Free Algebras And Pi Algebras

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Free Algebras and PI-algebras

Author : Vesselin S. Drensky
Publisher :
ISBN 13 :
Total Pages : 292 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Free Algebras and PI-algebras by : Vesselin S. Drensky

Download or read book Free Algebras and PI-algebras written by Vesselin S. Drensky and published by . This book was released on 2000 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book is devoted to the combinatorial theory of polynomial algebras, free associative and free Lie algebras, and algebras with polynomial identities. It also examines the structure of automorphism groups of free and relatively free algebras. It is based on graduate courses and short cycles of lectures presented by the author at several universities and its goal is to involve the reader as soon as possible in the research area, to make him or her able to read books and papers on the considered topics. It contains both classical and contemporary results and methods. A specific feature of the book is that it includes as its inseparable part more than 250 exercises and examples with detailed hints (50 % of the numbered statements), some of them treating serious mathematical results. The exposition is accessible for graduate and advanced undergraduate students with standard background on linear algebra and some elements of ring theory and group theory. The professional mathematician working in the field of algebra and other related topics also will find the book useful for his or her research and teaching. TOC:Introduction 1. Commutative, Associative and Lie Algebras: Basic properties of algebras; Free algebras; The Poincaré-Birkhoff-Witt theorem. 2. Algebras with Polynomial Identities: Definitions and examples of PI-Algebras; Varieties and relatively free algebras; The theorem of Birkhoff. 3. The Specht Problem: The finite basis property; Lie algebras in characteristic 2. 4. Numerical Invariants of T-Ideals: Graded vector spaces; Homogeneous and multilinear polynomial identities; Proper polynomial identities. 5. Polynomial Identities of Concrete Algebras: Polynomial identities of the Grassmann algebra; Polynomial identities of the upper triangular matrices. 6. Methods of Commutative Algebra: Rational Hilbert series; Nonmatrix polynomial identities; Commutative and noncommutative invariant theory. 7. Polynomial Identities of the Matrix Algebras: The Amitsur-Levitzki theorem; Generic matrices; Central polynomials; Various identities of matrices. 8. Multilinear Polynomial Identities: The codimension theorem of Regev; Algebras with polynomial growth of codimensions; The Nagata-Higman theorem; The theory of Kemer. 9. Finitely Generated PI-Algebras: The problems of Burnside and Kurosch; The Shirshov theorem; Growth of algebras and Gelfand-Kirillov dimension; Gelfand-Kirillov dimension of PI-Algebras. 10. Automorphisms of Free Algebras: Automorphisms of groups and algebras; The polynomial algebra in two variables; The free associative algebra of rank two; Exponential automorphisms; Automorphisms of relatively free algebras. 11. Free Lie Algebras and Their Automorphisms: Bases and subalgebras of free Lie algebras; Automorphisms of free Lie algebras; Automorphisms of relatively free Lie algebras. 12. The Method of Representation Theory: Representations of finite groups; The symmetric group; Multilinear polynomial identities; The action of the general linear group; Proper polynomial identities; Polynomial identities of matrices.

PI-Algebras

Author : N. Jacobson
Publisher : Springer
ISBN 13 : 3540374272
Total Pages : 120 pages
Book Rating : 4.5/5 (43 download)

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Book Synopsis PI-Algebras by : N. Jacobson

Download or read book PI-Algebras written by N. Jacobson and published by Springer. This book was released on 2006-11-15 with total page 120 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Pi-Algebras

Author : N. Jacobson
Publisher :
ISBN 13 : 9783662177952
Total Pages : 128 pages
Book Rating : 4.1/5 (779 download)

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Book Synopsis Pi-Algebras by : N. Jacobson

Download or read book Pi-Algebras written by N. Jacobson and published by . This book was released on 2014-01-15 with total page 128 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Polynomial Identities in Algebras

Author : Onofrio Mario Di Vincenzo
Publisher : Springer Nature
ISBN 13 : 3030631117
Total Pages : 421 pages
Book Rating : 4.0/5 (36 download)

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Book Synopsis Polynomial Identities in Algebras by : Onofrio Mario Di Vincenzo

Download or read book Polynomial Identities in Algebras written by Onofrio Mario Di Vincenzo and published by Springer Nature. This book was released on 2021-03-22 with total page 421 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the talks given at the INDAM workshop entitled "Polynomial identites in algebras", held in Rome in September 2019. The purpose of the book is to present the current state of the art in the theory of PI-algebras. The review of the classical results in the last few years has pointed out new perspectives for the development of the theory. In particular, the contributions emphasize on the computational and combinatorial aspects of the theory, its connection with invariant theory, representation theory, growth problems. It is addressed to researchers in the field.

