It became more and more usual, from, say, the 1970s, for each book on Module Theory, to point out and prove some (but in no more than 15 to 20 pages) generalizations to (mostly modular) lattices. This was justified by the nowadays widely accepted perception that the structure of a module over a ring is best understood in terms of the lattice struc ture of its submodule lattice. Citing Louis H. Rowen "this important example (the lattice of all the submodules of a module) is the raison d'etre for the study of lattice theory by ring theorists". Indeed, many module-theoretic results can be proved by using lattice theory alone. The purpose of this book is to collect and present all and only the results of this kind, although for this purpose one must develop some significant lattice theory. The results in this book are of the following categories: the folklore of Lattice Theory (to be found in each Lattice Theory book), module theoretic results generalized in (modular, and possibly compactly gen erated) lattices (to be found in some 6 to 7 books published in the last 20 years), very special module-theoretic results generalized in lattices (e. g. , purity in Chapter 9 and several dimensions in Chapter 13, to be found mostly in [27], respectively, [34] and [18]) and some new con cepts (e. g.

Symmetry is one of the most important organising principles in the natural sciences. The mathematical theory of symmetry has long been associated with group theory, but it is a basic premise of this book that there are aspects of symmetry which are more faithfully represented by a generalization of groups called inverse semigroups. The theory of inverse semigroups is described from its origins in the foundations of differential geometry through to its most recent applications in combinatorial group theory, and the theory tilings.

This book is concerned with the structure of linear semigroups, that is, subsemigroups of the multiplicative semigroup Mn(K) of n x n matrices over a field K (or, more generally, skew linear semigroups - if K is allowed to be a division ring) and its applications to certain problems on associative algebras, semigroups and linear representations. It is motivated by several recent developments in the area of linear semigroups and their applications. It summarizes the state of knowledge in this area, presenting the results for the first time in a unified form. The book's point of departure is a structure theorem, which allows the use of powerful techniques of linear groups. Certain aspects of a combinatorial nature, connections with the theory of linear representations and applications to various problems on associative algebras are also discussed.

In his 1974 seminal paper 'Elliptic modules', V G Drinfeld introduced objects into the arithmetic geometry of global function fields which are nowadays known as 'Drinfeld Modules'. They have many beautiful analogies with elliptic curves and abelian varieties. They study of their moduli spaces leads amongst others to explicit class field theory, Jacquet-Langlands theory, and a proof of the Shimura-Taniyama-Weil conjecture for global function fields.This book constitutes a carefully written instructional course of 12 lectures on these subjects, including many recent novel insights and examples. The instructional part is complemented by research papers centering around class field theory, modular forms and Heegner points in the theory of global function fields.The book will be indispensable for everyone who wants a clear view of Drinfeld's original work, and wants to be informed about the present state of research in the theory of arithmetic geometry over function fields.

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. It includes proofs of the main basic results (theorems of Hermite-Lindemann, Gelfond-Schneider, 6 exponentials theorem), an introduction to height functions and Lehmer's problem, several proofs of Baker's theorem as well as explicit measures of linear independence of logarithms. An original feature is the systematic use, in proofs, of Laurent's interpolation determinants. The most general result is the so-called Theorem of the Linear Subgroup, an effective version of which is also included. It yields new results of simultaneous approximation and of algebraic independence. Two chapters written by D. Roy provide complete and at the same time simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.

This book, an abridgment of Volumes I and II of the highly respected Group Theory in Physics, presents a carefully constructed introduction to group theory and its applications in physics. The book provides anintroduction to and description of the most important basic ideas and the role that they play in physical problems. The clearly written text contains many pertinent examples that illustrate the topics, even for those with no background in group theory.
This work presents important mathematical developments to theoretical physicists in a form that is easy to comprehend and appreciate. Finite groups, Lie groups, Lie algebras, semi-simple Lie algebras, crystallographic point groups and crystallographic space groups, electronic energy bands in solids, atomic physics, symmetry schemes for fundamental particles, and quantum mechanics are all covered in this compact new edition.
Key Features
* Covers both group theory and the theory of Lie algebras
* Includes studies of solid state physics, atomic physics, and fundamental particle physics
* Contains a comprehensive index
* Provides extensive examples

A systematic survey of all the basic results on the theory of discrete subgroups of Lie groups, presented in a convenient form for users. The book makes the theory accessible to a wide audience, and will be a standard reference for many years to come.

