Algebra Books

Book SynopsisThis balanced and comprehensive study presents the theory, methods and applications of matrix analysis in a new theoretical framework, allowing readers to understand second-order and higher-order matrix analysis in a completely new light. Alongside the core subjects in matrix analysis, such as singular value analysis, the solution of matrix equations and eigenanalysis, the author introduces new applications and perspectives that are unique to this book. The very topical subjects of gradient analysis and optimization play a central role here. Also included are subspace analysis, projection analysis and tensor analysis, subjects which are often neglected in other books. Having provided a solid foundation to the subject, the author goes on to place particular emphasis on the many applications matrix analysis has in science and engineering, making this book suitable for scientists, engineers and graduate students alike.Trade Review'Matrix analysis is the engine room of scientific computing since, inevitably, most computational problems are reduced to linear algebra. Xian-Da Zhang's monograph presents a thoroughly modern introduction to this important subject in a form suitable for its many users. Without taking liberties with mathematical content, the text privileges an algorithmic approach and ventures into new and important areas, e.g. tensor analysis. This is a valuable addition to any applied mathematician's bookshelf.' Arieh Iserles, University of Cambridge'This book provides various new topics on Matrix Analysis and Applications, which are not covered by some popular books on matrix analysis and computation. The text is well written, easy to be understood. There are several classical books on matrix analysis and computation on my bookshelf. But I will be happy to have this book on my bookshelf too.' Liqun Qi, Hong Kong Polytechnic UniversityTable of ContentsPreface; Notation; List of abbreviations; List of algorithms; Part I. Matrix Algebra: 1. Introduction to matrix algebra; 2. Special matrices; 3. Matrix differential; Part II. Matrix Analysis: 4. Gradient analysis and optimization; 5. Singular value analysis; 6. Solving matrix equations; 7. Eigenanalysis; 8. Subspace analysis and tracking; 9. Projection analysis; Part III. Higher-Order Matrix Analysis: 10. Tensor analysis; References; Index.

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Book SynopsisThis is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.Trade Review'This 2-volume book is a very welcome addition to the existing literature in modular representation theory. It contains a wealth of material much of which is here presented in textbook form for the first time. It gives an excellent overview of the state of the art in this fascinating subject and also of the many challenging and fundamental open problems. It is well written and will certainly become a standard reference.' Burkhard Kűlshammer, MathSciNetTable of ContentsIntroduction; 6. Blocks and source algebras; 7. Modules over finite p-groups; 8. Local structure; 9. Isometries and bimodules; 10. Structural results in block theory; 11. Blocks with cyclic defect groups; 12. Blocks with Klein four defect groups; Appendix; References; Index.

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Book SynopsisStarting with elementary operator theory and matrix analysis, this book introduces the basic properties of the numerical range and gradually builds up the whole numerical range theory. Over 400 assorted problems, ranging from routine exercises to published research results, give you the chance to put the theory into practice and test your understanding. Interspersed throughout the text are numerous comments and references, allowing you to discover related developments and to pursue areas of interest in the literature. Also included is an appendix on basic convexity properties on the Euclidean space. Targeted at graduate students as well as researchers interested in functional analysis, this book provides a comprehensive coverage of classic and recent works on the numerical range theory. It serves as an accessible entry point into this lively and exciting research area.Table of ContentsPreface; 0. Preliminaries in Operator Theory; 1. Numerical Range; 2. Numerical Ranges of Special Operators; 3. Numerical Contraction; 4. Algebraic and Essential Numerical Ranges 5. Numerical Range and Dilation; 6. Numerical Range of Finite Matrix; 7. Numerical Range of Sn-Matrix; 8. Generalized Numerical Ranges; Appendix: Convex Set.

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Book SynopsisOn its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the CurtisAlvis duality map and Mackey''s theorem and the results that can be deduced from it, before moving on to a discussion of DeligneLusztig induction and Lusztig''s Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples.Trade Review'… a useful resource for beginning graduate students in algebra as well as seasoned researchers.' Mathematical Reviews Clippings'… clearly written; there are useful examples, motivational comments, and exercises scattered throughout the text.' Mark Hunacek, The Mathematical GazetteTable of Contents1. Basic results on algebraic groups; 2. Structure theorems for reductive groups; 3. (B, N)-pairs; parabolic, Levi, and reductive subgroups; centralisers of semi-simple elements; 4. Rationality, the Frobenius endomorphism, the Lang–Steinberg theorem; 5. Harish–Chandra theory; 6. Iwahori–Hecke algebras; 7. The duality functor and the Steinberg character; 8. l-adic cohomology; 9. Deligne–Lusztig induction; the Mackey formula; 10. The character formula and other results on Deligne–Lusztig induction; 11. Geometric conjugacy and Lusztig series; 12. Regular elements; Gelfand–Graev representations; regular and semi-simple characters; 13. Green functions; 14. The decomposition of Deligne–Lusztig characters; References; Index.

