Groups and group theory Books
The University of Chicago Press Lie Algebras and Locally Compact Groups Chicago
Book SynopsisThis volume presents lecture notes based on the author's courses on Lie algebras and the solution of Hilbert's fifth problem. In chapter 1, Lie Algebras, the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, The Structure of Locally Compact Groups, deals with the solution of Hilbert's fifth problem given by Gleason, Montgomery, and Zipplin in 1952.
£38.00
Springer-Verlag New York Inc. Introduction to Mechanics and Symmetry
Book SynopsisSpecific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering.Trade Review"The book is self-contained.It remains a good and solid introduction to this subject."Nieuw Archief voor Wiskunde, March 2001 "... This book takes the reader on one of the greatest journeys in modern mathematics that has as its roots a subject that is more than 300 years old. Armed with this knowledge a reader is ready to pursue numerous topics of active mathematical research, from the more pure domains of symplectic geometry and topology to the geometric analysis of the limitless supply of examples from mechanics."Newsletter of the Newzealand Mathematical Society, No. 81, April 2001 Second Edition J.E. Marsden and T.S. Ratiu Introduction to Mechanics and Symmetry A Basic Exposition of Classical Mechanical Systems "As the name of the book implies, a consistent theme running through the book is that of symmetry. Indeed the latter half of the book focuses on Poisson manifolds, momentum maps, Lie-Poisson reduction, co-adjoint orbits and the integrability of the rigid body. The discussion of reduction must be the most comprehensive yet given. A pleasant feature of the book is that most of the theory that relates to finite-dimensional mechanical systems is illustrated concretely in terms of local coordinates, thereby making the book accessible even to beginners in the field."—MATHEMATICAL REVIEWSTable of ContentsPreface * About the Authors * 1 Introduction and Overview * 2 Hamiltonian Systems on Linear Symplectic Spaces * 3 An Introduction to Infinite-Dimensional Systems * 4 Manifolds, Vector Fields, and Differential Forms * 5 Hamiltonian Systems on Symplectic Manifolds * 6 Cotangent Bundles * 7 Lagrangian Mechanics * 8 Variational Principles, Constraints, and Rotating Systems * 9 An Introduction to Lie Groups * 10 Poisson Manifolds * 11 Momentum Maps *12 Computation and Properties of Momentum Maps * 13 Lie-Poisson and Euler-Poincare Reduction * 14 Coadjoint Orbits * 15 The Free Rigid Body * References
£67.49
John Wiley & Sons Inc Elementary Lie Group Analysis and Ordinary
Book SynopsisThis book presents ordinary differential equations based on Lie group analysis and related invariance principles. The author provides students and teachers with a text for one-semester undergraduate and graduate courses that spans a variety of topics, from the basic theory through to applications.Trade Review"…this is the first self-contained university text on ordinary differential equations…" (Zentralblatt Math, Vol.1047, No.22, 2004)Table of ContentsIntroduction to Differential Equations. Transformation Groups. Lie Group Analysis of Ordinary Differential Equations. Brief on Lie Algebras. First Order Differential Equations. Integration of Second Order Equations. Basic Theory of Linear Equations. Nonlinear Second Order Equations. Integration of Third Order Equations. Nonlinear Superposition Principle. Index.
£176.36
John Wiley & Sons Inc An Introduction to Grobner Bases
Book SynopsisAs algebra becomes more widely used in a variety of applications and computers are developed to allow efficient calculations in the field, so there becomes a need for new techniques to further this area of research. Grobner Bases is one topic which has recently become a very popular and important area of modern algebra.Table of ContentsRings. Fields, and Ideals. Monomial Ideals. Gröbner Bases. Algebraic Sets. Primary Decomposition of Ideals. Solving Systems of Polynomial Equations. Applications of Gröbner Bases. Homogeneous Algebras. Projective Varieties. The Associated Graded Ring. Hilbert Series. Variations of Gröbner Bases. Improvements to Buchberger's Algorithm. Software. Hints to Some Exercises. Answers to Exercises. Bibliography. Index.
£120.56
Princeton University Press The Ergodic Theory of Lattice Subgroups
Book SynopsisDevelops a systematic general approach to the proof of ergodic theorems for a large class of non-amenable locally compact groups and their lattice subgroups. This book formulates simple general conditions on the spectral theory of the group and the regularity of the averaging sets, which suffice to guarantee convergence to the ergodic mean.Table of ContentsPreface vii 0.1 Main objectives vii 0.2 Ergodic theory and amenable groups viii 0.3 Ergodic theory and nonamenable groups x Chapter 1. Main results: Semisimple Lie groups case 1 1.1 Admissible sets 1 1.2 Ergodic theorems on semisimple Lie groups 2 1.3 The lattice point-counting problem in admissible domains 4 1.4 Ergodic theorems for lattice subgroups 6 1.5 Scope of the method 8 Chapter 2. Examples and applications 11 2.1 Hyperbolic lattice points problem 11 2.2 Counting integral unimodular matrices 12 2.3 Integral equivalence of general forms 13 2.4 Lattice points in S-algebraic groups 15 2.5 Examples of ergodic theorems for lattice actions 16 Chapter 3. Definitions, preliminaries, and basic tools 19 3.1 Maximal and exponential-maximal inequalities 19 3.2 S-algebraic groups and upper local dimension 21 3.3 Admissible and coarsely admissible sets 21 3.4 Absolute continuity and examples of admissible averages 23 3.5 Balanced and well-balanced families on product groups 26 3.6 Roughly radial and quasi-uniform sets 27 3.7 Spectral gap and strong spectral gap 29 3.8 Finite-dimensional subrepresentations 30 Chapter 4. Main results and an overview of the proofs 33 4.1 Statement of ergodic theorems for S-algebraic groups 33 4.2 Ergodic theorems in the absence of a spectral gap: overview 35 4.3 Ergodic theorems in the presence of a spectral gap: overview 38 4.4 Statement of ergodic theorems for lattice subgroups 40 4.5 Ergodic theorems for lattice subgroups: overview 42 4.6 Volume regularity and volume asymptotics: overview 44 Chapter 5. Proof of ergodic theorems for S-algebraic groups 47 5.1 Iwasawa groups and spectral estimates 47 5.2 Ergodic theorems in the presence of a spectral gap 50 5.3 Ergodic theorems in the absence of a spectral gap, I 56 5.4 Ergodic theorems in the absence of a spectral gap, II 57 5.5 Ergodic theorems in the absence of a spectral gap, III 60 5.6 The invariance principle and stability of admissible averages 67 Chapter 6. Proof of ergodic theorems for lattice subgroups 71 6.1 Induced action 71 6.2 Reduction theorems 74 6.3 Strong maximal inequality 75 6.4 Mean ergodic theorem 78 6.5 Pointwise ergodic theorem 83 6.6 Exponential mean ergodic theorem 84 6.7 Exponential strong maximal inequality 87 6.8 Completion of the proofs 90 6.9 Equidistribution in isometric actions 91 Chapter 7. Volume estimates and volume regularity 93 7.1 Admissibility of standard averages 93 7.2 Convolution arguments 98 7.3 Admissible, well-balanced, and boundary-regular families 101 7.4 Admissible sets on principal homogeneous spaces 105 7.5 Tauberian arguments and Holder continuity 107 Chapter 8. Comments and complements 113 8.1 Lattice point-counting with explicit error term 113 8.2 Exponentially fast convergence versus equidistribution 115 8.3 Remark about balanced sets 116 Bibliography 117 Index 121
£42.50
Princeton University Press MumfordTate Groups and Domains
Book SynopsisMumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book provides a comprehensive exploration of Mumford-Tate groups and domains.Trade Review"The brilliance of the results and their broad spectrum of their applications makes this book an outstanding piece. Yet, there is more to write and to develop: the authors suggest the existence of future lines of research for a next book."--Jonathan Sanchez Hernandez, European Mathematical SocietyTable of ContentsIntroduction 1 I Mumford-Tate Groups 28 I.A Hodge structures 28 I.B Mumford-Tate groups 32 I.C Mixed Hodge structures and their Mumford-Tate groups 38 II Period Domains and Mumford-Tate Domains 45 II.A Period domains and their compact duals 45 II.B Mumford-Tate domains and their compact duals 55 II.C Noether-Lefschetz loci in period domains 61 III The Mumford-Tate Group of a Variation of Hodge Structure 67 III.A The structure theorem for variations of Hodge structures 69 III.B An application of Mumford-Tate groups 78 III.C Noether-Lefschetz loci and variations of Hodge structure .81 IV Hodge Representations and Hodge Domains 85 IV.A Part I: Hodge representations 86 IV.B The adjoint representation and characterization of which weights give faithful Hodge representations 109 IV.C Examples: The classical groups 117 IV.D Examples: The exceptional groups 126 IV.E Characterization of Mumford-Tate groups 132 IV.F Hodge domains 149 IV.G Mumford-Tate domains as particular homogeneous complex manifolds 168 Appendix: Notation from the structure theory of semisimple Lie algebras 179 V Hodge Structures with Complex Multiplication 187 V.A Oriented number fields 189 V.B Hodge structures with special endomorphisms 193 V.C A categorical equivalence 196 V.D Polarization and Mumford-Tate groups . 198 V.E An extended example 202 V.F Proofs of Propositions V.D.4 and V.D.5 in the Galois case 209 VI Arithmetic Aspects of Mumford-Tate Domains 213 VI.A Groups stabilizing subsets of D 215 VI.B Decomposition of Noether-Lefschetz into Hodge orientations 219 VI.C Weyl groups and permutations of Hodge orientations 231 VI.D Galois groups and fields of definition 234 Appendix: CM points in unitary Mumford-Tate domains 239 VII Classification of Mumford-Tate Subdomains 240 VII.A A general algorithm 240 VII.B Classification of some CM-Hodge structures 243 VII.C Determination of sub-Hodge-Lie-algebras 246 VII.D Existence of domains of type IV(f) 251 VII.E Characterization of domains of type IV(a) and IV(f) 253 VII.F Completion of the classification for weight 3 256 VII.G The weight 1 case 260 VII.H Algebro-geometric examples for the Noether-Lefschetzlocus types 265 VIII Arithmetic of Period Maps of Geometric Origin 269 VIII.A Behavior of fields of definition under the period Map -- image and preimage 270 VIII.B Existence and density of CM points in motivic VHS 275 Bibliography 277 Index 287
£170.00
Princeton University Press MumfordTate Groups and Domains
Book SynopsisMumford-Tate groups are the fundamental symmetry groups of Hodge theory, a subject which rests at the center of contemporary complex algebraic geometry. This book provides a comprehensive exploration of Mumford-Tate groups and domains.Trade Review"The brilliance of the results and their broad spectrum of their applications makes this book an outstanding piece. Yet, there is more to write and to develop: the authors suggest the existence of future lines of research for a next book."--Jonathan Sanchez Hernandez, European Mathematical SocietyTable of ContentsIntroduction 1 I Mumford-Tate Groups 28 I.A Hodge structures 28 I.B Mumford-Tate groups 32 I.C Mixed Hodge structures and their Mumford-Tate groups 38 II Period Domains and Mumford-Tate Domains 45 II.A Period domains and their compact duals 45 II.B Mumford-Tate domains and their compact duals 55 II.C Noether-Lefschetz loci in period domains 61 III The Mumford-Tate Group of a Variation of Hodge Structure 67 III.A The structure theorem for variations of Hodge structures 69 III.B An application of Mumford-Tate groups 78 III.C Noether-Lefschetz loci and variations of Hodge structure .81 IV Hodge Representations and Hodge Domains 85 IV.A Part I: Hodge representations 86 IV.B The adjoint representation and characterization of which weights give faithful Hodge representations 109 IV.C Examples: The classical groups 117 IV.D Examples: The exceptional groups 126 IV.E Characterization of Mumford-Tate groups 132 IV.F Hodge domains 149 IV.G Mumford-Tate domains as particular homogeneous complex manifolds 168 Appendix: Notation from the structure theory of semisimple Lie algebras 179 V Hodge Structures with Complex Multiplication 187 V.A Oriented number fields 189 V.B Hodge structures with special endomorphisms 193 V.C A categorical equivalence 196 V.D Polarization and Mumford-Tate groups . 198 V.E An extended example 202 V.F Proofs of Propositions V.D.4 and V.D.5 in the Galois case 209 VI Arithmetic Aspects of Mumford-Tate Domains 213 VI.A Groups stabilizing subsets of D 215 VI.B Decomposition of Noether-Lefschetz into Hodge orientations 219 VI.C Weyl groups and permutations of Hodge orientations 231 VI.D Galois groups and fields of definition 234 Appendix: CM points in unitary Mumford-Tate domains 239 VII Classification of Mumford-Tate Subdomains 240 VII.A A general algorithm 240 VII.B Classification of some CM-Hodge structures 243 VII.C Determination of sub-Hodge-Lie-algebras 246 VII.D Existence of domains of type IV(f) 251 VII.E Characterization of domains of type IV(a) and IV(f) 253 VII.F Completion of the classification for weight 3 256 VII.G The weight 1 case 260 VII.H Algebro-geometric examples for the Noether-Lefschetzlocus types 265 VIII Arithmetic of Period Maps of Geometric Origin 269 VIII.A Behavior of fields of definition under the period Map -- image and preimage 270 VIII.B Existence and density of CM points in motivic VHS 275 Bibliography 277 Index 287
