Introduction to Noncommutative Algebra

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.

Author: Matej Brešar

Publisher: Springer

ISBN: 9783319086934

Category: Mathematics

Page: 199

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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.

Topics in Noncommutative Algebra

The book assumes some undergraduate-level knowledge of algebra and analysis, but apart from that is self-contained. Part II of the monograph is devoted to the proofs of the algebraic background.

Author: Andrea Bonfiglioli

Publisher: Springer Science & Business Media

ISBN: 9783642225963

Category: Mathematics

Page: 539

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Motivated by the importance of the Campbell, Baker, Hausdorff, Dynkin Theorem in many different branches of Mathematics and Physics (Lie group-Lie algebra theory, linear PDEs, Quantum and Statistical Mechanics, Numerical Analysis, Theoretical Physics, Control Theory, sub-Riemannian Geometry), this monograph is intended to: fully enable readers (graduates or specialists, mathematicians, physicists or applied scientists, acquainted with Algebra or not) to understand and apply the statements and numerous corollaries of the main result, provide a wide spectrum of proofs from the modern literature, comparing different techniques and furnishing a unifying point of view and notation, provide a thorough historical background of the results, together with unknown facts about the effective early contributions by Schur, Poincaré, Pascal, Campbell, Baker, Hausdorff and Dynkin, give an outlook on the applications, especially in Differential Geometry (Lie group theory) and Analysis (PDEs of subelliptic type) and quickly enable the reader, through a description of the state-of-art and open problems, to understand the modern literature concerning a theorem which, though having its roots in the beginning of the 20th century, has not ceased to provide new problems and applications. The book assumes some undergraduate-level knowledge of algebra and analysis, but apart from that is self-contained. Part II of the monograph is devoted to the proofs of the algebraic background. The monograph may therefore provide a tool for beginners in Algebra.

Noncommutative Algebra

A Course in Homological Algebra. ... Introduction to Lie Algebras and Representation Theory. ... Geometric Topology in Dimensions 2 and 3. continued after index R. Keith Dennis Noncommutative Algebra With 13 Illustrations 8 Springer ...

Author: Benson Farb

Publisher: Springer Science & Business Media

ISBN: 9781461208891

Category: Mathematics

Page: 226

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About This Book This book is meant to be used by beginning graduate students. It covers basic material needed by any student of algebra, and is essential to those specializing in ring theory, homological algebra, representation theory and K-theory, among others. It will also be of interest to students of algebraic topology, functional analysis, differential geometry and number theory. Our approach is more homological than ring-theoretic, as this leads the to many important areas of mathematics. This ap student more quickly proach is also, we believe, cleaner and easier to understand. However, the more classical, ring-theoretic approach, as well as modern extensions, are also presented via several exercises and sections in Chapter Five. We have tried not to leave any gaps on the paths to proving the main theorem- at most we ask the reader to fill in details for some of the sideline results; indeed this can be a fruitful way of solidifying one's understanding.

Noncommutative Algebra and Geometry

AMS 317 (1990) 585–598 Lieven Le Bruyn and Stijn Symens, Partial desingularizations arising from non-commutative algebras, to appear [math.RA/0401?] Thierry Levasseur, Some properties of noncommutative regular graded rings Glasgow Math.

Author: Corrado De Concini

Publisher: CRC Press

ISBN: 9781420028102

Category: Mathematics

Page: 272

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A valuable addition to the Lecture Notes in Pure and Applied Mathematics series, this reference results from a conference held in St. Petersburg, Russia, in honor of Dr. Z. Borevich. This volume is mainly devoted to the contributions related to the European Science Foundation workshop, organized under the framework of noncommuntative geometry and integrated in the Borevich meeting. The topics presented, including algebraic groups and representations, algebraic number theory, rings, and modules, are a timely distillation of recent work in the field. Featuring a wide range of international experts as contributors, this book is an ideal reference for mathematicians in algebra and algebraic geometry.

Arithmetic Fundamental Groups and Noncommutative Algebra

1999 Von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999, Mathematical Sciences Research Institute, Berkeley, California Karen Redrobe Beckman, Von Neumann Conference on Arithmetic ...

Author: Karen Redrobe Beckman

Publisher: American Mathematical Soc.