Rings with Polynomial Identities and Finite Dimensional Representations of Algebras

Author : Eli Aljadeff
Publisher : American Mathematical Soc.
ISBN 13 : 1470451743
Total Pages : 630 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Rings with Polynomial Identities and Finite Dimensional Representations of Algebras by : Eli Aljadeff

Download or read book Rings with Polynomial Identities and Finite Dimensional Representations of Algebras written by Eli Aljadeff and published by American Mathematical Soc.. This book was released on 2020-12-14 with total page 630 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Polynomial Identities And Combinatorial Methods

Author : Antonio Giambruno
Publisher : CRC Press
ISBN 13 : 9780203911549
Total Pages : 442 pages
Book Rating : 4.9/5 (115 download)

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Book Synopsis Polynomial Identities And Combinatorial Methods by : Antonio Giambruno

Download or read book Polynomial Identities And Combinatorial Methods written by Antonio Giambruno and published by CRC Press. This book was released on 2003-05-20 with total page 442 pages. Available in PDF, EPUB and Kindle. Book excerpt: Polynomial Identities and Combinatorial Methods presents a wide range of perspectives on topics ranging from ring theory and combinatorics to invariant theory and associative algebras. It covers recent breakthroughs and strategies impacting research on polynomial identities and identifies new concepts in algebraic combinatorics, invariant and representation theory, and Lie algebras and superalgebras for novel studies in the field. It presents intensive discussions on various methods and techniques relating the theory of polynomial identities to other branches of algebraic study and includes discussions on Hopf algebras and quantum polynomials, free algebras and Scheier varieties.

Polynomial Identities and Asymptotic Methods

Author : A. Giambruno
Publisher : American Mathematical Soc.
ISBN 13 : 0821838296
Total Pages : 370 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Polynomial Identities and Asymptotic Methods by : A. Giambruno

Download or read book Polynomial Identities and Asymptotic Methods written by A. Giambruno and published by American Mathematical Soc.. This book was released on 2005 with total page 370 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives a state of the art approach to the study of polynomial identities satisfied by a given algebra by combining methods of ring theory, combinatorics, and representation theory of groups with analysis. The idea of applying analytical methods to the theory of polynomial identities appeared in the early 1970s and this approach has become one of the most powerful tools of the theory. A PI-algebra is any algebra satisfying at least one nontrivial polynomial identity. This includes the polynomial rings in one or several variables, the Grassmann algebra, finite-dimensional algebras, and many other algebras occurring naturally in mathematics. The core of the book is the proof that the sequence of co-dimensions of any PI-algebra has integral exponential growth - the PI-exponent of the algebra. Later chapters further apply these results to subjects such as a characterization of varieties of algebras having polynomial growth and a classification of varieties that are minimal for a given exponent.

Pi - Algebras. An Introduction

Author : Nathan Jacobson
Publisher :
ISBN 13 :
Total Pages : 116 pages
Book Rating : 4.:/5 (859 download)

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Book Synopsis Pi - Algebras. An Introduction by : Nathan Jacobson

Download or read book Pi - Algebras. An Introduction written by Nathan Jacobson and published by . This book was released on 1975 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Ideals of Identities of Associative Algebras

Author : Aleksandr Robertovich Kemer
Publisher : American Mathematical Soc.
ISBN 13 : 9780821898291
Total Pages : 96 pages
Book Rating : 4.8/5 (982 download)

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Book Synopsis Ideals of Identities of Associative Algebras by : Aleksandr Robertovich Kemer

Download or read book Ideals of Identities of Associative Algebras written by Aleksandr Robertovich Kemer and published by American Mathematical Soc.. This book was released on with total page 96 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book concerns the study of the structure of identities of PI-algebras over a field of characteristic zero. In the first chapter, the author brings out the connection between varieties of algebras and finitely-generated superalgebras. The second chapter examines graded identities of finitely-generated PI-superalgebras. One of the results proved concerns the decomposition of T-ideals, which is very useful for the study of specific varieties. In the fifth section of Chapter Two, the author solves Specht's problem, which asks whether every associative algebra over a field of characteristic zero has a finite basis of identities. The book closes with an application of methods and results established earlier: the author finds asymptotic bases of identities of algebras with unity satisfying all of the identities of the full algebra of matrices of order two.

