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
1.4.3 Siegel modular varieties
Part I Exceptional algebraic and Lie groups
2 Composition algebras and octonions
2.1 Alternative algebras
2.2 Composition algebras
2.3 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
2.8 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
3.5 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
4.3 Freudenthal’s construction
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
6.3 Some geometry of exceptional homogeneous spaces
6.4 Cohomology of the exceptional groups
7 Exceptional groups and projective planes
7.1 Real projective spaces
7.2 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
8.2 Division algebras and particles
8.3 Jordan algebras and the standard model
8.4 Exceptional homogeneous spaces arising from compactifications of supergravity
8.5 Some other occurrences of exceptional groups
9 Applications of exceptional groups in algebraic geometry
9.1 Unimodular surface singularities and orbits of Lie groups in the adjoint representation
9.2 Arrangements
9.3 The Weyl group W(F4) and related geometry
9.4 The Weyl group W(A5) and related geometry
9.5 The Weyl group W(E6) and related geometry
9.6 The Weyl group W(E7) and related geometry
9.7 The Weyl group W(E8) and related geometry
9.8 Solving algebraic equations and the resolvent degree
9.9 Del Pezzo surfaces
9.10 K3 surfaces
Part III Appendices
10 Root systems
11 Fiber bundles and homogeneous spaces
11.1 Topological results
11.2 Lie groups and representations
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
13.3 Algebraic groups
13.4 Moduli spaces
13.5 Ball quotients
14 Classical and quantum mechanics and field theory
14.1 Group representations and physics
14.2 Lagrangian and Hamiltonian mechanics
14.3 Electromagnetics
14.4 Gravity
14.5 Quantization
14.6 Gauge theories
14.7 Quantum field theory
14.8 Supersymmetry and supergravity
14.9 Superstrings and compactifications
14.10 Dualities
References
Index
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