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This is the second volume by the author, presenting the state of the art of the structure and classification of Lie algebras over fields of positive characteristic, an important topic in algebra. The contents is leading to the forefront of current research in this field. Leading to the forefront of current research in an important topic of algebra. ... Read more

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Through the 1990s, a circle of ideas emerged relating three very different kinds of objects associated to a complex semisimple Lie algebra: nilpotent orbits, representations of a Weyl group, and primitive ideals in an enveloping algebra. The principal aim of this book is to collect together the important results concerning the classification and properties of nilpotent orbits, beginning from the common ground of basic structure theory. The techniques used are elementary and in the toolkit of any graduate student interested in the harmonic analysis of representation theory of Lie groups. The book develops the Dynkin-Konstant and Bala-Carter classifications of complex nilpotent orbits, derives the Lusztig-Spaltenstein theory of induction of nilpotent orbits, discusses basic topological questions, and classifies real nilpotent orbits. The classical algebras are emphasized throughout; here the theory can be simplified by using the combinatorics of partitions and tableaux. The authors conclude with a survey of advanced topics related to the above circle of ideas. This book is the product of a two-quarter course taught at the University of Washington. ... Read more

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This textbook comprehensively introduces students and researchers to the application of continuous symmetries and their Lie algebras to ordinary and partial differential equations. Covering all the modern techniques in detail, it relates applications to cutting-edge research fields such as Yang Mills theory and string theory.Aimed at readers in applied mathematics and physics rather than pure mathematics, the material is ideally suited to students and researchers whose main interest lies in finding solutions to differential equations and invariants of maps.A large number of worked examples and challenging exercises help readers to work independently of teachers, and by including SymbolicC++ implementations of the techniques in each chapter, the book takes full advantage of the advancements in algebraic computation.Twelve new sections have been added in this edition, including: Haar measure, Sato's theory and sigma functions, universal algebra, anti-self dual Yang Mills equation, and discrete Painlevé equations. ... Read more

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The volume is almost entirely composed of the research andexpository papers by the participants of the International Workshop"Groups, Rings, Lie and Hopf Algebras", which was held at the MemorialUniversity of Newfoundland, St. John's, NF, Canada. All four areasfrom the title of the workshop are covered. In addition, some chapterstouch upon the topics, which belong to two or more areas at the sametime. Audience: The readership targeted includes researchers, graduateand senior undergraduate students in mathematics and its applications. ... Read more

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From the reviews: "..., the book must be of great help for a researcher who already has some idea of Lie theory, wants to employ it in his everyday research and/or teaching, and needs a source for customary reference on the subject. From my viewpoint, the volume is perfectly fit to serve as such a source, ... On the whole, it is quite a pleasure, after making yourself comfortable in that favourite office armchair of yours, just to keep the volume gently in your hands and browse it slowly and thoughtfully; and after all, what more on Earth can one expect of any book?" The New Zealand Mathematical Society Newsletter

"... Both parts are very nicely written and can be strongly recommended." European Mathematical Society ... Read more

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From the reviews of the French edition

"This is a rich and useful volume. The material it treats has relevance well beyond the theory of Lie groups and algebras, ranging from the geometry of regular polytopes and paving problems to current work on finite simple groups having a (B,N)-pair structure, or "Tits systems". A historical note provides a survey of the contexts in which groups generated by reflections have arisen. A brief introduction includes almost the only other mention of Lie groups and algebras to be found in the volume. Thus the presentation here is really quite independent of Lie theory. The choice of such an approach makes for an elegant, self-contained treatment of some highly interesting mathematics, which can be read with profit and with relative ease by a very wide circle of readers (and with delight by many, if the reviewer is at all representative)." (G.B. Seligman in MathReviews)