ISBN: 9780821820360

Category: Mathematics

Page: 569

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The arithmetic and geometry of moduli spaces and their fundamental groups are a very active research area. This book offers a complete overview of developments made over the last decade. The papers in this volume examine the geometry of moduli spaces of curves with a function on them. The main players in Part 1 are the absolute Galois group $G_{\mathbb Q}$ of the algebraic numbers and its close relatives. By analyzing how $G_{\mathbb Q}$ acts on fundamental groups defined by Hurwitz moduli problems, the authors achieve a grand generalization of Serre's program from the 1960s.Papers in Part 2 apply $\theta$-functions and configuration spaces to the study of fundamental groups over positive characteristic fields. In this section, several authors use Grothendieck's famous lifting results to give extensions to wildly ramified covers. Properties of the fundamental groups have brought collaborations between geometers and group theorists. Several Part 3 papers investigate new versions of the genus 0 problem. In particular, this includes results severely limiting possible monodromy groups of sphere covers. Finally, Part 4 papers treat Deligne's theory of Tannakian categories and arithmetic versions of the Kodaira-Spencer map. This volume is geared toward graduate students and research mathematicians interested in arithmetic algebraic geometry.

Commutative Algebra and Noncommutative Algebraic Geometry

the standard analog of normality for a noncommutative noetherian domain is the property of being a maximal order in ... References [Bavula 1992] V. V. Bavula, “Generalized Weyl algebras and their representations”, Algebra i Analiz 4:1 ...

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ISBN: 9781107149724

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Algebra II

In recent years modules over the Weyl algebra and sheaves of such modules became the foundation of the so-called microlocal analysis. The theory of operator algebras (Le.

Author: A.I. Kostrikin

Publisher: Springer Science & Business Media

ISBN: 9783642728990

Category: Mathematics

Page: 234

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The algebra of square matrices of size n ~ 2 over the field of complex numbers is, evidently, the best-known example of a non-commutative alge 1 bra • Subalgebras and subrings of this algebra (for example, the ring of n x n matrices with integral entries) arise naturally in many areas of mathemat ics. Historically however, the study of matrix algebras was preceded by the discovery of quatemions which, introduced in 1843 by Hamilton, found ap plications in the classical mechanics of the past century. Later it turned out that quaternion analysis had important applications in field theory. The al gebra of quaternions has become one of the classical mathematical objects; it is used, for instance, in algebra, geometry and topology. We will briefly focus on other examples of non-commutative rings and algebras which arise naturally in mathematics and in mathematical physics. The exterior algebra (or Grassmann algebra) is widely used in differential geometry - for example, in geometric theory of integration. Clifford algebras, which include exterior algebras as a special case, have applications in rep resentation theory and in algebraic topology. The Weyl algebra (Le. algebra of differential operators with· polynomial coefficients) often appears in the representation theory of Lie algebras. In recent years modules over the Weyl algebra and sheaves of such modules became the foundation of the so-called microlocal analysis. The theory of operator algebras (Le.

Noncommutative Localization in Algebra and Topology

[ 113 ] A. L. ROSENBERG , Noncommutative algebraic geometry and representations of quantized algebras , MAIA 330 , Kluwer 1995 . [ 114 ] A. L. ROSENBERG , Spectra of noncommutative spaces , MPI - 2003-110 , Bonn 2003 .

Author: Andrew Ranicki

Publisher: Cambridge University Press

ISBN: 052168160X

Category: Mathematics

Page: 313

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An introduction to noncommutative localization and an account of the state of the art suitable for researchers and graduate students.

Noncommutative Algebraic Geometry and Representations of Quantized Algebras

[L1] G. Lustig, Quantum deformations of certain simple modules over enveloping algebras, Adv.Math. 70 (1988), 237-249. ... [OV] F. Van Oystaeyen, A. Vershoren, Noncommutative algebraic geometry, Lecture Notes in Math.

Author: A. Rosenberg

Publisher: Springer Science & Business Media

ISBN: 9789401584302

Category: Mathematics

Page: 322

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This book is based on lectures delivered at Harvard in the Spring of 1991 and at the University of Utah during the academic year 1992-93. Formally, the book assumes only general algebraic knowledge (rings, modules, groups, Lie algebras, functors etc.). It is helpful, however, to know some basics of algebraic geometry and representation theory. Each chapter begins with its own introduction, and most sections even have a short overview. The purpose of what follows is to explain the spirit of the book and how different parts are linked together without entering into details. The point of departure is the notion of the left spectrum of an associative ring, and the first natural steps of general theory of noncommutative affine, quasi-affine, and projective schemes. This material is presented in Chapter I. Further developments originated from the requirements of several important examples I tried to understand, to begin with the first Weyl algebra and the quantum plane. The book reflects these developments as I worked them out in reallife and in my lectures. In Chapter 11, we study the left spectrum and irreducible representations of a whole lot of rings which are of interest for modern mathematical physics. The dasses of rings we consider indude as special cases: quantum plane, algebra of q-differential operators, (quantum) Heisenberg and Weyl algebras, (quantum) enveloping algebra ofthe Lie algebra sl(2) , coordinate algebra of the quantum group SL(2), the twisted SL(2) of Woronowicz, so called dispin algebra and many others.