£74.80
Princeton University Press Chow Rings Decomposition of the Diagonal and the
Book SynopsisProvides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. This title delves into arguments originating in Nori's work that have been further developed by others.Trade Review"This dense, fascinating book by Voisin is a report of some of the exciting discoveries she has made in the quest of the secrets of algebraic cycles."--Alberto Collino, Zentralblatt MATH "[An advanced] reader will find a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field. Any complex geometer interested in the interplay between algebraic cycles, Hodge theory and algebraic topology should have this book on his or her shelf."--C. A. M. Peters, Mathematical Reviews ClippingsTable of ContentsPreface vii 1Introduction 1 1.1 Decomposition of the diagonal and spread 3 1.2 The generalized Bloch conjecture 7 1.3 Decomposition of the small diagonal and application to the topology of families 9 1.4 Integral coefficients and birational invariants 11 1.5 Organization of the text 13 2Review of Hodge theory and algebraic cycles 15 2.1 Chow groups 15 2.2 Hodge structures 24 3Decomposition of the diagonal 36 3.1 A general principle 36 3.2 Varieties with small Chow groups 44 4Chow groups of large coniveau complete intersections 55 4.1 Hodge coniveau of complete intersections 55 4.2 Coniveau 2 complete intersections 64 4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67 4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86 5On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88 5.1 Tautological ring of a K3 surface 88 5.2 A decomposition of the small diagonal 96 5.3 Deligne's decomposition theorem for families of K3 surfaces 106 6Integral coefficients 123 6.1 Integral Hodge classes and birational invariants 123 6.2 Rationally connected varieties and the rationality problem 127 6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139 Bibliography 155 Index 163
£148.75
Princeton University Press Chow Rings Decomposition of the Diagonal and the
Book SynopsisProvides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. This title delves into arguments originating in Nori's work that have been further developed by others.Trade Review"This dense, fascinating book by Voisin is a report of some of the exciting discoveries she has made in the quest of the secrets of algebraic cycles."--Alberto Collino, Zentralblatt MATH "[An advanced] reader will find a rich collection of ideas as well as detailed machinery with which to attack difficult problems in the field. Any complex geometer interested in the interplay between algebraic cycles, Hodge theory and algebraic topology should have this book on his or her shelf."--C. A. M. Peters, Mathematical Reviews ClippingsTable of ContentsPreface vii 1Introduction 1 1.1 Decomposition of the diagonal and spread 3 1.2 The generalized Bloch conjecture 7 1.3 Decomposition of the small diagonal and application to the topology of families 9 1.4 Integral coefficients and birational invariants 11 1.5 Organization of the text 13 2Review of Hodge theory and algebraic cycles 15 2.1 Chow groups 15 2.2 Hodge structures 24 3Decomposition of the diagonal 36 3.1 A general principle 36 3.2 Varieties with small Chow groups 44 4Chow groups of large coniveau complete intersections 55 4.1 Hodge coniveau of complete intersections 55 4.2 Coniveau 2 complete intersections 64 4.3 Equivalence of generalized Bloch and Hodge conjectures for general complete intersections 67 4.4 Further applications to the Bloch conjecture on 0-cycles on surfaces 86 5On the Chow ring of K3 surfaces and hyper-Kahler manifolds 88 5.1 Tautological ring of a K3 surface 88 5.2 A decomposition of the small diagonal 96 5.3 Deligne's decomposition theorem for families of K3 surfaces 106 6Integral coefficients 123 6.1 Integral Hodge classes and birational invariants 123 6.2 Rationally connected varieties and the rationality problem 127 6.3 Integral decomposition of the diagonal and the structure of the Abel-Jacobi map 139 Bibliography 155 Index 163
£68.00
Princeton University Press Hodge Theory
Book SynopsisProvides an introduction to Hodge theory - one of the central and most vibrant areas of contemporary mathematics - from leading specialists on the subject. This book includes topics that range from the basic topology of algebraic varieties to the study of variations of mixed Hodge structure and the Hodge theory of maps.Trade Review"Charles and Schnell's chapter beautifully surveys the theory of absolute Hodge classes, giving in particular a complete proof of Deligne's theorem on absolute Hodge classes on abelian varieties... A welcome addition to the literature and should be useful to both graduate students and researchers working in Hodge theory."--Dan Petersen, MathSciNetTable of Contents*FrontMatter, pg. i*Contributors, pg. v*Contents, pg. vii*Preface, pg. xv*Chapter One. Introduction to Kahler Manifolds, pg. 1*Chapter Two. From Sheaf Cohomology to the Algebraic de Rham Theorem, pg. 70*Chapter Three. Mixed Hodge Structures, pg. 123*Chapter Four. Period Domains and Period Mappings, pg. 217*Chapter Five. The Hodge Theory of Maps, pg. 257*Chapter Six The Hodge Theory of Maps, pg. 273*Chapter Seven. Introduction to Variations of Hodge Structure, pg. 297*Chapter Eight. Variations of Mixed Hodge Structure, pg. 333*Chapter Nine. Lectures on Algebraic Cycles and Chow Groups, pg. 410*Chapter Ten. The Spread Philosophy in the Study of Algebraic Cycles, pg. 449*Chapter Eleven. Notes on Absolute Hodge Classes, pg. 469*Chapter Twelve. Shimura Varieties: A Hodge-Theoretic Perspective, pg. 531*Bibliography, pg. 574*Index, pg. 577
£78.20
Princeton University Press The padic Simpson Correspondence
Book SynopsisTrade Review"The authors give a very detailed introduction to the theory, smoothing out some difficulties by introducing new concepts."--Gerd Faltings, Zentralblatt MATHTable of Contents*Frontmatter, pg. i*Contents, pg. vii*Foreword, pg. ix*Chapter I. Representations of the fundamental group and the torsor of deformations. An overview, pg. 1*Chapter II. Representations of the fundamental group and the torsor of deformations. Local study, pg. 27*Chapter III. Representations of the fundamental group and the torsor of deformations. Global aspects, pg. 179*Chapter IV. Cohomology of Higgs isocrystals, pg. 307*Chapter V. Almost etale coverings, pg. 449*Chapter VI. Covanishing topos and generalizations, pg. 485*Facsimile : A p-adic Simpson correspondence, pg. 577*Bibliography, pg. 595*Indexes, pg. 599
£138.55
Springer New York Sphere Packings Lattices and Groups
Book SynopsisWith contributions by numerous expertsTrade ReviewThird Edition J.H. Conway and N.J.A. Sloane Sphere Packings, Lattices and Groups "This is the third edition of this reference work in the literature on sphere packings and related subjects. In addition to the content of the preceding editions, the present edition provides in its preface a detailed survey on recent developments in the field, and an exhaustive supplementary bibliography for 1988-1998. A few chapters in the main text have also been revised."—MATHEMATICAL REVIEWSTable of Contents1 Sphere Packings and Kissing Numbers.- 2 Coverings, Lattices and Quantizers.- 3 Codes, Designs and Groups.- 4 Certain Important Lattices and Their Properties.- 5 Sphere Packing and Error-Correcting Codes.- 6 Laminated Lattices.- 7 Further Connections Between Codes and Lattices.- 8 Algebraic Constructions for Lattices.- 9 Bounds for Codes and Sphere Packings.- 10 Three Lectures on Exceptional Groups.- 11 The Golay Codes and the Mathieu Groups.- 12 A Characterization of the Leech Lattice.- 13 Bounds on Kissing Numbers.- 14 Uniqueness of Certain Spherical Codes.- 15 On the Classification of Integral Quadratic Forms.- 16 Enumeration of Unimodular Lattices.- 17 The 24-Dimensional Odd Unimodular Lattices.- 18 Even Unimodular 24-Dimensional Lattices.- 19 Enumeration of Extremal Self-Dual Lattices.- 20 Finding the Closest Lattice Point.- 21 Voronoi Cells of Lattices and Quantization Errors.- 22 A Bound for the Covering Radius of the Leech Lattice.- 23 The Covering Radius of the Leech Lattice.- 24 Twenty-Three Constructions for the Leech Lattice.- 25 The Cellular Structure of the Leech Lattice.- 26 Lorentzian Forms for the Leech Lattice.- 27 The Automorphism Group of the 26-Dimensional Even Unimodular Lorentzian Lattice.- 28 Leech Roots and Vinberg Groups.- 29 The Monster Group and its 196884-Dimensional Space.- 30 A Monster Lie Algebra?.- Supplementary Bibliography.
£43.99
Springer New York Linear Algebraic Groups
Book SynopsisAGBackground Material From Algebraic Geometry.- 1. Some Topological Notions.- 2. Some Facts from Field Theory.- 3. Some Commutative Algebra.- 4. Sheaves.- 5. Affine K-Schemes, Prevarieties.- 6. Products; Varieties.- 7. Projective and Complete Varieties.- 8. Rational Functions; Dominant Morphisms.- 9. Dimension.- 10. Images and Fibres of a Morphism.- 11. k-structures on K-Schemes.- 12. k-Structures on Varieties.- 13. Separable points.- 14. Galois Criteria for Rationality.- 15. Derivations and Differentials.- 16. Tangent Spaces.- 17. Simple Points.- 18. Normal Varieties.- References.- IGeneral Notions Associated With Algebraic Groups.- 1. The Notion of an Algebraic Groups.- 2. Group Closure; Solvable and Nilpotent Groups.- 3. The Lie Algebra of an Algebraic Group.- 4. Jordan Decomposition.- II Homogeneous Spaces.- 5. Semi-Invariants.- 6. Homogeneous Spaces.- 7. Algebraic Groups in Characteristic Zero.- III Solvable Groups.- 8. Diagonalizable Groups and Tori.- 9. Conjugacy Classes and Centralizers of Semi-Simple Elements.- 10. Connected Solvable Groups.- IVBorel Subgroups; Reductive Groups.- 11. Borel Subgroups.- 12. Cartan Subgroups; Regular Elements.- 13. The Borel Subgroups Containing a Given Torus.- 14. Root Systems and Bruhat Decomposition in Reductive Groups.- VRationality Questions.- 15. Split Solvable Groups and Subgroups.- 16. Groups over Finite Fields.- 17. Quotient of a Group by a Lie Subalgebra.- 18. Cartan Subgroups over the Groundfield. Unirationality. Splitting of Reductive Groups.- 19. Cartan Subgroups of Solvable Groups.- 20. Isotropic Reductive Groups.- 21. Relative Root System and Bruhat Decomposition for Isotropic Reductive Groups.- 22. Central Isogenies.- 23. Examples.- 24. Survey of Some Other Topics.- A. Classification.- B. Linear Representations.- C. Real Reductive Groups.- References for Chapters I to V.- Index of Definition.- Index of Notation.Table of ContentsAG—Background Material From Algebraic Geometry.- §1. Some Topological Notions.- §2. Some Facts from Field Theory.- §3. Some Commutative Algebra.- §4. Sheaves.- §5. Affine K-Schemes, Prevarieties.- §6. Products; Varieties.- §7. Projective and Complete Varieties.- §8. Rational Functions; Dominant Morphisms.- §9. Dimension.- §10. Images and Fibres of a Morphism.- §11. k-structures on K-Schemes.- §12. k-Structures on Varieties.- §13. Separable points.- §14. Galois Criteria for Rationality.- §15. Derivations and Differentials.- §16. Tangent Spaces.- §17. Simple Points.- §18. Normal Varieties.- References.- I—General Notions Associated With Algebraic Groups.- §1. The Notion of an Algebraic Groups.- §2. Group Closure; Solvable and Nilpotent Groups.- §3. The Lie Algebra of an Algebraic Group.- §4. Jordan Decomposition.- II — Homogeneous Spaces.- §5. Semi-Invariants.- §6. Homogeneous Spaces.- §7. Algebraic Groups in Characteristic Zero.- III Solvable Groups.- §8. Diagonalizable Groups and Tori.- §9. Conjugacy Classes and Centralizers of Semi-Simple Elements.- §10. Connected Solvable Groups.- IV—Borel Subgroups; Reductive Groups.- §11. Borel Subgroups.- §12. Cartan Subgroups; Regular Elements.- §13. The Borel Subgroups Containing a Given Torus.- §14. Root Systems and Bruhat Decomposition in Reductive Groups.- V—Rationality Questions.- §15. Split Solvable Groups and Subgroups.- §16. Groups over Finite Fields.- §17. Quotient of a Group by a Lie Subalgebra.- §18. Cartan Subgroups over the Groundfield. Unirationality. Splitting of Reductive Groups.- §19. Cartan Subgroups of Solvable Groups.- §20. Isotropic Reductive Groups.- §21. Relative Root System and Bruhat Decomposition for Isotropic Reductive Groups.- §22. Central Isogenies.- §23. Examples.- §24. Survey of Some Other Topics.- A. Classification.- B. Linear Representations.- C. Real Reductive Groups.- References for Chapters I to V.- Index of Definition.- Index of Notation.