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Describing many of the most important aspects of Lie group theory, this book presents the subject in a 'hands on' way. Rather than concentrating on theorems and proofs, the book shows the applications of the material to physical sciences and applied mathematics. Many examples of Lie groups and Lie algebras are given throughout the text. The relation between Lie group theory and algorithms for solving ordinary differential equations is presented and shown to be analogous to the relation between Galois groups and algorithms for solving polynomial equations. Other chapters are devoted to differential geometry, relativity, electrodynamics, and the hydrogen atom. Problems are given at the end of each chapter so readers can monitor their understanding of the materials. This is a fascinating introduction to Lie groups for graduate and undergraduate students in physics, mathematics and electrical engineering, as well as researchers in these fields. ... Read more

Customer Reviews (2)

Very competent and realistic introduction
This book is intended as an introduction to the topic for students in physical and chemical disciplines. It should not be thought that this book is an abbreviated version of the previous one. The structure of this text is radically different from the 1974 book, which was more a compendium of group theoretical techniques, and presented very actual topics used in physics. This book, preserving the essential motivations, has been written to develop, step by step, the techniques and methods used when groups are applied to describe physical phenomena, with details and explanations that are usually omitted in most textbooks.

The book consists of sixteen chapters containing a large number of problems to be worked out by the reader. The results are presented in a very direct way, avoiding too technical developments and extracting the main facts. This philosophy is very convenient at a first level, because it focuses on the most important points and does not confuse the reader with involved proofs.

