theory of quantum enveloping algebras associated with Kac-Moody Lie algebras. These two independent theories eventually meet in Part 4, under the umbrella of Ringel-Hall algebras. Cartan matrices can also be used to define an important class of groups--Coxeter groups--and their associated Hecke algebras. Hecke algebras associated with symmetric groups give rise to an interesting class of quasi-hereditary algebras, the quantum Schur algebras. The structure of these finite dimensional algebras is used in Part 5 to build the entire quantum $\mathfrak{gl}_n$ through a completion process of a limit algebra (the Beilinson-Lusztig-MacPherson algebra). The book is suitable for advanced graduate students. Each chapter concludes with a series of exercises, ranging from the routine to sketches of proofs of recent results from the current literature."--Publisher's website.

Author : Dieter Happel
Publisher : Unknown
Page : 208 pages
File Size : 43,7 Mb
Release : 1988
Category : Electronic books
ISBN : 1107368448

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Triangulated Categories in the Representation Theory of Finite Dimensional Algebras by Dieter Happel Pdf

This book is an introduction to the use of triangulated categories in the study of representations of finite-dimensional algebras. In recent years representation theory has been an area of intense research and the author shows that derived categories of finite-dimensional algebras are a useful tool in studying tilting processes. Results on the structure of derived categories of hereditary algebras are used to investigate Dynkin algebras and interated tilted algebras. The author shows how triangulated categories arise naturally in the study of Frobenius categories. The study of trivial extension algebras and repetitive algebras is then developed using the triangulated structure on the stable category of the algebra's module category. With a comprehensive reference section, algebraists and research students in this field will find this an indispensable account of the theory of finite-dimensional algebras.

Author : Peter Gabriel,Andrei V. Roiter
Publisher : Springer
Page : 0 pages
File Size : 41,8 Mb
Release : 1997
Category : Mathematics
ISBN : 3642580971

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Representations of Finite-Dimensional Algebras by Peter Gabriel,Andrei V. Roiter Pdf

From the reviews: "... [Gabriel and Roiter] are pioneers in this subject and they have included proofs for statements which in their opinions are elementary, those which will help further understanding and those which are scarcely available elsewhere. They attempt to take us up to the point where we can find our way in the original literature. ..." --The Mathematical Gazette

Author : Neelacanta Sthanumoorthy
Publisher : Academic Press
Page : 512 pages
File Size : 53,5 Mb
Release : 2016-04-26
Category : Mathematics
ISBN : 9780128046838

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Introduction to Finite and Infinite Dimensional Lie (Super)algebras by Neelacanta Sthanumoorthy Pdf

Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras. Discusses the fundamental structure and all root relationships of Lie algebras and Lie superalgebras and their finite and infinite dimensional representation theory Closely describes BKM Lie superalgebras, their different classes of imaginary root systems, their complete classifications, root-supermultiplicities, and related combinatorial identities Includes numerous tables of the properties of individual Lie algebras and Lie superalgebras Focuses on Kac-Moody algebras

Author : Anonim
Publisher : Unknown
Page : 177 pages
File Size : 48,7 Mb
Release : 1982
Category : Electronic
ISBN : OCLC:1088823314

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Algebra VIII by Anonim Pdf

Author : Vlastimil Dlab,Peter Gabriel,Gerhard Michler
Publisher : Springer
Page : 356 pages
File Size : 49,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540397762

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Representation Theory I. Proceedings of the Fourth International Conference on Representations of Algebras, held in Ottawa, Canada, August 16-25, 1984 by Vlastimil Dlab,Peter Gabriel,Gerhard Michler Pdf

Author : Hiroyuki Tachikawa,Vlastimil Dlab,Canadian Mathematical Society
Publisher : American Mathematical Soc.
Page : 350 pages
File Size : 43,9 Mb
Release : 1991
Category : Mathematics
ISBN : 082186016X

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Representations of Finite Dimensional Algebras by Hiroyuki Tachikawa,Vlastimil Dlab,Canadian Mathematical Society Pdf

This volume contains the proceedings of the Tsukuba International Conference on Representations of Algebras and Related Topics (fifth ICRA), held at the University of Tsukuba, August 13-18, 1990. The conference focused on the rapid development of research on representations of finite-dimensional algebras and group representations. A subset of the fifty-seven lectures are collected here, together with a number of other papers not originally presented at the conference. With contributions by some of the world's leading experts in this area, this book provides a valuable overview of the frontier of research in representations of algebras.

Author : Ibrahim Assem,Flávio U. Coelho
Publisher : Springer Nature
Page : 318 pages
File Size : 46,8 Mb
Release : 2020-04-03
Category : Mathematics
ISBN : 9783030351182

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Basic Representation Theory of Algebras by Ibrahim Assem,Flávio U. Coelho Pdf

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.

Author : Vlastimil Dlab,Claus Michael Ringel
Publisher : American Mathematical Soc.
Page : 502 pages
File Size : 51,6 Mb
Release : 2004
Category : Associative algebras
ISBN : 9780821834169

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Representations of Finite Dimensional Algebras and Related Topics in Lie Theory and Geometry by Vlastimil Dlab,Claus Michael Ringel Pdf

These proceedings are from the Tenth International Conference on Representations of Algebras and Related Topics (ICRA X) held at The Fields Institute. In addition to the traditional ``instructional'' workshop preceding the conference, there were also workshops on ``Commutative Algebra, Algebraic Geometry and Representation Theory'', ``Finite Dimensional Algebras, Algebraic Groups and Lie Theory'', and ``Quantum Groups and Hall Algebras''. These workshops reflect the latest developments and the increasing interest in areas that are closely related to the representation theory of finite dimensional associative algebras. Although these workshops were organized separately, their topics are strongly interrelated. The workshop on Commutative Algebra, Algebraic Geometry and Representation Theory surveyed various recently established connections, such as those pertaining to the classification of vector bundles or Cohen-Macaulay modules over Noetherian rings, coherent sheaves on curves, or ideals in Weyl algebras. In addition, methods from algebraic geometry or commutative algebra relating to quiver representations and varieties of modules were presented. The workshop on Finite Dimensional Algebras, Algebraic Groups and Lie Theory surveyed developments in finite dimensional algebras and infinite dimensional Lie theory, especially as the two areas interact and may have future interactions. The workshop on Quantum Groups and Hall Algebras dealt with the different approaches of using the representation theory of quivers (and species) in order to construct quantum groups, working either over finite fields or over the complex numbers. In particular, these proceedings contain a quite detailed outline of the use of perverse sheaves in order to obtain canonical bases. The book is recommended for graduate students and researchers in algebra and geometry.