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This monograph deals with two aspects of the theory ofelliptic genus: its topological aspect involving elliptic functions,and its representation theoretic aspect involving vertex operatorsuper-algebras. For the second aspect, elliptic genera are shown tohave the structure of modules over certain vertex operatorsuper-algebras. The vertex operators corresponding to parallel tensorfields on closed Riemannian Spin Khler manifolds such as Riemanniantensors and Khler forms are shown to give rise to Virasoro algebrasand affine Lie algebras. This monograph is chiefly intended fortopologists and it includes accounts on topics outside of topologysuch as vertex operator algebras ... Read more

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This volume contains papers presented at the meetingDeformation Theory, Symplectic Geometry and Applications,held in Ascona, June 17-21, 1996.
The contents touch upon many frontier domains of modern mathematics,mathematical physics and theoretical physics and includeauthoritative, state-of-the-art contributions by leading scientists.New and important developments in the fields of symplectic geometry,deformation quantization, noncommutative geometry (NCG) and Lie theoryare presented. Among the subjects treated are: quantization of generalPoisson manifolds; new deformations needed for the quantization ofNambu mechanics; quantization of intersection cardinalities; quantumshuffles; new types of quantum groups and applications; quantumcohomology; strong homotopy Lie algebras; finite- andinfinite-dimensional Lie groups; and 2D field theories andapplications of NCG to gravity coupled with the standard model.
Audience: This book will be of interest to researchers andpost-graduate students of mathematical physics, global analysis,analysis on manifolds, topological groups, nonassociative rings andalgebras, and Lie algebras. ... Read more

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This book develops a new approach to the study ofinfinite-dimensional Jordan and Lie algebras and real associative*-algebras of operators on a Hilbert space. All these algebras arecanonically generated by involutive antiautomorphisms of von Neumannalgebras.
The first purpose of the book is to study the deep structure theoryfor Jordan operator algebras similar to (complex) von Neumann algebrastheory, such as type classification, traces, conjugacy ofautomorphisms and antiautomorphisms, injectivity, amenability, andsemidiscreteness. The second aim is to investigate pure algebraicproblems concerning Jordan and Lie structure in prime and simple ringswith involution in the frame work of operator algebras. These purealgebraic results give additional information on properties of singleoperators on a Hilbert space.
Audience: This volume will be of interest to postgraduatestudents and specialists in the field of operator algebras, andalgebraists whose work involves nonassociative andinfinite-dimensional rings. ... Read more

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Around 1978, a mutation of associative algebras was introducedto generalize the formalism of classical mechanics as well as quantummechanics.
This volume presents the first book devoted to a self-contained anddetailed treatment of the mathematical theory of mutation algebras,which is based on research in this subject over the past fifteenyears. The book also deals with a broader class of algebras, mutationsof alternative algebras, which are a natural generalization ofmutations of associative algebras. A complete structure theory,including automorphisms, derivations and certain representations, isgiven for mutations of artinian alternative algebras, and, inparticular, of Cayley--Dickson algebras. Since the mutation algebrasdo not form a variety, the structure theory explored in this volumetakes quite a different approach from the standard theory ofnonassociative algebras and provides an important interplay with thetheory of noncommutative (associative) algebras through mutationparameters. New simple algebras and open problems presented in thisbook will stimulate additional research and applications in the area.
This book will be valuable to graduate students, mathematicians andphysicists interested in applications of algebras.
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An exposition of a part of the theory of Jordan algebras, using linear algebraic groups starting with the basic theory of finite dimensional Jordan algebras with identity. Paper. ... Read more

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This monograph systematically develops and considers the so-called "dressing method" for solving differential equations (both linear and nonlinear), a means to generate new non-trivial solutions for a given equation from the (perhaps trivial) solution of the same or related equation. Throughout, the text exploits the "linear experience" of presentation, with special attention given to the algebraic aspects of the main mathematical constructions and to practical rules of obtaining new solutions.

