Geometry of Lie groups:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Dordrecht [u.a.]
Kluwer
1997
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| Series: | Mathematics and its applications
393 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVIII, 393 S. graph. Darst. |
| ISBN: | 0792343905 |
Staff View
MARC
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| 100 | 1 | |a Rozenfel'd, Boris A. |d 1917- |e Verfasser |0 (DE-588)124756638 |4 aut | |
| 245 | 1 | 0 | |a Geometry of Lie groups |c by Boris Rosenfeld |
| 264 | 1 | |a Dordrecht [u.a.] |b Kluwer |c 1997 | |
| 300 | |a XVIII, 393 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematics and its applications |v 393 | |
| 650 | 7 | |a Grupos de lie |2 larpcal | |
| 650 | 7 | |a Géométrie |2 ram | |
| 650 | 7 | |a Lie, Groupes de |2 ram | |
| 650 | 4 | |a Geometry | |
| 650 | 4 | |a Lie groups | |
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| adam_text | CONTENTS
Preface xvii
Chapter 0. Structures of Geometry 1
§0.1. Algebraic Structures 1
0.1.1. Mathematical structures (1). 0.1.2. Groups (1). 0.1.3.
Rings and Fields (2). 0.1.4. Subgroups, Subrings, and Subfields
(2). 0.1.5. Direct sums and products (3). 0.1.6. Linear Spaces (3).
0.1.7. Algebras (4).
§0.2. Topological Structures 5
0.2.1. Topological Spaces (5). 0.2.2. Subspaces of Topological
Spaces (5). 0.2.3. Continuous Mappings and Homeomorphisms
(6).
§0.3. Order Structures 6
0.3.1. Ordered Sets (6). 0.3.2. Lattices (6).
§0.4. Incidence Structures 7
0.4.1. Blocks and Incidences (7). 0.4.2. Affine Space (7). 0.4.3.
Projective space (7).
§0.5. Metric Structures 8
0.5.1. Metric Spaces (8). 0.5.2. Pseudometric Spaces (8). 0.5.3.
Euclidean Space (8). 0.5.4. Hyperspheres in Euclidean Space (10).
0.5.5. Natural Topology in Euclidean Space (10). 0.5.6. Pseudo
Euclidean Spaces (10). 0.5.7. Conformal Space (11). 0.5.8. Pseu
doconformal Spaces (11).
§0.6. Tensors and Linear Operators 11
0.6.1. Covectors (11). 0.6.2. Covectors in Affine and Projective
Spaces (12). 0.6.3. Tensors (12). 0.6.4. Linear Operators (12).
0.6.5. Tensor and Exterior Products of Vectors and Covectors (13).
0.6.6. Differentiable Scalar, Vector, and Tensor Fields (14).
§0.7. Riemannian Manifolds and Manifolds with Affine Connections 14
0.7.1. Topological Manifolds (14). 0.7.2. Differentiable Manifolds
(15). 0.7.3. Vectors and Tensors in Differentiable Manifolds (15).
0.7.4. Riemannian and Pseudo Riemannian Manifolds (15). 0.7.5.
Parallel Displacement (16). 0.7.6. Covariant Derivatives and Ab¬
solute Differentials (16). 0.7.7. Sectional Curvature (17). 0.7.8.
vii
viii CONTENTS
Manifolds with Affine Connections (18). 0.7.9. Arbitrary Frames
and Exterior Forms (19).
§0.8. Topological Groups and Lie Groups 21
0.8.1. Topological Groups (21). 0.8.2. Lie Groups (21).0.8.3. Solv¬
able, Simple, Semisimple, and Reductive Lie Groups (23). 0.8.4.
Lie Algebras (23). 0.8.5. Invariant Riemannian Metrics and Affine
Connections in Lie Groups (24). 0.8.6. Homogeneous Spaces (25).
0.8.7. Symmetric Spaces (26). 0.8.8. Reductive and fc Symmetric
Spaces (26). 0.8.9. Fundamental Groups of the Most Important
Homogeneous Spaces (26).
Chapter I. Algebras and Lie Groups 29
§1.1. Commutative Associative Algebras 29
1.1.1. Complex Numbers (29). 1.1.2. Split Complex Numbers (30).
1.1.3. Dual Numbers (31). 1.1.4. Automorphisms in 2 Algebras
(32). 1.1.5. Differentiable and Analytic Functions (33). 1.1.6.
