Lie Groups Beyond An Introduction

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Lie Groups Beyond an Introduction

Anthony W. Knapp
Lie Groups Beyond an Introduction Book PDF

Author : Anthony W. Knapp
Publisher : Springer Science & Business Media
Page : 622 pages
File Size : 53,9 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475724530

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Lie Groups Beyond an Introduction by Anthony W. Knapp Pdf

Lie Groups Beyond an Introduction takes the reader from the end of introductory Lie group theory to the threshold of infinite-dimensional group representations. Merging algebra and analysis throughout, the author uses Lie-theoretic methods to develop a beautiful theory having wide applications in mathematics and physics. A feature of the presentation is that it encourages the reader's comprehension of Lie group theory to evolve from beginner to expert: initial insights make use of actual matrices, while later insights come from such structural features as properties of root systems, or relationships among subgroups, or patterns among different subgroups.

An Introduction to Lie Groups and Lie Algebras

Alexander A. Kirillov
An Introduction to Lie Groups and Lie Algebras Book PDF

Author : Alexander A. Kirillov
Publisher : Cambridge University Press
Page : 237 pages
File Size : 51,8 Mb
Release : 2008-07-31
Category : Mathematics
ISBN : 9780521889698

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An Introduction to Lie Groups and Lie Algebras by Alexander A. Kirillov Pdf

Contemporary introduction to semisimple Lie algebras; concise and informal, with numerous exercises and examples

Complex Semisimple Lie Algebras

Jean-Pierre Serre
Complex Semisimple Lie Algebras Book PDF

Author : Jean-Pierre Serre
Publisher : Springer Science & Business Media
Page : 82 pages
File Size : 49,9 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781475739107

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Complex Semisimple Lie Algebras by Jean-Pierre Serre Pdf

These notes are a record of a course given in Algiers from lOth to 21st May, 1965. Their contents are as follows. The first two chapters are a summary, without proofs, of the general properties of nilpotent, solvable, and semisimple Lie algebras. These are well-known results, for which the reader can refer to, for example, Chapter I of Bourbaki or my Harvard notes. The theory of complex semisimple algebras occupies Chapters III and IV. The proofs of the main theorems are essentially complete; however, I have also found it useful to mention some complementary results without proof. These are indicated by an asterisk, and the proofs can be found in Bourbaki, Groupes et Algebres de Lie, Paris, Hermann, 1960-1975, Chapters IV-VIII. A final chapter shows, without proof, how to pass from Lie algebras to Lie groups (complex-and also compact). It is just an introduction, aimed at guiding the reader towards the topology of Lie groups and the theory of algebraic groups. I am happy to thank MM. Pierre Gigord and Daniel Lehmann, who wrote up a first draft of these notes, and also Mlle. Franr,:oise Pecha who was responsible for the typing of the manuscript.

Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics

Josi A. de Azcárraga,Josi M. Izquierdo
Lie Groups Lie Algebras Cohomology and Some Applications in Physics Book PDF

Author : Josi A. de Azcárraga,Josi M. Izquierdo
Publisher : Cambridge University Press
Page : 480 pages
File Size : 44,7 Mb
Release : 1998-08-06
Category : Mathematics
ISBN : 0521597005

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Lie Groups, Lie Algebras, Cohomology and Some Applications in Physics by Josi A. de Azcárraga,Josi M. Izquierdo Pdf

A self-contained introduction to the cohomology theory of Lie groups and some of its applications in physics.

Introduction to Lie Algebras and Representation Theory

J.E. Humphreys
Introduction to Lie Algebras and Representation Theory Book PDF

Author : J.E. Humphreys
Publisher : Springer Science & Business Media
Page : 189 pages
File Size : 54,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461263982

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Introduction to Lie Algebras and Representation Theory by J.E. Humphreys Pdf

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.

