Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Lev V. Sabinin (auth.)1402025440, 9781402025457, 9781402025440, 1402025459

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.

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Table of contents :
Preliminaries….Pages 3-8
Curvature Tensor of an Involutive Pair. Classical Inovolutive Pairs of Index 1….Pages 9-11
Iso-Involutive Sums of Lie Algebras….Pages 12-15
Iso-Involutive Base and Structure Equations….Pages 16-27
Iso-Involutive Sums of Types 1 and 2….Pages 28-33
Iso-Involutive Sums of Lower Index 1….Pages 34-44
Principal Central Involutive Automorphism of Type U….Pages 45-45
Principal Unitary Involutive Automorphism of Index 1….Pages 46-48
Hyper-Involutive Decomposition of a Simple Compact Lie Algebra….Pages 51-56
Some Auxiliary Results….Pages 57-59
Principal Involutive Automorphisms of Type O….Pages 60-68
Fundamental Theorem….Pages 69-76
Principal Di-Unitary Involutive Automorphism….Pages 77-86
Singular Principal Di-Unitary Involutive Automorphism….Pages 87-94
Mono-Unitary Non-Central Principal Involutive Automorphism….Pages 95-102
Exceptional Principal Involutive Automorphism of Types f and e….Pages 103-110
Classification of Simple Special Unitary Subalgebras….Pages 111-116
Hyper-Involutive Reconstructions of Basis Involutive Decompositions….Pages 117-124
Special Hyper-Involutive Sums….Pages 125-140
Notations, Definitions and Some Preliminaries….Pages 143-147
Symmetric Spaces of Rank 1….Pages 148-149
Principal Symmetric Spaces….Pages 150-154
Essentially Special Symmetric Spaces….Pages 155-157
Some Theorems on Simple Compact Lie Groups….Pages 158-162
Tri-Symmetric and Hyper-Tri-Symmetric Spaces….Pages 163-165
Tri-Symmetric Spaces with Exceptional Compact Groups….Pages 166-173
Tri-Symmetric Spaces with Groups of Motions SO(n), Sp(n), SU(n)….Pages 174-184
Subsymmetric Riemannian Homogeneous Spaces….Pages 187-191
Subsymmetric Homogeneous Spaces and Lie Algebras….Pages 192-197
Mirror Subsymmetric Lie Triplets of Riemannian Type….Pages 198-210
Mobile Mirrors. Iso-Involutive Decompositions….Pages 211-214
Homogeneous Riemannian Spaces with Two-Dimensional Mirrors….Pages 215-219
Homogeneous Riemannian Spaces with Groups SO(n), SU(3) and Two-Dimensional Mirrors….Pages 220-232
Homogeneous Riemannian Spaces with Simple Compact Lie Groups of Motions G≂SO(n), SU(3) and Two-dimensional Mirrors….Pages 233-235
Homogeneous Riemannian Spaces with Simple Compact Lie Groups of Motions and Two-Dimensional Immobile Mirrors….Pages 236-236