Part I of this book is a short review of the classical part of representation theory. The main chapters of representation theory are discussed: representations of finite and compact groups, finite- and infinite-dimensional representations of Lie groups. It is a typical feature of this survey that the structure of the theory is carefully exposed - the reader can easily see the essence of the theory without being overwhelmed by details. The final chapter is devoted to the method of orbits for different types of groups. Part II deals with representation of Virasoro and Kac-Moody algebra. The second part of the book deals with representations of Virasoro and Kac-Moody algebra. The wealth of recent results on representations of infinite-dimensional groups is presented.

Reflection groups and their invariant theory provide the main themes of this book and the first two parts focus on these topics. The first 13 chapters deal with reflection groups (Coxeter groups and Weyl groups) in Euclidean Space while the next thirteen chapters study the invariant theory of pseudo-reflection groups. The third part of the book studies conjugacy classes of the elements in reflection and pseudo-reflection groups. The book has evolved from various graduate courses given by the author over the past 10 years. It is intended to be a graduate text, accessible to students with a basic background in algebra.

A recurring theme in a traditional introductory graduate algebra course is the existence and consequences of relationships between different algebraic structures. This is also the theme of this book, an exposition of connections between representations of finite partially ordered sets and abelian groups. Emphasis is placed throughout on classification, a description of the objects up to isomorphism, and computation of representation type, a measure of when classification is feasible. David M. Arnold is the Ralph and Jean Storm Professor of Mathematics at Baylor University. He is the author of "Finite Rank Torsion Free Abelian Groups and Rings" published in the Springer-Verlag Lecture Notes in Mathematics series, a co-editor for two volumes of conference proceedings, and the author of numerous articles in mathematical research journals. His research interests are in abelian group theory and related topics, such as representations of partially ordered sets and modules over discrete valuation rings, subrings of algebraic number fields, and pullback rings. He received his Ph. D. from the University of Illinois, Urbana and was a member of the faculty at New Mexico State University for many years.

From the reviews:
.."., the book must be of great help for a researcher who already has some idea of Lie theory, wants to employ it in his everyday research and/or teaching, and needs a source for customary reference on the subject. From my viewpoint, the volume is perfectly fit to serve as such a source, ... On the whole, it is quite a pleasure, after making yourself comfortable in that favourite office armchair of yours, just to keep the volume gently in your hands and browse it slowly and thoughtfully; and after all, what more on Earth can one expect of any book?"
"The New Zealand Mathematical Society Newsletter"
..". Both parts are very nicely written and can be strongly recommended."
"European Mathematical Society"

This monograph presents a unified exposition of latin squares and mutually orthogonal sets of latin squares based on groups. Its focus is on orthomorphisms and complete mappings of finite groups, while also offering a complete proof of the Hall-Paige conjecture. The use of latin squares in constructions of nets, affine planes, projective planes, and transversal designs also motivates this inquiry. The text begins by introducing fundamental concepts, like the tests for determining whether a latin square is based on a group, as well as orthomorphisms and complete mappings. From there, it describes the existence problem for complete mappings of groups, building up to the proof of the Hall-Paige conjecture. The third part presents a comprehensive study of orthomorphism graphs of groups, while the last part provides a discussion of Cartesian projective planes, related combinatorial structures, and a list of open problems. Expanding the author's 1992 monograph, Orthomorphism Graphs of Groups, this book is an essential reference tool for mathematics researchers or graduate students tackling latin square problems in combinatorics. Its presentation draws on a basic understanding of finite group theory, finite field theory, linear algebra, and elementary number theory-more advanced theories are introduced in the text as needed.

This text provides an introduction to group theory with an emphasis on clear examples. The authors present groups as naturally occurring structures arising from symmetry in geometrical figures and other mathematical objects. Written in a 'user-friendly' style, where new ideas are always motivated before being fully introduced, the text will help readers to gain confidence and skill in handling group theory notation before progressing on to applying it in complex situations. An ideal companion to any first or second year course on the topic.

The theory of table algebras was introduced in 1991 by Z. Arad and H. Blau in order to treat, in a uniform way, products of conjugacy classes and irreducible characters of finite groups. Today, table algebra theory is a well-established branch of modern algebra with various applications, including the representation theory of finite groups, algebraic combinatorics and fusion rules algebras. This book presents the latest developments in this area. Its main goal is to give a classification of the Normalized Integral Table Algebras (Fusion Rings) generated by a faithful non-real element of degree 3. Divided into 4 parts, the first gives an outline of the classification approach, while remaining parts separately treat special cases that appear during classification. A particularly unique contribution to the field, can be found in part four, whereby a number of the algebras are linked to the polynomial irreducible representations of the group SL3(C). This book will be of interest to research mathematicians and PhD students working in table algebras, group representation theory, algebraic combinatorics and integral fusion rule algebras.

