Lectures on Lie Groups and Lie Algebras

Author: Roger William Carter

Publisher: N.A

ISBN: 9780521499224

Category: Mathematics

Page: 190

View: 8330

An excellent introduction to the theory of Lie groups and Lie algebras.
Posted in Mathematics

Compact Lie Groups

Author: Mark R. Sepanski

Publisher: Springer Science & Business Media

ISBN: 0387491589

Category: Mathematics

Page: 201

View: 9290

Blending algebra, analysis, and topology, the study of compact Lie groups is one of the most beautiful areas of mathematics and a key stepping stone to the theory of general Lie groups. Assuming no prior knowledge of Lie groups, this book covers the structure and representation theory of compact Lie groups. Coverage includes the construction of the Spin groups, Schur Orthogonality, the Peter-Weyl Theorem, the Plancherel Theorem, the Maximal Torus Theorem, the Commutator Theorem, the Weyl Integration and Character Formulas, the Highest Weight Classification, and the Borel-Weil Theorem. The book develops the necessary Lie algebra theory with a streamlined approach focusing on linear Lie groups.
Posted in Mathematics

An Introduction to Lie Groups and Lie Algebras

Author: Alexander Kirillov

Publisher: Cambridge University Press

ISBN: 0521889693

Category: Mathematics

Page: 222

View: 3600

This book is an introduction to semisimple Lie algebras; concise and informal, with numerous exercises and examples.
Posted in Mathematics

Undergraduate Commutative Algebra

Author: Miles Reid

Publisher: Cambridge University Press

ISBN: 9780521458894

Category: Mathematics

Page: 153

View: 8825

In this well-written introduction to commutative algebra, the author shows the link between commutative ring theory and algebraic geometry. In addition to standard material, the book contrasts the methods and ideology of modern abstract algebra with concrete applications in algebraic geometry and number theory. Professor Reid begins with a discussion of modules and Noetherian rings before moving on to finite extensions and the Noether normalization. Sections on the nullstellensatz and rings of fractions precede sections on primary decomposition and normal integral domains. This book is ideal for anyone seeking a primer on commutative algebra.
Posted in Mathematics

Representations of Lie Algebras

An Introduction Through gln

Author: Anthony Henderson

Publisher: Cambridge University Press

ISBN: 1139561367

Category: Mathematics

Page: N.A

View: 8475

This bold and refreshing approach to Lie algebras assumes only modest prerequisites (linear algebra up to the Jordan canonical form and a basic familiarity with groups and rings), yet it reaches a major result in representation theory: the highest-weight classification of irreducible modules of the general linear Lie algebra. The author's exposition is focused on this goal rather than aiming at the widest generality and emphasis is placed on explicit calculations with bases and matrices. The book begins with a motivating chapter explaining the context and relevance of Lie algebras and their representations and concludes with a guide to further reading. Numerous examples and exercises with full solutions are included. Based on the author's own introductory course on Lie algebras, this book has been thoroughly road-tested by advanced undergraduate and beginning graduate students and it is also suited to individual readers wanting an introduction to this important area of mathematics.
Posted in Mathematics

Lie Groups and Geometric Aspects of Isometric Actions

Author: Marcos M. Alexandrino,Renato G. Bettiol

Publisher: Springer

ISBN: 3319166131

Category: Mathematics

Page: 213

View: 394

This book provides quick access to the theory of Lie groups and isometric actions on smooth manifolds, using a concise geometric approach. After a gentle introduction to the subject, some of its recent applications to active research areas are explored, keeping a constant connection with the basic material. The topics discussed include polar actions, singular Riemannian foliations, cohomogeneity one actions, and positively curved manifolds with many symmetries. This book stems from the experience gathered by the authors in several lectures along the years and was designed to be as self-contained as possible. It is intended for advanced undergraduates, graduate students and young researchers in geometry and can be used for a one-semester course or independent study.
Posted in Mathematics

Parabolic Geometries: Background and general theory

Author: Andreas Cap,Jan Slovák

Publisher: American Mathematical Soc.

