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Author: Ioannis John Demetrius Vergados
Publisher: World Scientific
ISBN: 9811274770
Category : Science
Languages : en
Pages : 364
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Book Description
This book deals with the role played by symmetry in the understanding of the physical world, beginning with the notion of geometric symmetries of the ancient Greek philosophers and mathematicians. The recognition of the existence of symmetries led to the notion of transformations, which led from one state of the system to another. It was then realized that such transformations, under the operation of multiplication, constitute an interesting set, whose study led to the branch of mathematics known as Group Theory. With the emergence of quantum mechanics, this theory became much more interesting and led to some additional applications. The theory got another boost with the need for of the internal degrees of freedom in describing physical systems. This way the notion of symmetry is no longer purely geometric and evolved into a useful tool in the study of all physical sciences.For practical reasons as well as pedagogical reasons, group theory is usually split into two parts. The first deals with discrete groups, with the group elements being countable, usually finite in number, while the second deals with continuous groups, whose elements depend on continuous parameters. This volumefocuses the discussion on discrete groups. Given that group theory should be presented from a unified perspective, involving not only the mathematical rigor and beauty of symmetries, but also the ability to use it as a tool for applications, either currently popular or expected to become so in the future, this approach will surely be more beneficial to the dedicated reader. It is not intended for those who would like to just look up a formula or use the results of a table, without understanding their derivation.
Author: Ioannis John Demetrius Vergados
Publisher: World Scientific
ISBN: 9811274770
Category : Science
Languages : en
Pages : 364
Get Book Here
Book Description
This book deals with the role played by symmetry in the understanding of the physical world, beginning with the notion of geometric symmetries of the ancient Greek philosophers and mathematicians. The recognition of the existence of symmetries led to the notion of transformations, which led from one state of the system to another. It was then realized that such transformations, under the operation of multiplication, constitute an interesting set, whose study led to the branch of mathematics known as Group Theory. With the emergence of quantum mechanics, this theory became much more interesting and led to some additional applications. The theory got another boost with the need for of the internal degrees of freedom in describing physical systems. This way the notion of symmetry is no longer purely geometric and evolved into a useful tool in the study of all physical sciences.For practical reasons as well as pedagogical reasons, group theory is usually split into two parts. The first deals with discrete groups, with the group elements being countable, usually finite in number, while the second deals with continuous groups, whose elements depend on continuous parameters. This volumefocuses the discussion on discrete groups. Given that group theory should be presented from a unified perspective, involving not only the mathematical rigor and beauty of symmetries, but also the ability to use it as a tool for applications, either currently popular or expected to become so in the future, this approach will surely be more beneficial to the dedicated reader. It is not intended for those who would like to just look up a formula or use the results of a table, without understanding their derivation.
Author: Audrey Terras
Publisher: Cambridge University Press
ISBN: 9780521457187
Category : Mathematics
Languages : en
Pages : 456
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Book Description
It examines the theory of finite groups in a manner that is both accessible to the beginner and suitable for graduate research.
Author: Peter J. Nicholls
Publisher: Cambridge University Press
ISBN: 0521376742
Category : Mathematics
Languages : en
Pages : 237
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Book Description
The interaction between ergodic theory and discrete groups has a long history and much work was done in this area by Hedlund, Hopf and Myrberg in the 1930s. There has been a great resurgence of interest in the field, due in large measure to the pioneering work of Dennis Sullivan. Tools have been developed and applied with outstanding success to many deep problems. The ergodic theory of discrete groups has become a substantial field of mathematical research in its own right, and it is the aim of this book to provide a rigorous introduction from first principles to some of the major aspects of the theory. The particular focus of the book is on the remarkable measure supported on the limit set of a discrete group that was first developed by S. J. Patterson for Fuchsian groups, and later extended and refined by Sullivan.
