الصفحة 1
الصفحة 1
Naive Lie Theory
In this new textbook, acclaimed author John Stillwell presents a lucid introduction to Lie theory suitable for junior and senior level undergraduates. In order to achieve this, he focuses on the so-called "classical groups'' that capture the symmetries of real, complex, and quaternion spaces. These symmetry groups may be represented by matrices, which allows them to be studied by elementary methods from calculus and linear algebra.
Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces
As K. Nomizu has justly noted [K. Nomizu, 56], Differential Geometry ever will be initiating newer and newer aspects of the theory of Lie groups. This monograph is devoted to just some such aspects of Lie groups and Lie algebras. New differential geometric problems came into being in connection with so called subsymmetric spaces, subsymmetries, and mirrors introduced in our works dating back to 1957 [L.V. Sabinin, 58a,59a,59b]. In addition, the exploration of mirrors and systems of mirrors is of interest in the case of symmetric spaces. Geometrically, the most rich in content there appeared to be the homogeneous Riemannian spaces with systems of mirrors generated by commuting subsymmetries, in particular, so called tri-symmetric spaces introduced in [L.V. Sabinin, 61b]. As to the concrete geometric problem which needs be solved and which is solved in this monograph, we indicate, for example, the problem of the classification of all tri-symmetric spaces with simple compact groups of motions. Passing from groups and subgroups connected with mirrors and subsymmetries to the corresponding Lie algebras and subalgebras leads to an important new concept of the involutive sum of Lie algebras [L.V. Sabinin, 65]. This concept is directly concerned with unitary symmetry of elementary par- cles (see [L.V. Sabinin, 95,85] and Appendix 1). The first examples of involutive (even iso-involutive) sums appeared in the - ploration of homogeneous Riemannian spaces with and axial symmetry. The consideration of spaces with mirrors [L.V. Sabinin, 59b] again led to iso-involutive sums.
Metodi Matematici della Fisica = Mathematical Methods of Physics
This text draws its origin from my old notes, prepared for the course of Mathematical Methods of Physics and gradually arranged, refined and updated over the course of many years of teaching. The aim has always been to provide as simple and direct a presentation as possible of the mathematical methods relevant to Physics: Fourier series, Hilbert spaces, linear operators, functions of complex variables, Fourier and Laplace transforms, distributions. In addition to these basic topics, a brief introduction to the first notions of group theory, Lie algebras and symmetries in view of their applications to Physics is presented in the Appendix.
Introduction to Lie Algebras
This book provides an elementary introduction to Lie algebras. It starts with basic concepts. A section on low-dimensional Lie algebras provides readers with experience of some useful examples. This is followed by a discussion of solvable Lie algebras and a strategy towards a classification of finite-dimensional complex Lie algebras. The next chapters cover Engel's theorem, Lie's theorem and Cartan's criteria and introduce some representation theory. The root-space decomposition of a semisimple Lie algebra is discussed, and the classical Lie algebras studied in detail. The authors also classify root systems, and give an outline of Serre's construction of complex semisimple Lie algebras. An overview of further directions then concludes the book and shows the high degree to which Lie algebras influence present-day mathematics.The only prerequisite is some linear algebra and an appendix summarizes the main facts that are needed. The treatment is kept as simple as possible with no attempt at full generality.
Infinite dimensional algebras and quantum integrable systems
This volume presents the invited lectures of the workshop "Infinite Dimensional Algebras and Quantum Integrable Systems'' .ecent developments in the theory of infinite dimensional algebras and their applications to quantum integrable systems are reviewed by some of the leading experts in the field. The volume will be of interest to a broad audience from graduate students to researchers in mathematical physics and related fields.