Probability theory on compact Lie groups deals with the interaction between chance and symmetry, a beautiful area of mathematics of great interest in its own sake but which is now also finding increasing applications in statistics and engineering (particularly with respect to signal processing). The author gives a comprehensive introduction to some of the principle areas of study, with an emphasis on applicability. The most important topics presented are: the study of measures via the non-commutative Fourier transform, existence and regularity of densities, properties of random walks and convolution semigroups of measures and the statistical problem of deconvolution. The emphasis on compact (rather than general) Lie groups helps readers to get acquainted with what is widely seen as a difficult field but which is also justified by the wealth of interesting results at this level and the importance of these groups for applications.

The book is primarily aimed at researchers working in probability, stochastic analysis and harmonic analysis on groups. It will also be of interest to mathematicians working in Lie theory and physicists, statisticians and engineers who are working on related applications. A background in first year graduate level measure theoretic probability and functional analysis is essential; a background in Lie groups and representation theory is certainly helpful but the first two chapters also offer orientation in these subjects."

The Bia owie a workshops on Geometric Methods in Physics, taking place in the unique environment of the Bia owie a natural forest in Poland, are among the important meetings in the field. Every year some 80 to 100 participants both from mathematics and physics join to discuss new developments and to interchange ideas. The current volume was produced on the occasion of the XXXI meeting in 2012. For the first time the workshop was followed by a School on Geometry and Physics, which consisted of advanced lectures for graduate students and young researchers. Selected speakers of the workshop were asked to contribute, and additional review articles were added. The selection shows that despite its now long tradition the workshop remains always at the cutting edge of ongoing research. The XXXI workshop had as a special topic the works of the late Boris Vasilievich Fedosov (1938 2011) who is best known for a simple and very natural construction of a deformation quantization for any symplectic manifold, and for his contributions to index theory.

This book is a useful and accessible introduction to symmetry principles in particle physics. New ideas are explained in a way that throws considerable light on difficult concepts, such as Lie groups and their representations. This book begins with introdutions both to the types of symmetries known in physics and to group theory and representation theory. Successive chapters deal with the symmetric groups and their Young diagrams, braid groups, Lie groups and algebras, Cartan's classification of semi-simple groups, and the Lie groups most used in physics are treated in detail. Gauge groups are discussed, and applications to elementary particle physics and multiquark systems introduced throughout the book where appropriate. Many worked examples are also included. There is a growing interestinthe quatk structure of hadrons and in theories of particle interactions based on the principle of gauge symmetries. In this book the concepts of group theory are clearly explained and their applications to subnuclear physics brought up-to-date.

Finite reductive groups and their representations lie at the heart of group theory. This volume treats linear representations of finite reductive groups and their modular aspects together with Hecke algebras, complex reflection groups, quantum groups, arithmetic groups, Lie groups, symmetric groups and general finite groups.

The theory of operator algebras acting on a Hilbert space was initiated in thirties by papers of Murray and von Neumann. In these papers they have studied the structure of algebras which later were called von Neu mann algebras or W* -algebras. They are weakly closed complex *-algebras of operators on a Hilbert space. At present the theory of von Neumann algebras is a deeply developed theory with various applications. In the framework of von Neumann algebras theory the study of fac tors (i.e. W* -algebras with trivial centres) is very important, since they are comparatively simple and investigation of general W* -algebras can be reduced to the case of factors. Therefore the theory of factors is one of the main tools in the structure theory of von Neumann algebras. In the middle of sixtieth Topping [To 1] and Stormer [S 2] have ini tiated the study of Jordan (non associative and real) analogues of von Neumann algebras - so called JW-algebras, i.e. real linear spaces of self adjoint opera.tors on a complex Hilbert space, which contain the identity operator 1. closed with respect to the Jordan (i.e. symmetrised) product INTRODUCTION 2 x 0 y = ~(Xy + yx) and closed in the weak operator topology. The structure of these algebras has happened to be close to the struc ture of von Neumann algebras and it was possible to apply ideas and meth ods similar to von Neumann algebras theory in the study of JW-algebras.