Author: Ioannis John Demetrius Vergados
Publisher: World Scientific Publishing Company
ISBN: 9813202467
Category : Science
Languages : en
Pages : 349
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Book Description
This volume goes beyond the understanding of symmetries and exploits them in the study of the behavior of both classical and quantum physical systems. Thus it is important to study the symmetries described by continuous (Lie) groups of transformations. We then discuss how we get operators that form a Lie algebra. Of particular interest to physics is the representation of the elements of the algebra and the group in terms of matrices and, in particular, the irreducible representations. These representations can be identified with physical observables.This leads to the study of the classical Lie algebras, associated with unitary, unimodular, orthogonal and symplectic transformations. We also discuss some special algebras in some detail. The discussion proceeds along the lines of the Cartan-Weyl theory via the root vectors and root diagrams and, in particular, the Dynkin representation of the roots. Thus the representations are expressed in terms of weights, which are generated by the application of the elements of the algebra on uniquely specified highest weight states. Alternatively these representations can be described in terms of tensors labeled by the Young tableaux associated with the discrete symmetry Sn. The connection between the Young tableaux and the Dynkin weights is also discussed. It is also shown that in many physical systems the quantum numbers needed to specify the physical states involve not only the highest symmetry but also a number of sub-symmetries contained in them. This leads to the study of the role of subalgebras and in particular the possible maximal subalgebras. In many applications the physical system can be considered as composed of subsystems obeying a given symmetry. In such cases the reduction of the Kronecker product of irreducible representations of classical and special algebras becomes relevant and is discussed in some detail. The method of obtaining the relevant Clebsch-Gordan (C-G) coefficients for such algebras is discussed and some relevant algorithms are provided. In some simple cases suitable numerical tables of C-G are also included.The above exposition contains many examples, both as illustrations of the main ideas as well as well motivated applications. To this end two appendices of 51 pages — 11 tables in Appendix A, summarizing the material discussed in the main text and 39 tables in Appendix B containing results of more sophisticated examples are supplied. Reference to the tables is given in the main text and a guide to the appropriate section of the main text is given in the tables.
Author: Benjamin Steinberg
Publisher: Springer Science & Business Media
ISBN: 1461407761
Category : Mathematics
Languages : en
Pages : 166
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Book Description
This 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.
Author: Peter Webb
Publisher: Cambridge University Press
ISBN: 1107162394
Category : Mathematics
Languages : en
Pages : 339
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Book Description
This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.
Author: Gregori A. Margulis
Publisher: Springer Science & Business Media
ISBN: 9783540121794
Category : Mathematics
Languages : en
Pages : 408
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Book Description
Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi-simple Lie groups. Since the notion of "Lie group" is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e.g. classification results on abstract homomorphisms of semi-simple algebraic groups over global fields. The treatise of course contains a presentation of the author's fundamental rigidity and arithmeticity theorems. The work in this monograph requires the language and basic results from fields such as algebraic groups, ergodic theory, the theory of unitary representatons, and the theory of amenable groups. The author develops the necessary material from these subjects; so that, while the book is of obvious importance for researchers working in related areas, it is essentially self-contained and therefore is also of great interest for advanced students.
Author: A. Zee
Publisher: Princeton University Press
ISBN: 1400881188
Category : Science
Languages : en
Pages : 632
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Book Description
A concise, modern textbook on group theory written especially for physicists Although group theory is a mathematical subject, it is indispensable to many areas of modern theoretical physics, from atomic physics to condensed matter physics, particle physics to string theory. In particular, it is essential for an understanding of the fundamental forces. Yet until now, what has been missing is a modern, accessible, and self-contained textbook on the subject written especially for physicists. Group Theory in a Nutshell for Physicists fills this gap, providing a user-friendly and classroom-tested text that focuses on those aspects of group theory physicists most need to know. From the basic intuitive notion of a group, A. Zee takes readers all the way up to how theories based on gauge groups could unify three of the four fundamental forces. He also includes a concise review of the linear algebra needed for group theory, making the book ideal for self-study. Provides physicists with a modern and accessible introduction to group theory Covers applications to various areas of physics, including field theory, particle physics, relativity, and much more Topics include finite group and character tables; real, pseudoreal, and complex representations; Weyl, Dirac, and Majorana equations; the expanding universe and group theory; grand unification; and much more The essential textbook for students and an invaluable resource for researchers Features a brief, self-contained treatment of linear algebra An online illustration package is available to professors Solutions manual (available only to professors)
Author: Persi Diaconis
Publisher: Ims
ISBN:
Category : Mathematics
Languages : en
Pages : 212
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Book Description
Author: M. Aschbacher
Publisher: Cambridge University Press
ISBN: 9780521786751
Category : Mathematics
Languages : en
Pages : 320
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Book Description
During the last 40 years the theory of finite groups has developed dramatically. The finite simple groups have been classified and are becoming better understood. Tools exist to reduce many questions about arbitrary finite groups to similar questions about simple groups. Since the classification there have been numerous applications of this theory in other branches of mathematics. Finite Group Theory develops the foundations of the theory of finite groups. It can serve as a text for a course on finite groups for students already exposed to a first course in algebra. It could supply the background necessary to begin reading journal articles in the field. For specialists it also provides a reference on the foundations of the subject. This second edition has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises.
