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Exterior Algebras

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Exterior Algebras

Exterior Algebras Book
Author : Vincent Pavan
Publisher : Elsevier
Release : 2017-05-25
ISBN : 0081023480
Language : En, Es, Fr & De

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Book Description :

Exterior Algebras: Elementary Tribute to Grassmann's Ideas provides the theoretical basis for exterior computations. It first addresses the important question of constructing (pseudo)-Euclidian Grassmmann's algebras. Then, it shows how the latter can be used to treat a few basic, though significant, questions of linear algebra, such as co-linearity, determinant calculus, linear systems analyzing, volumes computations, invariant endomorphism considerations, skew-symmetric operator studies and decompositions, and Hodge conjugation, amongst others. Presents a self-contained guide that does not require any specific algebraic background Includes numerous examples and direct applications that are suited for beginners

Tensor Spaces and Exterior Algebra

Tensor Spaces and Exterior Algebra Book
Author : Takeo Yokonuma
Publisher : American Mathematical Soc.
Release : 1992
ISBN : 9780821827963
Language : En, Es, Fr & De

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Book Description :

This book explains, as clearly as possible, tensors and such related topics as tensor products of vector spaces, tensor algebras, and exterior algebras. You will appreciate Yokonuma's lucid and methodical treatment of the subject. This book is useful in undergraduate and graduate courses in multilinear algebra. Tensor Spaces and Exterior Algebra begins with basic notions associated with tensors. to facilitate understanding of the definitions, Yokonuma often presents two or more different ways of describing one object. Next, the properties and applications of tensors are developed, including the classical definition of tensors and the description of relative tensors. Also discussed are the algebraic foundations of tensor calculus and applications of exterior algebra to determinants and to geometry. This book closes with an examination of algebraic systems with bilinear multiplication. in particular, Yokonuma discusses the theory of replicas of Chevalley and several properties of Lie algebras deduced from them.

Linear Algebra Via Exterior Products

Linear Algebra Via Exterior Products Book
Author : Sergei Winitzki
Publisher : Sergei Winitzki
Release : 2009-07-30
ISBN : 140929496X
Language : En, Es, Fr & De

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Book Description :

This is a pedagogical introduction to the coordinate-free approach in basic finite-dimensional linear algebra. The reader should be already exposed to the array-based formalism of vector and matrix calculations. This book makes extensive use of the exterior (anti-commutative, "wedge") product of vectors. The coordinate-free formalism and the exterior product, while somewhat more abstract, provide a deeper understanding of the classical results in linear algebra. Without cumbersome matrix calculations, this text derives the standard properties of determinants, the Pythagorean formula for multidimensional volumes, the formulas of Jacobi and Liouville, the Cayley-Hamilton theorem, the Jordan canonical form, the properties of Pfaffians, as well as some generalizations of these results.

A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods

A Survey of Lie Groups and Lie Algebras with Applications and Computational Methods Book
Author : Johan G. F. Belinfante,Bernard Kolman
Publisher : SIAM
Release : 1989-01-01
ISBN : 9781611971330
Language : En, Es, Fr & De

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Book Description :

Introduces the concepts and methods of the Lie theory in a form accessible to the non-specialist by keeping mathematical prerequisites to a minimum. Although the authors have concentrated on presenting results while omitting most of the proofs, they have compensated for these omissions by including many references to the original literature. Their treatment is directed toward the reader seeking a broad view of the subject rather than elaborate information about technical details. Illustrations of various points of the Lie theory itself are found throughout the book in material on applications. In this reprint edition, the authors have resisted the temptation of including additional topics. Except for correcting a few minor misprints, the character of the book, especially its focus on classical representation theory and its computational aspects, has not been changed.

Algebra I

Algebra I Book
Author : N. Bourbaki
Publisher : Springer Science & Business Media
Release : 1998-08-03
ISBN : 9783540642435
Language : En, Es, Fr & De

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Book Description :

An exposition of the fundamentals of general, linear and multilinear algebra. The first chapter introduces the basic objects: groups, actions, rings, fields. The second chapter studies the properties of modules and linear maps, and the third investigatesalgebras, particularly tensor algebras.

