**Author**: Gijs M. Tuynman

**Publisher:** Springer Science & Business Media

**ISBN:** 1402022972

**Category : **Mathematics

**Languages : **en

**Pages : **416

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**Book Description**
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.

**Author**: Gijs M. Tuynman

**Publisher:** Springer Science & Business Media

**ISBN:** 1402022972

**Category : **Mathematics

**Languages : **en

**Pages : **416

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**Book Description**
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.

**Author**: Gijs M. Tuynman

**Publisher:** Springer

**ISBN:** 9789048100460

**Category : **Mathematics

**Languages : **en

**Pages : **416

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**Book Description**
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.

**Author**: C. Bartocci

**Publisher:** Springer Science & Business Media

**ISBN:** 9401135045

**Category : **Mathematics

**Languages : **en

**Pages : **242

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**Book Description**
'Et moi ... - si favait III mmment en revenir, One service mathematics has rendered the je n'y serais point aile:' human race. It has put CXlUImon sense back Iules Verne where it belongs. on the topmost shelf next to the dUlty canister lahelled 'discarded non- The series i. divergent; therefore we may be able to do something with it. Eric T. Bell O. Hesvi.ide Mathematics is a tool for thOUght. A highly necessary tool in a world where both feedback and non linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d't!tre of this series.

**Author**: T. Voronov

**Publisher:** CRC Press

**ISBN:** 9783718651993

**Category : **Mathematics

**Languages : **en

**Pages : **138

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**Book Description**
The author presents the first detailed and original account of his theory of forms on supermanifolds-a correct and non-trivial analogue of Cartan-de Rham theory based on new concepts. The paper develops the apparatus of supermanifold differential topology necessary for the integration theory. A key feature is the identification of a class of proper morphisms intimately connected with Berezin integration, which are of fundamental importance in various problems. The work also contains a condensed introduction to superanalysis and supermanifolds, free from algebraic formalism, which sets out afresh such challenging problems as the Berezin intgegral on a bounded domain.

**Author**: Rita Fioresi

**Publisher:** American Mathematical Soc.

**ISBN:** 0821853007

**Category : **Mathematics

**Languages : **en

**Pages : **64

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**Book Description**
In the framework of algebraic supergeometry, the authors give a construction of the scheme-theoretic supergeometric analogue of split reductive algebraic group-schemes, namely affine algebraic supergroups associated to simple Lie superalgebras of classical type. In particular, all Lie superalgebras of both basic and strange types are considered. This provides a unified approach to most of the algebraic supergroups considered so far in the literature, and an effective method to construct new ones. The authors' method follows the pattern of a suitable scheme-theoretic revisitation of Chevalley's construction of semisimple algebraic groups, adapted to the reductive case. As an intermediate step, they prove an existence theorem for Chevalley bases of simple classical Lie superalgebras and a PBW-like theorem for their associated Kostant superalgebras.

**Author**: S. Duplij

**Publisher:** Springer Science & Business Media

**ISBN:** 9401008361

**Category : **Science

**Languages : **en

**Pages : **484

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**Book Description**
A presentation of outstanding achievements and ideas, of both eastern and western scientists, both mathematicians and physicists. Their presentations of recent work on quantum field theory, supergravity, M-theory, black holes and quantum gravity, together with research into noncommutative geometry, Hopf algebras, representation theory, categories and quantum groups, take the reader to the forefront of the latest developments. Other topics covered include supergravity and branes, supersymmetric quantum mechanics and superparticles, (super) black holes, superalgebra representations, and SUSY GUT phenomenology. Essential reading for workers in the modern methods of theoretical and mathematical physics.

**Author**: Castellani Leonardo

**Publisher:** World Scientific Publishing Company

**ISBN:** 9814590738

**Category : **
**Languages : **en

**Pages : **2216

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**Book Description**
This book provides a self-contained presentation of supergravity theories from its fundamentals to its most recent union with string and superstring theories, which are also reviewed in a self-contained manner. The subject is presented consistently in a unified geometric formalism, relying on the calculus of exterior forms and the mathematics needed to develop the theory is explained in appropriate chapters.

**Author**: Jakob Schütt

**Publisher:**
**ISBN:**
**Category : **
**Languages : **en

**Pages : **
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**Book Description**
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**: Maria Gorelik

**Publisher:** Springer Science & Business

**ISBN:** 3319029525

**Category : **Mathematics

**Languages : **en

**Pages : **280

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**Book Description**
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**: Ivan Penkov

**Publisher:** Springer Nature

**ISBN:** 3030896609

**Category : **Mathematics

**Languages : **en

**Pages : **245

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**Book Description**
Originating from graduate topics courses given by the first author, this book functions as a unique text-monograph hybrid that bridges a traditional graduate course to research level representation theory. The exposition includes an introduction to the subject, some highlights of the theory and recent results in the field, and is therefore appropriate for advanced graduate students entering the field as well as research mathematicians wishing to expand their knowledge. The mathematical background required varies from chapter to chapter, but a standard course on Lie algebras and their representations, along with some knowledge of homological algebra, is necessary. Basic algebraic geometry and sheaf cohomology are needed for Chapter 10. Exercises of various levels of difficulty are interlaced throughout the text to add depth to topical comprehension. The unifying theme of this book is the structure and representation theory of infinite-dimensional locally reductive Lie algebras and superalgebras. Chapters 1-6 are foundational; each of the last 4 chapters presents a self-contained study of a specialized topic within the larger field. Lie superalgebras and flag supermanifolds are discussed in Chapters 3, 7, and 10, and may be skipped by the reader.