Computational Aspects of Polynomial Identities

Author : Alexei Kanel-Belov
Publisher : CRC Press
ISBN 13 : 1498720099
Total Pages : 418 pages
Book Rating : 4.4/5 (987 download)

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Book Synopsis Computational Aspects of Polynomial Identities by : Alexei Kanel-Belov

Download or read book Computational Aspects of Polynomial Identities written by Alexei Kanel-Belov and published by CRC Press. This book was released on 2015-10-22 with total page 418 pages. Available in PDF, EPUB and Kindle. Book excerpt: Computational Aspects of Polynomial Identities: Volume l, Kemer’s Theorems, 2nd Edition presents the underlying ideas in recent polynomial identity (PI)-theory and demonstrates the validity of the proofs of PI-theorems. This edition gives all the details involved in Kemer’s proof of Specht’s conjecture for affine PI-algebras in characteristic 0. The book first discusses the theory needed for Kemer’s proof, including the featured role of Grassmann algebra and the translation to superalgebras. The authors develop Kemer polynomials for arbitrary varieties as tools for proving diverse theorems. They also lay the groundwork for analogous theorems that have recently been proved for Lie algebras and alternative algebras. They then describe counterexamples to Specht’s conjecture in characteristic p as well as the underlying theory. The book also covers Noetherian PI-algebras, Poincaré–Hilbert series, Gelfand–Kirillov dimension, the combinatoric theory of affine PI-algebras, and homogeneous identities in terms of the representation theory of the general linear group GL. Through the theory of Kemer polynomials, this edition shows that the techniques of finite dimensional algebras are available for all affine PI-algebras. It also emphasizes the Grassmann algebra as a recurring theme, including in Rosset’s proof of the Amitsur–Levitzki theorem, a simple example of a finitely based T-ideal, the link between algebras and superalgebras, and a test algebra for counterexamples in characteristic p.

Polynomial Identity Rings

Author : Vesselin Drensky
Publisher : Birkhäuser
ISBN 13 : 3034879342
Total Pages : 197 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Polynomial Identity Rings by : Vesselin Drensky

Download or read book Polynomial Identity Rings written by Vesselin Drensky and published by Birkhäuser. This book was released on 2012-12-06 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: These lecture notes treat polynomial identity rings from both the combinatorial and structural points of view. The greater part of recent research in polynomial identity rings is about combinatorial questions, and the combinatorial part of the lecture notes gives an up-to-date account of recent research. On the other hand, the main structural results have been known for some time, and the emphasis there is on a presentation accessible to newcomers to the subject.

Introduction to Noncommutative Algebra

Author : Matej Brešar
Publisher : Springer
ISBN 13 : 3319086936
Total Pages : 227 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Introduction to Noncommutative Algebra by : Matej Brešar

Download or read book Introduction to Noncommutative Algebra written by Matej Brešar and published by Springer. This book was released on 2014-10-14 with total page 227 pages. Available in PDF, EPUB and Kindle. Book excerpt: Providing an elementary introduction to noncommutative rings and algebras, this textbook begins with the classical theory of finite dimensional algebras. Only after this, modules, vector spaces over division rings, and tensor products are introduced and studied. This is followed by Jacobson's structure theory of rings. The final chapters treat free algebras, polynomial identities, and rings of quotients. Many of the results are not presented in their full generality. Rather, the emphasis is on clarity of exposition and simplicity of the proofs, with several being different from those in other texts on the subject. Prerequisites are kept to a minimum, and new concepts are introduced gradually and are carefully motivated. Introduction to Noncommutative Algebra is therefore accessible to a wide mathematical audience. It is, however, primarily intended for beginning graduate and advanced undergraduate students encountering noncommutative algebra for the first time.