£42.49
Springer New York Linear Algebraic Groups 21 Graduate Texts in Mathematics
Book SynopsisHe has previously held posts at the University of Oregon and New York University. His main research interests include group theory and Lie algebras, and this graduate level text is an exceptionally well-written introduction to everything about linear algebraic groups.Trade ReviewJ.E. Humphreys Linear Algebraic Groups "Exceptionally well-written and ideally suited either for independent reading or as a graduate level text for an introduction to everything about linear algebraic groups."—MATHEMATICAL REVIEWSTable of ContentsI. Algebraic Geometry.- 0. Some Commutative Algebra.- 1. Affine and Projective Varieties.- 1.1 Ideals and Affine Varieties.- 1.2 Zariski Topology on Affine Space.- 1.3 Irreducible Components.- 1.4 Products of Affine Varieties.- 1.5 Affine Algebras and Morphisms.- 1.6 Projective Varieties.- 1.7 Products of Projective Varieties.- 1.8 Flag Varieties.- 2. Varieties.- 2.1 Local Rings.- 2.2 Prevarieties.- 2.3 Morphisms.- 2.4 Products.- 2.5 Hausdorff Axiom.- 3. Dimension.- 3.1 Dimension of a Variety.- 3.2 Dimension of a Subvariety.- 3.3 Dimension Theorem.- 3.4 Consequences.- 4. Morphisms.- 4.1 Fibres of a Morphism.- 4.2 Finite Morphisms.- 4.3 Image of a Morphism.- 4.4 Constructible Sets.- 4.5 Open Morphisms.- 4.6 Bijective Morphisms.- 4.7 Birational Morphisms.- 5. Tangent Spaces.- 5.1 Zariski Tangent Space.- 5.2 Existence of Simple Points.- 5.3 Local Ring of a Simple Point.- 5.4 Differential of a Morphism.- 5.5 Differential Criterion for Separability.- 6. Complete Varieties.- 6.1 Basic Properties.- 6.2 Completeness of Projective Varieties.- 6.3 Varieties Isomorphic to P1.- 6.4 Automorphisms of P1.- II. Affine Algebraic Groups.- 7. Basic Concepts and Examples.- 7.1 The Notion of Algebraic Group.- 7.2 Some Classical Groups.- 7.3 Identity Component.- 7.4 Subgroups and Homomorphisms.- 7.5 Generation by Irreducible Subsets.- 7.6 Hopf Algebras.- 8. Actions of Algebraic Groups on Varieties.- 8.1 Group Actions.- 8.2 Actions of Algebraic Groups.- 8.3 Closed Orbits.- 8.4 Semidirect Products.- 8.5 Translation of Functions.- 8.6 Linearization of Affine Groups.- III. Lie Algebras.- 9. Lie Algebra of an Algebraic Group.- 9.1 Lie Algebras and Tangent Spaces.- 9.2 Convolution.- 9.3 Examples.- 9.4 Subgroups and Lie Subalgebras.- 9.5 Dual Numbers.- 10. Differentiation.- 10.1 Some Elementary Formulas.- 10.2 Differential of Right Translation.- 10.3 The Adjoint Representation.- 10.4 Differential of Ad.- 10.5 Commutators.- 10.6 Centralizers.- 10.7 Automorphisms and Derivations.- IV. Homogeneous Spaces.- 11. Construction of Certain Representations.- 11.1 Action on Exterior Powers.- 11.2 A Theorem of Chevalley.- 11.3 Passage to Projective Space.- 11.4 Characters and Semi-Invariants.- 11.5 Normal Subgroups.- 12. Quotients.- 12.1 Universal Mapping Property.- 12.2 Topology of Y.- 12.3 Functions on Y.- 12.4 Complements.- 12.5 Characteristic 0.- V. Characteristic 0 Theory.- 13. Correspondence between Groups and Lie Algebras.- 13.1 The Lattice Correspondence.- 13.2 Invariants and Invariant Subspaces.- 13.3 Normal Subgroups and Ideals.- 13.4 Centers and Centralizers.- 13.5 Semisimple Groups and Lie Algebras.- 14. Semisimple Groups.- 14.1 The Adjoint Representation.- 14.2 Subgroups of a Semisimple Group.- 14.3 Complete Reducibility of Representations.- VI. Semisimple and Unipotent Elements.- 15. Jordan-Chevalley Decomposition.- 15.1 Decomposition of a Single Endomorphism.- 15.2 GL(n, K) and gl(n, K).- 15.3 Jordan Decomposition in Algebraic Groups.- 15.4 Commuting Sets of Endomorphisms.- 15.5 Structure of Commutative Algebraic Groups.- 16. Diagonalizable Groups.- 16.1 Characters and d-Groups.- 16.2 Tori.- 16.3 Rigidity of Diagonalizable Groups.- 16.4 Weights and Roots.- VII. Solvable Groups.- 17. Nilpotent and Solvable Groups.- 17.1 A Group-Theoretic Lemma.- 17.2 Commutator Groups.- 17.3 Solvable Groups.- 17.4 Nilpotent Groups.- 17.5 Unipotent Groups.- 17.6 Lie-Kolchin Theorem.- 18. Semisimple Elements.- 18.1 Global and Infinitesimal Centralizers.- 18.2 Closed Conjugacy Classes.- 18.3 Action of a Semisimple Element on a Unipotent Group.- 18.4 Action of a Diagonalizable Group.- 19. Connected Solvable Groups.- 19.1 An Exact Sequence.- 19.2 The Nilpotent Case.- 19.3 The General Case.- 19.4 Normalizer and Centralizer.- 19.5 Solvable and Unipotent Radicals.- 20. One Dimensional Groups.- 20.1 Commutativity of G.- 20.2 Vector Groups and e-Groups.- 20.3 Properties of p-Polynomials.- 20.4 Automorphisms of Vector Groups.- 20.5 The Main Theorem.- VIII. Borel Subgroups.- 21. Fixed Point and Conjugacy Theorems.- 21.1 Review of Complete Varieties.- 21.2 Fixed Point Theorem.- 21.3 Conjugacy of Borel Subgroups and Maximal Tori.- 21.4 Further Consequences.- 22. Density and Connectedness Theorems.- 22.1 The Main Lemma.- 22.2 Density Theorem.- 22.3 Connectedness Theorem.- 22.4 Borel Subgroups of CG(S).- 22.5 Cartan Subgroups: Summary.- 23. Normalizer Theorem.- 23.1 Statement of the Theorem.- 23.2 Proof of the Theorem.- 23.3 The variety G/B.- 23.4 Summary.- IX. Centralizers of Tori.- 24. Regular and Singular Tori.- 24.1 Weyl Groups.- 24.2 Regular Tori.- 24.3 Singular Tori and Roots.- 24.4 Regular 1-Parameter Subgroups.- 25. Action of a Maximal Torus on G/?.- 25.1 Action of a 1-Parameter Subgroup.- 25.2 Existence of Enough Fixed Points.- 25.3 Groups of Semisimple Rank 1.- 25.4 Weyl Chambers.- 26. The Unipotent Radical.- 26.1 Characterization of Ru(G).- 26.2 Some Consequences.- 26.3 The Groups U?.- X. Structure of Reductive Groups.- 27. The Root System.- 27.1 Abstract Root Systems.- 27.2 The Integrality Axiom.- 27.3 Simple Roots.- 27.4 The Automorphism Group of a Semisimple Group.- 27.5 Simple Components.- 28. Bruhat Decomposition.- 28.1 T-Stable Subgroups of Bu.- 28.2 Groups of Semisimple Rank 1.- 28.3 The Bruhat Decomposition.- 28.4 Normal Form in G.- 28.5 Complements.- 29. Tits Systems.- 29.1 Axioms.- 29.2 Bruhat Decomposition.- 29.3 Parabolic Subgroups.- 29.4 Generators and Relations for W.- 29.5 Normal Subgroups of G.- 30. Parabolic Subgroups.- 30.1 Standard Parabolic Subgroups.- 30.2 Levi Decompositions.- 30.3 Parabolic Subgroups Associated to Certain Unipotent Groups.- 30.4 Maximal Subgroups and Maximal Unipotent Subgroups.- XI. Representations and Classification of Semisimple Groups.- 31. Representations.- 31.1 Weights.- 31.2 Maximal Vectors.- 31.3 Irreducible Representations.- 31.4 Construction of Irreducible Representations.- 31.5 Multiplicities and Minimal Highest Weights.- 31.6 Contragredients and Invariant Bilinear Forms.- 32. Isomorphism Theorem.- 32.1 The Classification Problem.- 32.2 Extension of ?T to N(T).- 32.3 Extension of ?T to Z?.- 32.4 Extension of ?T to TU?.- 32.5 Extension of ?T to ?.- 32.6 Multiplicativity of ?.- 33. Root Systems of Rank 2.- 33.1 Reformulation of (?), (?), (?).- 33.2 Some Preliminaries.- 33.3 Type A2.- 33.4 Type B2.- 33.5 Type G2.- 33.6 The Existence Problem.- XII. Survey of Rationality Properties.- 34. Fields of Definition.- 34.1 Foundations.- 34.2 Review of Earlier Chapters.- 34.3 Tori.- 34.4 Some Basic Theorems.- 34.5 Borel-Tits Structure Theory.- 34.6 An Example: Orthogonal Groups.- 35. Special Cases.- 35.1 Split and Quasisplit Groups.- 35.2 Finite Fields.- 35.3 The Real Field.- 35.4 Local Fields.- 35.5 Classification.- Appendix. Root Systems.- Index of Terminology.- Index of Symbols.
£44.99
Springer Nature Switzerland AG Differential Geometry and Lie Groups: A
Book SynopsisThis textbook offers an introduction to differential geometry designed for readers interested in modern geometry processing. Working from basic undergraduate prerequisites, the authors develop manifold theory and Lie groups from scratch; fundamental topics in Riemannian geometry follow, culminating in the theory that underpins manifold optimization techniques. Students and professionals working in computer vision, robotics, and machine learning will appreciate this pathway into the mathematical concepts behind many modern applications.Starting with the matrix exponential, the text begins with an introduction to Lie groups and group actions. Manifolds, tangent spaces, and cotangent spaces follow; a chapter on the construction of manifolds from gluing data is particularly relevant to the reconstruction of surfaces from 3D meshes. Vector fields and basic point-set topology bridge into the second part of the book, which focuses on Riemannian geometry.Chapters on Riemannian manifolds encompass Riemannian metrics, geodesics, and curvature. Topics that follow include submersions, curvature on Lie groups, and the Log-Euclidean framework. The final chapter highlights naturally reductive homogeneous manifolds and symmetric spaces, revealing the machinery needed to generalize important optimization techniques to Riemannian manifolds. Exercises are included throughout, along with optional sections that delve into more theoretical topics.Differential Geometry and Lie Groups: A Computational Perspective offers a uniquely accessible perspective on differential geometry for those interested in the theory behind modern computing applications. Equally suited to classroom use or independent study, the text will appeal to students and professionals alike; only a background in calculus and linear algebra is assumed. Readers looking to continue on to more advanced topics will appreciate the authors’ companion volume Differential Geometry and Lie Groups: A Second Course.Trade Review“The book … is intended ‘for a wide audience ranging from upper undergraduate to advanced graduate students in mathematics, physics, and more broadly engineering students, especially in computer science.’ … The text’s coverage is extensive, its exposition clear throughout, and the color illustrations helpful. The authors are also familiar with many texts at a comparable level and have drawn on them in several places to include some of the most insightful proofs already in the literature.” (Jer-Chin Chuang, MAA Reviews, October 4, 2021)“The book is intended for incremental study and covers both basic concepts and more advanced ones. The former are thoroughly supported with theory and examples, and the latter are backed up with extensive reading lists and references. … Thanks to its design and approach style this is a timely and much needed addition that enables interdisciplinary bridges and the discovery of new applications for differential geometry.” (Corina Mohorian, zbMATH 1453.53001, 2021)Table of Contents1. The Matrix Exponential; Some Matrix Lie Groups.- 2. Adjoint Representations and the Derivative of exp.- 3. Introduction to Manifolds and Lie Groups.- 4. Groups and Group Actions.- 5. The Lorentz Groups ⊛.- 6. The Structure of O(p,q) and SO(p, q).- 7. Manifolds, Tangent Spaces, Cotangent Spaces.- 8. Construction of Manifolds From Gluing Data ⊛.- 9. Vector Fields, Integral Curves, Flows.- 10. Partitions of Unity, Covering Maps ⊛.- 11. Basic Analysis: Review of Series and Derivatives.- 12. A Review of Point Set Topology.-13. Riemannian Metrics, Riemannian Manifolds.- 14. Connections on Manifolds.- 15. Geodesics on Riemannian Manifolds.- 16. Curvature in Riemannian Manifolds.- 17. Isometries, Submersions, Killing Vector Fields.- 18. Lie Groups, Lie Algebra, Exponential Map.- 19. The Derivative of exp and Dynkin's Formula ⊛.- 20. Metrics, Connections, and Curvature of Lie Groups.- 21. The Log-Euclidean Framework.- 22. Manifolds Arising from Group Actions.
