Representations Of Semisimple Lie Algebras In The Bgg Category O
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Representations Of Semisimple Lie Algebras In The Bgg Category O
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Author : James E. Humphreys
language : en
Publisher: American Mathematical Soc.
Release Date : 2021-07-14
Representations Of Semisimple Lie Algebras In The Bgg Category O written by James E. Humphreys and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-07-14 with Education categories.
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.
Semi Simple Lie Algebras And Their Representations
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Author : Robert N. Cahn
language : en
Publisher: Courier Corporation
Release Date : 2014-06-10
Semi Simple Lie Algebras And Their Representations written by Robert N. Cahn and has been published by Courier Corporation this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-06-10 with Mathematics categories.
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.
Lectures On Real Semisimple Lie Algebras And Their Representations
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Author : A. L. Onishchik
language : en
Publisher: European Mathematical Society
Release Date : 2004
Lectures On Real Semisimple Lie Algebras And Their Representations written by A. L. Onishchik and has been published by European Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2004 with Mathematics categories.
The book begins with a simplified (and somewhat extended and corrected) exposition of the main results of F. Karpelevich's 1955 paper and relates them to the theory of Cartan-Iwahori. It concludes with some tables, where an involution of the Dynkin diagram that allows for finding self-conjugate representations is described and explicit formulas for the index are given. In a short addendum, written by J. V. Silhan, this involution is interpreted in terms of the Satake diagram.
Introduction To Lie Algebras And Representation Theory
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Author : J.E. Humphreys
language : en
Publisher: Springer Science & Business Media
Release Date : 2012-12-06
Introduction To Lie Algebras And Representation Theory written by J.E. Humphreys and has been published by Springer Science & Business Media this book supported file pdf, txt, epub, kindle and other format this book has been release on 2012-12-06 with Mathematics categories.
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.
Lectures On Real Semisimple Lie Algebras And Their Representations
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Author : ARKADY L. ONISHCHIK.
language : en
Publisher:
Release Date :
Lectures On Real Semisimple Lie Algebras And Their Representations written by ARKADY L. ONISHCHIK. and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on with Lie algebras categories.
In 1914, E. Cartan posed the problem to find all irreducible real linear Lie algebras. An updated exposition of his work was given by Iwahori (1959). This theory reduces the classification of irreducible real representations of a real Lie algebra to a description of the so-called self-conjugate irreducible complex representations of this algebra and to the calculation of an invariant of such a representation (with values +1 or -1) which is called the index. Moreover, these two problems were reduced to the case when the Lie algebra is simple and the highest weight of its irreducible complex representation is fundamental. A complete case-by-case classification for all simple real Lie algebras was given (without proof) in the tables of Tits (1967). But actually a general solution of these problems is contained in a paper of Karpelevich (1955) (written in Russian and not widely known), where inclusions between real forms induced by a complex representation were studied. We begin with a simplified (and somewhat extended and corrected) exposition of the main part of this paper and relate it to the theory of Cartan-Iwahori. We conclude with some tables, where an involution of the Dynkin diagram which allows us to find self-conjugate representations is described and explicit formulas for the index are given. In a short addendum, written by J. v. Silhan, this involution is interpreted in terms of the Satake diagram. The book is aimed at students in Lie groups, Lie algebras and their representations, as well as researchers in any field where these theories are used. The reader is supposed to know the classical theory of complex semisimple Lie algebras and their finite dimensional representation; the main facts are presented without proofs in Section 1. In the remaining sections the exposition is made with detailed proofs, including the correspondence between real forms and involutive automorphisms, the Cartan decompositions and the con ...
Connected At Infinity Ii A Selection Of Mathematics By Indians
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Author : Rajendra Bhatia
language : en
Publisher: Springer
Release Date : 2013-01-01
Connected At Infinity Ii A Selection Of Mathematics By Indians written by Rajendra Bhatia and has been published by Springer this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-01-01 with Mathematics categories.
Complex Semisimple Quantum Groups And Representation Theory
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Author : Christian Voigt
language : en
Publisher: Springer Nature
Release Date : 2020-09-24
Complex Semisimple Quantum Groups And Representation Theory written by Christian Voigt and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-09-24 with Mathematics categories.
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.
Advances In Lie Superalgebras
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Author : Maria Gorelik
language : en
Publisher: Springer Science & Business
Release Date : 2014-04-28
Advances In Lie Superalgebras written by Maria Gorelik and has been published by Springer Science & Business this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-04-28 with Mathematics categories.
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.
Semisimple Lie Algebras
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Author : Morikuni Goto
language : en
Publisher: CRC Press
Release Date : 2020-12-17
Semisimple Lie Algebras written by Morikuni Goto and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-17 with Mathematics categories.
This book provides an account of part of the theory of Lie algebras most relevant to Lie groups. It discusses the basic theory of Lie algebras, including the classification of complex semisimple Lie algebras, and the Levi, Cartan and Iwasawa decompositions.
Semisimple Lie Algebras
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Author : Morikuni Goto
language : en
Publisher: CRC Press
Release Date : 2020-12-18
Semisimple Lie Algebras written by Morikuni Goto and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-12-18 with Mathematics categories.
This book provides an account of part of the theory of Lie algebras most relevant to Lie groups. It discusses the basic theory of Lie algebras, including the classification of complex semisimple Lie algebras, and the Levi, Cartan and Iwasawa decompositions.