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Book SynopsisBased on the author''s university lecture courses, this book presents the many facets of one of the most important open problems in operator algebra theory. Central to this book is the proof of the equivalence of the various forms of the problem, including forms involving C*-algebra tensor products and free groups, ultraproducts of von Neumann algebras, and quantum information theory. The reader is guided through a number of results (some of them previously unpublished) revolving around tensor products of C*-algebras and operator spaces, which are reminiscent of Grothendieck''s famous Banach space theory work. The detailed style of the book and the inclusion of background information make it easily accessible for beginning researchers, Ph.D. students, and non-specialists alike.Trade Review'This is a very rich and detailed monograph on an enormously important subject. It is written in the crystal clear and elegant style that is the hallmark of its author, and it offers a lot of information to specialists and novices alike. The book will certainly become an authoritative guide.' Dirk Werner, London Mathematical Society Student Texts'This book is jam packed with information, and should be an invaluable guide to anyone interested in these ideas … For the complete picture and the recent advances, Pisier's book is the place to go.' Bulletin of the American Mathematical SocietyTable of ContentsIntroduction; 1. Completely bounded and completely positive maps: basics; 2. Completely bounded and completely positive maps: a tool kit; 3. C*-algebras of discrete groups; 4. C*-tensor products; 5. Multiplicative domains of c.p. maps; 6. Decomposable maps; 7. Tensorizing maps and functorial properties; 8. Biduals, injective von Neumann algebras and C*-norms; 9. Nuclear pairs, WEP, LLP and QWEP; 10. Exactness and nuclearity; 11. Traces and ultraproducts; 12. The Connes embedding problem; 13. Kirchberg's conjecture; 14. Equivalence of the two main questions; 15. Equivalence with finite representability conjecture; 16. Equivalence with Tsirelson's problem; 17. Property (T) and residually finite groups. Thom's example; 18. The WEP does not imply the LLP; 19. Other proofs that C(n) < n. Quantum expanders; 20. Local embeddability into ${\mathscr{C}}$ and non-separability of $(OS_n, d_{cb})$; 21. WEP as an extension property; 22. Complex interpolation and maximal tensor product; 23. Haagerup's characterizations of the WEP; 24. Full crossed products and failure of WEP for $\mathscr{B}\otimes_{\min}\mathscr{B}$; 25. Open problems; Appendix. Miscellaneous background; References; Index.

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Book SynopsisThis book is intended to present group representation theory at a level accessible to mature undergraduate students and beginning graduate students. This is achieved by mainly keeping the required background to the level of undergraduate linear algebra, group theory and very basic ring theory. Module theory and Wedderburn theory, as well as tensor products, are deliberately avoided. Instead, we take an approach based on discrete Fourier Analysis.Applications to the spectral theory of graphs are given to help the student appreciate the usefulness of the subject. A number of exercises are included. This book is intended for a 3rd/4th undergraduate course or an introductory graduate course on group representation theory. However, it can also be used as a reference for workers in all areas of mathematics and statistics.Trade ReviewFrom the reviews:“The aim of the author is to give an introductory text on (ordinary) representation theory of finite groups which is accessible for advanced undergraduates already. And Steinberg manages this outstandingly well. … a book which is ideally suited for beginners in math as well as physicists, engineers and so on, who need a concise, well conceived and easy comprehensible introduction into representation theory.” (G. Kowol, Monatshefte für Mathematik, Vol. 167 (3-4), September, 2012)“Steinberg … provides a one-semester course on representation theory with just linear algebra and a beginning course in abstract algebra (primarily group theory) as prerequisites. … the author covers most of the standard introductory topics in representation theory. The exercises provide more examples and further common results. It is the applications that Steinberg uses to motivate the subject that make this text both interesting and valuable. … Overall, a very user-friendly text with many examples and copious details. Summing Up: Recommended. Upper-division undergraduates through researchers/faculty.” (J. T. Zerger, Choice, Vol. 49 (11), August, 2012)“The book consists of 157 pages spread over 11 chapters. … This book is an introductory course and it could be used by mathematicians and students who would like to learn quickly about the representation theory and character theory of finite groups, and for non-algebraists, statisticians and physicists who use representation theory.” (Jamshid Moori, Mathematical Reviews, Issue 2012 j)“The required background as to this introductory course on group representations, is in the level of linear algebra, group theory and some ring theory. … the book under review is a welcome one for students at an advanced undergraduate or introductory graduate level course, also for those people like physicists, statisticians and non-algebraically oriented mathematicians who need representation theory in their work.” (R. W. van der Waall, Zentralblatt MATH, Vol. 1243, 2012)“The author has, by combining clear writing with an accessible and minimal-prerequisite approach to group representations, created a book that may well help bring group representation theory into the undergraduate curriculum. This is an impressive and useful text, and should be looked at by anybody with an interest in the subject.” (Mark Hunacek, The Mathematical Association of America, February, 2012)Table of Contents-Preface.-Introduction.-Review of Linear Algebra.-Group Representations.-Character Theory.-Fourier Analysis on Finite Groups.-Burnside's Theorem.-Permutation Representations.-Induced Representations.-Another Theorem of Burnside.-The Symmetric Group.-Bibliography.-Index.

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