Tensor Products of Commutative Algebras (34). 1.1.7. Cyclic,
Anticyclic and Plural Numbers (35). 1.1.8. Polynomial and Group
Algebras (36). 1.1.9. Simple and Semisimple Algebras (37). 1.1.10.
Quasisimple and r Quasisimple Algebras (37). 1.1.11. Hjelmslev
Algebras (38). 1.1.12. Frobenius Algebras (38).
§1.2. Noncommutative Associative Algebras 40
1.2.1. Quaternions (40). 1.2.2. Matrices (40). 1.2.3. Split Quater¬
nions (43). 1.2.4. Simple and Semisimple Algebras (45). 1.2.5.
Automorphisms in Simple Associative Algebras (45). 1.2.6. Semi
quaternions, Split Semiquaternions, and ^ Quaternions (46). 1.2.7.
Tensor Products of Algebras (48). 1.2.8. Alternions (48). 1.2.9.
Pseudoalternions (51). 1.2.10. Split Alternions and Bialternions
(52). 1.2.11. Quasisimple and r Quasisimple Noncommutative Al¬
gebras (52). 1.2.12. Grassmann Algebras (53). 1.2.13. Frobenius
Algebras (54).
§1.3. Alternative Algebras 54
1.3.1. Octonions (54). 1.3.2. Split Octonions and Bioctonions (55).
1.3.3. Zorn Vector Matrices (56). 1.3.4. Simple Alternative Alge¬
bras (57). 1.3.5. Automorphisms in Algebras of Octonions and
Split Octonions (57). 1.3.6. Tensor Products (60). 1.3.7. Semioc
tonions, Split Semioctonions, j Octonions, Split Octonions, |
Octonions (60).
§1.4. Lie Algebras and Lie Groups 61
1.4.1. Real and Complex Lie Groups and Algebras (61). 1.4.2.
Split Complex and Dual Lie Groups (61). 1.4.3. Root Vectors of
Simple Complex Lie Groups (62). 1.4.4. Dynkin Diagrams (67).
1.4.5. Cartan Matrices (68). 1.4.6. Isomorphisms of Simple Com¬
plex Lie Groups (69). 1.4.7. Spherical Weyl Groups for Simple
CONTENTS ix
Lie Groups (69). 1.4.8. Coxeter Transformations (72). 1.4.9. Ex¬
tended Dynkin Diagrams (73). 1.4.10. Affine Weyl Groups for Sim¬
ple Complex Lie Groups (75). 1.4.11. Root Vectors of Compact
Simple Real Lie Groups (76). 1.4.12. Root Vectors of Noncompact
Simple Real Lie Groups (80). 1.4.13. Satake Diagrams (81). 1.4.14.
Isomorphisms of Simple Real Lie Groups (82). 1.4.15. Topology
of Lie Groups (85). 1.4.16. Symmetric and Reductive Spaces (86).
1.4.17. Finite Groups of Lie Type (87).
§1.5. Jordan and Elastic Algebras 87
1.5.1. Commutative Jordan Algebras (87). 1.5.2. Elastic Alge¬
bras (91). 1.5.3. Algebras of Rectangular Matrices (91). 1.5.4.
Freudenthal Algebras (91). 1.5.5. Ternars (92).
§1.6. Linear Representations of Simple Lie Groups 93
1.6.1. Linear Representation of Lie Groups (93). 1.6.2. Linear
Representation of Semisimple Lie Groups (93). 1.6.3. Spinor Rep¬
resentations (97). 1.6.4. Fundamental and Parabolic Figures in
Spaces with Simple Fundamental Groups (101).
Chapter II. Affine and Projective Geometries 106
§2.1. Affine Geometries 106
2.1.1. Linear Spaces and Modules (106). 2.1.2. Real and Complex
Interpretations of Free Modules over Algebras (107). 2.1.3. Affine
Spaces (107). 2.1.4. Adjacent Points and Lines (108). 2.1.5. Affine
Theorem of Pappus (109). 2.1.6. Affine Theorem of Desargues
(109). 2.1.7. Real Interpretations of Affine Spaces Over Algebras
(110). 2.1.8. Applications of Segreans (112). 2.1.9. Other Real
Interpretations of Affine Spaces over Algebras (113).