From Vectors to Tensors

From Vectors to Tensors Book
Author : Juan R. Ruiz-Tolosa,Enrique Castillo
Publisher : Springer Science & Business Media
Release : 2006-03-30
ISBN : 3540270663
Language : En, Es, Fr & De

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Book Description :

This textbook deals with tensors that are treated as vectors. Coverage details such new tensor concepts as the rotation of tensors, the transposer tensor, the eigentensors, and the permutation tensor structure. The book covers an existing gap between the classic theory of tensors and the possibility of solving tensor problems with a computer. A complementary computer package, written in Mathematica, is available through the Internet.

Clifford Algebras and Spinor Structures

Clifford Algebras and Spinor Structures Book
Author : Rafal Ablamowicz,P. Lounesto
Publisher : Springer Science & Business Media
Release : 2013-06-29
ISBN : 9401584222
Language : En, Es, Fr & De

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Book Description :

This volume is dedicated to the memory of Albert Crumeyrolle, who died on June 17, 1992. In organizing the volume we gave priority to: articles summarizing Crumeyrolle's own work in differential geometry, general relativity and spinors, articles which give the reader an idea of the depth and breadth of Crumeyrolle's research interests and influence in the field, articles of high scientific quality which would be of general interest. In each of the areas to which Crumeyrolle made significant contribution - Clifford and exterior algebras, Weyl and pure spinors, spin structures on manifolds, principle of triality, conformal geometry - there has been substantial progress. Our hope is that the volume conveys the originality of Crumeyrolle's own work, the continuing vitality of the field he influenced, and the enduring respect for, and tribute to, him and his accomplishments in the mathematical community. It isour pleasure to thank Peter Morgan, Artibano Micali, Joseph Grifone, Marie Crumeyrolle and Kluwer Academic Publishers for their help in preparingthis volume.

Lectures on Exterior Algebras Over Commutative Rings

Lectures on Exterior Algebras Over Commutative Rings Book
Author : Robert B. Gardner
Publisher : Unknown
Release : 1972
ISBN : 0987650XXX
Language : En, Es, Fr & De

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Book Description :

Download Lectures on Exterior Algebras Over Commutative Rings book written by Robert B. Gardner, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Fundamental Concepts of Algebra

Fundamental Concepts of Algebra Book
Author : Anonim
Publisher : Academic Press
Release : 1957-01-01
ISBN : 9780080873152
Language : En, Es, Fr & De

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Book Description :

Fundamental Concepts of Algebra

Algebras Rings and Modules

Algebras  Rings and Modules Book
Author : Michiel Hazewinkel,Vladimir V. Kirichenko
Publisher : Springer Science & Business Media
Release : 2007
ISBN : 1402051409
Language : En, Es, Fr & De

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Book Description :

Download Algebras Rings and Modules book written by Michiel Hazewinkel,Vladimir V. Kirichenko, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Clifford Algebras and Lie Theory

Clifford Algebras and Lie Theory Book
Author : Eckhard Meinrenken
Publisher : Springer Science & Business Media
Release : 2013-02-28
ISBN : 3642362168
Language : En, Es, Fr & De

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Book Description :

This monograph provides an introduction to the theory of Clifford algebras, with an emphasis on its connections with the theory of Lie groups and Lie algebras. The book starts with a detailed presentation of the main results on symmetric bilinear forms and Clifford algebras. It develops the spin groups and the spin representation, culminating in Cartan’s famous triality automorphism for the group Spin(8). The discussion of enveloping algebras includes a presentation of Petracci’s proof of the Poincaré–Birkhoff–Witt theorem. This is followed by discussions of Weil algebras, Chern--Weil theory, the quantum Weil algebra, and the cubic Dirac operator. The applications to Lie theory include Duflo’s theorem for the case of quadratic Lie algebras, multiplets of representations, and Dirac induction. The last part of the book is an account of Kostant’s structure theory of the Clifford algebra over a semisimple Lie algebra. It describes his “Clifford algebra analogue” of the Hopf–Koszul–Samelson theorem, and explains his fascinating conjecture relating the Harish-Chandra projection for Clifford algebras to the principal sl(2) subalgebra. Aside from these beautiful applications, the book will serve as a convenient and up-to-date reference for background material from Clifford theory, relevant for students and researchers in mathematics and physics.

