Lie Algebras and Applications (Lecture Notes in Physics)

Book Description

This book, designed for advanced graduate students and post-graduate researchers, provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, a concise exposition is given of the basic concepts of Lie algebras, their representations and their invariants. The second part contains a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book contains many examples that help to elucidate the abstract algebraic definitions. It provides a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators and the dimensions of the representations of all classical Lie algebras.

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A welcomed unified approach of Lie algebras applied to Physics.......2006-12-11

This book is not a manual in the usual sense, but a compilation of facts concerning Lie algebras that continuously appear in physical problems. The material covered is the result of various seminars given by the author during many years, and synthetize the main facts that should be known to any physicist.

The material is divided into 12 chapters of variable length. The first two present the main theory of semisimple Lie algebras, enumerating the key results from root theory and Dynkin-Coxeter diagrams to classify the complex simple algebras. The real forms for the classical algebras are given in table form, without going into its detailed obtainment. It should also be taken into account that for Dynkin diagrams, the author does not distinguish between positively and negatively oriented angles, thus the angles between roots in equation (2.8) are reduced to five (unoriented) angles instead of the usual eight (oriented) angles.

Chapter three compiles the most important facts about Lie algebras of Lie groups, mainly focused on matrix groups. Important techniques like the exponential map and the covering of groups are nicely illustrated with the classical unitary algebra su(2) and the Lorentz group (in one dimension). I personally miss some comment on the left invariant vector fields or 1-forms (Maurer-Cartan equations), of importance in many applications to cosmology.

The fourth chapter is devoted to representation theory. Although the Weyl decomposition theorem is not included, it is assumed that any representation decomposes as a direct sum of irreducible modules (valid for semisimple Lie algebras). The fundamental representations are discussed for the classical algebras (symplectic, unitary and orthogonal), and for the latter, the spinor representations are also given. The dimension formulae are given, and the tensor products (Clebsch-Gordan problem) is developed by means of Young tableaux. This is applied to the branching rules of representations with respect to some chain and the missing label problem, illustrated by examples that are typical in the interacting boson model.

In chapter five, Casimir operators of Lie algebras are defined and obtained for the classical Lie algebras. Here the author uses the Perelomov-Popov approach of operators that can be identified with symmetric elements in the universal enveloping algebra. At the beginning of the chapter it is said that the number of Casimir operators equals the rank of the algebra. Again, this is only valid for semisimple Lie algebras, and generally false for arbitrary Lie algebras. The eigenvalue problem is presented using important examples, and the results resumed in a table at the end of the chapter.

The previous chapter is a nice motivation for tensor operators in general, which comprise essential techniques like the coupling and recoupling coefficients, how to determine them and their symmetries (much of this material was originally developed by Racah in his Princeton lectures of 1951). This chapter is of great importance for applications.

Chapters 8 and 9 are devoted to another technique of great relevance, the realizations of Lie algebras by means of creation and annihilation operators, divided into boson and fermion operators, according to commutation or anticommutation relations. Here the unitary case is exploited, and many subalgebra chains are analyzed with respect to these realizations. Of special interest are the sections concerning the L-S and j-j couplings used in spectroscopy of light nuclei and shell models, and where original examples have been used.

Chapter 9 presents another possibility for realizing Lie algebras, namely by differential operators. Although a short chapter, important topics like the Casimir operators as differential operators or the Laplace-Beltrami form is presented. In chapter 10, the classical matrix realizations (in fact representations by linear operators) are briefly recalled, and the classical interpretation of the Casimir operators is recovered (without using the Schur lemma).

The two last chapters deal with quite more specific topics, like dynamic symmetries, studied in both fermionic and bosonic systems, in the unitary algebras u(6) and u(4), in order to obtain mass and energy level diagrams. For the part of degeneracy algebras, the problems illustrated are the isotropic harmonic oscillator, the Coulomb problem and the Teller-Pöschl and Morse potentials. In all these problems the reader is referred to original articles to complete the information presented.

The chapters of the book do not develop the theory systematically, but rather focus on a type of problem or technique which is developed using the main Lie algebras appearing (mainly) in spectroscopy, atomic, nuclear and molecular physics, as well as quantum mechanics. No proofs are given, which prevents the reader from being distracted from the main objective of the lectures. To fill the gaps, the reader is led, at many places, to consult either original references or more formal books.

The book is written in an informal style, which simplifies its reading and makes it a suitable consultation work. The profusion of examples (many of them actually coming from original references) explains quite well the topics studied, and gives a concrete idea how to apply the techniques. It is a very welcomed addition to the literature that contains much topics treated for the first time in textbook form.

There are few misprints and mistakes in the text, which can however confuse the reader having no previous knowledge on Lie algebras. For example, on page 7, the definition of semidirect sum is confusing and wrong. It is actually not necessary that one of the algebras is an ideal in the other, it suffices that one of them acts by derivations on the other. The definition given in the book is incompatible with example 13 on the same page. Another confusing point is subsection 1.13. Here by derivations the author means the derived series of an algebra, determining whether it is solvable or not. Derivations are linear maps satisfying the Leibniz rule, and are completely independent on the solvable character. The notation is very confusing, since the derived subalgebra (commutator ideal) is denoted in the same manner as the Lie algebra of derivations (which is actually a linear Lie algebra).