§2.2. Projective Geometries 114
2.2.1. Projective Spaces (114). 2.2.2. Topology of Projective
Spaces (114). 2.2.3. Adjacent Points and Lines (118). 2.2.4. Pro¬
jective Theorems of Pappus and Desargues (118). 2.2.5. Axioms
of Kolmogorov (120). 2.2.6. Real and Complex Interpretations of
Projective Spaces Over Algebras (120).
§2.3. Affine and Projective Transformations 122
2.3.1. Affine Transformations (122). 2.3.2. Simple Ratio of Three
Points (124). 2.3.3. Affine Lines (124). 2.3.4. Collineations (125).
2.3.5. Correlations (126). 2.3.6. Cross Ratio of Four Points (127).
2.3.7. Projective Lines (128).
§2.4. Lines, m planes, and Hyperplanes 129
2.4.1. Matrix Coordinates of m Planes (129). 2.4.2. Intersection
of Two m Planes (130). 2.4.3. Projection onto an m Plane in the
Direction of an (n m 1) Plane (131). 2.4.4. m Pairs and Their
Cross Ratios (132). 2.4.5. Affine Matrix Coordinates (133). 2.4.6.
Duality Principle (135).
x CONTENTS
§2.5. Hyperquadrics 136
2.5.1. Hyperquadrics in Affine Spaces (136). 2.5.2. Hermitian Hy¬
perquadrics in Affine Spaces (136). 2.5.3. Center of a Hyperquadric
(137). 2.5.4. Conjugate Diameters (137). 2.5.5. Classification of
Hyperquadrics and Hermitian Hyperquadrics (139). 2.5.6. Poles
and Polar Hyperplanes (141). 2.5.7. Rectilinear and m Planar
Generators (143). 2.5.8. Hyperquadrics and Hermitian Hyper¬
quadrics in Projective Spaces (146). 2.5.9. Application of Matrix
Coordinates (149). 2.5.10. Varieties (151).
§2.6. Linear Complexes 152
2.6.1. Linear Complexes and Congruences of Lines (152). 2.6.2.
Null systems (153). 2.6.3. Hermitian Null Systems (153). 2.6.4.
Null m Planes of a Null System (154).
§2.7. Projective Configurations 155
2.7.1. Plane Configurations (155). 2.7.2. Space Configurations
(155). 2.7.3. Regular Configurations (156).
§2.8 Symmetry and Parabolic Figures 156
2.8.1. Symmetry Figures in Affine Spaces (156). 2.8.2. Symmetry
Figures in Projective Spaces (157). 2.8.3. Cosymmetry Figures in
Projective Spaces (159). 2.8.4. Fundamental and Parabolic Figures
in Projective Spaces (160).
§2.9. Finite Geometries 160
2.9.1. Finite Affine Geometries (160). 2.9.2. Finite Projective Ge¬
ometries (161). 2.9.3. Projective Spaces over Finite Rings (163).
2.9.4. Hyperquadrics and Hermitian Hyperquadrics in Finite Pro¬
jective Spaces (163). 2.9.5. Null systems in Finite Projective Spa¬
ces (166). 2.9.6. Affine and Projective Transformations in Finite
Spaces (166).
Chapter III. Euclidean, Pseudo Euclidean, Conformal and
Pseudoconformal Geometries 168
§3.1. Euclidean and Pseudo Euclidean Spaces 168
3.1.1. Quadratic Euclidean and Pseudo Euclidean Spaces (168).
3.1.2. Hermitian Euclidean and Pseudo Euclidean Spaces (169).
3.1.3. Real Metrics and Pseudometrics in Complex and Split Com¬
plex Quadratic Spaces (171). 3.1.4. Bicomplex, Biquaternionic,
and Quaterquaternionic Euclidean Spaces (171). 3.1.5. Applica¬
tions of Complex, Split Complex, and Dual Numbers to Real Plane
and Spherical Geometry (172).
§3.2. Motions and Similitudes 175
3.2.1. Motions (175). 3.2.2. Similitudes (176). 3.2.3. Antimotions
and Antisimilitudes (176).
