图书信息
图书目录
例外群及其几何(英文版)Exceptional groups and their geometry
暂无简介
作者：
[德]Bruce Hunt
定价：
299.00元
出版时间：
2025-04-29
ISBN：
978-7-04-063618-5
物料号：
63618-00
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
代数几何学
重点项目：
暂无
版面字数：
1200.00千字
开本：
16开
全书页数：
暂无
装帧形式：
精装
- 前辅文
- Introduction
- 1 The classical groups
- 1.1 The classical compact simple Lie groups
- 1.1.1 Maximal Tori and the Weyl group
- 1.1.2 Principal bundles and classifying spaces
- 1.1.3 Lie algebras of classical type
- 1.2 Real Lie algebras and groups of classical type
- 1.2.1 Involutions
- 1.2.2 Real forms
- 1.3 Q-Lie algebras and arithmetic groups of classical type
- 1.3.1 Classification theorem
- 1.3.2 The Q-forms for groups of classical type
- 1.3.3 Picard modular groups
- 1.3.4 Siegel modular groups
- 1.4 Arithmetic quotients of Riemannian symmetric spaces
- 1.4.1 Commensurability
- 1.4.2 Picard modular varieties
- 1.4.2.1 Compactification
- 1.4.2.2 Picard modular groups
- 1.4.3 Siegel modular varieties
- 1.1 The classical compact simple Lie groups
- Part I Exceptional algebraic and Lie groups
- 2 Composition algebras and octonions
- 2.1 Alternative algebras
- 2.2 Composition algebras
- 2.2.1 Quaternion algebras
- 2.2.2 Octonion algebras
- 2.3 The automorphism group of an octonion algebra
- 2.4 Derivations of an octonion algebra
- 2.5 Octonions and Clifford algebras
- 2.6 Triality
- 2.7 Lattices
- 2.7.1 The octavians, the ring of octonion integers
- 2.7.2 Octonions and the Leech lattice
- 2.8 The projective octonion line and Bott periodicity
- 3 Exceptional Jordan algebras and F4
- 3.1 Jordan algebras
- 3.2 Classification
- 3.3 Jordan triple systems
- 3.4 Albert algebras
- 3.4.1 Algebras of type f4 and groups of type F4
- 3.5 Orders in Jordan algebras
- 4 The exceptional complex Lie groups and their real forms
- 4.1 The Tits-Vinberg-Atsuyama constructions
- 4.2 Adams’ construction
- 4.2.1 Spin representations
- 4.2.2 Construction of E8
- 4.2.3 Other exceptional groups
- 4.3 Freudenthal’s construction
- 4.3.1 Type G2
- 4.3.2 Type E8
- 4.3.3 Type E7
- 4.3.4 Type E6
- 5 Q-forms and arithmetic subgroups of exceptional groups
- 5.1 Twisted composition algebras and exceptional D4
- 5.2 Descriptions of the Q-forms for E6, E7
- 6 Cohomology of exceptional Lie groups and homogeneous spaces
- 6.1 Generators of cohomology
- 6.2 Exceptional Hermitian symmetric spaces
- 6.2.1 The exceptional compact Hermitian symmetric spaces
- 6.2.2 The exceptional bounded symmetric domains
- 6.3 Some geometry of exceptional homogeneous spaces
- 6.3.1 G2
- 6.3.2 F4
- 6.3.3 E6
- 6.3.4 E7
- 6.3.5 E8
- 6.4 Cohomology of the exceptional groups
- 7 Exceptional groups and projective planes
- 7.1 Real projective spaces
- 7.2 Projective planes
- 7.2.1 Complex projective planes
- 7.2.2 Generalized projective planes
- 7.2.3 Quaternionic projective planes
- 7.2.4 Octonionic projective planes
- 2 Composition algebras and octonions
- Part II Applications of exceptional groups
- 8 Applications of octonions and exceptional Lie groups in theoretical physics
- 8.1 Division algebras and the standard model in theoretical physics
- 8.1.1 The geometry of division algebras and related algebras
- 8.1.1.1 The Hamiltonian quaternions
- 8.1.1.2 The Cayley-Graves algebra of octonions
- 8.1.2 Division algebras and Clifford algebras
- 8.1.3 The algebra of octonions
- 8.1.4 The physics of octonions
