# introduction to finite and infinite dimensional lie superalgebras

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**Author** | : Neelacanta Sthanumoorthy |

**Publisher** | : Academic Press |

**Release Date** | : 2016-04-26 |

**Pages** | : 512 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**Lie superalgebras are a natural generalization of Lie algebras, having applications in geometry, number theory, gauge field theory, and string theory. Introduction to Finite and Infinite Dimensional Lie Algebras and Superalgebras introduces the theory of Lie superalgebras, their algebras, and their representations. The material covered ranges from basic definitions of Lie groups to the classification of finite-dimensional representations of semi-simple Lie algebras. While discussing all classes of finite and infinite dimensional Lie algebras and Lie superalgebras in terms of their different classes of root systems, the book focuses on Kac-Moody algebras. With numerous exercises and worked examples, it is ideal for graduate courses on Lie groups and Lie algebras. Discusses the fundamental structure and all root relationships of Lie algebras and Lie superalgebras and their finite and infinite dimensional representation theory Closely describes BKM Lie superalgebras, their different classes of imaginary root systems, their complete classifications, root-supermultiplicities, and related combinatorial identities Includes numerous tables of the properties of individual Lie algebras and Lie superalgebras Focuses on Kac-Moody algebras

**Author** | : Minoru Wakimoto |

**Publisher** | : American Mathematical Soc. |

**Release Date** | : 2001 |

**Pages** | : 304 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**This volume begins with an introduction to the structure of finite-dimensional simple Lie algebras, including the representation of ${\widehat {\mathfrak {sl}}}(2, {\mathbb C})$, root systems, the Cartan matrix, and a Dynkin diagram of a finite-dimensional simple Lie algebra. Continuing on, the main subjects of the book are the structure (real and imaginary root systems) of and the character formula for Kac-Moody superalgebras, which is explained in a very general setting. Only elementary linear algebra and group theory are assumed. Also covered is modular property and asymptotic behavior of integrable characters of affine Lie algebras. The exposition is self-contained and includes examples. The book can be used in a graduate-level course on the topic.

**Author** | : Yuri Bahturin |

**Publisher** | : Walter de Gruyter |

**Release Date** | : 1992-01-01 |

**Pages** | : 260 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**The aim of the series is to present new and important developments in pure and applied mathematics. Well established in the community over two decades, it offers a large library of mathematics including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers wishing to thoroughly study the topic. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany

**Author** | : Gerard G. A. Bäuerle |

**Publisher** | : Academic Press |

**Release Date** | : 1990 |

**Pages** | : 554 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**This is the long awaited follow-up to Lie Algebras, Part I which covered a major part of the theory of Kac-Moody algebras, stressing primarily their mathematical structure. Part II deals mainly with the representations and applications of Lie Algebras and contains many cross references to Part I. The theoretical part largely deals with the representation theory of Lie algebras with a triangular decomposition, of which Kac-Moody algebras and the Virasoro algebra are prime examples. After setting up the general framework of highest weight representations, the book continues to treat topics as the Casimir operator and the Weyl-Kac character formula, which are specific for Kac-Moody algebras. The applications have a wide range. First, the book contains an exposition on the role of finite-dimensional semisimple Lie algebras and their representations in the standard and grand unified models of elementary particle physics. A second application is in the realm of soliton equations and their infinite-dimensional symmetry groups and algebras. The book concludes with a chapter on conformal field theory and the importance of the Virasoro and Kac-Moody algebras therein.

**Author** | : Victor G. Kac |

**Publisher** | : Springer Science & Business Media |

**Release Date** | : 2013-11-11 |

**Pages** | : 252 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:** **Author** | : Maria Gorelik |

**Publisher** | : Springer Science & Business |

**Release Date** | : 2014-04-28 |

**Pages** | : 280 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**The volume is the outcome of the conference "Lie superalgebras," which was held at the Istituto Nazionale di Alta Matematica, in 2012. The conference gathered many specialists in the subject, and the talks held provided comprehensive insights into the newest trends in research on Lie superalgebras (and related topics like vertex algebras, representation theory and supergeometry). The book contains contributions of many leading esperts in the field and provides a complete account of the newest trends in research on Lie Superalgebras.

