This book describes a new and original formalism based on mirror symmetries of Lie groups, Lie algebras and Homogeneous spaces. Special systems of mirrors are used for classification purposes and as an instrument for studies of Homogeneous spaces. Tri-symmetric and arbitrary Riemannian Homogeneous spaces can also be researched in this way. The book should be of particular interest to researchers in Lie Groups, Lie Algebras, Differential Geometry and their applications but it should also prove useful for other postgraduate and advanced graduate students in mathematics.
Author(s): Lev V. Sabinin
Series: Mathematics and Its Applications 573
Edition: 1
Publisher: Springer
Year: 2004
Language: English
Pages: 335
Table of contents......Page 10
On the artistic and poetic fragments of the book......Page 12
Introduction......Page 14
PART ONE......Page 24
I.1. Preliminaries......Page 26
I.2. Curvature tensor of involutive pair. Classical involutive pairs of index 1......Page 32
I.3. Iso-involutive sums of Lie algebras......Page 35
I.4. Iso-involutive base and structure equations......Page 39
I.5. Iso-involutive sums of types 1 and 2......Page 51
I.6. Iso-involutive sums of lower index 1......Page 57
I.7. Principal central involutive automorphism of type U......Page 68
I.8. Principal unitary involutive automorphism of index 1......Page 69
PART TWO......Page 72
II.1. Hyper-involutive decomposition of a simple compact Lie algebra......Page 74
II.2. Some auxiliary results......Page 80
II.3. Principal involutive automorphisms of type O......Page 83
II.4. Fundamental theorem......Page 92
II.5. Principal di-unitary involutive automorphism......Page 100
II.6. Singular principal di-unitary involutive automorphism......Page 110
II.7. Mono-unitary non-central principal involutive automorphism......Page 118
II.8. Principal involutive automorphism of types f and e......Page 126
II.9. Classification of simple special unitary subalgebras......Page 134
II.10. Hyper-involutive reconstruction of basic decompositions......Page 140
II.11. Special hyper-involutive sums......Page 148
PART THREE......Page 164
III.1. Notations, definitions and some preliminaries......Page 166
III.2. Symmetric spaces of rank 1......Page 171
III.3. Principal symmetric spaces......Page 173
III.4. Essentially special symmetric spaces......Page 178
III.5. Some theorems on simple compact Lie groups......Page 181
III.6. Tri-symmetric and hyper-tri-symmetric spaces......Page 186
III.7. Tri-symmetric spaces with exceptional compact groups......Page 189
III.8. Tri-symmetric spaces with groups of motions SO(n), Sp(n), SU(n)......Page 197
PART FOUR......Page 208
IV.1. Subsymmetric Riemannian homogeneous spaces......Page 210
IV.2. Subsymmetric homogeneous spaces and Lie algebras......Page 215
IV.3. Mirror Subsymmetric Lie triplets of Riemannian type......Page 221
IV.4. Mobile mirrors. Iso-involutive decompositions......Page 234
IV.5. Homogeneous Riemannian spaces with two-dimensional mirrors......Page 238
IV.6. Homogeneous Riemannian space with groups SO(n), SU(3) and two-dimensional mirrors......Page 243
IV.7. Homogeneous Riemannian spaces with simple compact Lie groups G [omitted] SO(n), SU(3) and two-dimensional mirrors......Page 256
IV.8. Homogeneous Riemannian spaces with simple compact Lie group of motions and two-dimensional immobile mirrors......Page 259
Appendix 1. On the structure of T, U, V-isospins in the theory of higher symmetry......Page 260
Appendix 2. Description of contents......Page 268
Appendix 3. Definitions......Page 278
Appendix 4. Theorems......Page 292
Bibliography......Page 328
Index......Page 332