- 8.1.4.1 Symmetry groups
- 8.1.4.2 Field content
- 8.1.4.3 Gauge fields
- 8.1.4.4 Symmetry breaking
- 8.1.4.5 Three generations
- 8.1.1 The geometry of division algebras and related algebras
- 8.2 Division algebras and particles
- 8.2.1 The Pauli algebra
- 8.2.2 The algebra
- 8.2.3 More Clifford algebras and the Dixon algebra T
- 8.2.4 Three generations
- 8.3 Jordan algebras and the standard model
- 8.3.1 H2(H) and the electro-weak force
- 8.3.2 H2(O): one generation of standard model particles
- 8.3.3 H3(O) and the strong force
- 8.3.4 Triality and three generations
- 8.4 Exceptional homogeneous spaces arising from compactifications of supergravity
- 8.4.1 Dimensional reduction and Kaluza-Klein theory
- 8.4.2 Normed division algebras and physics in dimensions D = 3,4,6,10
- 8.4.3 Super Yang-Mills
- 8.4.4 Magic pyramids
- 8.4.5 D = 11 supergravity expressed in octonions
- 8.4.6 Symmetric spaces and supergravity theories
- 8.4.7 Exceptional field theory (ExFT)
- 8.4.7.1 The bosonic sector
- 8.4.7.2 The fermionic sector
- 8.4.7.3 Embedding other known theories in E(8)8 ExFT
- 8.5 Some other occurrences of exceptional groups
- 8.5.1 1976: GUT and exceptional groups
- 8.5.2 1985: The heterotic string, Virasoro algebra and Kac-Moody algebras
- 8.5.3 1995: Exceptional structures in the context of string dualities
- 8.5.4 2010: Cayley plane bundles and fields in M-theory
- 8.5.5 2022: Fitting the standard model in E8(-24)
- 8.5.6 Condensed matter physics: E8 and the Ising model
- 8.1 Division algebras and the standard model in theoretical physics
- 9 Applications of exceptional groups in algebraic geometry
- 9.1 Unimodular surface singularities and orbits of Lie groups in the adjoint representation
- 9.1.1 Rational double points
- 9.1.2 Orbits of the adjoint representation
- 9.1.2.1 Kähler homogeneous spaces
- 9.1.2.2 The adjoint quotient
- 9.1.2.3 Regular, subregular and unipotent elements
- 9.1.3 Simultaneous resolution of the adjoint quotient
- 9.1.4 Singularities of the subregular orbit
- 9.1.5 Semiuniversal deformations of rational double points
- 9.1.6 Deformations of exceptional singularities
- 9.2 Arrangements
- 9.2.1 Geometry of an arrangement
- 9.2.2 Projective arrangements
- 9.2.3 The Fermat covers associated with a projective arrangement
- 9.2.4 Arrangements defined by reflection groups
- 9.3 The Weyl group W(F4) and related geometry
- 9.3.1 The arrangement
- 9.3.2 Invariants
- 9.3.3 A ball quotient derived from theW(F4) arrangement
- 9.4 The Weyl group W(A5) and related geometry
- 9.4.1 The arrangement
- 9.4.2 Symmetric varieties
- 9.5 The Weyl group W(E6) and related geometry
- 9.5.1 The 27 lines on a smooth cubic surface
- 9.5.2 The arrangement of W(E6)
- 9.5.3 Invariant varieties
- 9.6 The Weyl group W(E7) and related geometry
- 9.6.1 The 28 bitangents of a smooth quartic in the plane
- 9.6.2 The 27 lines and the 28 bitangents
- 9.6.3 The root system of E7
- 9.6.4 The arrangement of W(E7)
- 9.6.4.1 63 hyperplanes and 63 points
- 9.6.4.2 28 hyperplanes and 28 points
- 9.6.4.3 336 P4’s and 336 lines
- 9.6.4.4 315 P3’s and 315 P2’s
- 9.6.4.5 378 lines
- 9.6.5 Invariants of W(E7)
- 9.7 The Weyl group W(E8) and related geometry
- 9.7.1 The 240 sections of a rational elliptic surface
- 9.7.2 The root system of E8
- 9.7.3 The arrangement of W(E8)
- 9.7.3.1 120 P6’s and 120 points
- 9.7.3.2 1120 P5’s and 1120 lines
