Representations Of Semisimple Lie Algebras In The Bgg Category O PDF Download
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Author | : James E. Humphreys |
Publisher | : American Mathematical Soc. |
Total Pages | : 289 |
Release | : 2021-07-14 |
Genre | : Education |
ISBN | : 1470463261 |
Download Representations of Semisimple Lie Algebras in the BGG Category O Book in PDF, ePub and Kindle
This is the first textbook treatment of work leading to the landmark 1979 Kazhdan–Lusztig Conjecture on characters of simple highest weight modules for a semisimple Lie algebra g g over C C. The setting is the module category O O introduced by Bernstein–Gelfand–Gelfand, which includes all highest weight modules for g g such as Verma modules and finite dimensional simple modules. Analogues of this category have become influential in many areas of representation theory. Part I can be used as a text for independent study or for a mid-level one semester graduate course; it includes exercises and examples. The main prerequisite is familiarity with the structure theory of g g. Basic techniques in category O O such as BGG Reciprocity and Jantzen's translation functors are developed, culminating in an overview of the proof of the Kazhdan–Lusztig Conjecture (due to Beilinson–Bernstein and Brylinski–Kashiwara). The full proof however is beyond the scope of this book, requiring deep geometric methods: D D-modules and perverse sheaves on the flag variety. Part II introduces closely related topics important in current research: parabolic category O O, projective functors, tilting modules, twisting and completion functors, and Koszul duality theorem of Beilinson–Ginzburg–Soergel.
Author | : J.E. Humphreys |
Publisher | : Springer Science & Business Media |
Total Pages | : 189 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 1461263980 |
Download Introduction to Lie Algebras and Representation Theory Book in PDF, ePub and Kindle
This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.
Author | : Diximier |
Publisher | : Newnes |
Total Pages | : 393 |
Release | : 2009-02-10 |
Genre | : Business & Economics |
ISBN | : 0444110771 |
Download Enveloping Algebras Book in PDF, ePub and Kindle
Enveloping Algebras
Author | : Robert N. Cahn |
Publisher | : Courier Corporation |
Total Pages | : 180 |
Release | : 2014-06-10 |
Genre | : Mathematics |
ISBN | : 0486150313 |
Download Semi-Simple Lie Algebras and Their Representations Book in PDF, ePub and Kindle
Designed to acquaint students of particle physiME already familiar with SU(2) and SU(3) with techniques applicable to all simple Lie algebras, this text is especially suited to the study of grand unification theories. Author Robert N. Cahn, who is affiliated with the Lawrence Berkeley National Laboratory in Berkeley, California, has provided a new preface for this edition. Subjects include the killing form, the structure of simple Lie algebras and their representations, simple roots and the Cartan matrix, the classical Lie algebras, and the exceptional Lie algebras. Additional topiME include Casimir operators and Freudenthal's formula, the Weyl group, Weyl's dimension formula, reducing product representations, subalgebras, and branching rules. 1984 edition.
Author | : Christian Voigt |
Publisher | : Springer Nature |
Total Pages | : 382 |
Release | : 2020-09-24 |
Genre | : Mathematics |
ISBN | : 3030524639 |
Download Complex Semisimple Quantum Groups and Representation Theory Book in PDF, ePub and Kindle
This book provides a thorough introduction to the theory of complex semisimple quantum groups, that is, Drinfeld doubles of q-deformations of compact semisimple Lie groups. The presentation is comprehensive, beginning with background information on Hopf algebras, and ending with the classification of admissible representations of the q-deformation of a complex semisimple Lie group. The main components are: - a thorough introduction to quantized universal enveloping algebras over general base fields and generic deformation parameters, including finite dimensional representation theory, the Poincaré-Birkhoff-Witt Theorem, the locally finite part, and the Harish-Chandra homomorphism, - the analytic theory of quantized complex semisimple Lie groups in terms of quantized algebras of functions and their duals, - algebraic representation theory in terms of category O, and - analytic representation theory of quantized complex semisimple groups. Given its scope, the book will be a valuable resource for both graduate students and researchers in the area of quantum groups.
Author | : Rajendra Bhatia |
Publisher | : Springer |
Total Pages | : 193 |
Release | : 2013-01-01 |
Genre | : Mathematics |
ISBN | : 9386279568 |
Download Connected at infinity II: a selection of mathematics by Indians Book in PDF, ePub and Kindle
Author | : Ben Elias |
Publisher | : Springer Nature |
Total Pages | : 588 |
Release | : 2020-09-26 |
Genre | : Mathematics |
ISBN | : 3030488268 |
Download Introduction to Soergel Bimodules Book in PDF, ePub and Kindle
This book provides a comprehensive introduction to Soergel bimodules. First introduced by Wolfgang Soergel in the early 1990s, they have since become a powerful tool in geometric representation theory. On the one hand, these bimodules are fairly elementary objects and explicit calculations are possible. On the other, they have deep connections to Lie theory and geometry. Taking these two aspects together, they offer a wonderful primer on geometric representation theory. In this book the reader is introduced to the theory through a series of lectures, which range from the basics, all the way to the latest frontiers of research. This book serves both as an introduction and as a reference guide to the theory of Soergel bimodules. Thus it is intended for anyone who wants to learn about this exciting field, from graduate students to experienced researchers.
Author | : Maria Gorelik |
Publisher | : Springer Science & Business |
Total Pages | : 281 |
Release | : 2014-04-28 |
Genre | : Mathematics |
ISBN | : 3319029525 |
Download Advances in Lie Superalgebras Book in PDF, ePub and Kindle
The volume is the outcome of the conference "Lie superalgebras," which was held at the Istituto Nazionale di Alta Matematica, in 2012. The conference gathered many specialists in the subject, and the talks held provided comprehensive insights into the newest trends in research on Lie superalgebras (and related topics like vertex algebras, representation theory and supergeometry). The book contains contributions of many leading esperts in the field and provides a complete account of the newest trends in research on Lie Superalgebras.
Author | : Henning Krause |
Publisher | : Cambridge University Press |
Total Pages | : 518 |
Release | : 2021-11-18 |
Genre | : Mathematics |
ISBN | : 1108985815 |
Download Homological Theory of Representations Book in PDF, ePub and Kindle
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.
Author | : Erhard Neher |
Publisher | : American Mathematical Soc. |
Total Pages | : 226 |
Release | : 2011 |
Genre | : Mathematics |
ISBN | : 082185237X |
Download Geometric Representation Theory and Extended Affine Lie Algebras Book in PDF, ePub and Kindle
Lie theory has connections to many other disciplines such as geometry, number theory, mathematical physics, and algebraic combinatorics. The interaction between algebra, geometry and combinatorics has proven to be extremely powerful in shedding new light on each of these areas. This book presents the lectures given at the Fields Institute Summer School on Geometric Representation Theory and Extended Affine Lie Algebras held at the University of Ottawa in 2009. It provides a systematic account by experts of some of the exciting developments in Lie algebras and representation theory in the last two decades. It includes topics such as geometric realizations of irreducible representations in three different approaches, combinatorics and geometry of canonical and crystal bases, finite $W$-algebras arising as the quantization of the transversal slice to a nilpotent orbit, structure theory of extended affine Lie algebras, and representation theory of affine Lie algebras at level zero. This book will be of interest to mathematicians working in Lie algebras and to graduate students interested in learning the basic ideas of some very active research directions. The extensive references in the book will be helpful to guide non-experts to the original sources.