§3.3. Lines, m Planes and Hyperplanes 177
3.3.1. Matrix Coordinates of m Planes (177). 3.3.2. Shortest Dis¬
tance Between Two m Planes (177). 3.3.3. Stationary Angles Be
CONTENTS xi
tween Two m Planes (178). 3.3.4. Application of Complex, Split
Complex, and Dual Numbers to the Geometry of Real 3 Spaces
and 3 Spheres (179). 3.3.5. Symmetry Figures in Euclidean and
Pseudo Euclidean Spaces (181). 3.3.6. Holomorphy Angle, Holo
morphic and Antiholomorphic Real 2 Directions (182).
§3.4. Polyhedra 183
3.4.1. Real Polyhedra (183). 3.4.2. Regular Polyhedra (184). 3.4.3.
Regular Honeycombs (185).
§3.5. Hyperquadrics 186
3.5.1. Principal Axes of Hyperquadrics (186). 3.5.2. Invariants of
Equations of Hyperquadrics (187).
§3.6. Hyperspheres 188
3.6.1. Hyperspheres and Hermitian Hyperspheres (188). 3.6.2. An¬
gle Between Hyperspheres (189). 3.6.3. Spherical Trigonometry
(190). 3.6.4. Volumes of Spherical Figures (193). 3.6.5. Areas of
Spherical Triangles and Volumes of Spherical Simplexes (195).
§3.7. Sliding Vectors 198
3.7.1. Sliding Vectors and Systems of Them (198). 3.7.2. Main
Vector and Main Moment of a System of Sliding Vectors (199).
3.7.3. Canonical Forms of Systems of Sliding Vectors (201). 3.7.4.
Sliding Vectors in Pseudo Euclidean Spaces (201). 3.7.5. Spherical
Sliding Vectors in Euclidean Spaces (202). 3.7.6. Spherical Sliding
Vectors in Pseudo Euclidean Spaces (203).
§3.8. Conformal and Pseudoconformal Spaces 204
3.8.1 Conformal Space and Conformal Transformations (204).
3.8.2. Projective Interpretation of Conformal Space (205). 3.8.3.
(n + 2) Spherical Coordinates of Hyperspheres (206). 3.8.4. Con¬
formal Configurations (207). 3.8.5. Pseudoconformal Spaces (209).
3.8.6. Geometry of m Spheres (211). 3.8.7. Application of Com¬
plex Numbers, Quaternions, Split Complex Numbers, and Split
Quaternions to the Geometry of Conformal and Pseudoconformal
Planes and 4 Spaces (212).
§3.9. Finite Geometries 213
3.9.1. Finite Conformal and Pseudoconformal Spaces (213). 3.9.2.
The Steiner Triple Systems (214). 3.9.3. Matthieu Planes (214).
§3.10. Applications to Physics 215
3.10.1. Space time of Special Relativity (215). 3.10.2. Addition
of Velocities in Special Relativity (216). 3.10.3. Electromagnetic
Field (217).
Chapter IV. Elliptic, Hyperbolic, Pseudoelliptic, and Pseudo
hyperbolic Geometries 219
§4.1. Elliptic, Hyperbolic, Pseudoelliptic, and Pseudohyperbolic Spaces 219
4.1.1. Quadratic Elliptic, Hyperbolic, Pseudoelliptic, and Pseu¬
dohyperbolic Spaces (219). 4.1.2. Hermitian Elliptic, Hyperbolic,
xii CONTENTS
Pseudoelliptic, and Pseudohyperbolic Spaces (222). 4.1.3. Bicom
plex, Biquaternionic, and Quaterquaternionic Elliptic and Hyper¬
bolic Spaces (225). 4.1.4. Complex, Split Complex, Quaternionic
and Split Quaternionic Hermitian Lines (226).
§4.2. Motions 227
4.2.1. Motions in Elliptic, Hyperbolic, Pseudoelliptic, and Pseudo
hyperbolic Spaces (227). 4.2.2. Groups of Motions (228). 4.2.3.
Isomorphisms of the Groups of Motions (228).
§4.3. Lines, m Planes and Hyperplanes 229
4.3.1. Matrix Coordinates of m Planes (229). 4.3.2. Stationary
Distances Between Two m Planes (229). 4.3.3. Paratactic Con¬
gruences (230).