- 9.7.3.3 7560 P4’s and 7560 P2’s
- 9.7.3.4 24 192 P3’s and skew lines on the cubic surface
- 9.7.3.5 3150 P3’s and the tritangents of a cubic surface
- 9.7.3.6 8640 points and E8-Steiner complexes
- 9.7.3.7 1080 points
- 9.7.3.8 3360 points
- 9.8 Solving algebraic equations and the resolvent degree
- 9.8.1 Modern formulation
- 9.8.2 Versal G-spaces for exceptional Weyl groups
- 9.9 Del Pezzo surfaces
- 9.9.1 Point sets
- 9.9.2 The Weyl group of a generalized del Pezzo variety
- 9.9.3 Del Pezzo surfaces
- 9.9.4 Moduli spaces
- 9.10 K3 surfaces
- 9.1 Unimodular surface singularities and orbits of Lie groups in the adjoint representation
- 8 Applications of octonions and exceptional Lie groups in theoretical physics
- Part III Appendices
- 10 Root systems
- 11 Fiber bundles and homogeneous spaces
- 11.1 Topological results
- 11.1.1 Invariants
- 11.1.2 Hopf’s theorem on the cohomology of Lie groups
- 11.1.3 G-spaces
- 11.1.4 The Leray spectral sequence
- 11.2 Lie groups and representations
- 11.2.1 The Lie algebra
- 11.2.2 Representations
- 11.2.2.1 Unitary representations
- 11.2.2.2 Weight and root lattices
- 11.2.2.3 Classification: highest weights and Weyl’s unitary trick
- 11.2.2.4 Induced representations
- 11.2.3 Maximal subgroups of semi-simple Lie groups
- 11.1 Topological results
- 12 Clifford algebras
- 12.1 Algebraic formulation
- 12.2 Minkowski space
- 12.3 Bott periodicity
- 12.4 Spin, semispin and orthogonal groups
- 12.5 Spin representations
- 12.6 Gamma matrices
- 13 Some algebraic geometry
- 13.1 Plane curves
- 13.2 Singularities and resolutions
- 13.2.1 Singularities
- 13.2.2 Resolutions
- 13.3 Algebraic groups
- 13.4 Moduli spaces
- 13.4.1 The notion of moduli space
- 13.4.2 Abelian varieties
- 13.4.3 Bounded symmetric domains and Hermitian symmetric spaces
- 13.5 Ball quotients
- 13.5.1 The Yau inequality
- 13.5.2 Criteria for ball quotients
- 13.5.3 Höfer’s theory
- 14 Classical and quantum mechanics and field theory
- 14.1 Group representations and physics
- 14.2 Lagrangian and Hamiltonian mechanics
- 14.2.1 Classical mechanics
- 14.2.2 Field theories
- 14.3 Electromagnetics
- 14.3.1 Maxwell’s equations
- 14.3.2 Potentials
- 14.4 Gravity
- 14.4.1 Minkowski space
- 14.4.2 Unitary representations of the Poincar´e group
- 14.4.3 Special relativity
- 14.4.4 Relativistic electromagnetism
- 14.4.5 Gravity
- 14.5 Quantization
- 14.5.1 Postulates of quantum mechanics
- 14.5.2 Angular momentum, spin and the Pauli equation
- 14.5.3 Relativistic quantum mechanics
- 14.6 Gauge theories
- 14.6.1 Local and global symmetries
- 14.6.2 Symmetry breaking
- 14.7 Quantum field theory
- 14.7.1 Physics intuition
- 14.7.2 Mathematical formulation
- 14.7.3 The standard model
- 14.7.3.1 Particle content
- 14.7.3.2 Lagrangian
- 14.7.3.3 Gauge bosons and symmetry breaking
- 14.7.3.4 Problems with the standard model
- 14.8 Supersymmetry and supergravity
- 14.8.1 The algebra of supersymmetry
- 14.8.1.1 Representations of the supersymmetry algebra
- 14.8.1.2 Transformation properties
- 14.8.2 Supergravity
- 14.8.3 D = 11 supergravity
- 14.8.1 The algebra of supersymmetry
- 14.9 Superstrings and compactifications
- 14.9.1 The bosonic string
- 14.9.2 The Virasoro algebra
- 14.9.3 Quantization
- 14.9.4 Superstrings and the super-Virasoro algebra
- 14.9.5 The 5 superstring theories
- 14.10 Dualities
- References
- Index
- 插图