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The 12 lectures presented in Representation Theories andAlgebraic Geometry focus on the very rich and powerfulinterplay between algebraic geometry and the representation theoriesof various modern mathematical structures, such as reductive groups,quantum groups, Hecke algebras, restricted Lie algebras, and theircompanions. This interplay has been extensively exploited duringrecent years, resulting in great progress in these representationtheories. Conversely, a great stimulus has been given to thedevelopment of such geometric theories as D-modules, perverse sheafsand equivariant intersection cohomology.
The range of topics covered is wide, from equivariant Chow groups,decomposition classes and Schubert varieties, multiplicity freeactions, convolution algebras, standard monomial theory, and canonicalbases, to annihilators of quantum Verma modules, modularrepresentation theory of Lie algebras and combinatorics ofrepresentation categories of Harish-Chandra modules. ... Read more

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Based on invited lectures at the 1992 Canadian Algebra Seminar,this volume represents an up-to-date and unique report onfinite-dimensional algebras as a subject with many seriousinteractions with other mathematical disciplines,including algebraicgroups and Lie theory, automorphic forms, sheaf theory, finite groups,and homological algebra. It will interest mathematicians and graduatestudents in these and related subjects as an introduction to researchin an area of increasing relevance and importance.
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This book gives a systematic introduction to the theory ofclasses of groups, including research subjects, major (recent)research achievements, and directions for future research. It clearlyand concisely treats a wealth of topics, such as a brief introductionto the fundamental knowledge of group theory; the classical part ofthe theory of classes of groups covering mainly F-covering subgroups,F-projectors, F-injectors and F-normalisers; local formations; Schunckclasses; Fitting classes; properties of local formations; formationconstructions of finite groups and related applications; and thealgebra of formations. Audience: This volume will be of interest to mathematiciansinvolved in group theory and generalisations, algebras, order,lattices, ordered algebraic structures, general mathematical systemsand the mathematics of physics and chemistry. ... Read more

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This monograph tells the story of a philosophy of J-P. Serre and his vision of relating that philosophy to problems in affine algebraic geometry. It gives a lucid presentation of the Quillen-Suslin theorem settling Serre's conjecture. The central topic of the book is the question of whether a curve in $n$-space is as a set an intersection of $(n-1)$ hypersurfaces, depicted by the central theorems of Ferrand, Szpiro, Cowsik-Nori, Mohan Kumar, Boratýnsk.

The book gives a comprehensive introduction to basic commutative algebra, together with the related methods from homological algebra, which will enable students who know only the fundamentals of algebra to enjoy the power of using these tools. At the same time, it also serves as a valuable reference for the research specialist and as

potential course material, because the authors present, for the first time in book form, an approach here that is an intermix of classical algebraic K-theory and complete intersection techniques, making connections with the famous results of Forster-Swan and Eisenbud-Evans. A study of projective modules and their connections with topological vector bundles in a form due to Vaserstein is included. Important subsidiary results appear in the copious exercises.

Even this advanced material, presented comprehensively, keeps in mind the young student as potential reader besides the specialists of the subject.

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This volume has developed from courses given at Moscow StateUniversity. The main purpose of the material presented is to introducethe concepts, results and problems of contemporary algebra, assumingsome knowledge of the standard theory of linear algebra and vectorspaces. One important aspect is also to demonstrate how the conceptsdiscussed relate to each other and how they work in practice. The bookbegins with an introduction to the fundamental concepts of groups,rings, fields and modules and their representations. The sevenchapters which follow are devoted respectively to the followingtopics: commutative algebra; groups; associative rings; Lie algebras;homological algebra; algebraic groups; and varieties of algebras. Thevolume concludes with a supplement dealing with set theory, referencesand indices.
The book is as self-contained as possible.
For graduate students and researchers wishing to obtain a goodintroduction to the concepts of contemporary algebra.
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This book presents a systematic study on the structures ofvertex operator superalgebras and their modules. Related theories ofself-dual codes and lattices are included, as well as recentachievements on classifications of certain simple vertex operatorsuperalgebras and their irreducible twisted modules, constructions ofsimple vertex operator superalgebras from graded associative algebrasand their anti-involutions, self-dual codes and lattices.
Audience: This book is of interest to researchers and graduatestudents in mathematics and mathematical physics. ... Read more

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The Concise Encyclopedia of Supersymmetry presents outstanding achievements and ideas, both well-established and more recent, in quantum field theory, supersymmetry, supergravity, M-theory, black holes and quantum gravity, noncommutative geometry, representation theory, categories and quantum groups, which pushes the reader at the forefront of the latest their developments.

The Concise Encyclopedia of Supersymmetry consists of more than 700 clear and informative, but comprehensive, authored articles and definitions from more than 250 high level experts and well-known scientists including Nobel Prize Winner Gerard 't Hooft, Fields Medal winner Alain Connes and Stephen Adler.

