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This book is a collection of a series of lectures givenby Prof. V Kac at Tata Institute, India in Dec '85 and Jan '86. Theselectures focus on the idea of a highest weight representation, whichgoes through four different incarnations.

The first is the canonical commutation relations of theinfinite-dimensional Heisenberg Algebra (= ocillator algebra). Thesecond is the highest weight representations of the Lie algebra gl¥of infinite matrices, along with their applications to the theory ofsoliton equations, discovered by Sato and Date, Jimbo, Kashiwara andMiwa. The third is the unitary highest weight representations of thecurrent (= affine Kac-Moody) algebras. These algebras appear in thelectures twice, in the reduction theory of soliton equations (KP ®KdV) and in the Sugawara construction as the main tool in the study ofthe fourth incarnation of the main idea, the theory of the highestweight representations of the Virasoro algebra.

This book should be very useful for both mathematicians andphysicists. To mathematicians, it illustrates the interaction of thekey ideas of the representation theory of infinite-dimensional Liealgebras; and to physicists, this theory is turning into an importantcomponent of such domains of theoretical physics as soliton theory,theory of two-dimensional statistical models, and string theory. ... Read more

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This volume is the proceedings of the Ramanujan International Symposium on Kac-Moody Lie algebras and their applications. The symposium provided researchers in mathematics and physics with the opportunity to discuss new developments in this rapidly-growing area of research. The book contains several excellent articles with new and significant results. It is suitable for graduate students and researchers working in Kac-Moody Lie algebras, their applications, and related areas of research. ... Read more

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This book is a collection of papers dedicated to Richard E. Block, whose research has been largely devoted to the study of Lie algebras of prime characteristic (specifically the classification of simple Lie algebras). The volume presents proceedings of a conference held at the University of California at Riverside in February 1995 on the occasion of his retirement. The conference focused on the interplay between the theory of Lie algebras of prime characteristic, quantum groups, and Lie superalgebras. ... Read more

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During the academic year 1987-1988 the University of Wisconsin in Madison hosted a Special Year of Lie Algebras. A Workshop on Lie Algebras, of which these are the proceedings, inaugurated the special year. The principal focus of the year and of the workshop was the long-standing problem of classifying the simple finite-dimensional Lie algebras over algebraically closed field of prime characteristic. However, other lectures at the workshop dealt with the related areas of algebraic groups, representation theory, and Kac-Moody Lie algebras. Fourteen papers were presented and nine of these (eight research articles and one expository article) make up this volume. ... Read more

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These proceedings are from the Tenth International Conference on Representations of Algebras and Related Topics (ICRA X) held at The Fields Institute. In addition to the traditional "instructional" workshop preceding the conference, there were also workshops on "Commutative Algebra, Algebraic Geometry and Representation Theory", "Finite Dimensional Algebras, Algebraic Groups and Lie Theory", and "Quantum Groups and Hall Algebras". These workshops reflect the latest developments and the increasing interest in areas that are closely related to the representation theory of finite dimensional associative algebras. Although these workshops were organized separately, their topics are strongly interrelated.

The workshop on Commutative Algebra, Algebraic Geometry and Representation Theory surveyed various recently established connections, such as those pertaining to the classification of vector bundles or Cohen-Macaulay modules over Noetherian rings, coherent sheaves on curves, or ideals in Weyl algebras. In addition, methods from algebraic geometry or commutative algebra relating to quiver representations and varieties of modules were presented.

The workshop on Finite Dimensional Algebras, Algebraic Groups and Lie Theory surveyed developments in finite dimensional algebras and infinite dimensional Lie theory, especially as the two areas interact and may have future interactions.

The workshop on Quantum Groups and Hall Algebras dealt with the different approaches of using the representation theory of quivers (and species) in order to construct quantum groups, working either over finite fields or over the complex numbers. In particular, these proceedings contain a quite detailed outline of the use of perverse sheaves in order to obtain canonical bases.