**Author** | : Shun-Jen Cheng |

**Publisher** | : American Mathematical Soc. |

**Release Date** | : 2012 |

**Pages** | : 302 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**This book gives a systematic account of the structure and representation theory of finite-dimensional complex Lie superalgebras of classical type and serves as a good introduction to representation theory of Lie superalgebras. Several folklore results are rigorously proved (and occasionally corrected in detail), sometimes with new proofs. Three important dualities are presented in the book, with the unifying theme of determining irreducible characters of Lie superalgebras. In order of increasing sophistication, they are Schur duality, Howe duality, and super duality. The combinatorics of symmetric functions is developed as needed in connections to Harish-Chandra homomorphism as well as irreducible characters for Lie superalgebras. Schur-Sergeev duality for the queer Lie superalgebra is presented from scratch with complete detail. Howe duality for Lie superalgebras is presented in book form for the first time. Super duality is a new approach developed in the past few years toward understanding the Bernstein-Gelfand-Gelfand category of modules for classical Lie superalgebras. Super duality relates the representation theory of classical Lie superalgebras directly to the representation theory of classical Lie algebras and thus gives a solution to the irreducible character problem of Lie superalgebras via the Kazhdan-Lusztig polynomials of classical Lie algebras.

**Author** | : Jain Surender K |

**Publisher** | : World Scientific |

**Release Date** | : 1993-09-30 |

**Pages** | : 392 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**This invaluable book deals with the many-electron theory of the solid state. Mastery of the material in it will equip the reader for research in areas such as high-temperature superconductors and the fractional quantum Hall effect. The whole book has been designed to provide the diligent reader with a wide variety of approaches to many-electron theory.The level of the book is suitable for research workers and higher-degree students in a number of disciplines, embracing theoretical physics, materials science and solid-state chemistry. It should be useful not only to theorists in these areas but also to experimental scientists who desire to orient their programmes to address outstanding questions raised by many-body theory.

**Author** | : Urmie Ray |

**Publisher** | : Springer Science & Business Media |

**Release Date** | : 2007-03-06 |

**Pages** | : 278 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**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.

**Author** | : N. H. March |

**Publisher** | : Academic Press |

**Release Date** | : 2013-10-22 |

**Pages** | : 628 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**Recent devopments, particularly in high-energy physics, have projected group theory and symmetry consideration into a central position in theoretical physics. These developments have taken physicists increasingly deeper into the fascinating world of pure mathematics. This work presents important mathematical developments of the last fifteen years in a form that is easy to comprehend and appreciate.

**Author** | : Xiaoping Xu |

**Publisher** | : Springer Science & Business Media |

**Release Date** | : 1998-09-30 |

**Pages** | : 356 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**Vertex algebra was introduced by Boreherds, and the slightly revised notion "vertex oper ator algebra" was formulated by Frenkel, Lepowsky and Meurman, in order to solve the problem of the moonshine representation of the Monster group - the largest sporadie group. On the one hand, vertex operator algebras ean be viewed as extensions of eertain infinite-dimensional Lie algebras such as affine Lie algebras and the Virasoro algebra. On the other hand, they are natural one-variable generalizations of commutative associative algebras with an identity element. In a certain sense, Lie algebras and commutative asso ciative algebras are reconciled in vertex operator algebras. Moreover, some other algebraie structures, such as integral linear lattiees, Jordan algebras and noncommutative associa tive algebras, also appear as subalgebraic structures of vertex operator algebras. The axioms of vertex operator algebra have geometrie interpretations in terms of Riemman spheres with punctures. The trace functions of a certain component of vertex operators enjoy the modular invariant properties. Vertex operator algebras appeared in physies as the fundamental algebraic structures of eonformal field theory, whieh plays an important role in string theory and statistieal meehanies. Moreover,eonformalfieldtheoryreveals animportantmathematiealproperty,the so called "mirror symmetry" among Calabi-Yau manifolds. The general correspondence between vertex operator algebras and Calabi-Yau manifolds still remains mysterious. Ever since the first book on vertex operator algebras by Frenkel, Lepowsky and Meur man was published in 1988, there has been a rapid development in vertex operator su peralgebras, which are slight generalizations of vertex operator algebras.

