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This collection brings together papers related to the classicalideas of Sophus Lie. The present work reflects the interests ofscientists associated with the International Sophus Lie Center, andprovides up-to-date results in Lie groups and Lie algebras, quantummathematics, hypergroups, homogeneous spaces, Lie superalgebras, thetheory of representations and applications to differential equationsand integrable systems. Among the topics that are treated are quantization of Poissonstructures, applications of multivalued groups, noncommutative aspectsof hypergroups, homology invariants of homogeneous spaces,generalisations of the Godbillon--Vey invariant, relationsbetween classical problems of linear analysis and representationtheory and the geometry of current groups. Audience: This volume will be of interest to mathematicians andphysicists specialising in the theory and applications of Lie groupsand Lie algebras, quantum groups, hypergroups and homogeneous spaces. ... Read more

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The Fourier transforms of invariant functions on finite reductive Lie algebras are due to T.A. Springer (1976) in connection with the geometry of nilpotent orbits. In this book the author studies Fourier transforms using Deligne-Lusztig induction and the Lie algebra version of Lusztig’s character sheaves theory. He conjectures a commutation formula between Deligne-Lusztig induction and Fourier transforms that he proves in many cases. As an application the computation of the values of the trigonometric sums (on reductive Lie algebras) is shown to reduce to the computation of the generalized Green functions and to the computation of some fourth roots of unity.

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This volume presents contributions by leading experts in the field. The articles are dedicated to D. B. Fuchs on the occasion of his 60th birthday. Contributors to the book were directly influenced by Professor Fuchs and include his students, friends, and professional colleagues. In addition to their research, they offer personal reminicences about Professor Fuchs, giving insight into the history of Russian mathematics.

The main topics addressed in this unique work are infinite-dimensional Lie algebras with applications (vertex operator algebras, conformal field theory, quantum integrable systems, etc.) and differential topology. The volume provides an excellent introduction to current research in the field. ... Read more

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Unique source of data
This book is the only place to find lots of theorems on exceptional Lie algebras (and therefore also on exceptional algebraic groups).For example, it is the only place I know of where it is proven that two27-dimensional exceptional Jordan algebras are isotopic if and only iftheir norm forms are similar.As Wallach said, this book also explains thedescription of the roots systems for the exceptional algebras found inJacobson's other book "Lie algebras".

The description of theD_4 (Spin_8) as a subalgebra of F_4 is quite beautiful. ... Read more

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There is a fruitful and fascinating interaction between infinite dimensional operator theory (particularly decomposable, scalar and spectral generalized operator theory due to C. Foias and I. Colojoara) and Lie algebra theory.

The present book is the first devoted to this field, ranging from some short historical notes to the most recent developments. Nilpotence criteria, infinite dimensional variants of Lie's theorem for solvable systems of bounded operators, spectral properties of elements of semisimple Lie algebras and simultaneous triangularisation are expounded.The book is self-contained and features an extensive bibliography. It is aimed at postgraduate students and researchers who are introduced to an interesting recent area of research and will learn some new methods useful for both of the domains - operator theory and Lie algebra theory. ... Read more

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This much-needed new book is the first to specifically detail free Lie algebras.Lie polynomials appeared at the turn of the century and were identified with the free Lie algebra by Magnus and Witt some thirty years later.Many recent, important developments have occurred in the field--especially from the point of view of representation theory--that have necessitated a thorough treatment of the subject. This timely book covers all aspects of the field, including characterization of Lie polynomials and Lie series, subalgebras and automorphisms, canonical projections, Hall bases, shuffles and subwords, circular words, Lie representations of the symmetric group, related symmetric functions, descent algebra, and quasisymmetric functions.With its emphasis on the algebraic and combinatorial point of view as well as representation theory, this book will be welcomed by students and researchers in mathematics and theoretical computer science. ... Read more

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All you need to know about free Lie algebras
With free Lie algebras becoming increasingly important in topology and computer science, this book is an invaluable resource. It is the only book on the market today which collects together the essential facts about free Lie algebras, which were previously widely scattered across numerous journals and known only to experts, and presents them in an accessible manner. This book works its way up from the fundamentals to what was known around 1993, which provides a strong basis for current research in the subject. ... Read more

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In this work the author analyse various structural properties of indecomposable nine dimensional real Lie algebras having a non-trivial Levi decomposition. These algebras, classified in the early 90s by Turkowski, actually show patterns that do not occur in Lie algebras of lower dimension. The main objective is to determine a fundamental system of invariants for the co-adjoint representation. Due to the large number of essential parameters, a direct analytical approach is not useful, due to the quantity of special cases to be analysed. In order to integrate the corresponding systems of PDEs, the author consider the theory of semi-invariants developed by Trofimov and a reduction of the problem to certain subalgebras, the invariants of which can be computed easily or are already known. The procedure is based in the labelling of representations using subgroup chains (the so called missing label operator problem), which allows to express the invariants of the total algebra as functions of invariant of subalgebras. The method also provides those Lie algebras that arise as an extension of some subalgebra, therefore providing information on the trivial cohomology. ... Read more

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This practical treatise is an introduction to the mathematics and physics of affine Kac-Moody algebras. It is the result of an unusual interdisciplinary effort by two physicists and two mathematicians to make this field understandable to a broad readership and to illuminate the connections among seemingly disparate domains of mathematics and physics that are tantalizingly suggested by the ubiquity of Lie theory. The book will be useful to Lie algebraists, high energy physicists, statistical mechanics, and number theorists. Volume One contains a description of Kac-Moody Lie algebras, and especially the affine algebras and their representations; the results of extensive computations follow in Volume Two, which is spiral bound for easy reference. ... Read more

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This book contains the refereed proceedings of the symposium on Lie algebras and representation theory which was held at Seoul National University (Korea) in January 1995. The symposium was sponsored by the Global Analysis Research Center of Seoul National University.

Over the past 30 years, exciting developments in diverse areas of the theory of Lie algebras and their representations have been observed. The symposium covered topics such as Lie algebras and combinatorics, crystal bases for quantum groups, quantum groups and solvable lattice models, and modular and infinite-dimensional Lie algebras. In this volume, readers will find several excellent expository articles and research papers containing many significant new results in this area. Consequently, this book can serve both as an introduction to various aspects of the theory of Lie algebras and their representations and as a good reference work for further research. ... Read more

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This book is directed to researchers in Lie theory and in the theory of linear algebra, associative or otherwise, and to graduate students who have had some background in one or more of these areas. ... Read more