ISBN: 0821826816

Category: Mathematics

Page: 628

View: 8728

Parabolic geometries encompass a very diverse class of geometric structures, including such important examples as conformal, projective, and almost quaternionic structures, hypersurface type CR-structures and various types of generic distributions. The characteristic feature of parabolic geometries is an equivalent description by a Cartan geometry modeled on a generalized flag manifold (the quotient of a semisimple Lie group by a parabolic subgroup). Background on differential geometry, with a view towards Cartan connections, and on semisimple Lie algebras and their representations, which play a crucial role in the theory, is collected in two introductory chapters. The main part discusses the equivalence between Cartan connections and underlying structures, including a complete proof of Kostant's version of the Bott-Borel-Weil theorem, which is used as an important tool. For many examples, the complete description of the geometry and its basic invariants is worked out in detail. The constructions of correspondence spaces and twistor spaces and analogs of the Fefferman construction are presented both in general and in several examples. The last chapter studies Weyl structures, which provide classes of distinguished connections as well as an equivalent description of the Cartan connection in terms of data associated to the underlying geometry. Several applications are discussed throughout the text.
Posted in Mathematics

Complex Semisimple Lie Algebras

Author: Jean-Pierre Serre

Publisher: Springer Science & Business Media

ISBN: 9783540678274

Category: Mathematics

Page: 74

View: 9867

These short notes, already well-known in their original French edition, present the basic theory of semisimple Lie algebras over the complex numbers. The author begins with a summary of the general properties of nilpotent, solvable, and semisimple Lie algebras. Subsequent chapters introduce Cartan subalgebras, root systems, and linear representations. The last chapter discusses the connection between Lie algebras, complex groups and compact groups. The book is intended to guide the reader towards further study.
Posted in Mathematics

Groups, Representations and Physics

Author: H.F Jones

Publisher: CRC Press

ISBN: 9781420050295

Category: Mathematics

Page: 340

View: 7561

Illustrating the fascinating interplay between physics and mathematics, Groups, Representations and Physics, Second Edition provides a solid foundation in the theory of groups, particularly group representations. For this new, fully revised edition, the author has enhanced the book's usefulness and widened its appeal by adding a chapter on the Cartan-Dynkin treatment of Lie algebras. This treatment, a generalization of the method of raising and lowering operators used for the rotation group, leads to a systematic classification of Lie algebras and enables one to enumerate and construct their irreducible representations. Taking an approach that allows physics students to recognize the power and elegance of the abstract, axiomatic method, the book focuses on chapters that develop the formalism, followed by chapters that deal with the physical applications. It also illustrates formal mathematical definitions and proofs with numerous concrete examples.
Posted in Mathematics

Lectures on Buildings

Updated and Revised

Author: Mark Ronan

Publisher: University of Chicago Press

ISBN: 0226724999

Category: Mathematics

Page: 228

View: 9866

In mathematics, “buildings” are geometric structures that represent groups of Lie type over an arbitrary field. This concept is critical to physicists and mathematicians working in discrete mathematics, simple groups, and algebraic group theory, to name just a few areas. Almost twenty years after its original publication, Mark Ronan’s Lectures on Buildings remains one of the best introductory texts on the subject. A thorough, concise introduction to mathematical buildings, it contains problem sets and an excellent bibliography that will prove invaluable to students new to the field. Lectures on Buildings will find a grateful audience among those doing research or teaching courses on Lie-type groups, on finite groups, or on discrete groups. “Ronan’s account of the classification of affine buildings [is] both interesting and stimulating, and his book is highly recommended to those who already have some knowledge and enthusiasm for the theory of buildings.”—Bulletin of the London Mathematical Society
Posted in Mathematics

An Introduction to General Relativity

Author: L. P. Hughston,K. P. Tod

Publisher: Cambridge University Press

ISBN: 9780521339438

Category: Mathematics

Page: 183

View: 9367

This textbook provides an introduction to general relativity for mathematics undergraduates or graduate physicists. After a review of Cartesian tensor notation and special relativity the concepts of Riemannian differential geometry are introducted. More emphasis is placed on an intuitive grasp of the subject and a calculational facility than on a rigorous mathematical exposition. General relativity is then presented as a relativistic theory of gravity reducing in the appropriate limits to Newtonian gravity or special relativity. The Schwarzchild solution is derived and the gravitational red-shift, time dilation and classic tests of general relativity are discussed. There is a brief account of gravitational collapse and black holes based on the extended Schwarzchild solution. Other vacuum solutions are described, motivated by their counterparts in linearised general relativity. The book ends with chapters on cosmological solutions to the field equations. There are exercises attached to each chapter, some of which extend the development given in the text.
Posted in Mathematics

Lectures on Quantum Groups

Author: Jens Carsten Jantzen

Publisher: American Mathematical Soc.