In the first chapter, the author presents the historical motivation that led S. Lie to develop the theory of continuous groups, the Galois theory. This serves mostly to motivate the study of Lie groups, but presents no specific interest to the physicist. Chapters 2 and 3 are devoted to the main properties of matrix Lie groups, which are the main object of study in this book and correspond to the types usually encountered in applications. In this sense, the different classical groups are presented as those subjected to different constraints, motivating the geometrical interpretation of these groups.
In chapter 4 the discussion of Lie algebras begins. First of all, it is illustrated why these structures serve to simplify a lot the analysis, since Lie algebras correspond to the linear approximation to the group at a given point. The exponentiation map is shortly introduced, without going yet into more involved questions like the local determination of the group from the Lie algebra. Important facts like the adjoint representation, the Killing and invariant metrics are introduced. This leads to a first insight into the structure of Lie algebras. This analysis continues in chapter 5, where the Lie algebras of the classical groups are derived from the corresponding constraints. The role played by the Killing form is studied in these examples, constituting a first approximation to the well known characterization of semisimple algebras. Chapter six is devoted to the usual techniques to deal with Lie algebras in physical applications, namely, the realizations by creation and annihilation operators and the realizations by vector fields. Although a very short section, the problems illustrate important topics like the angular momentum by means of Schwinger representations used in Quantum Mechanics. The seventh chapter reconsiders the problem of exponentiation in a more technical way. The limitations of the procedure and the isomorphism problem are developed having in mind the important su(2) case. The main result, the covering theorem, is presented graphically, illustrating quite well the general pattern of the theory. The Campbell-Hausdorff formula is introduced motivated by the non-trivial reparameterization problem. The informal way chosen to present this deep result is quite adequate, since it focuses on the meaning of the theorem instead of presenting a technical proof that it not trivial. Once the basic material has been presented, chapter 8 begins with the systematic study of the structure of Lie algebras. The main types of algebras, abelian, nilpotent, solvable, simple and semisimple are defined using the properties of the adjoint representation. Although not explicitely stated, this corresponds actually to the Levi decomposition. One important point should be clarified here: in section 2.3, the "canonical" form of solvable algebras is presented, according to the well known flag space technique of the Lie theorem. However, upper (respectively lower) triangular matrices are the model for solvable Lie algebras only for the complex base field (the Lie theorem being false in general for real solvable Lie algebras). At no point this crucial point is mentioned, which could lead to confusion to the non-expert. Chapters nine and ten concentrate on the classification problem of complex semsimple Lie algebras. This part is a shortened version of the material contained in the previous book of the author, presenting only the indispensable facts. The graphics of root systems help a lot to understand the general situation and the motivation of the classification of Dynkin diagrams. I miss however some comments on the Cartan matrix, which is the natural link between the (fundamental) roots and the corresponding diagram. The next chapter focuses on the real forms of simple complex Lie algebras. The main idea of its obtention is studied, as well as the main steps of the Cartan method to determine the non-equivalent real forms. The material of this section is crucial for applications, since many important models are based on non-compact Lie algebras. Being a quite delicate question, I agree with the author in the decision of leaving out the notions of inner and outer involutive endomorphisms used in their classification.
These first eleven chapters cover the main facts about Lie theory that any student in either physics or chemistry should master for a full comprehension of more technical. Chapter 12 reviews Riemannian symmetric spaces, a very important type of manifolds. Here the geometrical role of the exponentiation map is exploited, helping to understand the implications of the choice of real form and its consequences in the geometry and topology of the corresponding manifold. The material is again presented and commented using important examples, instead of developing cumbersome theoretical argumentations, which can be found in the cited literature. The results are complemented by carefully chosen problems of physical nature, pointing out the relevance of symmetric spaces in applications. Chapter 13 introduces a more sophisticated technique, the contractions of Lie groups. This procedure, of essential importance in physics, is developed following the classical method of Inönü and Wigner. How to use contractions in limiting processes of other objects is illustrated in the different sections. However, I believe that focusing only on Inönü-Wigner contractions gives a quite restrictive view of this technique (even if this constitutes a very important class of contractions, as shown by their applications to symmetry breaking).
Chapter 14 constitutes an introduction to the study of symmetry in physical systems. This is an important part, since many textbooks usually assume the reader is aware of the different notions of symmetries used. To this extent, the author chooses a classical and vital example, the hydrogen atom. The different types of symmetry (geometrical, dynamical, spectrum generating algebra) not only point out the different physical properties to be described by means of symmetry, but also the importance of how to embed a Lie group into another. The detailed description made by the author will surely clarify some aspects that are generally quickly reviewed, and therefore constitute a difficulty for the unexperienced reader.
The Maxwell equations are derived in chapter 15 using the properties of two fundamental groups in Physics: the Lorentz group SO(1,3) and the Poincaré group. Although it may appear that this chapter is disconnected from the rest, it actually has been placed in the right place. On one hand, the Maxwell equations are connected to the most important physical groups,.and further, these are closely related to the conformal group previously introduced, being a natural link to justify the importance of symmetries of differential equations.
The last chapter connects with the first in the sense that Lie groups are used to determine whether a differential equation can be solved by quadratures or not. Since this is a large and complicated theory, only the basic elements that show how Lie groups are used to simplify the integration of differential equations are presented.

This book constitutes a very comprehensive introduction to Lie theory in physics, dealing with the most important features that students will encounter. The problems help not only to understand the material presented, but also exhibit real physical situations where Lie groups are usedThis book further solves some difficulties encountered by beginners in other books, usually written at a more specialized level.

Best introductory text on Lie Groups so far.
I am so happy with this book that I could not wait to finish one chapter and then post a review. This is my initial review but maybe I will extend it.

I am a theoretical physics studentand so far I have read one section on lie algebra and the approach is very clear and not at all In mathematical language and notations ( that you are required to master first to understand the underlying mathematics generally).

Robert Gilmore done a very good job on this introductory book which fits with the title. He explains the ideas in very clear and concise way for non mathematical students. First he explained lie groups briefly and then came to lie algebra and explain why this is done. All most all authors forget to mention why they introduced lie algebra. I have many other books on group theory and lie groups e.g.Sternberg, Fuchs & Schweigert , Wu-Ki Tung, Georgi etcand the main point to be noted is that many authors do a good job explaining " how" but they forget to mention "why". This is where Robert Gilmore comes in.It is a pity that he did not write a book on Group theory as a whole including other topics in group theory as well.