Inspite of these minor details, the book will certainly be of great use for students or specialists that want to refresh their knowledge on Lie algebras applied to physics. The list of references is quite complete and provides a deeper insight into the problems where these structures appear. However, there are also some surprising absences in the references, such as the books of J. F. Cornwell or H. Lipkin, in my opinion two classicals on group theory in physics. Among the original articles, I miss for example the relevant review article by R. Slansky [Phys. Rep. 79 (1981), 1-128], although it is clear that giving a complete reference list is impossible.

Resuming, the book by Iachello constitutes an excellent reference for those interested in the practical application and techniques of Lie algebras to physics, and that try to avoid the often embarrassing theoretical works. It should also be mentioned that much of the material is divided into hundreds of original articles, and therefore a unified presentation will be of great use for the physical community.

Representations of Algebraic Groups, Quantum Groups, and Lie Algebras (Contemporary Mathematics)

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Representations of Algebraic Groups, Quantum Groups, and Lie Algebras (Contemporary Mathematics)

Manufacturer: American Mathematical Society
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The book contains several well-written, accessible survey papers in many interrelated areas of current research. These areas cover various aspects of the representation theory of Lie algebras, finite groups of Lie types, Hecke algebras, and Lie superalgebras. Geometric methods have been instrumental in representation theory, and these proceedings include surveys on geometric as well as combinatorial constructions of the crystal basis for representations of quantum groups. Humphreys' paper outlines intricate connections among irreducible representations of certain blocks of reduced enveloping algebras of semi-simple Lie algebras in positive characteristic, left cells in two sided cells of affine Weyl groups, and the geometry of the nilpotent orbits. All these papers provide the reader with a broad picture of the interaction of many different research areas and should be helpful to those who want to have a glimpse of current research involving representation theory.

Studies in Lie Theory: Dedicated to A. Joseph on his Sixtieth Birthday (Progress in Mathematics)

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Studies in Lie Theory: Dedicated to A. Joseph on his Sixtieth Birthday (Progress in Mathematics)

Manufacturer: Birkhäuser Boston
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ASIN: 0817643427

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Dedicated to Anthony Joseph, this volume contains surveys and invited articles by leading specialists in representation theory. The focus here is on semisimple Lie algebras and quantum groups, where the impact of Joseph's work has been seminal and has changed the face of the subject.

Two introductory biographical overviews of Joseph's contributions in classical representation theory (the theory of primitive ideals in semisimple Lie algebras) and quantized representation theory (the study of the quantized enveloping algebra) are followed by 16 research articles covering a number of varied and interesting topics in representation theory.

Contributors: J. Alev; A. Beilinson; A. Braverman; I. Cherednik; J. Dixmier; F. Dumas; P. Etingof; D. Farkas; D. Gaitsgory; F. Ivorra; A. Joseph; D. Joseph; M. Kashiwara; D. Kazhdan; A.A. Kirillov; B. Kostant; S. Kumar; G. Letzter; T. Levasseur; G. Lusztig; L. Makar-Limanov; W. McGovern; M. Nazarov; K-H. Neeb; L.G. Rybnikov; P. Schapira; V. Schechtman; A. Sergeev; J.T. Stafford; Ya. Varshavsky; N. Wallach; and I. Waschkies.

Quantum Groups and Lie Theory (London Mathematical Society Lecture Note Series)

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Quantum Groups and Lie Theory (London Mathematical Society Lecture Note Series)

Manufacturer: Cambridge University Press
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ASIN: 0521010403

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To take stock and to discuss the most fruitful directions for future research, many of the world's leading figures met at the Durham Symposium on Quantum Groups in the summer of 1999, and this volume provides an excellent overview of the material presented there. It includes important surveys of both cyclotomic Hecke algebras and the dynamical Yang-Baxter equation. Plus contributions which treat the construction and classification of quantum groups or the associated solutions of the quantum Yang-Baxter equation. The representation theory of quantum groups is discussed, as is the function algebra approach to quantum groups, and there is a new look at the origins of quantum groups in the theory of integrable systems.

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Since its genesis in the early 1980s, the subject of quantum groups has grown rapidly. By the late 1990s most of the foundational issues had been resolved and many of the outstanding problems clearly formulated. To take stock and to discuss the most fruitful directions for future research many of the world's leading figures in this area met at the Durham Symposium on Quantum Groups in the summer of 1999, and this volume provides an excellent overview of the material presented there. It includes important surveys of both cyclotomic Hecke algebras and the dynamical Yang-Baxter equation. Plus contributions which treat the construction and classification of quantum groups or the associated solutions of the quantum Yang-Baxter equation. The representation theory of quantum groups is discussed, as is the function algebra approach to quantum groups, and there is a new look at the origins of quantum groups in the theory of integrable systems.

Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory (Cambridge Monographs on Mathematical Physics)

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Affine Lie Algebras and Quantum Groups: An Introduction, with Applications in Conformal Field Theory (Cambridge Monographs on Mathematical Physics)
Jürgen A. Fuchs
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Symmetries, Lie Algebras and Representations: A Graduate Course for Physicists (Cambridge Monographs on Mathematical Physics)
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ASIN: 052148412X

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This is an introduction to the theory of affine Lie algebras and to the theory of quantum groups. It is unique in discussing these two subjects in a unified manner, which is made possible by discussing their respective applications in conformal field theory. The description of affine algebras covers the classification problem, the connection with loop algebras, and representation theory including modular properties. The necessary background from the theory of semisimple Lie algebras is also provided. The discussion of quantum groups concentrates on deformed enveloping algebras and their representation theory, but other aspects such as R-matrices and matrix quantum groups are also dealt with.

Algebraic Approach to Simple Quantum Systems: With Applications to Perturbation Theory/Book and Disk

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Algebraic Approach to Simple Quantum Systems: With Applications to Perturbation Theory/Book and Disk
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Algebraic Methods in Quantum Chemistry and Physics (Mathematical Chemistry Series)

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Algebraic Methods in Quantum Chemistry and Physics (Mathematical Chemistry Series)
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Algebraic Methods in Quantum Chemistry and Physics provides straightforward presentations of selected topics in theoretical chemistry and physics, including Lie algebras and their applications, harmonic oscillators, bilinear oscillators, perturbation theory, numerical solutions of the Schrödinger equation, and parameterizations of the time-evolution operator. The mathematical tools described in this book are presented in a manner that clearly illustrates their application to problems arising in theoretical chemistry and physics. The application techniques are carefully explained with step-by-step instructions that are easy to follow, and the results are organized to facilitate both manual and numerical calculations. Algebraic Methods in Quantum Chemistry and Physics demonstrates how to obtain useful analytical results with elementary algebra and calculus and an understanding of basic quantum chemistry and physics.

Coherent States, Wavelets, and Their Generalizations (Graduate Texts in Contemporary Physics)

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A demanding work by true experts and enthusiasts

Coherent States, Wavelets, and Their Generalizations (Graduate Texts in Contemporary Physics)
Syed T. Ali , Jean-Pierre Antoine , and Jean-Perre Gazeau
Manufacturer: Springer
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ASIN: 0387989080

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This book presents a survey of the theory of coherent states, wavelets, and some of their generalizations, emphasizing mathematical structures. The point of view is that both the theories of both wavelets and coherent states can be subsumed into a single analytic structure. Starting from the standard theory of coherent states over Lie groups, the authors generalize the formalism by associating coherent states to group representations that are square integrable over a homogeneous space; a further step allows one to dispense with the group context altogether. In this context, wavelets can be generated from coherent states of the affine group of the real line, and higher-dimensional wavelets arise from coherent states of other groups. The unified background makes transparent otherwise obscure properties of wavelets and of coherent states. Many concrete examples, such as semisimple Lie groups, the relativity group, and several kinds of wavelets, are discussed in detail. The book concludes with physical applications, centering on the quantum measurement problem and the quantum-classical transition. Intended as an introduction to current research for graduate students and others entering the field, the mathematical discussion is self- contained. With its extensive references to the research literature, the book will also be a useful compendium of recent results for physicists and mathematicians already active in the field.

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A demanding work by true experts and enthusiasts.......2007-10-05

The authors of this book are probably the world's most knowledgeable experts on the topic. They have a tremendous enthusiasm for their subject and there is great beauty in the way they present it. Coherent states arise from quantum mechanics and allow the user to take a classical (non-quantum) view of quantum states in physics. Wavelets, on the other hand, arise from signal and image processing and provide a time-frequency view of a signal, showing which frequencies are present in a signal at each time point (or which spatial scales are present in an image at each location). The book ties these two theories together very elegantly, thereby showing the full power of coherent states. The book assumes extensive knowledge of quantum mechanics and pure mathematics, and demands a lot of the reader.

Conjugacy of Altb5s and Sl(2, 5) Subgroups of Eb8S(C (Memoirs of the American Mathematical Society, No. 634)

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Conjugacy of Altb5s and Sl(2, 5) Subgroups of Eb8S(C (Memoirs of the American Mathematical Society, No. 634)
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Exceptional complex Lie groups have become increasingly important in various fields of mathematics and physics. As a result, there has been interest in expanding the representation theory of finite groups to include embeddings into the exceptional Lie groups. Cohen, Griess, Lisser, Ryba, Serre and Wales have pioneered this area, classifying the finite simple and quasisimple subgroups that embed in the exceptional complex Lie groups.

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This volume is composed of invited expository articles by well-known mathematicians in differential geometry and mathematical physics that have been arranged in celebration of Hideki Omori's recent retirement from Tokyo University of Science and in honor of his fundamental contributions to these areas.

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