§4.4. Interpretations of Quadratic and Hermitian Spaces 231
4.4.1. The Conformai Interpretations (231). 4.4.2. Interpreta¬
tions of Quadratic and Hermitian Hyperbolic Spaces in Euclidean
Spaces (232). 4.4.3. The Interpretations of Kotelnikov Study
Fubini (235). 4.4.4. Interpretations of Manifolds of Lines and
3 Planes (236). 4.4.5. Real and Quaternionic Projective Geome¬
tries as Hermitian Elliptic Geometries (237). 4.4.6. Interpreta¬
tion of Pseudoelliptic, Hyperbolic, and Pseudohyperbolic spaces by
Means of Hyperspheres (238). 4.4.7. The Interpretations of Hesse
and Pliicker (238). 4.4.8. Interpretations of Hermitian 3 Spaces in
Real 5 Spaces (239). 4.4.9. Connections Between Isomorphisms of
Simple Lie Groups and Spinor Representations (240).
§4.5. Trigonometry 241
4.5.1. Trigonometry in the Spaces Sn and Hn (241). 4.5.2. Holo
morphy Angles and Holomorphic and Antiholomorphic Real 2
Directions in the Spaces CS , Cff , ES , and Hff (241). 4.5.3.
Trigonometry in the Spaces CS and ES (242). 4.5.4. Trigonom¬
etry in the Spaces Cff and Hff (244). 4.5.5. Trigonometry in
the Grassmann Manifolds (245).
§4.6. Sectional Curvature in Hermitian Spaces 245
4.6.1. Sectional Curvature in Symmetric Spaces (245). 4.6.2. Sec¬
tional Curvature in Complex and Split Complex Spaces (246).
4.6.3. Sectional Curvature in Quaternionic and Split Quaternionic
Spaces (248).
§4.7. Polyhedra, Hyperquadrics, and Hyperspheres 251
4.7.1. Polyhedra (251). 4.7.2. Centers of Hyperquadrics and Her¬
mitian Hyperquadrics (251). 4.7.3. Hyperspheres and m Equidis
tant Hyperquadrics (252). 4.7.4. Horohyperspheres (253). 4.7.5.
Hermitian Hyperspheres, Equidistant Hyperquadrics, and Horohy¬
perspheres (254). 4.7.6. Heisenberg Group (255). 4.7.7. Classi¬
fication of Conies in the Hyperbolic Plane (257). 4.7.8. Topol¬
ogy of Manifolds of Generators of Maximal Dimension of Real Hy
CONTENTS xiii
perquadrics (259). 4.7.9. Triality Principle in Real Quadratic 7
Spaces (260). 4.7.10. Circles, Equidistant Conies, and Horocycles
(261). 4.7.11. The Quadrangles of Khayyam Saccheri and Ibn al
Haytham Lambert (262).
§4.8. Interpretations of Skopets and Popovic 264
4.8.1. Skopets First Interpretation (264). 4.8.2. Skopets Second
Interpretation (264). 4.8.3. Interpretation of Popovic (264).
§4.9. Regular Polyhedra and Honeycombs 265
4.9.1. Regular Polyhedra and Honeycombs in Elliptic Space (265).
4.9.2. Regular Hoheycombs in the Hyperbolic Plane (265). 4.9.3.
Regular Polyhedra and Honeycombs in Hyperbolic Space (266).
4.9.4. Fuchsian Groups (267). 4.9.5. Kleinian Groups (268). 4.9.6.
Groups of Motions of Hyperbolic Spaces Generated by Reflections
(268).
§4.10. Symmetry and Parabolic Figures 269
4.10.1. Symmetry Figures (269). 4.10.2. Absolutes and Superab
solutes of Symmetric Spaces (271). 4.10.3. Fundamental and Par¬
abolic Figures (276). 4.10.4. Local Absolutes and Superabsolutes
in Parabolic Spaces (277). 4.10.5. Representations of a Simple Lie
Group by Fractional Linear Transformations in a Jordan Algebra
(278).
§4.11. Space Forms 278
4.11.1. Space Forms of Clifford Klein (278). 4.11.2. 2 Dimensional
Space Forms (279). 4.11.3. 3 Dimensional and Multidimensional
Space Forms (280).
