Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

ISBN-13: 9789048166763 / Angielski / Miękka / 2010 / 312 str.

Lev V. Sabinin

Opis

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.

Spis treści

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.