£55.24
Springer Nature Switzerland AG Geometry, Lie Theory and Applications: The Abel
Book SynopsisThis book consists of contributions from the participants of the Abel Symposium 2019 held in Ålesund, Norway. It was centered about applications of the ideas of symmetry and invariance, including equivalence and deformation theory of geometric structures, classification of differential invariants and invariant differential operators, integrability analysis of equations of mathematical physics, progress in parabolic geometry and mathematical aspects of general relativity.The chapters are written by leading international researchers, and consist of both survey and research articles. The book gives the reader an insight into the current research in differential geometry and Lie theory, as well as applications of these topics, in particular to general relativity and string theory.Table of ContentsFour-dimensional homogeneous generalizations of Einstein Metrics.- Conformal and isometric embeddings of gravitational instantons.- Recent results on closed G2-structures, by Anna Fino and Alberto Raffero.- Almost Zoll affine surfaces.- Distinguished curves and fist integrals on Poincare-Einstein and other conformally singular geometries.- A car as parabolic geometry.- Legendrian cone structures and contact prolongations.- The search for solitons on homogeneous spaces.- On Ricci negative Lie groups.- Semi-Riemannian cones.- Building new Einstein spaces by deforming symmetric Einstein spaces.- Remarks on highly supersymmetric backgrounds of 11-dimensional supergravity.- Krichever-Novikov type algebras.
£112.49
Springer Nature Switzerland AG Representation Theory and Algebraic Geometry: A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg
Book SynopsisThe chapters in this volume explore the influence of the Russian school on the development of algebraic geometry and representation theory, particularly the pioneering work of two of its illustrious members, Alexander Beilinson and Victor Ginzburg, in celebration of their 60th birthdays. Based on the work of speakers and invited participants at the conference “Interactions Between Representation Theory and Algebraic Geometry”, held at the University of Chicago, August 21-25, 2017, this volume illustrates the impact of their research and how it has shaped the development of various branches of mathematics through the use of D-modules, the affine Grassmannian, symplectic algebraic geometry, and other topics. All authors have been deeply influenced by their ideas and present here cutting-edge developments on modern topics. Chapters are organized around three distinct themes: Groups, algebras, categories, and representation theory D-modules and perverse sheaves Analogous varieties defined by quivers Representation Theory and Algebraic Geometry will be an ideal resource for researchers who work in the area, particularly those interested in exploring the impact of the Russian school.Table of ContentsPart I: Groups, algebras, categories, and their representation theory.- On semisimplification of tensor categories.- Total aspherical parameters for Cherednik algebras.- Microlocal approach to Lusztig's symmetries.- Part II: D-modules and perverse sheaves, particularly on flag varieties and their generalizations.- Fourier-Sato Transform on hyperplane arrangements.- A quasi-coherent description of the category D-mod(Gr GL(n)).- The semi-infinite intersection cohomology sheaf--II: the Ran space version.- A topological approach to Soergel theory.- Part III: Varieties associated to quivers and relations to representation theory and symplectic geometry.- Loop Grassmannians of quivers and affine quantum groups.- Symplectic resolutions for multiplicative quiver varieties and character varieties for punctured surfaces.
£84.99
Springer International Publishing AG Analysis and Quantum Groups
Book SynopsisThis volume presents a completely self-contained introduction to the elaborate theory of locally compact quantum groups, bringing the reader to the frontiers of present-day research. The exposition includes a substantial amount of material on functional analysis and operator algebras, subjects which in themselves have become increasingly important with the advent of quantum information theory. In particular, the rather unfamiliar modular theory of weights plays a crucial role in the theory, due to the presence of ‘Haar integrals’ on locally compact quantum groups, and is thus treated quite extensively The topics covered are developed independently, and each can serve either as a separate course in its own right or as part of a broader course on locally compact quantum groups. The second part of the book covers crossed products of coactions, their relation to subfactors and other types of natural products such as cocycle bicrossed products, quantum doubles and doublecrossed products. Induced corepresentations, Galois objects and deformations of coactions by cocycles are also treated. Each section is followed by a generous supply of exercises. To complete the book, an appendix is provided on topology, measure theory and complex function theory.Table of ContentsPreface.- Set theoretic preliminaries.- Banach spaces.- Bases in Banach spaces.- Operators on Hilbert spaces.- Spectral theory.- States and representations.- Types of von Neumann algebras.- Tensor products.- Unbounded operators.- Tomita-Takesaki theory.- Spectra and type III factors.- Quantum groups and duality.- Special cases.- Classical crossed products.- Crossed products for quantum groups.- Generalized and continuous crossed products.- Basic construction and quantum groups.- Galois objects and cocycle deformations.- Doublecrossed products of quantum groups.- Induction.-Appendix.- Bibliography.- Index.- Exercises.
£67.49
Springer International Publishing AG Excursions in Number Theory, Algebra, and
Book SynopsisThis textbook originates from a course taught by the late Ken Ireland in 1972. Designed to explore the theoretical underpinnings of undergraduate mathematics, the course focused on interrelationships and hands-on experience. Readers of this textbook will be taken on a modern rendering of Ireland’s path of discovery, consisting of excursions into number theory, algebra, and analysis. Replete with surprising connections, deep insights, and brilliantly curated invitations to try problems at just the right moment, this journey weaves a rich body of knowledge that is ideal for those going on to study or teach mathematics. A pool of 200 ‘Dialing In’ problems opens the book, providing fuel for active enquiry throughout a course. The following chapters develop theory to illuminate the observations and roadblocks encountered in the problems, situating them in the broader mathematical landscape. Topics cover polygons and modular arithmetic; the fundamental theorems of arithmetic and algebra; irrational, algebraic and transcendental numbers; and Fourier series and Gauss sums. A lively accompaniment of examples, exercises, historical anecdotes, and asides adds motivation and context to the theory. Return trips to the Dialing In problems are encouraged, offering opportunities to put theory into practice and make lasting connections along the way. Excursions in Number Theory, Algebra, and Analysis invites readers on a journey as important as the destination. Suitable for a senior capstone, professional development for practicing teachers, or independent reading, this textbook offers insights and skills valuable to math majors and high school teachers alike. A background in real analysis and abstract algebra is assumed, though the most important prerequisite is a willingness to put pen to paper and do some mathematics.Trade Review“Rather than being a book that one reads from cover to cover, Excursions is a curated collection problems followed by expository material aimed at providing background material useful for solving these problems. I imagine it would be a great experience to have a course taught out of this book. The second author clearly enjoyed the experience of studying this material under the guidance of the first author and wanted to make that experience available to others.” (John D. Cook, MAA Reviews, June 17, 2023)Table of ContentsPreface.- 1. Dialing In Problems.- 2. Polygons and Modular Arithmetic.- 3. The Fundamental Theorem of Arithmetic.- 4. The Fundamental Theorem of Algebra.- 5. Irrational, Algebraic and Transcendental Numbers.- 6. Fourier Series and Gauss Sums.- Epilogue.- Notation.- Bibliography.- Index.
£42.49
Springer International Publishing AG Hypergroups
Book SynopsisThis book provides a comprehensive algebraic treatment of hypergroups, as defined by F. Marty in 1934. It starts with structural results, which are developed along the lines of the structure theory of groups. The focus then turns to a number of concrete classes of hypergroups with small parameters, and continues with a closer look at the role of involutions (modeled after the definition of group-theoretic involutions) within the theory of hypergroups. Hypergroups generated by involutions lead to the exchange condition (a genuine generalization of the group-theoretic exchange condition), and this condition defines the so-called Coxeter hypergroups. Coxeter hypergroups can be treated in a similar way to Coxeter groups. On the other hand, their regular actions are mathematically equivalent to buildings (in the sense of Jacques Tits). A similar equivalence is discussed for twin buildings. The primary audience for the monograph will be researchers working in Algebra and/or Algebraic Combinatorics, in particular on association schemes.Table of Contents1 Basic Facts : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11.1 Neutral Elements and Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Complex Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Thin Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.5 Groups and Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Actions of Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Hypergroups Admitting Regular Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.8 Association Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Closed Subsets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 272.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Dedekind Modularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Generating Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Commutators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372.5 Conjugation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.6 The Thin Radical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.7 Foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423 Elementary Structure Theory: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 473.1 Centralizers and Normalizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.2 Su cient Conditions for Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Strong Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.5 Computations in Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.6 Homomorphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.7 The Homomorphism Theorem and the Isomorphism Theorems . . . . . . . . . . 714 Subnormality and Thin Residues : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 794.1 Subnormal Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Composition Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.3 The Thin Residue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.4 Thin Residues of Thin Residues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.5 Residually Thin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.6 Finite Residually Thin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.7 Solvable Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045 Tight Hypergroups : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1075.1 Tight Hypergroup Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.2 The Set S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3 The Sets a b \ Fc and Sa;b(Fc) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.4 The Sets bf1b \ Fa and Sb;(f1;:::;fn)(Fa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.5 Structure Constants of Finite Tight Hypergroups . . . . . . . . . . . . . . . . . . . . . 1225.6 Rings Arising from Certain Finite Tight Hypergroups . . . . . . . . . . . . . . . . . 1265.7 Finite Metathin Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1285.8 Finite Metathin Hypergroups with Restricted Thin Residue . . . . . . . . . . . . 1326 Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1376.1 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1386.2 Cosets of Closed Subsets Generated by an Involution, I . . . . . . . . . . . . . . . . 1426.3 Cosets of Closed Subsets Generated by an Involution, II . . . . . . . . . . . . . . . 1456.4 Cosets of Closed Subsets Generated by an Involution, III . . . . . . . . . . . . . . . 1476.5 Length Functions De ned by Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . 1526.6 Hypergroups Generated by Two Distinct Involutions . . . . . . . . . . . . . . . . . . 1566.7 Dichotomy and the Exchange Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1616.8 Projective Hypergroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1647 Hypergroups with a Small Number of Elements : : : : : : : : : : : : : : : : : : : : : : 1717.1 Hypergroups of Cardinality at Most 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1727.2 Non-Symmetric Hypergroups of Cardinality 4 . . . . . . . . . . . . . . . . . . . . . . . . 1797.3 Hypergroups of Cardinality 6 with a Non-Normal Closed Subset, I . . . . . . 1907.4 Hypergroups of Cardinality 6 with a Non-Normal Closed Subset, II . . . . . . 2027.5 Non-Normal Closed Subsets Missing Four Elements . . . . . . . . . . . . . . . . . . . 2157.6 Non-Normal Closed Subsets Missing Four Elements and Thin Elements . . 2218 Constrained Sets of Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2238.1 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2248.2 Constrained Sets of Involutions and Cosets . . . . . . . . . . . . . . . . . . . . . . . . . . . 2288.3 Constrained Sets of Involutions and the Thin Radical . . . . . . . . . . . . . . . . . . 2308.4 Constrained Sets of Involutions and Dichotomy . . . . . . . . . . . . . . . . . . . . . . . 2338.5 Constrained Sets of Non-Thin Involutions and Dichotomy . . . . . . . . . . . . . . 2398.6 Constrained Sets of Involutions and Foldings . . . . . . . . . . . . . . . . . . . . . . . . . 2448.7 Dichotomic Constrained Sets of Involutions and Foldings . . . . . . . . . . . . . . . 2489 Coxeter Sets of Involutions : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 2519.1 General Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2529.2 The Sets V1(U) for Subsets U of Coxeter Sets V of Involutions . . . . . . . . . . 2569.3 The Sets V����1(U) for Subsets U of Coxeter Sets V of Involutions . . . . . . . . . 2639.4 Sets of Subsets of Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . . . 2659.5 Spherical Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2689.6 Subsets of Spherical Coxeter Sets of Involutions . . . . . . . . . . . . . . . . . . . . . . . 2739.7 Coxeter Sets of Involutions and Foldings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2779.8 Coxeter Sets of Involutions and Their Coxeter Numbers . . . . . . . . . . . . . . . . 2809.9 Coxeter Sets of Involutions and Type Preserving Bijections . . . . . . . . . . . . . 28610 Regular Actions of (Twin) Coxeter Hypergroups: : : : : : : : : : : : : : : : : : : : : 29310.1 Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29310.2 Twin Buildings, I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29810.3 Twin Buildings, II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30110.4 Regular Actions of Coxeter Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30510.5 Regular Actions of Twin Coxeter Hypergroups . . . . . . . . . . . . . . . . . . . . . . . . 315References : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 333
£89.99
Springer International Publishing AG Completely Regular Semigroup Varieties
Book SynopsisSince any in-depth study of the lattice of varieties requires an understanding of free completely regular semigroups, the book begins by describing the free object on countably infinite sets and the properties of the lattice of fully invariant congruences on the free object.