§4.12. Sliding Vectors 281
4.12.1. Sliding Vectors in Elliptic Space (281). 4.12.2. Sliding
Vectors in Hyperbolic Space (281).
§4.13. Finite Geometries 282
4.13.1. Finite Quadratic Spaces (282). 4.13.2. Finite Hermitian
Space (282).
§4.14. Applications to Physics 282
4.14.1. Application of Hyperbolic Geometry (282). 4.14.2. Twistor
Program of R. Penrose (283).
Chapter V. Quasielliptic, Quasihyperbolic, and Quasi Eucli¬
dean Geometries 284
§5.1. Quasielliptic, Quasihyperbolic, and Quasi Euclidean Spaces 284
5.1.1. Co Euclidean and Copseudo Euclidean Spaces (284). 5.1.2.
Quasielliptic, Quasihyperbolic, Quasipseudoelliptic, and Quasi
pseudohyperbolic Spaces (285). 5.1.3. Quasi Euclidean and Quasi
pseudo Euclidean Spaces (288). 5.1.4. Quasisimple and r Quasi
simple Lie Groups (289). 5.1.5. Quasimetrics of Order One (290).
5.1.6. Invariant Metrics, Pseudometrics, and Quasimetrics in the
Groups of Motions in Planes (291).
xiv CONTENTS
§5.2. r Quasielliptic, r Quasihyperbolic, and r Quasi Euclidean Spaces 291
5.2.1. r Quasielliptic, r Quasihyperbolic, r Quasipseudoelliptic,
and r Quasipseudohyperbolic Spaces (291). 5.2.2. r Quasi Eucli
dean and r Quasipseudo Euclidean Spaces (295). 5.2.3. Quasimet
rics of Higher Orders (296).
§5.3. Hyperquadrics, Hyperspheres, and Hypercycles 296
5.3.1. Centers of Hyperquadrics and Hermitian Hyperquadrics
(296). 5.3.2. Hyperspheres (297). 5.3.3. Hypercycles (297). 5.3.4.
Cycles in the Flag Plane (298). 5.3.5. Angles Between Hypercy¬
cles (298). 5.3.6. Conformal Transformations and Quasiconformal,
Quasipseudoconformal, r Quasiconformal, and r Quasipseudocon
formal Spaces (299).
§5.4. Lines, m Planes and Symmetry Figures 299
5.4.1. Elliptic, Hyperbolic, and Parabolic Lines (299). 5.4.2. El¬
liptic, Pseudoelliptic, and Parabolic m Planes (300). 5.4.3. Matrix
Coordinates of m Planes (300). 5.4.4. Stationary Distances of
Two Elliptic or Pseudoelliptic m Planes (300). 5.4.5. Application
of Affine Matrix Coordinates for Quasielliptic or Quasipseudoel
liptic Space (301). 5.4.6. Lines and m Planes in Hermitian r
Quasielliptic and r Quasipseudoelliptic Spaces (302). 5.4.7. The
Interpretations of Manifolds of Lines in 3 Spaces (302). 5.4.8.
Symmetry Figures in Quasielliptic and Quasipseudoelliptic Spaces
(303). 5.4.9. Cosymmetry Figures in Quasielliptic and Quasipseu¬
doelliptic Spaces (304). 5.4.10. Symmetry and Cosymmetry Fig¬
ures in Quasielliptic 3 Space in the Interpretation on Two Eu¬
clidean Planes (304).
§5.5 m Horospheres in Pseudoelliptic and Pseudohyperbolic Spaces 305
5.5.1. m Horospheres in Quadratic Spaces (305). 5.5.2. m Horo
spheres in Hermitian Spaces (305).
§5.6. Sliding Vectors 305
5.6.1. Sliding Vectors in Quasielliptic Spaces (305). 5.6.2. Sliding
Vectors in Quasihyperbolic, Quasipseudoelliptic, and Quasipseudo
hyperbolic Spaces (306). 5.6.3. Sliding Vectors in r Quasielliptic, r
Quasihyperbolic, r Quasipseudoelliptic, and r Quasipseudohyper
bolic Spaces (307).
