目录
 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
 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
 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
   8.2 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
  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
 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
  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
   14.9 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
 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
 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
 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
   8.2 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
  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
 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
  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
   14.9 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