£33.24
Springer Completely Regular Semigroup Varieties
Book SynopsisGeneral Sublattices of L(CR).- Neutrality and Intervals.- Free Objects.- Constructions.- Canonical Varieties.
£33.24
Springer The Geometry and Topology of Coxeter Groups
Book SynopsisChapter 1. Introduction and preview.- Chapter 2. Some basic notions in geometric group theory.- Chapter 3. Coxeter groups.- Chapter 4. More combinatorics of Coxeter groups.- Chapter 5. The basic construction.- Chapter 6. Geometric reflection groups.- Chapter 7. The complex E.- Chapter 8. The algebraic topology of U and of E.- Chapter 9. The fundamental group and the fundamental group at infinity.- Chapter 10. Actions on manifolds.- Chapter 11. The reflection group trick.- Chapter 12. E is CAT(0).- Chapter 13. Rigidity.- Chapter 14. Free quotients and surface subgroups.- Chapter 15. Another look at (co)homology.- Chapter 16. The Euler characteristic.- Chapter 17. Growth series.- Chapter 18. Artin Groups.- Chapter 19. L2-Betti numbers of Artin groups.- Chapter 20. Buildings.- Chapter 21. Hecke - von Neumann algebras.- Chapter 22. Weighted L2- (co)homology.
£134.99
Springer International Publishing AG Handbook of Geometry and Topology of Singularities VIII
£125.99
Birkhauser Verlag AG Representations of Finite Groups: Local
Book SynopsisThe seminar focuses on a recent solution, by the authors, of a long standing problem concerning the stable module category (of not necessarily finite dimensional representations) of a finite group. The proof draws on ideas from commutative algebra, cohomology of groups, and stable homotopy theory. The unifying theme is a notion of support which provides a geometric approach for studying various algebraic structures. The prototype for this has been Daniel Quillen’s description of the algebraic variety corresponding to the cohomology ring of a finite group, based on which Jon Carlson introduced support varieties for modular representations. This has made it possible to apply methods of algebraic geometry to obtain representation theoretic information. Their work has inspired the development of analogous theories in various contexts, notably modules over commutative complete intersection rings and over cocommutative Hopf algebras. One of the threads in this development has been the classification of thick or localizing subcategories of various triangulated categories of representations. This story started with Mike Hopkins’ classification of thick subcategories of the perfect complexes over a commutative Noetherian ring, followed by a classification of localizing subcategories of its full derived category, due to Amnon Neeman. The authors have been developing an approach to address such classification problems, based on a construction of local cohomology functors and support for triangulated categories with ring of operators. The book serves as an introduction to this circle of ideas.Trade ReviewFrom the reviews:“The book is aimed at a readership with a solid background in algebra, in particular representation theory, commutative algebra and homological algebra. The volume comprises five chapters and an appendix, and each chapter is divided into four sections. Each chapter consists of the lecture material and the exercises handled during one day at the Oberwolfach Seminar (in 2010) with the same title. … The book ends with an appendix … and there is a comprehensive bibliography.” (Nadia P. Mazza, Mathematical Reviews, March, 2013)“The manuscript under review provides a quite nice introduction to the tools used in these classification theorems and offers an excellent starting point for someone new to the area. The manuscript is based on a week-long series of lectures given by the authors to introduce people to the ideas involved in the proof of the classification of localising subcategories of Mod(kG).” (Christopher P. Bendel, Zentralblatt MATH, Vol. 1246, 2012)Table of ContentsPreface.- 1 Monday.- 1.1 Overview.- 1.2 Modules over group algebras.- 1.3 Triangulated categories.- 1.4 Exercises.- 2 Tuesday.- 2.1 Perfect complexes over commutative rings.- 2.2 Brown representability and localization.- 2.3 The stable module category of a finite group.- 2.4 Exercises.- 3 Wednesday.- 3.1.- 3.2 Koszul objects and support.- 3.3 The homotopy category of injectives.- 3.4 Exercises.- 4 Thursday.- 4.1 Stratifying triangulated categories.- 4.2 Consequences of stratification.- 4.3 The Klein four group.- 4.4 Exercises.- 5 Friday.- 5.1 Localising subcategories of D(A).- 5.2 Elementary abelian 2-groups.- 5.3 Stratification for arbitrary finite groups.- 5.4 Exercises.- A Support for modules over commutative rings.- Bibliography.- Index.
£22.99
De Gruyter Algebra in the Stone-Cech Compactification:
Book SynopsisThis is the second revised and extendededition of the successful book on the algebraic structure of the Stone-Čech compactification of a discrete semigroup and its combinatorial applications, primarily in the field known as Ramsey Theory. There has been very active research in the subject dealt with by the book in the 12 years which is now included in this edition. This book is a self-contained exposition of the theory of compact right semigroupsfor discrete semigroups and the algebraic properties of these objects. The methods applied in the book constitute a mosaic of infinite combinatorics, algebra, and topology. The reader will find numerous combinatorial applications of the theory, including the central sets theorem, partition regularity of matrices, multidimensional Ramsey theory, and many more.Trade ReviewThe present book is the first devoted to an extensive study of the algebraic structure of sS and the many applications thereof; it is an exciting book, written - and very well written - by two mathematicians who are eminently qualified two write it, and it is essentially self-contained, requiring only that the reader come to it with the basic concepts of first graduate courses in algebra, analysis and topology. I recommend this book highly; it will be very useful, both to researchers and to students. Its index, list of symbols and up-to-date bibliography are very helpful. Paul Milnes, Zentralblatt MATH / 1998 The authors present a self-contained exposition. The book under review is written by two mathematicians who have contributed in a decisive way to this rapidly expanding area and provides a unique opportunity to obtain a 'colorful' panoramic view of the subject. Michael Tkacenko, MathSciNet / 1999
£64.60
De Gruyter Endliche Gruppen: Eine Einführung
Book SynopsisDieses Lehrbuch ist eine Einführung in die Techniken der Gruppentheorie und behandelt alle wichtigen Begriffe aus diesem Gebiet, wobei der Schwerpunkt im Bereich der endlichen Gruppen liegt. Es beginnt dort, wo die Gruppentheorie beginnt: bei den Permutationsgruppen. Danach werden wesentliche Strukturen und Methoden, wie das Arbeiten mit Kommutatoren und die Konstruktion von neuen aus gegebenen Gruppen behandelt. Nächstes Ziel sind die Fittinggruppe und ihre Verallgemeinerung, wozu nilpotente Gruppen studiert werden. Danach wendet sich der Text den einfachen Gruppen zu. Zu guter Letzt wird zunächst die Einfachheit der projektiven linearen Gruppen bewiesen und ein Überblick über orthogonale, symplektische und unitäre Gruppen gegeben. Weiter werden die sporadischen Mathieu-Gruppen und die Higman-Sims-Gruppe konstruiert. Das Buch ist geschrieben für Studierende im Bachelor- und Masterstudium. Es setzt den Besuch der üblichen Algebra-Vorlesungen und somit nur allgemeine Kenntnisse über Gruppen voraus.
£25.65
De Gruyter Discrete, Finite and Lie Groups: Comprehensive Group Theory in Geometry and Analysis
Book SynopsisIn a self contained and exhaustive work the author covers Group Theory in its multifaceted aspects, treating its conceptual foundations in a proper logical order. First discrete and finite group theory, that includes the entire chemical-physical field of crystallography is developed self consistently, followed by the structural theory of Lie Algebras with a complete exposition of the roots and Dynkin diagrams lore. A primary on Fibre-Bundles, Connections and Gauge fields, Riemannian Geometry and the theory of Homogeneous Spaces G/H is also included and systematically developed.
£81.90
De Gruyter Topics in Infinite Group Theory
Book Synopsis
£81.90
Springer International Publishing AG Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
Book SynopsisThis textbook treats Lie groups, Lie algebras and their representations in an elementary but fully rigorous fashion requiring minimal prerequisites. In particular, the theory of matrix Lie groups and their Lie algebras is developed using only linear algebra, and more motivation and intuition for proofs is provided than in most classic texts on the subject.In addition to its accessible treatment of the basic theory of Lie groups and Lie algebras, the book is also noteworthy for including: a treatment of the Baker–Campbell–Hausdorff formula and its use in place of the Frobenius theorem to establish deeper results about the relationship between Lie groups and Lie algebras motivation for the machinery of roots, weights and the Weyl group via a concrete and detailed exposition of the representation theory of sl(3;C) an unconventional definition of semisimplicity that allows for a rapid development of the structure theory of semisimple Lie algebras a self-contained construction of the representations of compact groups, independent of Lie-algebraic arguments The second edition of Lie Groups, Lie Algebras, and Representations contains many substantial improvements and additions, among them: an entirely new part devoted to the structure and representation theory of compact Lie groups; a complete derivation of the main properties of root systems; the construction of finite-dimensional representations of semisimple Lie algebras has been elaborated; a treatment of universal enveloping algebras, including a proof of the Poincaré–Birkhoff–Witt theorem and the existence of Verma modules; complete proofs of the Weyl character formula, the Weyl dimension formula and the Kostant multiplicity formula.Review of the first edition:This is an excellent book. It deserves to, and undoubtedly will, become the standard text for early graduate courses in Lie group theory ... an important addition to the textbook literature ... it is highly recommended.— The Mathematical GazetteTrade Review“The first edition of this book was very good; the second is even better, and more versatile. This text remains one of the most attractive sources available from which to learn elementary Lie group theory, and is highly recommended.” (Mark Hunacek, The Mathematical Gazette, Vol. 101 (551), July, 2017)Table of ContentsPart I: General Theory.-Matrix Lie Groups.- The Matrix Exponential.- Lie Algebras.- Basic Representation Theory.- The Baker–Campbell–Hausdorff Formula and its Consequences.- Part II: Semisimple Lie Algebras.- The Representations of sl(3;C).-Semisimple Lie Algebras.- Root Systems.- Representations of Semisimple Lie Algebras.- Further Properties of the Representations.- Part III: Compact lie Groups.- Compact Lie Groups and Maximal Tori.- The Compact Group Approach to Representation Theory.- Fundamental Groups of Compact Lie Groups.- Appendices.