**Author** | : Petr P. Kulish |

**Publisher** | : Springer Science & Business Media |

**Release Date** | : 2006-01-17 |

**Pages** | : 263 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**This volume presents the invited lectures of the workshop "Infinite Dimensional Algebras and Quantum Integrable Systems" held in July 2003 at the University of Algarve, Faro, Portugal, as a satellite workshop of the XIV International Congress on Mathematical Physics. In it, recent developments in the theory of infinite dimensional algebras, and their applications to quantum integrable systems, are reviewed by leading experts in the field.

**Author** | : Chat Yin Ho |

**Publisher** | : Walter de Gruyter |

**Release Date** | : 2004-01-01 |

**Pages** | : 432 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**Dieser Band ist eine Sammlung von Forschungsartikeln zu endlichen Gruppen. Die behandelten Themen umfassen die Klassifikation von endlichen einfachen Gruppen, die Theorie der p-Gruppen, die Kohomologie von Gruppen, die Darstellungstheorie und die Theorie der Gebäude und der Geometrie.

**Author** | : Noga Alon |

**Publisher** | : Springer Science & Business Media |

**Release Date** | : 2011-03-31 |

**Pages** | : 454 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**"Visions in Mathematics - Towards 2000" was one of the most remarkable mathematical meetings in recent years. It was held in Tel Aviv from August 25th to September 3rd, 1999, and united some of the leading mathematicians worldwide. The goals of the conference were to discuss the importance, the methods, the past and the future of mathematics as we enter the 21st century and to consider the connection between mathematics and related areas. The aims of the conference are reflected in the present set of survey articles, documenting the state of art and future prospects in many branches of mathematics of current interest. This is the first part of a two-volume set that will serve any research mathematician or advanced student as an overview and guideline through the multifaceted body of mathematical research in the present and near future.