My advice is if you need an introduction to lie groups and lie algebra and tired of authors who only try to impress other authors instead of the student then invest on this book.You won't be disappointed and maybe this one goes into your collection.

PS: for student of particle physics, try also Lie algebras from Howard Georgi.
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In 1914, E. Cartan posed the problem of finding all irreducible real linear Lie algebras. Iwahori gave an updated exposition of Cartan's work in 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 in the tables of Tits (1967). But actually a general solution of these problems is contained in a paper of Karpelevich (1955) that was written in Russian and not widely known.

The book begins with a simplified (and somewhat extended and corrected) exposition of the main results of Karpelevich's 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.

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. Readers should 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 conjugacy of maximal compact subgroups of the automorphism group.

Published by the European Mathematical Society and distributed within the Americas by the American Mathematical Society. ... Read more

Customer Reviews (1)

An excellent introduction to the topic of real simple Lie algebras
The main purpose of these notes is to give a self-contained and complete exposition of the representation theory of real semisimple Lie algebras. Although various texts on the topic exist, the originality of this small book is the elegance in the exposition and the presentation of some important facts that are absent in other treatises or only enumerated without further comment. Written by a prestigious expert in Lie theory, the text only demands a standard knowledge in the theory of complex Lie algebras and groups, and constitutes therefore an excellent text as a complement to an advanced course on the classification of complex semisimple Lie algebras and their representation theory.

The problem of classifying real simple and semisimple Lie algebras and their representations arises from the geometry of homogeneous spaces, and the first results in this direction were developed by E. Cartan himself in 1914. Using the more standardized algebraic theory and the work of Weyl, the study of real simple Lie algebras and groups was later expanded by various authors in order to develop a self-contained theory in analogy to the complex case. This work accomplishes this objective perfectly, and also pays homage to the important work of the late Fridrikh Izrailevich Karpelevich , who already solved many problems in the representation theory of real simple Lie algebras. However, these papers are unfortunately not widely known in the literature, and various of his results were later rediscovered by other authors.

The text is divided into nine sections, which present the main results with detailed proofs and illustrated with examples using the special simple algebra sl(n,C). The choice of this algebra is justified by the role it plays in the characterization of self-dual complex irreducible representations of real forms. For the remaining algebras the reader is led to the references.

The first section reviews the classical theory of semisimple complex Lie algebras, and fixes the notation that will be used in later chapters. The main material on compact groups that will be applied in the obtainment of real forms is also briefly presented, such as the theorem of Weyl. As recopilatory chapter, no proofs are given at this stage.
The second section deals with the complexification and realification of real and complex Lie algebras, respectively. Two important examples of real forms of complex semisimple Lie algebras are introduced: the real normal form, which can intuitively be interpreted as the algebra obtained by restriction of scalarsand the compact form, which will be central for the construction of the remaining non-compact real forms. The first structural results concerning real forms are presented, namely, that real forms of simple complex algebras are simple, while complexification of real simple algebras are either simple or semisimple complex algebras [the insertion of the classical Lorentz algebra would have been welcomed after example 4]. The third section introduces the main tool used in the classification of real forms, the involutive automorphisms of a complex semisimple algebra and its correspondence with the real forms. It follows in particular that the compact form is unique. In order to describe this correspondence, the next section is devoted to various technical results concerning the automorphisms of complex semisimple algebras. Endowed with this machinery, the Cartan decomposition is discussed in detail. The conjugacy theorem of maximal compact subgroups of the adjoint linear group Int(g) is proved. Section 6 is devoted to an important problem which often appears in representation theory: given a homomorphism of complex semisimple Lie algebras f: ĝ →ĥ, which real forms of ĥ contain the image by f of some real form of ĝ? A satisfactory answer to this problem is given by means of the involutive automorphisms corresponding to the real forms. The material of this section follows the original work of F. I. Karpelevich in the beginning fifties . The material previously developed in chapter 3 concerning hermitean vector spaces is applied. Introducing a special class of morphisms, denoted S-homomorphisms , the result is sharpened. The seventh section devotes to the extension problem for irreducible representations for the case of the special linear Lie algebra sl(n,R). Special attention is devoted to the Karpelevich index, and the original formulae for computing this invariant are generalized to arbitrary involutive automorphisms. These results are applied in section 8 to classify explicitly the irreducible real representations in terms of highest weights, following the outline used by Iwahori in 1959. More precisely, real representations divide into two classes depending on the existence or not of an invariant complex structure. The last section, written by J. ?ilhan, presents an alternative classification by means of Satake diagrams, i.e., a generalization of the classical Dynkin diagram based on the introduction of two colors and arrows relating vertices of one color. It is described how to read off the involutions using these diagrams, and a characterization of self-dual complex irreducible representations is obtained. Additional material is presented in tabulated form at the end of this section.