Non-Associative Algebras and Related Topics

Author : Helena Albuquerque
Publisher : Springer Nature
ISBN 13 : 3031327071
Total Pages : 305 pages
Book Rating : 4.0/5 (313 download)

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Book Synopsis Non-Associative Algebras and Related Topics by : Helena Albuquerque

Download or read book Non-Associative Algebras and Related Topics written by Helena Albuquerque and published by Springer Nature. This book was released on 2023-07-28 with total page 305 pages. Available in PDF, EPUB and Kindle. Book excerpt: This proceedings volume presents a selection of peer-reviewed contributions from the Second Non-Associative Algebras and Related Topics (NAART II) conference, which was held at the University of Coimbra, Portugal, from July 18–22, 2022. The conference was held in honor of mathematician Alberto Elduque, who has made significant contributions to the study of non-associative structures such as Lie, Jordan, and Leibniz algebras. The papers in this volume are organized into four parts: Lie algebras, superalgebras, and groups; Leibniz algebras; associative and Jordan algebras; and other non-associative structures. They cover a variety of topics, including classification problems, special maps (automorphisms, derivations, etc.), constructions that relate different structures, and representation theory. One of the unique features of NAART is that it is open to all topics related to non-associative algebras, including octonion algebras, composite algebras, Banach algebras, connections with geometry, applications in coding theory, combinatorial problems, and more. This diversity allows researchers from a range of fields to find the conference subjects interesting and discover connections with their own areas, even if they are not traditionally considered non-associative algebraists. Since its inception in 2011, NAART has been committed to fostering cross-disciplinary connections in the study of non-associative structures.

The Concise Handbook of Algebra

Author : Alexander V. Mikhalev
Publisher : Springer Science & Business Media
ISBN 13 : 9401732671
Total Pages : 629 pages
Book Rating : 4.4/5 (17 download)

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Book Synopsis The Concise Handbook of Algebra by : Alexander V. Mikhalev

Download or read book The Concise Handbook of Algebra written by Alexander V. Mikhalev and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 629 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is by no means clear what comprises the "heart" or "core" of algebra, the part of algebra which every algebraist should know. Hence we feel that a book on "our heart" might be useful. We have tried to catch this heart in a collection of about 150 short sections, written by leading algebraists in these areas. These sections are organized in 9 chapters A, B, . . . , I. Of course, the selection is partly based on personal preferences, and we ask you for your understanding if some selections do not meet your taste (for unknown reasons, we only had problems in the chapter "Groups" to get enough articles in time). We hope that this book sets up a standard of what all algebraists are supposed to know in "their" chapters; interested people from other areas should be able to get a quick idea about the area. So the target group consists of anyone interested in algebra, from graduate students to established researchers, including those who want to obtain a quick overview or a better understanding of our selected topics. The prerequisites are something like the contents of standard textbooks on higher algebra. This book should also enable the reader to read the "big" Handbook (Hazewinkel 1999-) and other handbooks. In case of multiple authors, the authors are listed alphabetically; so their order has nothing to do with the amounts of their contributions.

Methods in Ring Theory

Author : Vesselin Drensky
Publisher : CRC Press
ISBN 13 : 9780824701833
Total Pages : 332 pages
Book Rating : 4.7/5 (18 download)

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Book Synopsis Methods in Ring Theory by : Vesselin Drensky

Download or read book Methods in Ring Theory written by Vesselin Drensky and published by CRC Press. This book was released on 1998-03-27 with total page 332 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Furnishes important research papers and results on group algebras and PI-algebras presented recently at the Conference on Methods in Ring Theory held in Levico Terme, Italy-familiarizing researchers with the latest topics, techniques, and methodologies encompassing contemporary algebra."

Encyclopaedia of Mathematics

Author : M. Hazewinkel
Publisher : Springer
ISBN 13 : 1489937919
Total Pages : 932 pages
Book Rating : 4.4/5 (899 download)

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Book Synopsis Encyclopaedia of Mathematics by : M. Hazewinkel

Download or read book Encyclopaedia of Mathematics written by M. Hazewinkel and published by Springer. This book was released on 2013-12-01 with total page 932 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Encyclopaedia of Mathematics

Author : Michiel Hazewinkel
Publisher : Springer Science & Business Media
ISBN 13 : 940151237X
Total Pages : 506 pages
Book Rating : 4.4/5 (15 download)

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Book Synopsis Encyclopaedia of Mathematics by : Michiel Hazewinkel

Download or read book Encyclopaedia of Mathematics written by Michiel Hazewinkel and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: This ENCYCLOPAEDIA OF MA THEMA TICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.