Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Abstract

Table of contents. On the artistic and poetic fragments of the book. Introduction. PART ONE. 1.1. Preliminaries, 1.2. Curvature tensor of involutive pair. Classical involutive pairs of index, 1.3. Iso-involutive sums of Lie algebras. 1.4. Iso-involutive base and structure equations, 1.5. Iso-involutive sums of types 1 and 2, 1.6. Iso-inolutive sums of lower index 1, 1.7. Principal central involutive automorphism of type U, 1.8. Principal unitary involutive automorphism of index I. PART TWO. 11.1. Hyper-involutive decomposition of a simple compact Lie algebra, 11.2. Some auxiliary results, 11.3. Principal involutive automorphisms of type 0, 11.4. Fundamental theorem, 11.5. Principal di-unitary involutive automorphism, 11.6. Singular principal di-unitary involutive automorphism, 11.7. Mono-unitary non-central principal involutive automorphism, 11.8. Principal involutive automorphism of types f and e, 11.9. Classification of simple special unitary subalgebras, 11.10. Hyper-involutive reconstruction of basic decompositions 11.11. Special hyper-involutive sums. PART THREE, 111.1. Notations, definitions and some preliminaries, 111.2. Symmetric spaces of rank 1, 111.3. Principal symmetric spaces, 111.4. Essentially special symmetric spaces, 111.5. Some theorems on simple compact Lie groups, 111.6. Tn-symmetric and hyper-tri-symmetric spaces, 111. 7. Tn-symmetric spaces with exceptional compact groups, 111.8. Tn-symmetric spaces with groups of motions SO(n), Sp(n), SU(n). PART FOUR, IV.1. Subsymmetric Riemannian homogeneous spaces, IV.2. Subsymmetric homogeneous spaces and Lie algebras, IV.3. Mirror subsymmetric Lie triplets of Riemannian type , IV.4. Mobile mirrors. Iso-involutive decompositions, IV.5. Homogeneous Riemannian spaces with two-dimensional mirrors, IV.6. Homogeneous Riemannian space with groups SO(n), SU(3) and two-dimensional mirrors, IV.7. Homogeneous Riemannian spaces with simple compact Lie groups G SO(n), SU(3) and two-dimensional mirrors, IV.8. Homogeneous Riemannian spaces with simple compact Lie group of motions and two-dimensional immobile mirrors . Appendix 1. On the structure of T, U, V-isospins in the theory of higher symmetry, Appendix 2. Description of contents, Appendix 3. Definitions, Appendix 4. Theorems Bibliography, Index.