Groupes et algèbres de Lie : Chapitres 4 à 6 = Lie groups and algebras : Chapters 4 to 6
Nicolas BOURBAKI's Elements of Mathematics aim to provide a rigorous, systematic presentation without prerequisites of mathematics from their foundations.This third volume of the Book on Groups and Lie Algebras, ninth Book of the treatise, is devoted to the structures of root systems , Coxeter groups and Tits systems, which appear naturally in the study of analytic or algebraic Lie groups
Groupes et algèbres de Lie : Chapitres 2 et 3 = Lie groups and algebras : Chapters 2 and 3
Nicolas BOURBAKI's Elements of Mathematics aim to provide a rigorous, systematic and prerequisite presentation of mathematics from their foundations. Chapter 2 continues the presentation of the fundamental notions of Lie algebras with the introduction of free Lie algebras and the series by Hausdorff. Chapter 3 is devoted to the basic concepts for the groups of Lies on an Archimedean or ultrametric body.
Groupes et algèbres de Lie : Chapitre 9, Groupes de Lie réels compacts = Lie groups and algebras : Chapter 9, Compact real Lie groups
Nicolas BOURBAKI's Elements of Mathematics aim to provide a rigorous, systematic and un-prerequisite presentation of mathematics from their foundations. This ninth chapter of the Book on Groups and Lie Algebras, ninth Book of the treatise, includes the paragraphs, Compact Lie Algebras ; Maximum tori of compact Lie groups; Compact fromes of complex semi-simple Lie algebras; Root system associated with a compact group; Conjugation classes; Integration into compact Lie groups; Irreducible representations of connected compact Lie groups; Fourier transformation; Operation of compact Lie groups on manifolds.
Groupes et algèbres de Lie : Chapitre 1 = Lie groups and algebras : Chapter 1
Nicolas BOURBAKI's Elements of Mathematics aim to provide a rigorous, systematic and un-prerequisite presentation of mathematics from their foundations. This ninth chapter of the Book on Groups and Lie Algebras, ninth Book of the treatise, includes the paragraphs, Compact Lie Algebras ; Maximum tori of compact Lie groups; Compact fromes of complex semi-simple Lie algebras; Root system associated with a compact group; Conjugation classes; Integration into compact Lie groups; Irreducible representations of connected compact Lie groups; Fourier transformation; Operation of compact Lie groups on manifolds.
Functional Identities
The theory of functional identities (FIs) is a relatively new one - the first results were published at the beginning of the 1990s, and this is the first book on this subject. An FI can be informally described as an identical relation involving arbitrary elements in an associative ring together with arbitrary (unknown) functions. The goal of the general FI theory is to describe these functions, or, when this is not possible, to describe the structure of the ring admitting the FI in question. This abstract theory has turned out to be a powerful tool for solving a variety of problems in ring theory, Lie algebras, Jordan algebras, linear algebra, and operator theory.
Fourier Transforms of Invariant Functions on Finite Reductive Lie Algebras
In this book the author studies Fourier transforms using Deligne-Lusztig induction and the Lie algebra version of Lusztig’s character sheaves theory. He conjectures a commutation formula between Deligne-Lusztig induction and Fourier transforms that he proves in many cases. As an application the computation of the values of the trigonometric sums (on reductive Lie algebras) is shown to reduce to the computation of the generalized Green functions and to the computation of some fourth roots of unity.
Factorization Method in Quantum Mechanics
Introduces the factorization method in quantum mechanics at an advanced level with an aim to put mathematical and physical concepts and techniques like the factorization method, Lie algebras, matrix elements and quantum control at the Reader’s disposal.
Evolution Algebras and their Applications
Behind genetics and Markov chains, there is an intrinsic algebraic structure. It is defined as a type of new algebra: as evolution algebra. This concept lies between algebras and dynamical systems. Algebraically, evolution algebras are non-associative Banach algebras; dynamically, they represent discrete dynamical systems. Evolution algebras have many connections with other mathematical fields including graph theory, group theory, stochastic processes, dynamical systems, knot theory, 3-manifolds, and the study of the Ihara-Selberg zeta function. In this volume the foundation of evolution algebra theory and applications in non-Mendelian genetics and Markov chains is developed, with pointers to some further research topics.