Naive Lie Theory

John Stillwell
Naive Lie Theory Book PDF

Author : John Stillwell
Publisher : Springer Science & Business Media
Page : 230 pages
File Size : 52,7 Mb
Release : 2008-12-15
Category : Mathematics
ISBN : 9780387782157

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Naive Lie Theory by John Stillwell Pdf

In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra. This naive approach to Lie theory is originally due to von Neumann, and it is now possible to streamline it by using standard results of undergraduate mathematics. To compensate for the limitations of the naive approach, end of chapter discussions introduce important results beyond those proved in the book, as part of an informal sketch of Lie theory and its history. John Stillwell is Professor of Mathematics at the University of San Francisco. He is the author of several highly regarded books published by Springer, including The Four Pillars of Geometry (2005), Elements of Number Theory (2003), Mathematics and Its History (Second Edition, 2002), Numbers and Geometry (1998) and Elements of Algebra (1994).

Lie Groups

Harriet Suzanne Katcher Pollatsek
Lie Groups Book PDF

Author : Harriet Suzanne Katcher Pollatsek
Publisher : MAA
Page : 194 pages
File Size : 51,8 Mb
Release : 2009-09-24
Category : Mathematics
ISBN : 0883857596

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Lie Groups by Harriet Suzanne Katcher Pollatsek Pdf

This textbook is a complete introduction to Lie groups for undergraduate students. The only prerequisites are multi-variable calculus and linear algebra. The emphasis is placed on the algebraic ideas, with just enough analysis to define the tangent space and the differential and to make sense of the exponential map. This textbook works on the principle that students learn best when they are actively engaged. To this end nearly 200 problems are included in the text, ranging from the routine to the challenging level. Every chapter has a section called 'Putting the pieces together' in which all definitions and results are collected for reference and further reading is suggested.

Differential Geometry and Lie Groups for Physicists

Marián Fecko
Differential Geometry and Lie Groups for Physicists Book PDF

Author : Marián Fecko
Publisher : Cambridge University Press
Page : 11 pages
File Size : 42,9 Mb
Release : 2006-10-12
Category : Science
ISBN : 9781139458030

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Differential Geometry and Lie Groups for Physicists by Marián Fecko Pdf

Covering subjects including manifolds, tensor fields, spinors, and differential forms, this textbook introduces geometrical topics useful in modern theoretical physics and mathematics. It develops understanding through over 1000 short exercises, and is suitable for advanced undergraduate or graduate courses in physics, mathematics and engineering.

Representations of Semisimple Lie Algebras in the BGG Category O

James E. Humphreys
Representations of Semisimple Lie Algebras in the BGG Category O Book PDF

Author : James E. Humphreys
Publisher : American Mathematical Soc.
Page : 289 pages
File Size : 44,8 Mb
Release : 2021-07-14
Category : Education
ISBN : 9781470463267

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Representations of Semisimple Lie Algebras in the BGG Category O by James E. Humphreys Pdf

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.

Analysis on Lie Groups with Polynomial Growth

Nick Dungey,A.F.M. (Tom) ter Elst,Derek William Robinson
Analysis on Lie Groups with Polynomial Growth Book PDF

Author : Nick Dungey,A.F.M. (Tom) ter Elst,Derek William Robinson
Publisher : Springer Science & Business Media
Page : 315 pages
File Size : 54,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461220626

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Analysis on Lie Groups with Polynomial Growth by Nick Dungey,A.F.M. (Tom) ter Elst,Derek William Robinson Pdf

Analysis on Lie Groups with Polynomial Growth is the first book to present a method for examining the surprising connection between invariant differential operators and almost periodic operators on a suitable nilpotent Lie group. It deals with the theory of second-order, right invariant, elliptic operators on a large class of manifolds: Lie groups with polynomial growth. In systematically developing the analytic and algebraic background on Lie groups with polynomial growth, it is possible to describe the large time behavior for the semigroup generated by a complex second-order operator with the aid of homogenization theory and to present an asymptotic expansion. Further, the text goes beyond the classical homogenization theory by converting an analytical problem into an algebraic one. This work is aimed at graduate students as well as researchers in the above areas. Prerequisites include knowledge of basic results from semigroup theory and Lie group theory.

Group Theory

Predrag Cvitanović
Group Theory Book PDF

Author : Predrag Cvitanović
Publisher : Princeton University Press
Page : 278 pages
File Size : 46,9 Mb
Release : 2020-05-26
Category : Mathematics
ISBN : 9780691202983

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Group Theory by Predrag Cvitanović Pdf

If classical Lie groups preserve bilinear vector norms, what Lie groups preserve trilinear, quadrilinear, and higher order invariants? Answering this question from a fresh and original perspective, Predrag Cvitanovic takes the reader on the amazing, four-thousand-diagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional. The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, "birdtracks" notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as "negative dimensional" relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.