Resuming, this book is very welcome reference on real simple Lie algebras, and has the innovation of presenting material that is distributed in many technical papers in a compact and effective way. It should be expected that this work will become a classic on the topic among the specialists in Lie algebras. ... Read more

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This book deals with central simple Lie algebras over arbitrary fields of characteristic zero. It aims to give constructions of the algebras and their finite-dimensional modules in terms that are rational with respect to the given ground field. All isotropic algebras with non-reduced relative root systems are treated, along with classical anisotropic algebras. The latter are treated by what seems to be a novel device, namely by studying certain modules for isotropic classical algebras in which they are embedded. In this development, symmetric powers of central simple associative algebras, along with generalized even Clifford algebras of involutorial algebras, play central roles. Considerable attention is given to exceptional algebras. The pace is that of a rather expansive research monograph. The reader who has at hand a standard introductory text on Lie algebras, such as Jacobson or Humphreys, should be in a position to understand the results. More technical matters arise in some of the detailed arguments. The book is intended for researchers and students of algebraic Lie theory, as well as for other researchers who are seeking explicit realizations of algebras or modules. It will probably be more useful as a resource to be dipped into, than as a text to be worked straight through. ... Read more

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Imparts a self-contained development of the algebraic theory of Kac-Moody algebras, their representations and close relatives—the Virasoro and Heisenberg algebras. Focuses on developing the theory of triangular decompositions and part of the Kac-Moody theory not specific to the affine case. Also covers lattices, and finite root systems, infinite-dimensional theory, Weyl groups and conjugacy theorems. ... Read more

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This work presents the proceedings of the 2nd Columbus Monster Conference, held in Ohio in 1996. The papers discuss the various aspects of group theory and Lie algebra theory with the monster group as the underlying central subject. Topics covered include vertex operator algebras and the application in conformal field theory and elliptical cohomology, the net group, modular Lie algebras, affine Lie algebras, quantum groups, and applications of Hopf algebras in the study of Lie algebras. ... Read more

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This monograph focuses on extending theorems for the classical Lie algebras in order to determine the structure and representation theory for Lie algebras of Cartan type. More specifically, Nakano investigates the block theory for the restricted universal enveloping algebras of the Lie algebras of Cartan type. The first section employs techniques developed by Holmes and Nakano to prove a Brauer-Humphreys reciprocity law for graded restricted Lie algebras and also to find the decompositions for the intermediate (Verma) modules used in the reciprocity law. The second section uses this information to investigate the structure of projective modules for the Lie algebras of types W and K. The restricted enveloping algebras for these Lie algebras are shown to have one block. Furthermore, Nakano provides a procedure for computing the Cartan invariants for Lie algebras of types W and K, given knowledge about the decomposition of the generalized Verma modules and about the Jantzen matrix of the classical/reductive zero component. Noteworthy for its readability and the continuity of its theme and purpose, this monograph appeals to graduate students and researchers interested in Lie algebras. ... Read more