The book is recommended for graduate students and researchers in algebra and geometry. ... Read more

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This Ergebnisse volume is aimed at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations and construct the algebraic tori on which they linearize; at the same time, it is, for the student, a playing ground to applying algebraic geometry and Lie theory. The book is meant to be reasonably self-contained and presents numerous examples. The latter appear throughout the text to illustrate the ideas, and make up the core of the last part of the book. The first part of the book contains the basic tools from Lie groups, algebraic and differential geometry to understand the main topic.

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This work is about extended affine Lie algebras (EALA's) and their root systems. EALA's were introduced by Høegh-Krohn and Torresani under the name irreducible quasi-simple Lie algebras. The major objective is to develop enough theory to provide a firm foundation for further study of EALA's.

The first chapter of the paper is devoted to establishing some basic structure theory. It includes a proof of the fact that, as conjectured by Kac, the invariant symmetric bilinear form on an EALA can be scaled so that its restriction to the real span of the root system is positive semi-definite.

The second chapter studies extended affine root systems (EARS) which are an axiomatized version of the root systems arising from EALA's. The concept of a semilattice is used to give a complete description of EARS. In the final chapter, a number of new examples of extended affine Lie algebras are given.

The concluding appendix contains an axiomatic characterization of the nonisotropic roots in an EARS in a more general context than the one used in the rest of the paper.

Features:

Provides a foundation for the study of an important class of Lie algebras that generalizes the class of affine Kac-Moody Lie algebras

Includes material on Lie algebras and on root systems that can be read independently. ... Read more

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This classic paper pertaining to the theory of the Yang-Baxter equations presents results on the applications of methods of the theory of Lie algebras to the classification of solutions of the Yang-Baxter equations. ... Read more

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This book deals mainly with the relevance of integral manifoldsassociated with a Lie algebra with singularities for studying systemsof first order partial differential equations, stochastic differentialequations and nonlinear control systems. The analysis is based on thealgebraic representation of gradient systems in a Lie algebra,allowing the recovery of the original vector fields and the associatedLie algebra as well. Special attention is paid to nonlinear controlsystems encompassing specific problems of this theory and theirsignificance for stochastic differential equations. The work iswritten in a self-contained manner, presupposing only some basicknowledge of algebra, geometry and differential equations. Audience: This volume will be of interest to mathematicians andengineers working in the field of applied geometric and algebraicmethods in differential equations. It can also be recommended as asupplementary text for postgraduate students. ... Read more

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As the Proceedings of the 1984 Canadian Mathematical Society's Summer Seminar, this book focuses on some advances in the theory of semisimple Lie algebras and some direct outgrowths of that theory. The following papers are of particular interest: an important survey article by R. Block and R. Wilson on restricted simple Lie algebras, a survey of universal enveloping algebras of semisimple Lie algebras by W. Borho, a course on Kac-Moody Lie algebras by I. G. Macdonald with an extensive bibliography of this field by Georgia Benkart, and a course on formal groups by M. Hazewinkel.

Because of the expository surveys and courses, the book will be especially useful to graduate students in Lie theory, as well as to researchers in the field. ... Read more

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The aim of the present work is two-fold. Firstly it aims at a giving an account of many existing algorithms for calculating with finite-dimensional Lie algebras. Secondly, the book provides an introduction into the theory of finite-dimensional Lie algebras. These two subject areas are intimately related. First of all, the algorithmic perspective often invites a different approach to the theoretical material than the one taken in various other monographs (e.g., [42], [48], [77], [86]). Indeed, on various occasions the knowledge of certain algorithms allows us to obtain a straightforward proof of theoretical results (we mention the proof of the Poincaré-Birkhoff-Witt theorem and the proof of Iwasawa's theorem as examples). Also proofs that contain algorithmic constructions are explicitly formulated as algorithms (an example is the isomorphism theorem for semisimple Lie algebras that constructs an isomorphism in case it exists). Secondly, the algorithms can be used to arrive at a better understanding of the theory. Performing the algorithms in concrete examples, calculating with the concepts involved, really brings the theory of life.
... Read more