Lie Algebras and Algebraic Groups

ISBN-13: 9783540241706 / Angielski / Twarda / 2005 / 656 str.

Patrice Tauvel;Rupert W. T. Yu

Opis

The theory of groups and Lie algebras is interesting for many reasons. In the mathematical viewpoint, it employs at the same time algebra, analysis and geometry. On the other hand, it intervenes in other areas of science, in particularindi?erentbranchesofphysicsandchemistry.Itisanactivedomain of current research. Oneofthedi?cultiesthatgraduatestudentsormathematiciansinterested in the theory come across, is the fact that the theory has very much advanced, andconsequently, theyneedtoreadavastamountofbooksandarticlesbefore they could tackle interesting problems. One of the goals we wish to achieve with this book is to assemble in a single volume the basis of the algebraic aspects of the theory of groups and Lie algebras. More precisely, we have presented the foundation of the study of ?nite-dimensional Lie algebras over an algebraically closed ?eld of characteristic zero. Here, the geometrical aspect is fundamental, and consequently, we need to use the notion of algebraic groups. One of the main di?erences between this book and many other books on the subject is that we give complete proofs for the relationships between algebraic groups and Lie algebras, instead of admitting them. We have also given the proofs of certain results on commutative al- bra and algebraic geometry that we needed so as to make this book as se- contained as possible. We believe that in this way, the book can be useful for both graduate students and mathematicians working in this area. Let us give a brief description of the material treated in this book

Recenzje

From the reviews:

"As Tauvel and Yu focus on algebraic groups, they approach Lie theory via algebraic geometry and even develop that subject from scratch ... . For the purpose at hand, Tauvel and Yu's work compares favorably ... . Summing Up: Highly recommended. Upper-division undergraduates through professionals." (D.V. Feldman, Choice, 43:10, June 2006)

"The sheer volume of material covered herein should make this book an invaluable reference for people interested in, or teaching, Lie algebras or algebraic groups. It truly provides 'one stop shopping' for someone needing a result or hard-to-find proof. ... I cannot even begin to imagine how much work must have gone into creating such a thorough and comprehensive reference, and I have no doubt it will be an important and useful addition to the literature on this subject." (Mark Hunacek, The Mathematical Gazette, 90:19, 2006)

"The focus of this book is the study of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. ... the book is largely self-contained. ... the authors are extremely knowledgeable in their subjects and the reader can profit from the wealth of material contained in this book. Therefore this book is an ideal reference source and research guide for graduate students and mathematicians working in this area." (Benjamin Cahen, Zentralblatt MATH, Vol. 1068, 2005)

"The theory of Lie algebras and algebraic groups has been an area of active research for the last 50 years. ... The aim of this book is to assemble in a single volume the algebraic aspects of the theory, so as to present the foundations of the theory in characteristic zero. Detailed proofs are included and some recent results are discussed in the final chapters. All the prerequisites on commutative algebra and algebraic geometry are included." (L'Enseignement Mathematique, Vol. 51 (3-4), 2006)

"This introduction to Lie algebras and algebraic groups aims to provide a full background to the subject. ... The book has an encyclopedic character, offering much else besides the actual subject." (Mathematika, Vol. 52, 2005)

"The stated goal of the authors is to provide a 'foundation for the study of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero' in a self-contained work that will be useful to 'both graduate students and mathematicians working in this area'. ... the book contains a wealth of detail and takes the reader from the basic classical concepts to the modern borders of this still-active area. Complete proofs are given and the authors present their material clearly and concisely throughout." (Duncan Melville, MathDL, March, 2006)

"This book offers ... complete presentation of the theory of the topics in its title over an algebraically closed field of characteristic zero. Assuming only an undergraduate background in abstract algebra, it covers in detail all the prerequisites that one needs for the theory of Lie algebras and algebraic groups together with the foundations of that theory. ... The book is well written and easy to follow ... ." (William M. McGovern, SIAM Reviews, Vol. 48 (1), 2006)

"The theory of algebraic groups and Lie algebras is a deeply advanced and developed area of modern mathematics. ... The text is clearly written and the material is well organized and considered, so the present book may be strongly recommended both to a beginner looking for a self-contained introduction to the theory of algebraic groups and Lie algebras, and to a specialist who wants to have a systematic presentation of the theory." (Ivan V. Arzhantsev, Mathematical Reviews, Issue, 2006 c)

Spis treści

Preface 1. Results on topological spaces 1.1 Irreducible sets and spaces 1.2 Dimension 1.3 Noetherian spaces 1.4 Constructible sets 1.5 Gluing topological spaces 2. Rings and modules 2.1 Ideals 2.2 Prime and maximal ideals 2.3 Rings of fractions and localization 2.4 Localization of modules 2.5 Radical of an ideal 2.6 Local rings 2.7 Noetherian rings and modules 2.8 Derivations 2.9 Module of differentials 3. Integral extensions 3.1 Integral dependence 3.2 Integrally closed rings 3.3 Extensions of prime ideals 4. Factorial rings 4.1 Generalities 4.2 Unique factorization 4.3 Principal ideal domains and Euclidean domains 4.4 Polynomial and factorial rings 4.5 Symmetric polynomials 4.6 Resultant and discriminant 5. Field extensions 5.1 Extensions 5.2 Algebraic and transcendental elements 5.3 Algebraic extensions 5.4 Transcendence basis 5.5 Norm and trace 5.6 Theorem of the primitive element 5.7 Going Down Theorem 5.8 Fields and derivations 5.9 Conductor 6. Finitely generated algebras 6.1 Dimension 6.2 Noether’s Normalization Theorem 6.3 Krull’s Principal Ideal Theorem 6.4 Maximal ideals 6.5 Zariski topology 7. Gradings and filtrations 7.1 Graded rings and graded modules 7.2 Graded submodules 7.3 Applications 7.4 Filtrations 7.5 Grading associated to a filtration 8. Inductive limits 8.1 Generalities 8.2 Inductive systems of maps 8.3 Inductive systems of magmas, groups and rings 8.4 An example 8.5 Inductive systems of algebras 9. Sheaves of functions 9.1 Sheaves 9.2 Morphisms 9.3 Sheaf associated to a presheaf 9.4 Gluing 9.5 Ringed space 10. Jordan decomposition and some basic results on groups 10.1 Jordan decomposition 10.2 Generalities on groups 10.3 Commutators 10.4 Solvable groups 10.5 Nilpotent groups 10.6 Group actions 10.7 Generalities on representations 10.8 Examples 11. Algebraic sets 11.1 Affine algebraic sets 11.2 Zariski topology 11.3 Regular functions 11.4 Morphisms 11.5 Examples of morphisms 11.6 Abstract algebraic sets 11.7 Principal open subsets 11.8 Products of algebraic sets 12. Prevarieties and varieties 12.1 Structure sheaf 12.2 Algebraic prevarieties 12.3 Morphisms of prevarieties 12.4 Products of prevarieties 12.5 Algebraic varieties 12.6 Gluing 12.7 Rational functions 12.8 Local rings of a variety 13. Projective varieties 13.1 Projective spaces 13.2 Projective spaces and varieties 13.3 Cones and projective varieties 13.4 Complete varieties 13.5 Products 13.6 Grassmannian variety 14. Dimension 14.1 Dimension of varieties 14.2 Dimension and the number of equations 14.3 System of parameters 14.4 Counterexamples 15. Morphisms

Blurb

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.