£43.99
Springer International Publishing AG Quantum Theory, Groups and Representations: An Introduction
Book SynopsisThis text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.Trade Review“The book presents a large variety of important subjects, including the basic principles of quantum mechanics … . This good book is recommended for mathematicians, physicists, philosophers of physics, researchers, and advanced students in mathematics and physics, as well as for readers with some elementary physics, multivariate calculus and linear algebra courses.” (Michael M. Dediu, Mathematical Reviews, June, 2018)Table of ContentsPreface.- 1 Introduction and Overview.- 2 The Group U(1) and its Representations.- 3 Two-state Systems and SU(2).- 4 Linear Algebra Review, Unitary and Orthogonal Groups.- 5 Lie Algebras and Lie Algebra Representations.- 6 The Rotation and Spin Groups in 3 and 4 Dimensions.- 7 Rotations and the Spin 1/2 Particle in a Magnetic Field.- 8 Representations of SU(2) and SO(3).- 9 Tensor Products, Entanglement, and Addition of Spin.- 10 Momentum and the Free Particle.- 11 Fourier Analysis and the Free Particle.- 12 Position and the Free Particle.- 13 The Heisenberg group and the Schrödinger Representation.- 14 The Poisson Bracket and Symplectic Geometry.- 15 Hamiltonian Vector Fields and the Moment Map.- 16 Quadratic Polynomials and the Symplectic Group.- 17 Quantization.- 18 Semi-direct Products.- 19 The Quantum Free Particle as a Representation of the Euclidean Group.- 20 Representations of Semi-direct Products.- 21 Central Potentials and the Hydrogen Atom.- 22 The Harmonic Oscillator.- 23 Coherent States and the Propagator for the Harmonic Oscillator.- 24 The Metaplectic Representation and Annihilation and Creation Operators, d = 1.- 25 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d.- 26 Complex Structures and Quantization.- 27 The Fermionic Oscillator.- 28 Weyl and Clifford Algebras.- 29 Clifford Algebras and Geometry.- 30 Anticommuting Variables and Pseudo-classical Mechanics.- 31 Fermionic Quantization and Spinors.- 32 A Summary: Parallels Between Bosonic and Fermionic Quantization.- 33 Supersymmetry, Some Simple Examples.- 34 The Pauli Equation and the Dirac Operator.- 35 Lagrangian Methods and the Path Integral.- 36 Multi-particle Systems: Momentum Space Description.- 37 Multi-particle Systems and Field Quantization.- 38 Symmetries and Non-relativistic Quantum Fields.- 39 Quantization of Infinite dimensional Phase Spaces.- 40 Minkowski Space and the Lorentz Group.- 41 Representations of the Lorentz Group.- 42 The Poincaré Group and its Representations.- 43 The Klein-Gordon Equation and Scalar Quantum Fields.- 44 Symmetries and Relativistic Scalar Quantum Fields.- 45 U(1) Gauge Symmetry and Electromagnetic Field.- 46 Quantization of the Electromagnetic Field: the Photon.- 47 The Dirac Equation and Spin-1/2 Fields.- 48 An Introduction to the Standard Model.- 49 Further Topics.- A Conventions.- B Exercises.- Index.
£54.39
Springer International Publishing AG Quantum Theory, Groups and Representations: An Introduction
Book SynopsisThis text systematically presents the basics of quantum mechanics, emphasizing the role of Lie groups, Lie algebras, and their unitary representations. The mathematical structure of the subject is brought to the fore, intentionally avoiding significant overlap with material from standard physics courses in quantum mechanics and quantum field theory. The level of presentation is attractive to mathematics students looking to learn about both quantum mechanics and representation theory, while also appealing to physics students who would like to know more about the mathematics underlying the subject. This text showcases the numerous differences between typical mathematical and physical treatments of the subject. The latter portions of the book focus on central mathematical objects that occur in the Standard Model of particle physics, underlining the deep and intimate connections between mathematics and the physical world. While an elementary physics course of some kind would be helpful to the reader, no specific background in physics is assumed, making this book accessible to students with a grounding in multivariable calculus and linear algebra. Many exercises are provided to develop the reader's understanding of and facility in quantum-theoretical concepts and calculations.Trade Review“The book presents a large variety of important subjects, including the basic principles of quantum mechanics … . This good book is recommended for mathematicians, physicists, philosophers of physics, researchers, and advanced students in mathematics and physics, as well as for readers with some elementary physics, multivariate calculus and linear algebra courses.” (Michael M. Dediu, Mathematical Reviews, June, 2018)Table of ContentsPreface.- 1 Introduction and Overview.- 2 The Group U(1) and its Representations.- 3 Two-state Systems and SU(2).- 4 Linear Algebra Review, Unitary and Orthogonal Groups.- 5 Lie Algebras and Lie Algebra Representations.- 6 The Rotation and Spin Groups in 3 and 4 Dimensions.- 7 Rotations and the Spin 1/2 Particle in a Magnetic Field.- 8 Representations of SU(2) and SO(3).- 9 Tensor Products, Entanglement, and Addition of Spin.- 10 Momentum and the Free Particle.- 11 Fourier Analysis and the Free Particle.- 12 Position and the Free Particle.- 13 The Heisenberg group and the Schrödinger Representation.- 14 The Poisson Bracket and Symplectic Geometry.- 15 Hamiltonian Vector Fields and the Moment Map.- 16 Quadratic Polynomials and the Symplectic Group.- 17 Quantization.- 18 Semi-direct Products.- 19 The Quantum Free Particle as a Representation of the Euclidean Group.- 20 Representations of Semi-direct Products.- 21 Central Potentials and the Hydrogen Atom.- 22 The Harmonic Oscillator.- 23 Coherent States and the Propagator for the Harmonic Oscillator.- 24 The Metaplectic Representation and Annihilation and Creation Operators, d = 1.- 25 The Metaplectic Representation and Annihilation and Creation Operators, arbitrary d.- 26 Complex Structures and Quantization.- 27 The Fermionic Oscillator.- 28 Weyl and Clifford Algebras.- 29 Clifford Algebras and Geometry.- 30 Anticommuting Variables and Pseudo-classical Mechanics.- 31 Fermionic Quantization and Spinors.- 32 A Summary: Parallels Between Bosonic and Fermionic Quantization.- 33 Supersymmetry, Some Simple Examples.- 34 The Pauli Equation and the Dirac Operator.- 35 Lagrangian Methods and the Path Integral.- 36 Multi-particle Systems: Momentum Space Description.- 37 Multi-particle Systems and Field Quantization.- 38 Symmetries and Non-relativistic Quantum Fields.- 39 Quantization of Infinite dimensional Phase Spaces.- 40 Minkowski Space and the Lorentz Group.- 41 Representations of the Lorentz Group.- 42 The Poincaré Group and its Representations.- 43 The Klein-Gordon Equation and Scalar Quantum Fields.- 44 Symmetries and Relativistic Scalar Quantum Fields.- 45 U(1) Gauge Symmetry and Electromagnetic Field.- 46 Quantization of the Electromagnetic Field: the Photon.- 47 The Dirac Equation and Spin-1/2 Fields.- 48 An Introduction to the Standard Model.- 49 Further Topics.- A Conventions.- B Exercises.- Index.
£54.39
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Group Theory: Application to the Physics of
Book SynopsisThis concise, class-tested book was refined over the authors’ 30 years as instructors at MIT and the University Federal of Minas Gerais (UFMG) in Brazil. The approach centers on the conviction that teaching group theory along with applications helps students to learn, understand and use it for their own needs. Thus, the theoretical background is confined to introductory chapters. Subsequent chapters develop new theory alongside applications so that students can retain new concepts, build on concepts already learned, and see interrelations between topics. Essential problem sets between chapters aid retention of new material and consolidate material learned in previous chapters.Trade ReviewFrom the reviews:"It was developed for a graduate course taught mostly by Millie Dresselhaus at MIT for more than 30 years, with many revisions of lecture notes. Very much a graduate text or specialist monograph, the book covers a wealth of applications across solid-state physics. … The book can be warmly recommended to students and researchers in solid-state physics, either to serve as a text for an advanced lecture course or for individual study … ." (Volker Heine, Physics Today, November, 2008)"This textbook is based on the authors’ pedagogical experience during their 30 years at MIT. … the book develops all of the relevant mathematics (linear algebra) and the necessary physics (quantum mechanics), it is eminently suitable to a wide audience in physics, chemistry and materials science." (Barry R. Masters, Optics and Photonics News, July/August, 2009)“This is an excellent text … . originates from lectures by Charles Kittel and J. H. van Vleck in the 1950s and much of the material was presented in courses by the authors over the last three decades. The material is meant for Electrical Engineering and Physics students at the graduate level … . has exercises at the end of each chapter and an extensive set of appendices. The exposition is clear and detailed. This is a very good book for its target audience.” (W. Miller Jr., Zentralblatt MATH, Vol. 1175, 2010)“The goal of the book under review is to teach group theory in close connection to applications. … Every chapter of the book finishes with several selected problems. Specific to this book is the feature that every abstract theoretical group concept is introduced and applied in a concrete physical way. This is why the book is very useful for anyone interested in applications of group theory to the wide range of condensed matter phenomena.” (Oktay K. Pashaev, Mathematical Reviews, Issue 2010 i)“It is highly welcomed because of its well-thought structuring and the plenty of non-trivial examples. The authors develop those parts of the theory of groups which are interesting for physicists, from chapter to chapter offering nearly at any step one or more informative application.” (G. Kowol, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)Table of ContentsBasic Mathematics.- Basic Mathematical Background: Introduction.- Representation Theory and Basic Theorems.- Character of a Representation.- Basis Functions.- Introductory Application to Quantum Systems.- Splitting of Atomic Orbitals in a Crystal Potential.- Application to Selection Rules and Direct Products.- Molecular Systems.- Electronic States of Molecules and Directed Valence.- Molecular Vibrations, Infrared, and Raman Activity.- Application to Periodic Lattices.- Space Groups in Real Space.- Space Groups in Reciprocal Space and Representations.- Electron and Phonon Dispersion Relation.- Applications to Lattice Vibrations.- Electronic Energy Levels in a Cubic Crystals.- Energy Band Models Based on Symmetry.- Spin–Orbit Interaction in Solids and Double Groups.- Application of Double Groups to Energy Bands with Spin.- Other Symmetries.- Time Reversal Symmetry.- Permutation Groups and Many-Electron States.- Symmetry Properties of Tensors.
£56.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Geometric Invariant Theory
Book SynopsisThis standard reference on applications of invariant theory to the construction of moduli spaces is a systematic exposition of the geometric aspects of classical theory of polynomial invariants. This new, revised edition is completely updated and enlarged with an additional chapter on the moment map by Professor Frances Kirwan. It includes a fully updated bibliography of work in this area.Table of Contents0. Preliminaries.- 1. Definitions.- 2. First properties.- 3. Good and bad actions.- 4. Further properties.- 5. Resumé of some results of Grothendieck.- 1. Fundamental theorems for the actions of reductive groups.- 1. Definitions.- 2. The affine case.- 3. Linearization of an invertible sheaf.- 4. The general case.- 5. Functional properties.- 2. Analysis of stability.- 1. A numeral criterion.- 2. The flag complex.- 3. Applications.- 3. An elementary example.- 1. Pre-stability.- 2. Stability.- 4. Further examples.- 1. Binary quantics.- 2. Hypersurfaces.- 3. Counter-examples.- 4. Sequences of linear subspaces.- 5. The projective adjoint action.- 6. Space curves.- 5. The problem of moduli — 1st construction.- 1. General discussion.- 2. Moduli as an orbit space.- 3. First chern classes.- 4. Utilization of 4.6.- 6. Abelian schemes.- 1. Duals.- 2. Polarizations.- 3. Deformations.- 7. The method of covariants — 2nd construction.- 1. The technique.- 2. Moduli as an orbit space.- 3. The covariant.- 4. Application to curves.- 8. The moment map.- 1. Symplectic geometry.- 2. Symplectic quotients and geometric invariant theory.- 3. Kähler and hyperkähler quotients.- 4. Singular quotients.- 5. Geometry of the moment map.- 6. The cohomology of quotients: the symplectic case.- 7. The cohomology of quotients: the algebraic case.- 8. Vector bundles and the Yang-Mills functional.- 9. Yang-Mills theory over Riemann surfaces.- Appendix to Chapter 1.- Appendix to Chapter 2.- Appendix to Chapter 3.- Appendix to Chapter 4.- Appendix to Chapter 5.- Appendix to Chapter 7.- References.- Index of definitions and notations.
£134.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Algebra I: Chapters 1-3
Book SynopsisThis softcover reprint of the 1974 English translation of the first three chapters of Bourbaki’s Algebre gives a thorough exposition of the fundamentals of general, linear, and multilinear algebra. The first chapter introduces the basic objects, such as groups and rings. The second chapter studies the properties of modules and linear maps, and the third chapter discusses algebras, especially tensor algebras.Table of ContentsAlgebraic Structures.- Linear Algebra.- Tensor Algebras, Exterior Algebras.- Symmetric Algebras.- Historical Notes.
£46.74
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG The Classical Groups and K-Theory
Book SynopsisIt is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).Table of ContentsNotation and Conventions.- 1. General Linear Groups, Steinberg Groups, and K-Groups.- 2. Linear Groups over Division Rings.- 3. Isomorphism Theory for the Linear Groups.- 4. Linear Groups over General Classes of Rings.- 5. Unitary Groups, Unitary Steinberg Groups, and Unitary K-Groups.- 6. Unitary Groups over Division Rings.- 7. Clifford Algebras and Orthogonal Groups over Commutative Rings.- 8. Isomorphism Theory for the Unitary Groups.- 9. Unitary Groups over General Classes of Form Rings.- Concluding Remarks.- Index of Concepts.- Index of Symbols.
