Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

1. Front Matter

   Pages i-xvii

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2. Part One

   1. Preliminaries

      Pages 3-8

   2. Curvature Tensor of an Involutive Pair. Classical Inovolutive Pairs of Index 1

      Pages 9-11

   3. Iso-Involutive Sums of Lie Algebras

      Pages 12-15

   4. Iso-Involutive Base and Structure Equations

      Pages 16-27

   5. Iso-Involutive Sums of Types 1 and 2

      Pages 28-33

   6. Iso-Involutive Sums of Lower Index 1

      Pages 34-44

   7. Principal Central Involutive Automorphism of Type U

      Pages 45-45

   8. Principal Unitary Involutive Automorphism of Index 1

      Pages 46-48

3. Part Two

   1. Hyper-Involutive Decomposition of a Simple Compact Lie Algebra

      Pages 51-56

   2. Some Auxiliary Results

      Pages 57-59

   3. Principal Involutive Automorphisms of Type O

      Pages 60-68

   4. Fundamental Theorem

      Pages 69-76

   5. Principal Di-Unitary Involutive Automorphism

      Pages 77-86

   6. Singular Principal Di-Unitary Involutive Automorphism

      Pages 87-94

   7. Mono-Unitary Non-Central Principal Involutive Automorphism

      Pages 95-102

   8. Exceptional Principal Involutive Automorphism of Types f and e

      Pages 103-110

   9. Classification of Simple Special Unitary Subalgebras

      Pages 111-116

   10. Hyper-Involutive Reconstructions of Basis Involutive Decompositions

       Pages 117-124

   11. Special Hyper-Involutive Sums

       Pages 125-140

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.

• Lie algebra
• Lie group
• Riemannian manifold
• brandonwiskunde
• curvature
• differential geometry

• Faculty of Science, Morelos State University, Morelos, Mexico

  Lev V. Sabinin

• Friendship University, Moscow, Russia

  Lev V. Sabinin

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