§5.7. Quasi Riemannian, Quasipseudo Riemannian, r Quasi Riemanni
an, and r Quasipseudo Riemannian Manifolds and Symmetric Spaces 307
5.7.1. Quasi Riemannian, Quasipseudo Riemannian, r Quasi Rie
mannian, and r Quasipseudo Riemannian Manifolds (307). 5.7.2.
Quasi Riemannian, Quasipseudo Riemannian, r Quasi Riemanni
an, and r Quasipseudo Riemannian Symmetric Spaces (308).
§5.8. Applications to Physics 309
5.8.1. Space time of Classical Mechanics (309). 5.8.2. Appli¬
cations of Pseudo Riemannian, Quasipseudo Riemannian and r
Quasipseudo Riemannian Geometries (309).
CONTENTS xv
Chapter VI. Symplectic and Quasisymplectic Geometries 311
§6.1. Symplectic Spaces 311
6.1.1. Quadratic Symplectic Spaces (311). 6.1.2. Another Qua¬
dratic Symplectic Space (313). 6.1.3. Hermitian Symplectic Spaces
(313).
§6.2. Interpretations of Symplectic Spaces 315
6.2.1. Interpretations of Quadratic Line and 3 Space (315). 6.2.2.
Real Symplectic Geometry as Hermitian Elliptic Geometry (315).
6.2.3. Real Interpretation of the Split Quaternionic Hermitian
Symplectic Space (317). 6.2.4. Interpretations of Hermitian Line,
Plane, and 3 Space (318). 6.2.5. Connections between the Inter¬
pretations of Symplectic Spaces and Spinor Representations (320).
§6.3. Quasisymplectic and r Quasisymplectic Spaces 321
6.3.1. Quasisymplectic Spaces (321). 6.3.2. Interpretations of
Quasisymplectic Spaces (321). 6.3.3. Semiquaternionic Hermitian
Symplectic Spaces (322). 6.3.4. r Quasisymplectic Spaces (322).
§6.4. Symmetry and Parabolic Figures 323
6.4.1. Symmetry Figures (323). 6.4.2. Fundamental and Parabolic
Figures (324).
§6.5. Symplectic and Quasisymplectic Connections 326
6.5.1. Symplectic Connections (326). 6.5.2. Hermitian Riemann
ian and Kahlerian Manifolds (326). 6.5.3. Quasisymplectic Con¬
nections (326).
§6.6. Finite Geometry 326
6.6.1. Symplectic Space over a Galois Field (326).
§6.7. Applications to Physics 327
6.7.1. Hamiltonian Equations (327). 6.7.2. Duality and Stability
(328). 6.7.3. Dualities in Philosophy and Mathematics (329).
Chapter VII. Geometries of Exceptional Lie Groups. Meta
symplectic Geometries 331
§7.1. Geometry of the Groups G2 331
7.1.1. The Simple Lie Groups G2 (331). 7.1.2. G Elliptic, G
Pseudoelliptic, and G Pseudohyperbolic 6 Spaces (331). 7.1.3. Qua
sisimple Lie Groups G2. G quasielliptic, G quasipseudoelliptic, and
G quasipseudohyperbolic 6 Spaces (332).
§7.2. Geometry of the Groups F4 and E6 333
7.2.1. The Simple Lie Groups F4 and E6 (333). 7.2.2. Octonionic
and Split Octonionic Projective Planes (333). 7.2.3. Octonionic
and Split Octonionic Hermitian Planes (336). 7.2.4. The Quasisim
ple Lie Groups F4 (338). 7.2.5. Holomorphy Angles and Holomor
phic and Antiholomorphic Real 2 Directions in the Planes OS ,
OH2, and 052 and Trigonometry in These Planes (339). 7.2.6.
xvi CONTENTS
Sectional Curvature of Octonionic and Split Octonionic Hermitian
Planes (339).
§7.3. Geometry of the Groups E6, E7, and E8 340
7.3.1. The Simple Lie Groups E6,E7, and E8 (340). 7.3.2. Ge¬
ometry of all Simple Groups E6 (340). 7.3.3. Geometry of Simple
Groups E7 and E8 (344). 7.3.4. The Quasisimple Lie Groups
E6,E7, and Es (349).
