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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.

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.

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.

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.

Applied Exterior Calculus

Applied Exterior Calculus Book
Author : Dominic G. B. Edelen
Publisher : Courier Corporation
Release : 1985
ISBN : 0486438716
Language : En, Es, Fr & De

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Book Description :

This text begins with the essentials, advancing to applications and studies of physical disciplines, including classical and irreversible thermodynamics, electrodynamics, and the theory of gauge fields. Geared toward advanced undergraduates and graduate students, it develops most of the theory and requires only a familiarity with upper-division algebra and mathematical analysis. "Essential." — SciTech Book News. 1985 edition.

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.

Hochschild Cohomology for Algebras

Hochschild Cohomology for Algebras Book
Author : Sarah J. Witherspoon
Publisher : American Mathematical Soc.
Release : 2019-12-10
ISBN : 1470449315
Language : En, Es, Fr & De

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Book Description :

This book gives a thorough and self-contained introduction to the theory of Hochschild cohomology for algebras and includes many examples and exercises. The book then explores Hochschild cohomology as a Gerstenhaber algebra in detail, the notions of smoothness and duality, algebraic deformation theory, infinity structures, support varieties, and connections to Hopf algebra cohomology. Useful homological algebra background is provided in an appendix. The book is designed both as an introduction for advanced graduate students and as a resource for mathematicians who use Hochschild cohomology in their work.

Modules and Rings

Modules and Rings Book
Author : John Dauns
Publisher : Cambridge University Press
Release : 1994-10-28
ISBN : 9780521462587
Language : En, Es, Fr & De

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Book Description :

This book on modern module and non-commutative ring theory is ideal for beginning graduate students. It starts at the foundations of the subject and progresses rapidly through the basic concepts to help the reader reach current research frontiers. Students will have the chance to develop proofs, solve problems, and to find interesting questions. The first half of the book is concerned with free, projective, and injective modules, tensor algebras, simple modules and primitive rings, the Jacobson radical, and subdirect products. Later in the book, more advanced topics, such as hereditary rings, categories and functors, flat modules, and purity are introduced. These later chapters will also prove a useful reference for researchers in non-commutative ring theory. Enough background material (including detailed proofs) is supplied to give the student a firm grounding in the subject.

The Algebraic Theory of Spinors and Clifford Algebras

The Algebraic Theory of Spinors and Clifford Algebras Book
Author : Claude Chevalley
Publisher : Springer Science & Business Media
Release : 1996-12-13
ISBN : 9783540570639
Language : En, Es, Fr & De

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Book Description :

In 1982, Claude Chevalley expressed three specific wishes with respect to the publication of his Works. First, he stated very clearly that such a publication should include his non technical papers. His reasons for that were two-fold. One reason was his life long commitment to epistemology and to politics, which made him strongly opposed to the view otherwise currently held that mathematics involves only half of a man. As he wrote to G. C. Rota on November 29th, 1982: "An important number of papers published by me are not of a mathematical nature. Some have epistemological features which might explain their presence in an edition of collected papers of a mathematician, but quite a number of them are concerned with theoretical politics ( . . . ) they reflect an aspect of myself the omission of which would, I think, give a wrong idea of my lines of thinking". On the other hand, Chevalley thought that the Collected Works of a mathematician ought to be read not only by other mathematicians, but also by historians of science.

Rings with Polynomial Identities and Finite Dimensional Representations of Algebras

Rings with Polynomial Identities and Finite Dimensional Representations of Algebras Book
Author : Eli Aljadeff,Antonio Giambruno,Claudio Procesi,Amitai Regev
Publisher : American Mathematical Soc.
Release : 2020-12-14
ISBN : 1470451743
Language : En, Es, Fr & De

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Book Description :

A polynomial identity for an algebra (or a ring) A A is a polynomial in noncommutative variables that vanishes under any evaluation in A A. An algebra satisfying a nontrivial polynomial identity is called a PI algebra, and this is the main object of study in this book, which can be used by graduate students and researchers alike. The book is divided into four parts. Part 1 contains foundational material on representation theory and noncommutative algebra. In addition to setting the stage for the rest of the book, this part can be used for an introductory course in noncommutative algebra. An expert reader may use Part 1 as reference and start with the main topics in the remaining parts. Part 2 discusses the combinatorial aspects of the theory, the growth theorem, and Shirshov's bases. Here methods of representation theory of the symmetric group play a major role. Part 3 contains the main body of structure theorems for PI algebras, theorems of Kaplansky and Posner, the theory of central polynomials, M. Artin's theorem on Azumaya algebras, and the geometric part on the variety of semisimple representations, including the foundations of the theory of Cayley–Hamilton algebras. Part 4 is devoted first to the proof of the theorem of Razmyslov, Kemer, and Braun on the nilpotency of the nil radical for finitely generated PI algebras over Noetherian rings, then to the theory of Kemer and the Specht problem. Finally, the authors discuss PI exponent and codimension growth. This part uses some nontrivial analytic tools coming from probability theory. The appendix presents the counterexamples of Golod and Shafarevich to the Burnside problem.