This text is about the geometric theory of discrete groups and the associated tesselations of the underlying space. The theory of Möbius transformations in n-dimensional Euclidean space is developed. These transformations are discussed as isometries of hyperbolic space and are then identified with the elementary transformations of complex analysis. A detailed account of analytic hyperbolic trigonometry is given, and this forms the basis of the subsequent analysis of tesselations of the hyperbolic plane. Emphasis is placed on the geometrical aspects of the subject and on the universal constraints which must be satisfied by all tesselations.

I don't know who Gigerenzer is, but he wrote something very clever that I saw quoted in a popular glossy magazine: "Evolution has tuned the way we think to frequencies of co-occurances, as with the hunter who remembers the area where he has had the most success killing game." This sanguine thought explains my obsession with the division algebras. Every effort I have ever made to connect them to physics - to the design of reality - has succeeded, with my expectations often surpassed. Doubtless this strong statement is colored by a selective memory, but the kind of game I sought, and still seek, seems to frowst about this particular watering hole in droves. I settled down there some years ago and have never feIt like Ieaving. This book is about the beasts I selected for attention (if you will, to ren der this metaphor politically correct, let's say I was a nature photographer), and the kind of tools I had to develop to get the kind of shots Iwanted (the tools that I found there were for my taste overly abstract and theoretical). Half of thisbook is about these tools, and some applications thereof that should demonstrate their power. The rest is devoted to a demonstration of the intimate connection between the mathematics of the division algebras and the Standard Model of quarks and leptons with U(l) x SU(2) x SU(3) gauge fields, and the connection of this model to lO-dimensional spacetime implied by the mathematics."

First year, undergraduate, mathematics students in Japan have for many years had the opportunity of a unique experience---an introduction, at an elementary level, to some very advanced ideas in mathematics from one of the leading mathematicians of the world. Michio Kuga s lectures on Group Theory and Differential Equations are a realization of two dreams---one to see Galois groups used to attack the problems of differential equations---the other to do so in such a manner as to take students from a very basic level to an understanding of the heart of this fascinating mathematical problem. English reading students now have the opportunity to enjoy this lively presentation, from elementary ideas to cartoons to funny examples, and to follow the mind of an imaginative and creative mathematician into a world of enduring mathematical creations."

Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics.

Special relativity and quantum mechanics are likely to remain the two most important languages in physics for many years to come. The underlying language for both disciplines is group theory. Eugene P. Wigner's 1939 paper on the Unitary Representations of the Inhomogeneous Lorentz Group laid the foundation for unifying the concepts and algorithms of quantum mechanics and special relativity. In view of the strong current interest in the space-time symmetries of elementary particles, it is safe to say that Wigner's 1939 paper was fifty years ahead of its time. This edited volume consists of Wigner's 1939 paper and the major papers on the Lorentz group published since 1939. . This volume is intended for graduate and advanced undergraduate students in physics and mathematics, as well as mature physicists wishing to understand the more fundamental aspects of physics than are available from the fashion-oriented theoretical models which come and go. The original papers contained in this volume are useful as supplementary reading material for students in courses on group theory, relativistic quantum mechanics and quantum field theory, relativistic electrodynamics, general relativity, and elementary particle physics. This reprint collection is an extension of the textbook by the present editors entitled "Theory and Applications of the Poincare Group." Since this book is largely based on the articles contained herein, the present volume should be viewed as a reading for the previous work. continuation of and supplementary We would like to thank Professors J. Bjorken, R. Feynman, R. Hofstadter, J.

Chapter 1 introduces some of the terminology and notation used later and indicates prerequisites. Chapter 2 gives a reasonably thorough account of all finite subgroups of the orthogonal groups in two and three dimensions. The presentation is somewhat less formal than in succeeding chapters. For instance, the existence of the icosahedron is accepted as an empirical fact, and no formal proof of existence is included. Throughout most of Chapter 2 we do not distinguish between groups that are "geo metrically indistinguishable," that is, conjugate in the orthogonal group. Very little of the material in Chapter 2 is actually required for the sub sequent chapters, but it serves two important purposes: It aids in the development of geometrical insight, and it serves as a source of illustrative examples. There is a discussion offundamental regions in Chapter 3. Chapter 4 provides a correspondence between fundamental reflections and funda mental regions via a discussion of root systems. The actual classification and construction of finite reflection groups takes place in Chapter 5. where we have in part followed the methods of E. Witt and B. L. van der Waerden. Generators and relations for finite reflection groups are discussed in Chapter 6. There are historical remarks and suggestions for further reading in a Post lude."