§7.4. Symplectic and Metasymplectic Geometries 350
7.4.1. Symplectic 5 Spaces (350). 7.4.2. Metasymplectic Geome¬
tries and the Preudenthal Magic Square (350).
§7.5. Symmetry Figures and Symmetric Spaces 352
7.5.1. Symmetry Figures (352). 7.5.2. Local Absolutes in Sym¬
metric Spaces (354).
§7.6. Parabolic Figures and Fundamental Representations 356
7.6.1. Fundamental Figures (356). 7.6.2. Parabolic Figures of the
Groups G2 (356). 7.6.3. Parabolic Figures of the Group F4 (357).
7.6.4. Parabolic Figures of the Groups E6 (358). 7.6.5. Parabolic
Figures of the Groups E7 (360). 7.6.6. Parabolic Figures of the
Groups Eg (362). 7.6.7. Construction of Manifolds of Fundamen¬
tal Figures (365). 7.6.8. Fundamental Linear Representations of
Exceptional Lie Groups (367).
§7.7. Finite Geometries 368
7.7.1. Exceptional Simple Finite Groups of Lie Type (368) 7.7.2.
Finite Algebras over Galois Fields (368). 7.7.3. Finite Geometries
with Fundamental Exceptional Simple Finite Groups of Lie Type
(368).
§7.8. Applications to Physics 368
7.8.1. Geometry of Exceptional Lie Group and Supergravity The¬
ory (368).
References 370
Index of Persons 381
Index of Subjects 384
|
| any_adam_object | 1 |
| author | Rozenfel'd, Boris A. 1917- |
| author_GND | (DE-588)124756638 |
| author_facet | Rozenfel'd, Boris A. 1917- |
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| bvnumber | BV011572900 |
| callnumber-first | Q - Science |
| callnumber-label | QA387 |
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| classification_rvk | SK 340 |
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| dewey-full | 512/.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512/.55 |
| dewey-search | 512/.55 |
| dewey-sort | 3512 255 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011572900 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:12:03Z |
| institution | BVB |
| isbn | 0792343905 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-007792492 |
| oclc_num | 36083558 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-384 DE-824 DE-11 |
| owner_facet | DE-703 DE-20 DE-384 DE-824 DE-11 |
| physical | XVIII, 393 S. graph. Darst. |
| publishDate | 1997 |
| publishDateSearch | 1997 |
| publishDateSort | 1997 |
| publisher | Kluwer |
| record_format | marc |
| series | Mathematics and its applications |
| series2 | Mathematics and its applications |
| spelling | Rozenfel'd, Boris A. 1917- Verfasser (DE-588)124756638 aut Geometry of Lie groups by Boris Rosenfeld Dordrecht [u.a.] Kluwer 1997 XVIII, 393 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematics and its applications 393 Grupos de lie larpcal Géométrie ram Lie, Groupes de ram Geometry Lie groups Lie-Gruppe (DE-588)4035695-4 gnd rswk-swf Lie-Gruppe (DE-588)4035695-4 s DE-604 Mathematics and its applications 393 (DE-604)BV008163334 393 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007792492&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rozenfel'd, Boris A. 1917- Geometry of Lie groups Mathematics and its applications Grupos de lie larpcal Géométrie ram Lie, Groupes de ram Geometry Lie groups Lie-Gruppe (DE-588)4035695-4 gnd |
| subject_GND | (DE-588)4035695-4 |
| title | Geometry of Lie groups |
| title_auth | Geometry of Lie groups |
| title_exact_search | Geometry of Lie groups |
| title_full | Geometry of Lie groups by Boris Rosenfeld |
| title_fullStr | Geometry of Lie groups by Boris Rosenfeld |
| title_full_unstemmed | Geometry of Lie groups by Boris Rosenfeld |
| title_short | Geometry of Lie groups |
| title_sort | geometry of lie groups |
| topic | Grupos de lie larpcal Géométrie ram Lie, Groupes de ram Geometry Lie groups Lie-Gruppe (DE-588)4035695-4 gnd |
| topic_facet | Grupos de lie Géométrie Lie, Groupes de Geometry Lie groups Lie-Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007792492&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV008163334 |
| work_keys_str_mv | AT rozenfeldborisa geometryofliegroups |