Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

322 Pages·2005·11.31 MB·English

by  Lev V. Sabinin (auth.)

Mirror Geometry of Lie Algebras, Lie Groups and
Homogeneous Spaces
Mathematics and Its Applications
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Volume 573
Mirror Geometry of Lie
Algebras, Lie Groups and
Homogeneous Spaces
by
Lev V. Sabinin
Faculty of Science,
Morelos State University, Morelos, Cuernavaca, Mexico
and Friendship University,
Moscow, Russia
KLUWER ACADEMIC PUBLISHERS
NEW YORK,BOSTON, DORDRECHT, LONDON, MOSCOW
eBookISBN: 1-4020-2545-9
Print ISBN: 1-4020-2544-0
©2005 Springer Science + Business Media, Inc.
Print ©2004 Kluwer Academic Publishers
Dordrecht
All rights reserved
No part of this eBook maybe reproducedor transmitted inanyform or byanymeans,electronic,
mechanical, recording, or otherwise, without written consent from the Publisher
Created in the United States of America
Visit Springer's eBookstore at: http://ebooks.springerlink.com
and the Springer Global Website Online at: http://www.springeronline.com

TABLE OF CONTENTS
Table of contents v
On the artistic and poetic fragments of the book vii
Introduction xi
PART ONE 1
I.1. Preliminaries 3
I.2. Curvature tensor of involutive pair. Classical involutive
pairs of index 1 9
I.3. Iso-involutive sums of Lie algebras 12
I.4. Iso-involutive base and structure equations 16
I.5. Iso-involutive sums of types 1 and 2 28
I.6. Iso-inolutive sums of lower index 1 34
I.7. Principal central involutive automorphism of type U 45
I.8. Principal unitary involutive automorphism of index 1 46
PART TWO 49
II.1. Hyper-involutive decomposition of a simple compact
Lie algebra 51
II.2. Some auxiliary results 57
II.3. Principal involutive automorphisms of type O 60
II.4. Fundamental theorem 69
II.5. Principal di-unitary involutive automorphism 77
II.6. Singular principal di-unitary involutive automorphism 87
II.7. Mono-unitary non-central principal involutive
automorphism 95
II.8. Principal involutive automorphism of types f and e 103
II.9. Classification of simple special unitary subalgebras 111
II.10. Hyper-involutive reconstruction of basic
decompositions 117
v
vi LEV SABININ
II.11. Special hyper-involutive sums 125
PART THREE 141
III.1. Notations, definitions and some preliminaries 143
III.2. Symmetric spaces of rank 1 148
III.3. Principal symmetric spaces 150
III.4. Essentially special symmetric spaces 155
III.5. Some theorems on simple compact Lie groups 158
III.6. Tri-symmetric and hyper-tri-symmetric spaces 163
III. 7. Tri-symmetric spaces with exceptional compact groups 166
III.8. Tri-symmetric spaces with groups of motions
SO(n), Sp(n), SU(n) 174
PART FOUR 185
IV.1. Subsymmetric Riemannian homogeneous spaces 187
IV.2. Subsymmetric homogeneous spaces and Lie algebras 192
IV.3. Mirror Subsymmetric Lie triplets of Riemannian type 198
IV.4. Mobile mirrors. Iso-involutive decompositions 211
IV.5. Homogeneous Riemannian spaces with two-dimensional
mirrors 215
IV.6. Homogeneous Riemannian space with groups SO(n), SU(3)
and two-dimensional mirrors 220
IV.7. Homogeneous Riemannian spaces with simple compact Lie
groups SU(3) and two-dimensional mirrors 233
IV.8. Homogeneous Riemannian spaces with simple compact Lie
group of motions and two-dimensional immobile mirrors 236
Appendix 1. On the structure of T, U, V-isospins in the theory of
higher symmetry 237
Appendix 2. Description of contents 245
Appendix 3. Definitions 255
Appendix 4. Theorems 269
Bibliography 305
Index 309
On the artistic and poetic fragments of the book
It is evident to us that the way of writing a mathematical treatise in a mo-
notonous logically-didactic manner subsumed in the contemporary world is very
harmful and belittles the greatness of Mathematics, which is authentic basis of the
Transcendental Being of Universe.
Everyone touched by Mathematical Creativity knows that Images, Words,
Sounds, and Colours fly above the ocean of logic in the process of exploration, con-
stituting the real body of concepts, theorems, and proofs. But all this abundance
disappears and, alas, does so tracelessly for the reader of a modern mathematical
treatise.
Therefore the attempts at attracting a reader to this majestic Irrationality
appear to be natural and justified.
Thus the inclusion of poetic inscriptions into mathematical works has already
(and long ago) been used by different authors. Attempts to draw (not to illustrate
only) Mathematics is already habitual amongst intellectuals. In this connection let
us note the remarkable pathological-topological-anatomical graphics of the Moscow
topologist A.T. Fomenko.
In our treatise we also make use of graphics and drawings (mental images-faces
of super-mathematical reality, created by sweet dreams of mirages of pure logic )
and words (poetic inscriptions) which Eternity whispered during our aspirations
to learn the beauty of Irrationality.
The reader should not search for any direct relations between our artistic poetic
substance and certain parts and sections of the treatise. It is to be considered as
some general super-mathematical-philosophical body of the treatise as a whole.
Lev Sabinin
vii