Elements of Mathematics : Lie Groups and Lie Algebras
Provide a formal, systematic presentation of mathematics from their beginning. This volume concludes the book on Lie Groups and Lie Algebras by covering the structure and representation theory of semi-simple Lie algebras and compact Lie groups. It contains the following chapters: 7. Cartan Subalgebras and Regular Elements / 8. Split Semi-Simple Lie Algebras / 9. Compact Real Lie Groups
Eléments dhistoire des mathématiques = Elements of the history of mathematics
Brings together the historical notes published in the various books of mathematics elements by the author. They therefore concern all the matters covered in this treaty: set theory, algebra, topology, functions of a variable real, topological vector spaces, integration, commutative algebra, groups and Lie algebras. Composed of initially separate studies, this work does not claim to sketch a followed and complete history of development of mathematics. The interweaving of the different themes and the unity of the point of view ensure the deep consistency.
Dirac Operators in Representation Theory
This monograph presents a comprehensive treatment of important new ideas on Dirac operators and Dirac cohomology. Dirac operators are widely used in physics, differential geometry, and group-theoretic settings (particularly, the geometric construction of discrete series representations). The related concept of Dirac cohomology, which is defined using Dirac operators, is a far-reaching generalization that connects index theory in differential geometry to representation theory. Using Dirac operators as a unifying theme, the authors demonstrate how some of the most important results in representation theory fit together when viewed from this perspective.
Computer algebra and geometric algebra with applications ; 6th International Workshop, IWMM 2004, Shanghai, China, May 19-21, 2004 and International Workshop, GIAE 2004, Xian, China, May 24-28, 2004.Revised Selected Papers
MathematicsMechanization consistsoftheory,softwareandapplicationofc- puterized mathematical activities such as computing, reasoning and discovering. ItsuniquefeaturecanbesuccinctlydescribedasAAA(Algebraization,Algori- mization, Application). The name “Mathematics Mechanization” has its origin in the work of Hao Wang (1960s), one of the pioneers in using computers to do research in mathematics, particularly in automated theorem proving. Since the 1970s, this research direction has been actively pursued and extensively dev- oped by Prof. Wen-tsun Wu and his followers. It di?ers from the closely related disciplines like Computer Mathematics, Symbolic Computation and Automated Reasoning in that its goal is to make algorithmic studies and applications of mathematics the major trend of mathematics development in the information age.
Lie Groups : An Approach through Invariants and Representations
Lie groups has been an increasing area of focus and rich research since the middle of the 20th century. Procesi's masterful approach to Lie groups through invariants and representations gives the reader a comprehensive treatment of the classical groups along with an extensive introduction to a wide range of topics associated with Lie groups: symmetric functions, theory of algebraic forms, Lie algebras, tensor algebra and symmetry, semisimple Lie algebras, algebraic groups, group representations, invariants, Hilbert theory, and binary forms with fields ranging from pure algebra to functional analysis.
Lie Algebras and Applications
This book, designed for advanced graduate students and post-graduate researchers, provides an introduction to Lie algebras and some of their applications to the spectroscopy of molecules, atoms, nuclei and hadrons. In the first part, a concise exposition is given of the basic concepts of Lie algebras, their representations and their invariants. The second part contains a description of how Lie algebras are used in practice in the treatment of bosonic and fermionic systems. Physical applications considered include rotations and vibrations of molecules (vibron model), collective modes in nuclei (interacting boson model), the atomic shell model, the nuclear shell model, and the quark model of hadrons. One of the key concepts in the application of Lie algebraic methods in physics, that of spectrum generating algebras and their associated dynamic symmetries, is also discussed. The book contains many examples that help to elucidate the abstract algebraic definitions. It provides a summary of many formulas of practical interest, such as the eigenvalues of Casimir operators and the dimensions of the representations of all classical Lie algebras.
Lie Algebras and Algebraic Groups
The theory of Lie algebras and algebraic groups has been an area of active research in the last 50 years. It intervenes in many different areas of mathematics : for example invariant theory, Poisson geometry, harmonic analysis, mathematical physics. The aim of this book is to assemble in a single volume the algebraic aspects of the theory so as to present the foundation of the theory in characteristic zero. Detailed proofs are included and some recent results are discussed in the last chapters. All the prerequisites on commutative algebra and algebraic geometry are included.