Abstract Algebra

Abstract Algebra Book
Author : Pierre Antoine Grillet
Publisher : Springer Science & Business Media
Release : 2007-07-21
ISBN : 0387715681
Language : En, Es, Fr & De

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Book Description :

A completely reworked new edition of this superb textbook. This key work is geared to the needs of the graduate student. It covers, with proofs, the usual major branches of groups, rings, fields, and modules. Its inclusive approach means that all of the necessary areas are explored, while the level of detail is ideal for the intended readership. The text tries to promote the conceptual understanding of algebra as a whole, doing so with a masterful grasp of methodology. Despite the abstract subject matter, the author includes a careful selection of important examples, together with a detailed elaboration of the more sophisticated, abstract theories.

Clifford Algebras and Spinors

Clifford Algebras and Spinors Book
Author : Pertti Lounesto
Publisher : Cambridge University Press
Release : 2001-05-03
ISBN : 0521005515
Language : En, Es, Fr & De

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Book Description :

In this book, Professor Lounesto offers a unique introduction to Clifford algebras and spinors. The initial chapters could be read by undergraduates; vectors, complex numbers and quaternions are introduced with an eye on Clifford algebras. The next chapters will also interest physicists, and include treatments of the quantum mechanics of the electron, electromagnetism and special relativity with a flavour of Clifford algebras. This book also gives the first comprehensive survey of recent research on Clifford algebras. A new classification of spinors is introduced, based on bilinear covariants of physical observables. This reveals a new class of spinors, residing between the Weyl, Majorana and Dirac spinors. Scalar products of spinors are classified by involutory anti-automorphisms of Clifford algebras. This leads to the chessboard of automorphism groups of scalar products of spinors. On the analytic side, Brauer-Wall groups and Witt rings are discussed, and Caucy's integral formula is generalized to higher dimensions.

Linear Algebra and Geometry

Linear Algebra and Geometry Book
Author : Igor R. Shafarevich,Alexey O. Remizov
Publisher : Springer Science & Business Media
Release : 2012-08-23
ISBN : 3642309941
Language : En, Es, Fr & De

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Book Description :

This book on linear algebra and geometry is based on a course given by renowned academician I.R. Shafarevich at Moscow State University. The book begins with the theory of linear algebraic equations and the basic elements of matrix theory and continues with vector spaces, linear transformations, inner product spaces, and the theory of affine and projective spaces. The book also includes some subjects that are naturally related to linear algebra but are usually not covered in such courses: exterior algebras, non-Euclidean geometry, topological properties of projective spaces, theory of quadrics (in affine and projective spaces), decomposition of finite abelian groups, and finitely generated periodic modules (similar to Jordan normal forms of linear operators). Mathematical reasoning, theorems, and concepts are illustrated with numerous examples from various fields of mathematics, including differential equations and differential geometry, as well as from mechanics and physics.

Building Bridges Between Algebra and Topology

Building Bridges Between Algebra and Topology Book
Author : Wojciech Chachólski,Tobias Dyckerhoff,John Greenlees,Greg Stevenson
Publisher : Birkhäuser
Release : 2018-03-31
ISBN : 3319701576
Language : En, Es, Fr & De

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Book Description :

This volume presents an elaborated version of lecture notes for two advanced courses: (Re)Emerging methods in Commutative Algebra and Representation Theory and Building Bridges Between Algebra and Topology, held at the CRM in the spring of 2015. Homological algebra is a rich and ubiquitous area; it is both an active field of research and a widespread toolbox for many mathematicians. Together, these notes introduce recent applications and interactions of homological methods in commutative algebra, representation theory and topology, narrowing the gap between specialists from different areas wishing to acquaint themselves with a rapidly growing field. The covered topics range from a fresh introduction to the growing area of support theory for triangulated categories to the striking consequences of the formulation in the homotopy theory of classical concepts in commutative algebra. Moreover, they also include a higher categories view of Hall algebras and an introduction to the use of idempotent functors in algebra and topology.