Introduction to Lie Algebras

K. Erdmann,Mark J. Wildon
Introduction to Lie Algebras Book PDF

Author : K. Erdmann,Mark J. Wildon
Publisher : Springer Science & Business Media
Page : 251 pages
File Size : 50,9 Mb
Release : 2006-09-28
Category : Mathematics
ISBN : 9781846284908

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Introduction to Lie Algebras by K. Erdmann,Mark J. Wildon Pdf

Lie groups and Lie algebras have become essential to many parts of mathematics and theoretical physics, with Lie algebras a central object of interest in their own right. This book provides an elementary introduction to Lie algebras based on a lecture course given to fourth-year undergraduates. The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality. Numerous worked examples and exercises are provided to test understanding, along with more demanding problems, several of which have solutions. Introduction to Lie Algebras covers the core material required for almost all other work in Lie theory and provides a self-study guide suitable for undergraduate students in their final year and graduate students and researchers in mathematics and theoretical physics.

Lie Groups, Lie Algebras, and Representations

Brian C. Hall
Lie Groups Lie Algebras and Representations Book PDF

Author : Brian C. Hall
Publisher : Springer Science & Business Media
Page : 376 pages
File Size : 42,5 Mb
Release : 2003-08-07
Category : Mathematics
ISBN : 0387401229

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Lie Groups, Lie Algebras, and Representations by Brian C. Hall Pdf

This book provides an introduction to Lie groups, Lie algebras, and repre sentation theory, aimed at graduate students in mathematics and physics. Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that I hope will make it a useful addition to the literature. First, it treats Lie groups (not just Lie alge bras) in a way that minimizes the amount of manifold theory needed. Thus, I neither assume a prior course on differentiable manifolds nor provide a con densed such course in the beginning chapters. Second, this book provides a gentle introduction to the machinery of semi simple groups and Lie algebras by treating the representation theory of SU(2) and SU(3) in detail before going to the general case. This allows the reader to see roots, weights, and the Weyl group "in action" in simple cases before confronting the general theory. The standard books on Lie theory begin immediately with the general case: a smooth manifold that is also a group. The Lie algebra is then defined as the space of left-invariant vector fields and the exponential mapping is defined in terms of the flow along such vector fields. This approach is undoubtedly the right one in the long run, but it is rather abstract for a reader encountering such things for the first time.

Lie Groups

Wulf Rossmann
Lie Groups Book PDF

Author : Wulf Rossmann
Publisher : Oxford University Press, USA
Page : 290 pages
File Size : 55,7 Mb
Release : 2006
Category : Business & Economics
ISBN : 0199202516

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Lie Groups by Wulf Rossmann Pdf

This book is an introduction to the theory of Lie groups and their representations at the advanced undergraduate or beginning graduate level. It covers the essentials of the subject starting from basic undergraduate mathematics. The correspondence between linear Lie groups and Lie algebras is developed in its local and global aspects. The classical groups are analyzed in detail, first with elementary matrix methods, then with the help of the structural tools typical of the theory of semisimple groups, such as Cartan subgroups, root, weights and reflections. The fundamental groups of the classical groups are worked out as an application of these methods. Manifolds are introduced when needed, in connection with homogeneous spaces, and the elements of differential and integral calculus on manifolds are presented, with special emphasis on integration on groups and homogeneous spaces. Representation theory starts from first principles, such as Schur's lemma and its consequences, and proceeds from there to the Peter-Weyl theorem, Weyl's character formula, and the Borel-Weil theorem, all in the context of linear groups.

Compact Lie Groups

Mark R. Sepanski
Compact Lie Groups Book PDF

Author : Mark R. Sepanski
Publisher : Springer Science & Business Media
Page : 208 pages
File Size : 49,7 Mb
Release : 2007-04-05
Category : Mathematics
ISBN : 9780387491585

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Compact Lie Groups by Mark R. Sepanski Pdf

Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. Coverage includes the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. The book develops the necessary Lie algebra theory with a streamlined approach focusing on linear Lie groups.