£74.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Collected Papers - Gesammelte Abhandlungen
Book SynopsisFrom the preface: "Hopf algebras, Hopf fibration of spheres, Hopf-Rinow complete Riemannian manifolds, Hopf theorem on the ends of groups - can one imagine modern mathematics without all this? Many other concepts and methods, fundamental in various mathematical disciplines, also go back directly or indirectly to the work of Heinz Hopf: homological algebra, singularities of vector fields and characteristic classes, group-like spaces, global differential geometry, and the whole algebraisation of topology with its influence on group theory, analysis and algebraic geometry. It is astonishing to realize that this oeuvre of a whole scientific life consists of only about 70 writings. Astonishing also the transparent and clear style, the concreteness of the problems, and how abstract and far-reaching the methods Hopf invented."Trade Review Heinz Hopf (1894-1981) is rightly considered to be one of the outstanding and most influential mathematicians of the XXth century. He was a pioneer in algebraic topology as well as in differential geometry. He is widely known as having studied the ‘Hopf fibration’. The very general abstract notion of Hopf algebra was introduced as tracing in Hopf’s works; he may be considered to have been a forerunner of the creation of homological algebra. He found a noncontractible map of the 3-sphere into the 2-sphere; that result was an essential step towards the concept of ‘Hopf invariant’ and the popularization of the homotopy group notion due to Hurewicz. Heinz Hopf was born in Wroclaw (Breslau), in the then German part of Poland. He studied in his home town, in Heidelberg and in Berlin, visited Göttingen, Princeton University, and finally settled at ETH in Zürich, where he became Weyl’s successor. The Heinz Hopf Selecta published in 1964 contained an important – although far from being complete – part of Hopf’s mathematical production. So this volume presenting Hopf’s collected works is welcome. As one may expect, the organisational achievement by Beno Eckmann, Hopf’s student and friend, is high class. Two important articles are translated from German into English. This book of over 1200 pages featuring 71 items constitutes an essential reference for the development of mathematics during the XXth century. Jean-Paul Pier (Zbl. MATH 980, 01027)Table of ContentsTable of Contents.- List of Publications of Heinz Hopf.- Editor's Preface.- Papers of Heinz Hopf.- Heinz Hopf Selecta.
£52.24
Springer Fachmedien Wiesbaden Endliche Permutationsgruppen
Book SynopsisDieses Buch über Permutationsgruppen bietet neben modernen Beweisen klassischer Ergebnisse, die bislang nicht in Buchform erschienen sind, einen Zugang zur Klassifikation der primitiven Gruppen. Symmetriebetrachtungen von geometrischen Objekten spielen in vielen Naturwissenschaften eine bedeutende Rolle und lassen sich mathematisch durch Permutationsgruppen modellieren. Nachdem wir in diesem Buch eine beliebige Permutationsgruppe in ihre primitiven Bestandteile zerlegt haben, beweisen wir den wichtigen Klassifikationssatz von Aschbacher-O'Nan-Scott, wonach jede primitive Gruppe zu genau einer von fünf Familien gehört. Dieses Resultat erlaubt es zum Beispiel die 2-transitiven Gruppen explizit anzugeben, sodass wir uns im Folgenden auf die primitiven Gruppen, die nicht 2-transitiv sind, konzentrieren können. Die hierfür entwickelte Theorie der Subgrade ermöglicht uns als Anwendung einen Spezialfall des Satzes von Feit-Thompson zu beweisen. Neben zahlreichen Informationen über aktuelle Entwicklungen stehen dem Studierenden über 100 Übungsaufgaben mit vollständigen Lösungen zur Selbstkontrolle zur Verfügung. Vorausgesetzt werden lediglich Kenntnisse einer Algebra-Vorlesung, wobei wir die Grundlagen der elementaren Gruppentheorie im ersten Kapitel wiederholen. Abgerundet wird das Werk durch einen Anhang mit alternativen Beweisen und Quellcodes für die Computeralgebrasysteme GAP und MAGMA.Table of ContentsGrundlagen.- Operationen auf Mengen.- Abelsche Normalteiler in primitiven Gruppen.- Mehrfach transitive Gruppen.- Konstruktion primitiver Gruppen mit vorgegebenem Sockel.- Klassifikation der primitiven Gruppen.- p-Elemente in primitiven Gruppen.- Transitive Gruppen mit Primzahlgrad.- Subgrade.- Operationen auf Gruppen.- Gruppen ungerader Ordnung.- Rubiks Zauberwürfel.- Anhang.- Lösungen der Aufgaben.
£23.74
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Algebraic Theory of Locally Nilpotent Derivations
Book SynopsisThis book explores the theory and application of locally nilpotent derivations, a subject motivated by questions in affine algebraic geometry and having fundamental connections to areas such as commutative algebra, representation theory, Lie algebras and differential equations. The author provides a unified treatment of the subject, beginning with 16 First Principles on which the theory is based. These are used to establish classical results, such as Rentschler's Theorem for the plane and the Cancellation Theorem for Curves.More recent results, such as Makar-Limanov's theorem for locally nilpotent derivations of polynomial rings, are also discussed. Topics of special interest include progress in classifying additive actions on three-dimensional affine space, finiteness questions (Hilbert's 14th Problem), algorithms, the Makar-Limanov invariant, and connections to the Cancellation Problem and the Embedding Problem.A lot of new material is included in this expanded second edition, such as canonical factorization of quotient morphisms, and a more extended treatment of linear actions. The reader will also find a wealth of examples and open problems and an updated resource for future investigations.Table of ContentsIntroduction.- 1 First Principles.- 2 Further Properties of LNDs.- 3 Polynomial Rings.- 4 Dimension Two.- 5 Dimension Three.- 6 Linear Actions of Unipotent Groups.- 7 Non-Finitely Generated Kernels.- 8 Algorithms.- 9 Makar-Limanov and Derksen Invariants.- 10 Slices, Embeddings and Cancellation.- 11 Epilogue.- References.- Index.
£95.99
Springer-Verlag Berlin and Heidelberg GmbH & Co. KG Visual Group Theory
Book Synopsis
£999.99
Hindustan Book Agency Lie Groups and Lie Algebras
Book SynopsisThis is a textbook meant to be used at the advanced undergraduate or graduate level. It is an introduction to the theory of Lie groups and Lie algebras. The book treats real and p-adic groups in a unified manner. The first chapter outlines preliminary material that is used in the rest of the book. The second chapter is on analytic functions and is of an elementary nature; this material is included to cater to students who may not be familiar with p-adic fields. The third chapter introduces analytic manifolds and contains standard material; the only notable feature being that it covers both real and p-adic analytic manifolds. Chapters 4 and 5 are on Lie groups. All the standard results on Lie groups are proved here. Some of the proofs are different from those in the earlier literature. The last two chapters are on Lie algebras and cover their structure theory as found in the first of the Bourbaki volumes on the subject. Some proofs here are new.
£44.80
Springer Verlag, Singapore Leavitt Path Algebras and Classical K-Theory
Book SynopsisThe book offers a comprehensive introduction to Leavitt path algebras (LPAs) and graph C*-algebras. Highlighting their significant connection with classical K-theory—which plays an important role in mathematics and its related emerging fields—this book allows readers from diverse mathematical backgrounds to understand and appreciate these structures. The articles on LPAs are mostly of an expository nature and the ones dealing with K-theory provide new proofs and are accessible to interested students and beginners of the field. It is a useful resource for graduate students and researchers working in this field and related areas, such as C*-algebras and symbolic dynamics. Table of ContentsChapter 1. Morita Equivalent Leavitt Path Algebras.- Chapter 2. A survey on the ideal structure of Leavitt path algebras.- Chapter 3. The injective and projective Leavitt complexes.- Chapter 4. Graph C*-algebras.- Chapter 5. Steinberg Algebras.- Chapter 6. Leavitt path algebras.- Chapter 7. Relating the principles of Quillen-Suslin theory.- Chapter 8. Action on Alternating matrices and Compound matrices.- Chapter 9. On the relative Quillen-Suslin Local Global Principle.- Chapter 10. On the non-injectivity of the Vaserstein symbol for real threefolds.- Chapter 11. The quotient Unimodular Vector group is nilpotent.- Chapter 12. Symplectic linearization of an alternating polynomial matrix.- Chapter 13. On a theorem of Suslin.- Chapter 14. On a group structure on unimodular rows of length three over a two dimensional ring.- Chapter 15. On an algebraic analogue of the Mayer-Vietoris sequence.- Chapter 16. On the completability of unimodular rows of length three.- Chapter 17. Sandwich classification for classical-like groups over commutative rings.- Chapter 18. A Survey on applications of K-theory in affine algebraic geometry.- Chapter 19. On the non-infectivity of the Vaserstein Symbol in dimension three.- Chapter 20. A survey on affine monoids and K-theory.- Chapter 21. A Survey on the elementary orthogonal groups.
£97.49
Springer Verlag, Singapore Topological Dynamics and Topological Data
Book SynopsisThis book collects select papers presented at the International Workshop and Conference on Topology & Applications, held in Kochi, India, from 9–11 December 2018. The book discusses topics on topological dynamical systems and topological data analysis. Topics are ranging from general topology, algebraic topology, differential topology, fuzzy topology, topological dynamical systems, topological groups, linear dynamics, dynamics of operator network topology, iterated function systems and applications of topology. All contributing authors are eminent academicians, scientists, researchers and scholars in their respective fields, hailing from around the world. The book is a valuable resource for researchers, scientists and engineers from both academia and industry.Table of ContentsH. Bruin, An Overview of Unimodal Inverse Limit Spaces.- B. Barany, M. Rams, K. Simon, Dimension Theory of Some Non Markovian Rapellers Part I: A General Introduction.- B. Barany, M. Rams, K. Simon, Dimension Theory of Some Non Markovian Repellers: Part II: Dynamically Defined Function Graphs.- K. Lesniak, Iterated Function Systems – A Topological Approach Attractors.- H. Kato, Zero Dimensional Covers of Dynamical Systems.- H. Kato, Chaotic Continua in Chaotic Dynamical Systems.- R. L. Devaney, S. M. Marotta, Mandelpinski Necklaces in the Parameter Planes of Rational Maps.- Kit C Chan, Some Examples of Hypercyclic Operators and Universal Sequences of Operators.- Kit C Chan, Some Basic Properties of Hypercyclic Operators.- Kit C Chan, The Testing Ground of Weighted Shift Operators for Hypercyclicity.- D. Drozdov, M. Samuel, A. Tetenov, On -deformations of Polygonal Dendrites.- A. Tetenov, K. Kamalutdinov, V. Aseev, General Position Theorems and its Applications.- A. Raj P, V. Kumar P B, The nth iterate of a map with dense orbit.- Aswathy R K, S. Mathew, Finite Products of Irregular Iterated Function Systems and Their Separation Properties.- A. Akbar, Mubeena T, Periodic Points of N-dimensional Toral Automorphisms.- S. Jose, V. Kumar P B, Julia Sets in Topological Spaces.- K U Sreeja, V. Kumar P B, Ramkumar P B, Julia, Sets of Some Graphs Using Independence Polynomials.- P. Frosini, An Introduction to the Notion of Natural Pseudo Distance in Topological Data Analysis.- A. Cerri, P. Frosini, A Brief Introduction to Multidimensional Persistent Betti Numbers.- N. Quercioli, Some New Methods to Build Group Equivariant Non Expansive Operators in TDA.- Y. Dabaghian, Topological Stability of the Hippocampal Spatial Map and Synaptic Transience.- A. Jacob, Ramkumar P B, Intuitionistic Fuzzy Graph Morphological Topology.- A. G. Pillai, Ramkumar P B, Some Properties of the Bitopological Space Associated with the 3-Uniform Semigraph of Cycle graph.- D. Chandran R, Ramkumar P B, Hypergraph Topology.