**Author** | : Aleksander Strasburger |

**Publisher** | : |

**Release Date** | : 2002 |

**Pages** | : 357 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:** **Author** | : Jakob Schütt |

**Publisher** | : |

**Release Date** | : 2018 |

**Pages** | : |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**In dieser Arbeit stellen wir eine zugängliche Einführung in die Theorie lokalkonvexer Supermannigfaltigkeiten im Rahmen des kategoriellen Ansatzes vor. Hierbei wird ein besonderer Schwerpunkt auf Lie-Supergruppen und die Supergruppe der Superdiffeomorphismen gelegt. In diesem Zugang ist eine Supermannigfaltigkeit ein Funktor von der Kategorie der Grassmann-Algebren in die Kategorie der lokalkonvexen Mannigfaltigkeiten, der bestimmte lokale Modelle besitzt, die etwas wie einen Atlas bilden. Wir zeigen, dass die Werte dieser Funktoren die Struktur sogenannter multilinearer Bündel haben. Wir nutzen dies aus um einen treuen Funktor von der Kategorie der Supermannigfaltigkeiten in die Kategorie der Mannigfaltigkeiten zu konstruieren. Dieser Funktor erhält Produkte, vertauscht mit dem jeweiligen Tangentialfunktor und erhält die jeweilige Hausdorff Eigenschaft. Auf diese Weise können wir Supermannigfaltigkeiten als eine besondere Art von unendlich-dimensionalen Faserbündeln auffassen. Mittels ähnlicher Techniken erhalten wir einige nützliche Trivialisierungen von Lie-Supergruppen, sowie eine kanonische Zerlegung in einen rein geraden und einen rein ungeraden Teil. Dies erlaubt uns die klassische Äquivalenz zwischen Lie-Supergruppen und Super-Harish-Chandra-Paaren auf den Fall lokalkonvexer Lie-Supergruppen zu verallgemeinern. Die Supergruppe der Superdiffeomorphismen einer Supermannigfaltigkeit M ist ein Set-wertiger Funktor SDiff(M), der gewisse Aspekte gerader und ungerader Transformationen von M beschreibt. Wir zeigen, dass SDiff(M) sich im Wesentlichen genau wie eine Lie-Supergruppe zerlegen lässt. Falls M eine Banach-Supermannigfaltigkeit mit sigma-kompakter, endlich-dimensionaler Basis ist, gelingt es uns der Supergruppe der kompakt getragenen Superdiffeomorphismen die Struktur einer Lie-Supergruppe zu geben. - In this thesis, we provide an accessible introduction to the theory of locally convex supermanifolds in the categorical approach with a focus on Lie supergroups and the supergroup of superdiffeomorphisms. In this setting, a supermanifold is a functor from the category of Grassmann algebras to the category of locally convex manifolds that has certain local models, forming something akin to an atlas. We show that the values that these functors take have the structure of a so called multilinear bundle. We use this fact to construct a faithful functor from the category of supermanifolds to the category of manifolds. This functor respects products, commutes with the respective tangent functor and retains the respective Hausdorff property. In this way, supermanifolds can be seen as a particular kind of infinite-dimensional fiber bundles. For Lie supergroups, we use similar techniques to show several useful trivializations and construct a canonical decomposition into purely even and purely odd parts. Using this, we are able to generalize the classical equivalence between Lie supergroups and super Harish-Chandra pairs to the case of arbitrary locally convex Lie supergroups. The supergroup of superdiffeomorphisms of a supermanifold M is a certain set-valued functor SDiff(M) from the category of Grassmann algebras that captures even and odd aspects of supersmooth transformations of M. We show that SDiff(M) has essentially the same decompositions as a Lie supergroup for an arbitrary supermanifold M. If M is a Banach supermanifold with finite-dimensional and sigma-compact base manifold, we are able to turn the supergroup of superdiffeomorphisms with compact support into a Lie supergroup.

**Author** | : |

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**Release Date** | : 2004 |

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**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:** **Author** | : |

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**Release Date** | : 2007 |

**Pages** | : |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:** **Author** | : Nihon Sūgakkai |

**Publisher** | : |

**Release Date** | : 2005 |

**Pages** | : |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:** **Author** | : James Lepowsky |

**Publisher** | : Springer Science & Business Media |

**Release Date** | : 2004 |

**Pages** | : 318 |

**ISBN** | : |

**Available Language** | : English, Spanish, And French |

**EBOOK SYNOPSIS:**The deep and relatively new field of vertex operator algebras is intimately related to a variety of areas in mathematics and physics: for example, the concepts of "monstrous moonshine," infinite-dimensional Lie theory, string theory, and conformal field theory. This book introduces the reader to the fundamental theory of vertex operator algebras and its basic techniques and examples. Beginning with a detailed presentation of the theoretical foundations and proceeding to a range of applications, the text includes a number of new, original results and also highlights and brings fresh perspective to important works of many researchers. After introducing the elementary "formal calculus'' underlying the subject, the book provides an axiomatic development of vertex operator algebras and their modules, expanding on the early contributions of R. Borcherds, I. Frenkel, J. Lepowsky, A. Meurman, Y.-Z. Huang, C. Dong, Y. Zhu and others. The concept of a "representation'' of a vertex (operator) algebra is treated in detail, following and extending the work of H. Li; this approach is used to construct important families of vertex (operator) algebras and their modules. Requiring only a familiarity with basic algebra, Introduction to Vertex Operator Algebras and Their Representations will be useful for graduate students and researchers in mathematics and physics. The booka??s presentation of the core topics will equip readers to embark on many active research directions related to vertex operator algebras, group theory, representation theory, and string theory.