As an Introduction it contains the "SUSY story", a collection of informal articles by founders of the field of supersymmetry and precursors with their rarest extraordinary memories and phenomenal explanations how and why they started with the idea of supersymmetry in 60th-70th. For convenience and searching the SUSY Encyclopedia contains a very detailed Subject/Article three level index (more than 50 three-column pages) and an Author index. The Encyclopedia will be an essential and indispensable reference book, handbook and guide for research workers, Ph.D. students, specialists and professionals in supersymmetry, quantum groups, noncommutative geometry and other modern methods of theoretical and mathematical physics. ... Read more

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Supermanifolds and Supergroups explains the basic ingredients of super manifolds and super Lie groups. It starts with super linear algebra and follows with a treatment of super smooth functions and the basic definition of a super manifold. When discussing the tangent bundle, integration of vector fields is treated as well as the machinery of differential forms. For super Lie groups the standard results are shown, including the construction of a super Lie group for any super Lie algebra. The last chapter is entirely devoted to super connections.

The book requires standard undergraduate knowledge on super differential geometry and super Lie groups.

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Invited articles by top notch experts

Focus is on topics in representation theory of algebraic groups and quantum groups

Of interest to graduate students and researchers in representation theory, group theory, algebraic geometry, quantum theory and math physics

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This book provides the reader with the tools to understand the ongoing classification and construction project of Lie superalgebras. It presents the material in as simple terms as possible. Coverage specifically details Borcherds-Kac-Moody superalgebras. The book examines the link between the above class of Lie superalgebras and automorphic form and explains their construction from lattice vertex algebras. It also includes all necessary background information.

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The four real division algebras (reals, complexes, quaternionsand octonions) are the most obvious signposts to a rich and intricaterealm of select and beautiful mathematical structures. Using the newtool of adjoint division algebras, with respect to which the divisionalgebras themselves appear in the role of spinor spaces, some of thesestructures are developed, including parallelizable spheres,exceptional Lie groups, and triality. In the case of triality the useof adjoint octonions greatly simplifies its investigation. Motivatingthis work, however, is a strong conviction that the design of ourphysical reality arises from this select mathematical realm. Acompelling case for that conviction is presented, a derivation of thestandard model of leptons and quarks.
The book will be of particular interest to particle and high energytheorists, and to applied mathematicians.
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Customer Reviews (2)

I am waiting for Dixon 's Octonians...
I have not yet received my command. P. MERAT

Mathematics behind physics
This is an excellent book for those who want to study Hamilton's quaternions, and other algebraic structures, used in modern physics. Dixon believes that octonions and triality of Spin(8) are essential in understanding particle physics. This clear exposition contains many ideas which have gone unnoticed from other researchers. The book is a treasure trove for mathematical physicists. The author also compares the Cayley algebra of octonions to other algebraic systems used in physics: matrices and Clifford algebras, in particular the Dirac algebra. ... Read more

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Devoted to the theory of Lie algebras and algebraic groups, this book includes a large amount of commutative algebra and algebraic geometry so as to make it as self-contained as possible. The aim of the book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included, and some recent results are discussed in the final chapters.

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This volume contains articles by 20 leading workers in the field of algebraic groups and related finite groups.

Articles on representation theory are written by Andersen on tilting modules, Carter on canonical bases, Cline, Parshall and Scott on endomorphism algebras, James and Kleshchev on the symmetric group, Littelmann on the path model, Lusztig on homology bases, McNinch on semisimplicity in prime characteristic, Robinson on block theory, Scott on Lusztig's character formula, and Tanisaki on highest weight modules.
Articles on subgroup structure are written by Seitz and Brundan on double cosets, Liebeck on exceptional groups, Saxl on subgroups containing special elements, and Guralnick on applications of subgroup structure.
Steinberg gives a new, short proof of the isomorphism and isogeny theorems for reductive groups.
Aschbacher discusses the classification of quasithin groups and Borovik the classification of groups of finite Morley rank.

Audience: The book contains accounts of many recent advances and will interest research workers and students in the theory of algebraic groups and related areas of mathematics.

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This book surveys recent developments and outlines researchprospects in various fields, the fundamental questions of which can bestated in the language of symmetric functions. Interdisciplinaryinterconnections are emphasized. ... Read more