Spectra and the Steenrod Algebra

Spectra and the Steenrod Algebra Book
Author : H.R. Margolis
Publisher : Elsevier
Release : 2011-08-18
ISBN : 0080960170
Language : En, Es, Fr & De

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Book Description :

I have intended this book to be more than just the sum of its chapters, and the introduction is, in part, an attempt to spell out what the more is. Algebraic topology is the study of topological problems by algebraic means. More precisely, this has come to be framed as the study of topological categories by means of functors to algebraic categories. Beyond the basic definitions and structure, the focus is often on particular problems, for example, Adams’ use of K-theory to solve the vector fields on spheres problem. On the other hand, there are contributions of a more global nature yielding insight into the overall structure of some topological category, for example, Quillen’s work on rational homotopy type. This book is intended primarily as a contribution of this latter sort. So while there will be a variety of particular examples and computations, and although the structure being developed has significant application to many specific problems (some of which are considered here), the major thrust of the text is toward understanding the global structure and linkage of the topological and algebraic categories considered: the stable homotopy category and the category of modules over the Steenrod algebra.

Advances in Representation Theory of Algebras

Advances in Representation Theory of Algebras Book
Author : Ibrahim Assem,Christof Geiß,Sonia Trepode
Publisher : American Mathematical Soc.
Release : 2021-01-06
ISBN : 147045159X
Language : En, Es, Fr & De

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Book Description :

The Seventh ARTA (“Advances in Representation Theory of Algebras VII”) conference took place at the Instituto de Matemáticas of the Universidad Nacional Autónoma de México, in Mexico City, from September 24–28, 2018, in honor of José Antonio de la Peña's 60th birthday. Papers in this volume cover topics Professor de la Peña worked on, such as covering theory, tame algebras, and the use of quadratic forms in representation theory. Also included are papers on the categorical approach to representations of algebras and relations to Lie theory, Cohen–Macaulay modules, quantum groups and other algebraic structures.

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.

Orthogonal and Symplectic Clifford Algebras

Orthogonal and Symplectic Clifford Algebras Book
Author : A. Crumeyrolle
Publisher : Springer Science & Business Media
Release : 2013-06-29
ISBN : 940157877X
Language : En, Es, Fr & De

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Book Description :

Download Orthogonal and Symplectic Clifford Algebras book written by A. Crumeyrolle, available in PDF, EPUB, and Kindle, or read full book online anywhere and anytime. Compatible with any devices.

Clifford Algebras and their Applications in Mathematical Physics

Clifford Algebras and their Applications in Mathematical Physics Book
Author : Rafal Ablamowicz,Bertfried Fauser
Publisher : Springer Science & Business Media
Release : 2012-12-06
ISBN : 1461213681
Language : En, Es, Fr & De

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Book Description :

The plausible relativistic physical variables describing a spinning, charged and massive particle are, besides the charge itself, its Minkowski (four) po sition X, its relativistic linear (four) momentum P and also its so-called Lorentz (four) angular momentum E # 0, the latter forming four trans lation invariant part of its total angular (four) momentum M. Expressing these variables in terms of Poincare covariant real valued functions defined on an extended relativistic phase space [2, 7J means that the mutual Pois son bracket relations among the total angular momentum functions Mab and the linear momentum functions pa have to represent the commutation relations of the Poincare algebra. On any such an extended relativistic phase space, as shown by Zakrzewski [2, 7], the (natural?) Poisson bracket relations (1. 1) imply that for the splitting of the total angular momentum into its orbital and its spin part (1. 2) one necessarily obtains (1. 3) On the other hand it is always possible to shift (translate) the commuting (see (1. 1)) four position xa by a four vector ~Xa (1. 4) so that the total angular four momentum splits instead into a new orbital and a new (Pauli-Lubanski) spin part (1. 5) in such a way that (1. 6) However, as proved by Zakrzewski [2, 7J, the so-defined new shifted four a position functions X must fulfill the following Poisson bracket relations: (1.

Clifford Algebras And Zeons Geometry To Combinatorics And Beyond

Clifford Algebras And Zeons  Geometry To Combinatorics And Beyond Book
Author : George Stacey Staples
Publisher : World Scientific
Release : 2019-09-24
ISBN : 9811202591
Language : En, Es, Fr & De

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Book Description :

Clifford algebras have many well-known applications in physics, engineering, and computer graphics. Zeon algebras are subalgebras of Clifford algebras whose combinatorial properties lend them to graph-theoretic applications such as enumerating minimal cost paths in dynamic networks. This book provides a foundational working knowledge of zeon algebras, their properties, and their potential applications in an increasingly technological world.As a graduate-level or advanced undergraduate-level mathematics textbook, it is suitable for self-study by researchers interested in new approaches to existing combinatorial problems and applications (wireless networks, Boolean satisfiability, coding theory, etc.).As the first textbook to explore algebraic and combinatorial properties of zeon algebras in depth, it is suitable for interdisciplinary study in analysis, algebra, and combinatorics. The material is complemented by the CliffMath software package for Mathematica, which is freely available through the book's webpage.

Hopf Algebras in Noncommutative Geometry and Physics

Hopf Algebras in Noncommutative Geometry and Physics Book
Author : Stefaan Caenepeel,Fred Van Oystaeyen
Publisher : CRC Press
Release : 2019-05-07
ISBN : 0429530072
Language : En, Es, Fr & De

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Book Description :

This comprehensive reference summarizes the proceedings and keynote presentations from a recent conference held in Brussels, Belgium. Offering 1155 display equations, this volume contains original research and survey papers as well as contributions from world-renowned algebraists. It focuses on new results in classical Hopf algebras as well as the

Linear Algebra over Commutative Rings

Linear Algebra over Commutative Rings Book
Author : Bernard R. McDonald
Publisher : CRC Press
Release : 2020-11-25
ISBN : 1000110613
Language : En, Es, Fr & De

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Book Description :

This monograph arose from lectures at the University of Oklahoma on topics related to linear algebra over commutative rings. It provides an introduction of matrix theory over commutative rings. The monograph discusses the structure theory of a projective module.

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.