£119.99
Springer Verlag, Singapore A First Course in Group Theory
Book SynopsisThis textbook provides a readable account of the examples and fundamental results of groups from a theoretical and geometrical point of view. Topics on important examples of groups (like cyclic groups, permutation groups, group of arithmetical functions, matrix groups and linear groups), Lagrange’s theorem, normal subgroups, factor groups, derived subgroup, homomorphism, isomorphism and automorphism of groups have been discussed in depth. Covering all major topics, this book is targeted to undergraduate students of mathematics with no prerequisite knowledge of the discussed topics. Each section ends with a set of worked-out problems and supplementary exercises to challenge the knowledge and ability of the reader.Trade Review“Advanced school students and well-motivated undergraduates can profitably read it, and it is a very useful general reference for the history of substantial parts of mathematics, placed in the context of contemporary social and political events. … as a readable … and refreshingly detailed account of the whole sweep of ‘infinitesimal methods’ from antiquity to the 1990s, this book is highly recommended.” (Peter Giblin, The Mathematical Gazette, Vol. 107 (570), November, 2023)“Davvaz's book, on the other hand, features many excellent discussions of groups of matrices. Indeed, matrix groups are used not just as examples of groups, but to help clarify and add depth to Davvaz's discussion of other families of groups. … . It is also, in my opinion, the highlight of the book.” (Benjamin Linowitz, MAA Reviews, February 20, 2022)Table of ContentsPreliminaries Notions.- Symmetries of Shapes.- Binary Operations.- Cyclic Groups.- Inverse Functions and Permutations.- Group of Arithmetical Functions.- Matrix Groups.- Translation and Scaling Matrices.- Cosets of Subgroups and Lagrange’s Theorem.- Normal Subgroups and Factor Groups.- Some Special Subgroups.- Commutators and Derived Subgroups.- Maximal Subgroups.- Group Homomorphisms.- Homomorphisms and Their Properties.- Cayley’s Theorem.- Another View of Linear Groups.
£38.24
Springer Verlag, Singapore Lie Theory and Its Applications in Physics:
Book SynopsisThis volume presents modern trends in the area of symmetries and their applications based on contributions to the Workshop "Lie Theory and Its Applications in Physics" held in Sofia, Bulgaria (on-line) in June 2021.Traditionally, Lie theory is a tool to build mathematical models for physical systems. Recently, the trend is towards geometrization of the mathematical description of physical systems and objects. A geometric approach to a system yields in general some notion of symmetry which is very helpful in understanding its structure. Geometrization and symmetries are meant in their widest sense, i.e., representation theory, algebraic geometry, number theory, infinite-dimensional Lie algebras and groups, superalgebras and supergroups, groups and quantum groups, noncommutative geometry, symmetries of linear and nonlinear partial differential operators, special functions, and others. Furthermore, the necessary tools from functional analysis are included. This is a big interdisciplinary and interrelated field.The topics covered in this Volume are the most modern trends in the field of the Workshop: Representation Theory, Symmetries in String Theories, Symmetries in Gravity Theories, Supergravity, Conformal Field Theory, Integrable Systems, Quantum Computing and Deep Learning, Entanglement, Applications to Quantum Theory, Exceptional quantum algebra for the standard model of particle physics, Gauge Theories and Applications, Structures on Lie Groups and Lie Algebras.This book is suitable for a broad audience of mathematicians, mathematical physicists, and theoretical physicists, including researchers and graduate students interested in Lie Theory.Table of ContentsPlenary Talks: T. Kobayashi, Multiplicity in Restricting Minimal Representations.- Yang-Hui He, From the String Landscape to the Mathematical Landscape: a Machine-Learning Outlook.- I. Todorov, Octonionic Clifford Algebra for the Internal Space of the Standard Model.- P. Vitale, The Jacobi Sigma Model.- P. Aschieri, Levi-Civita Connections on Braided Algebras.- N. Bobev, Notes on AdS4 Holography and Higher-Derivative Supergravity.- T. Brzezinski, Homothetic Rota-Baxter Systems and Dyckm-Algebras.- M. Henkel, Quantum Dynamics Far from Equilibrium: a Case Study in the Spherical Model.- Hankyung Ko and V. Mazorchuk, On First Extensions in S -Subcategories of O.- Robert de Mello Koch and Sanjaye Ramgoolam, Higher Dimensional CFTs as 2D Conformally-Equivariant Topological Field Theories.- G. Manolakos, G. Patellis and G. Zoupanos, Reducing the N = 1, 10D E8 Gauge Theory over a Modified Flag Manifold.- String Theories, (Super-)Gravity, Cosmology: Andre Alves Lima, Galen M. Sotkov and Marian Stanishkov, Ramond States of the D1-D5 CFT Away from the Free Orbifold Point.- L. Anguelova, Primordial Black Hole Generation in a Two-field Inflationary Model.- D. Staicova, Late Time Cosmic Acceleration with Uncorrelated Baryon Acoustic Oscillations.- L. Ravera, On the Hidden Symmetries of D = 11 Supergravity.- F. Nieri, Defects at the Intersection: the Supergroup Side.- T. Masuda, A New S-matrix Formula and Extension of the State Space in Open String Field Theory.- E. Boffo, Dual Dilaton with R and Q Fluxes.- Representation Theory: E. Poletaeva, On 1-Dimensional Modules over the Super-Yangian of the Superalgebra Q(1).- N. I. Stoilova and Joris Van der Jeugt, A Klein Operator for Paraparticles.- G. Sengor and C. Skordis, Principal and Complementary Series Representations at the Late-Time Boundary of de Sitter.- S. Aoki, Janos Balog, T. Onogi, and S. Yokoyama, Bulk Reconstruction from a Scalar CFT at the Boundary by the Smearing with the Flow Equation.- Y. Wang and Chih-Hao Fu, Building Momentum Kernel from Shapovalov Form.- Ilia Smilga, Action of w0 on VL for Orthogonal and Exceptional Groups.- Ood Shabtai, Pairs of Spectral Projections of Spin Operators.- Integrable Systems: Jean-Emile Bourgine, Algebraic Engineering and Integrable Hierarchies.- Cestm ˇ ´ır Burd´ık and O. Navratil, Nested Bethe Ansatz for RTT–Algebra An.- O. Vaneeva, O. Magda and A. Zhalij, Lie Reductions and Exact Solutions of Generalized Kawahara Equations.- Y. Nasuda, Several Exactly Solvable Quantum Mechanical Systems and the SWKB Quantization Condition.- A. Pribytok, Automorphic Symmetries and AdSn Integrable Deformations.- Applications to Quantum Theory: M. Kirchbach, T. Popov, and J.-A. Vallejo, The Conformal-Symmetry–Color-Neutrality Connection in Strong Interaction.- I. Salom and N. Manojlovic, sℓ(2) Gaudin Model with General Boundary Terms.- T. Barron and A. Kazachek, Entanglement of Mixed States in Kahler Quantization.- J. Alnefjord, A. Lifson, C. Reuschle, and M. Sjodahl, The Chirality-Flow Formalism for Standard Model Calculations.- F. Kuipers, Spacetime Stochasticity and Second Order Geometry.- Special Mathematical Results: P. Moylan, Velocity Reciprocity in Flat and Curved Space-Time.- S. Stoimenov and M. Henkel, Meta-Schrodinger Transformations.- Hulya Arg ¨ uz¨, The Quantum Mirror to the Quartic del Pezzo Surface.- A. Ganchev, Bidirectional Processes - in Category Theory, Physics, Engineering.- Gauge Theories and Applications: Richard S. Garavuso, Nonholomorphic Superpotentials in Heterotic Landau-Ginzburg Models.- F. Feruglio, Automorphic Forms and Fermion Masses.- T. Ishibashi, Wilson Lines and Their Laurent Positivity.- Maro Cvitan, Predrag Dominis Prester, Stefano Gregorio Giaccari, Mateo Paulisiˇ c´, and Ivan Vukovic´, Gauging Higher-Spin-Like Symmetries Using the Moyal Product.- N. Ikeda and S. Sasaki, Integration of Double Field Theory Algebroids and Pre-rackoid in Doubled Geometry.- H. Mori, S. Sasaki, K. Shiozawa, Doubled Aspects of Algebroids and Gauge Symmetry in Double Field Theory.- C. Anghel and D. Cheptea, Lie Algebroids and Weight Systems.- Structures on Lie Groups and Lie Algebras: K. Arashi, Visible Actions of Certain Affine Transformation Groups of a Siegel Domain of the Second Kind.- A. Brus, Jiˇr´ı Hrivnak´ and L. Motlochova´, Quantum Particle on Lattices in Weyl Alcoves.- A. Latorre and L. Ugarte, Abelian J-Invariant Ideals on Nilpotent Lie Algebras.- Alexis Langlois-Remillard, The Dihedral Dunkl–Dirac Symmetry Algebra with Negative Clifford Signature.- Tekin Karadag˘, Lie Structure on Hopf Algebra Cohomology.- Esther Garcıa, Miguel Gomez Lozano, and Ruben Munoz Alcazar, Filtration Associated to an Abelian Inner Ideal and the Speciality of the Subquotient of a Lie Algebra.- Esther Garc´ıa, Miguel Gomez Lozano, and Guillermo Vera de Salas, Nilpotent Inner Derivations in Prime Superalgebras.
£159.99
Birkhäuser Symmetry in Geometry and Analysis Volume 2
Book SynopsisThe source operator method: an overview (Salem Ben Said, Jean-Louis Clerc and Khalid Koufany).- Some mixed norm bounds for the spectral projections of the Heisenberg sublaplacian (Valentina Casarino and Paolo Ciatti).- Four variations on the Rankin-Cohen brackets (Jean-Louis Clerc).- Restricting holomorphic discrete series representations to a compact dual pair (Jan Frahm and Quentin Labriet).- Nets of standard subspaces on non-compactly causal symmetric spaces (Jan Frahm, Karl-Hermann Neeb, and Gestur Ólafsson).- Heisenberg parabolically induced representations of Hermitian Lie groups, Part II: Next-to-minimal representations and branching rules (Jan Frahm, Clemens Weiske and Genkai Zhang).- Quantum-Classical Correspondences for Locally Symmetric Spaces (Joachim Hilgert).- Classification of K-type formulas for the Heisenberg ultrahyperbolic operator ?s for ?? ??(??, R) and tridiagonal determinants for local Heun functions (Toshihisa Kubo and Bent Ørsted).- Gauss--Berezin integral operators, spinors over orthosymplectic supergroups, and Lagrangian super-Grassmannians (Yury A. Neretin).- Towards Gan-Gross-Prasad type conjecture for discrete series representations of symmetric spaces (Bent Ørsted and Birgit Speh).- Pseudo-dual pairs and branching of Discrete Series (Bent Ørsted and Jorge A. Vargas).- Integral transformations of hypergeometric functions with several variables (Toshio Oshima).
£151.99
Springer London Ltd Matrix Groups: An Introduction to Lie Group
Book SynopsisThis book offers a first taste of the theory of Lie groups, focusing mainly on matrix groups: closed subgroups of real and complex general linear groups. The first part studies examples and describes classical families of simply connected compact groups. The second section introduces the idea of a lie group and explores the associated notion of a homogeneous space using orbits of smooth actions. The emphasis throughout is on accessibility. Trade ReviewFrom the reviews of the first edition: MATHEMATICAL REVIEWS "This excellent book gives an easy introduction to the theory of Lie groups and Lie algebras by restricting the material to real and complex matrix groups. This provides the reader not only with a wealth of examples, but it also makes the key concepts much more concrete. This combination makes the material in this book more easily accessible for the readers with a limited background…The book is very easy to read and suitable for an elementary course in Lie theory aimed at advanced undergraduates or beginning graduate students…To summarize, this is a well-written book, which is highly suited as an introductory text for beginning graduate students without much background in differential geometry or for advanced undergraduates. It is a welcome addition to the literature in Lie theory." "This book is an introduction to Lie group theory with focus on the matrix case. … This book can be recommended to students, making Lie group theory more accessible to them." (A. Akutowicz, Zentralblatt MATH, Vol. 1009, 2003)Table of ContentsI. Basic Ideas and Examples.- 1. Real and Complex Matrix Groups.- 2. Exponentials, Differential Equations and One-parameter Subgroups.- 3. Tangent Spaces and Lie Algebras.- 4. Algebras, Quaternions and Quaternionic Symplectic Groups.- 5. Clifford Algebras and Spinor Groups.- 6. Lorentz Groups.- II. Matrix Groups as Lie Groups.- 7. Lie Groups.- 8. Homogeneous Spaces.- 9. Connectivity of Matrix Groups.- III. Compact Connected Lie Groups and their Classification.- 10. Maximal Tori in Compact Connected Lie Groups.- 11. Semi-simple Factorisation.- 12. Roots Systems, Weyl Groups and Dynkin Diagrams.- Hints and Solutions to Selected Exercises.
£29.99
Oxford University Press ARITHMETICAL SIMILAR OMM C Prime Decomposition and Finite Group Theory Oxford Mathematical Monographs
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£137.50