An Introduction to Clifford Algebras and Spinors

An Introduction to Clifford Algebras and Spinors Book
Author : Jayme Vaz Jr.,Roldão da Rocha Jr.
Publisher : Oxford University Press
Release : 2016
ISBN : 0198782926
Language : En, Es, Fr & De

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Book Description :

This work is unique compared to the existing literature. It is very didactical and accessible to both students and researchers, without neglecting the formal character and the deep algebraic completeness of the topic along with its physical applications.

Commutative Algebra

Commutative Algebra Book
Author : David Eisenbud
Publisher : Springer Science & Business Media
Release : 2013-12-01
ISBN : 1461253500
Language : En, Es, Fr & De

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Book Description :

This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. The book gives a concise treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Many exercises included.

Quadratic Mappings and Clifford Algebras

Quadratic Mappings and Clifford Algebras Book
Author : Jacques Helmstetter,Artibano Micali
Publisher : Springer Science & Business Media
Release : 2008-05-24
ISBN : 3764386061
Language : En, Es, Fr & De

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Book Description :

After general properties of quadratic mappings over rings, the authors more intensely study quadratic forms, and especially their Clifford algebras. To this purpose they review the required part of commutative algebra, and they present a significant part of the theory of graded Azumaya algebras. Interior multiplications and deformations of Clifford algebras are treated with the most efficient methods.

Handbook of Mathematics

Handbook of Mathematics Book
Author : Thierry Vialar
Publisher : BoD - Books on Demand
Release : 2016-12-07
ISBN : 2955199001
Language : En, Es, Fr & De

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Book Description :

The book consists of XI Parts and 28 Chapters covering all areas of mathematics. It is a tool for students, scientists, engineers, students of many disciplines, teachers, professionals, writers and also for a general reader with an interest in mathematics and in science. It provides a wide range of mathematical concepts, definitions, propositions, theorems, proofs, examples, and numerous illustrations. The difficulty level can vary depending on chapters, and sustained attention will be required for some. The structure and list of Parts are quite classical: I. Foundations of Mathematics, II. Algebra, III. Number Theory, IV. Geometry, V. Analytic Geometry, VI. Topology, VII .Algebraic Topology, VIII. Analysis, IX. Category Theory, X. Probability and Statistics, XI. Applied Mathematics. Appendices provide useful lists of symbols and tables for ready reference. The publisher’s hope is that this book, slightly revised and in a convenient format, will serve the needs of readers, be it for study, teaching, exploration, work, or research.

Hasse Schmidt Derivations on Grassmann Algebras

Hasse Schmidt Derivations on Grassmann Algebras Book
Author : Letterio Gatto,Parham Salehyan
Publisher : Springer
Release : 2016-07-08
ISBN : 331931842X
Language : En, Es, Fr & De

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Book Description :

This book provides a comprehensive advanced multi-linear algebra course based on the concept of Hasse-Schmidt derivations on a Grassmann algebra (an analogue of the Taylor expansion for real-valued functions), and shows how this notion provides a natural framework for many ostensibly unrelated subjects: traces of an endomorphism and the Cayley-Hamilton theorem, generic linear ODEs and their Wronskians, the exponential of a matrix with indeterminate entries (Putzer's method revisited), universal decomposition of a polynomial in the product of two monic polynomials of fixed smaller degree, Schubert calculus for Grassmannian varieties, and vertex operators obtained with the help of Schubert calculus tools (Giambelli's formula). Significant emphasis is placed on the characterization of decomposable tensors of an exterior power of a free abelian group of possibly infinite rank, which then leads to the celebrated Hirota bilinear form of the Kadomtsev-Petviashvili (KP) hierarchy describing the Plücker embedding of an infinite-dimensional Grassmannian. By gathering ostensibly disparate issues together under a unified perspective, the book reveals how even the most advanced topics can be discovered at the elementary level.