ISBN: 9780821872345

Category: Mathematics

Page: 266

View: 2876

Starting with the quantum analog of sl2, the author carefully leads the reader through all the details necessary for full understanding of the subject, particularly emphasizing similarities and differences with the classical theory. The final chapters of the book describe the Kashiwara-Lusztig theory of so-called crystal (or canonical) bases in representations of complex semisimple Lie algebra.
Posted in Mathematics

Lectures on Profinite Topics in Group Theory

Author: Benjamin Klopsch,Nikolay Nikolov,Christopher Voll

Publisher: Cambridge University Press

ISBN: 1139495658

Category: Mathematics

Page: N.A

View: 9676

In this book, three authors introduce readers to strong approximation methods, analytic pro-p groups and zeta functions of groups. Each chapter illustrates connections between infinite group theory, number theory and Lie theory. The first introduces the theory of compact p-adic Lie groups. The second explains how methods from linear algebraic groups can be utilised to study the finite images of linear groups. The final chapter provides an overview of zeta functions associated to groups and rings. Derived from an LMS/EPSRC Short Course for graduate students, this book provides a concise introduction to a very active research area and assumes less prior knowledge than existing monographs or original research articles. Accessible to beginning graduate students in group theory, it will also appeal to researchers interested in infinite group theory and its interface with Lie theory and number theory.
Posted in Mathematics

Applicable Differential Geometry

Author: M. Crampin,F. A. E. Pirani

Publisher: Cambridge University Press

ISBN: 9780521231909

Category: Mathematics

Page: 394

View: 1677

An introduction to geometrical topics used in applied mathematics and theoretical physics.
Posted in Mathematics

Analytic Pro-P Groups

Author: J. D. Dixon,M. P. F. Du Sautoy,A. Mann,D. Segal

Publisher: Cambridge University Press

ISBN: 9780521542180

Category: Mathematics

Page: 388

View: 3776

An up-to-date treatment of analytic pro-p groups for graduate students and researchers.
Posted in Mathematics

Permutation Groups

Author: Donald S. Passman

Publisher: Courier Corporation

ISBN: 0486310914

Category: Mathematics

Page: 160

View: 7402

Lecture notes by a prominent authority provide a self-contained account of classification theorems. Includes work of Zassenhaus on Frobenius elements and sharply transitive groups, Huppert's theorem, more. 1968 edition.
Posted in Mathematics

Groups, Graphs and Random Walks

Author: Tullio Ceccherini-Silberstein,Maura Salvatori,Ecaterina Sava-Huss

Publisher: Cambridge University Press

ISBN: 1316604403

Category: Mathematics

Page: 542

View: 8466

An accessible and panoramic account of the theory of random walks on groups and graphs, stressing the strong connections of the theory with other branches of mathematics, including geometric and combinatorial group theory, potential analysis, and theoretical computer science. This volume brings together original surveys and research-expository papers from renowned and leading experts, many of whom spoke at the workshop 'Groups, Graphs and Random Walks' celebrating the sixtieth birthday of Wolfgang Woess in Cortona, Italy. Topics include: growth and amenability of groups; Schrdinger operators and symbolic dynamics; ergodic theorems; Thompson's group F; Poisson boundaries; probability theory on buildings and groups of Lie type; structure trees for edge cuts in networks; and mathematical crystallography. In what is currently a fast-growing area of mathematics, this book provides an up-to-date and valuable reference for both researchers and graduate students, from which future research activities will undoubtedly stem.
Posted in Mathematics

Ergodic Theory

with a view towards Number Theory

Author: Manfred Einsiedler,Thomas Ward

Publisher: Springer Science & Business Media

ISBN: 9780857290212

Category: Mathematics

Page: 481

View: 5569

This text is a rigorous introduction to ergodic theory, developing the machinery of conditional measures and expectations, mixing, and recurrence. Beginning by developing the basics of ergodic theory and progressing to describe some recent applications to number theory, this book goes beyond the standard texts in this topic. Applications include Weyl's polynomial equidistribution theorem, the ergodic proof of Szemeredi's theorem, the connection between the continued fraction map and the modular surface, and a proof of the equidistribution of horocycle orbits. Ergodic Theory with a view towards Number Theory will appeal to mathematicians with some standard background in measure theory and functional analysis. No background in ergodic theory or Lie theory is assumed, and a number of exercises and hints to problems are included, making this the perfect companion for graduate students and researchers in ergodic theory, homogenous dynamics or number theory.
Posted in Mathematics

The Finite Simple Groups

Author: Robert Wilson

Publisher: Springer Science & Business Media

ISBN: 1848009879

Category: Mathematics

Page: 298

View: 8444

Here, a thorough grounding in the theory of alternating and classical groups is followed by discussion of exceptional groups (classed as automorphism groups of multilinear forms), sporadic and Chevalley groups, as well as the theory of Lie algebras.
Posted in Mathematics