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Cover image for Mirror geometry of lie algebras, lie groups and homogeneous spaces
Title:
Mirror geometry of lie algebras, lie groups and homogeneous spaces
Series:
Mathematics and its applications
Publication Information:
Dordrecht : Kluwer Academic Pubs, 2004
ISBN:
9781402025440

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30000004994533 QA252.3 S22 2004 Open Access Book Book
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Summary

Summary

As K. Nomizu has justly noted [K. Nomizu, 56], Differential Geometry ever will be initiating newer and newer aspects of the theory of Lie groups. This monograph is devoted to just some such aspects of Lie groups and Lie algebras. New differential geometric problems came into being in connection with so called subsymmetric spaces, subsymmetries, and mirrors introduced in our works dating back to 1957 [L.V. Sabinin, 58a,59a,59b]. In addition, the exploration of mirrors and systems of mirrors is of interest in the case of symmetric spaces. Geometrically, the most rich in content there appeared to be the homogeneous Riemannian spaces with systems of mirrors generated by commuting subsymmetries, in particular, so called tri-symmetric spaces introduced in [L.V. Sabinin, 61b]. As to the concrete geometric problem which needs be solved and which is solved in this monograph, we indicate, for example, the problem of the classification of all tri-symmetric spaces with simple compact groups of motions. Passing from groups and subgroups connected with mirrors and subsymmetries to the corresponding Lie algebras and subalgebras leads to an important new concept of the involutive sum of Lie algebras [L.V. Sabinin, 65]. This concept is directly concerned with unitary symmetry of elementary par- cles (see [L.V. Sabinin, 95,85] and Appendix 1). The first examples of involutive (even iso-involutive) sums appeared in the - ploration of homogeneous Riemannian spaces with and axial symmetry. The consideration of spaces with mirrors [L.V. Sabinin, 59b] again led to iso-involutive sums.


Table of Contents

On the artistic and poetic fragments of the book
Introduction
Part1
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 2
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 3
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 4
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
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