Smooth Quasigroups and Loops

262 Pages·1999·21.67 MB·English

by  Lev V. Sabinin (auth.)

Smooth Quasigroups and Loops
Mathematics and Its Applications
ManagingEditor:
M. HAZEWINKEL
CentreforMathematicsandComputerScience, Amsterdam, TheNetherlands
Volume 492
Smooth Quasigroups and
Loops
by
Lev V. Sabinin
Russian Frientiship of Nations University.
Mo.rcow, Russia
alld
Michoacon University,
Mexico
SPRINGER-SCIENCE+BUSINESS MEDIA, B.Y.
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN 978-94-010-5921-3 ISBN 978-94-011-4491-9 (eBook)
DOI 10.1007/978-94-011-4491-9
Printed (}n addlree paper
AII Rights Reserved
© 1999 Springer Science+Business Media Dordrecht
Originally published by Kluwer Academic Publishers in 1999
Softcover reprint ofthe hardcover Ist edition 1999
No part of the material protected by this copyright notice may be reproduced or
utilized in any form or by any means, electronic or mechanical,
inc\uding photocopying, recording or by any information storage and
retrieval system. without written permis sion from the copyright owner.
TABLE OF CONTENTS
PREFACE ix
INTRODUCTION xiii
CHAPTER O. INTRODUCTORY SURVEY:
QUASIGROUPS, LOOPUSCULAR GEOMETRY
AND NONLINEAR GEOMETRIC ALGEBRA 1
PART ONE. FUNDAMENTAL STRUCTURES
OF NONLINEAR GEOMETRIC ALGEBRA 21
CHAPTER 1. BASIC ALGEBRAIC STRUCTURES 23
A. Quasigroups, loops, odules, diodules 23
B. Loopuscular, odular, diodular algebras 25
C. Holonomial and geometric odules 29
D. Antiproducts, Chern algebras, geodetic spaces 30
E. Geoodular axiomatics ofaffine spaces 31
CHAPTER 2. SEMIDIRECT PRODUCTS OF
A QUASIGROUP BY ITS TRANSASSOCIANTS 36
CHAPTER 3. BASIC SMOOTH STRUCTURES 47
A. Smooth universal algebras 47
B. Maximal partial algebras 50
C. Smooth odules and odular structures 51
D. Canonical odules and odular structures 55
v
PART TWO. SMOOTH LOOPS AND
HYPERALGEBRAS 57
CHAPTER 4. INFINITESIMAL THEORY
OF SMOOTH LOOPS 59
A. General theory 59
B. Smooth local geometric odules 72
C. Smooth holonomial odules 77
D. Additional differential equations 84
CHAPTER 5. SMOOTH BOL LOOPS
AND BOL ALGEBRAS 87
A. Bol algebras 87
B. General theory 88
C. Smooth Bol loops and homogeneous spaces 97
D. Triple Lie systems of vector fields 99
E. Pseudoderivatives of Bol algebras 100
F. Enveloping Lie algebras of a Bol algebra 102
G. Infinitesimal theory 103
H. Final notices 104
CHAPTER 6. SMOOTH MOUFANG LOOPS
AND MAL'CEV ALGEBRAS 105
CHAPTER 7. SMOOTH HYPOREDUCTIVE
AND PSEUDOREDUCTIVE LOOPS 111
A. Smooth hyporeductive loops 111
B. Smooth pseudoreductive loops 123
PART THREE. LOOPUSCULAR GEOMETRY 129
CHAPTER 8. AFFINE CONNECTIONS
AND LOOPUSCULAR STRUCTURES 131
A. Tangent affine connections of loopuscular structures 131
B. Natural geoodular (linear geodiodular) structures of
an affinely connected manifold 135
C. Flat geoodular manifolds 140
D. Differential geometry ofright monoalternative loops 143
E. Final remarks 145
vi
CHAPTER 9. REDUCTIVE GEOODULAR SPACES 146
CHAPTER 10. SYMMETRIC GEOODULAR
SPACES 155
CHAPTER 11. s-SPACES 166
A. General theory 166
B. Perfect s-structures 172
CHAPTER 12. GEOMETRY OF SMOOTH BOL
AND MOUFANG LOOPS 175
A. Differential geometry of smooth Bol loops 175
B. Main structure theorem ofthe theory ofsmooth Bol
loops 177
C. Differential geometry ofsmooth Moufang loops 180
APPENDICES 183
APPENDIX 1. LIE TRIPLE ALGEBRAS AND
REDUCTIVE SPACES 185
APPENDIX 2. LEFT F-QUASIGROUPS.
LOOPUSCULAR APPROACH 189
APPENDIX 3. LEFT F-QUASIGROUPS AND
REDUCTIVE SPACES 199
APPENDIX 4. GEOMETRY OF
TRANSSYMMETRIC SPACES 205
APPENDIX 5. HALF BOL LOOPS 210
APPENDIX 6. ALMOST SYMMETRIC AND
ANTISYMMETRIC MANIFOLDS 218
APPENDIX 7. RIGHT ALTERNATIVE LOCAL
ANALYTIC LOOPS 226
BIBLIOGRAPHY 229
INDEX 245
vii
PREFACE
During the last twenty-five years quite remarkable relations between nonas
sociative algebra and differential geometry have been discovered in our work.
Such exotic structures of algebra as quasigroups and loops were obtained from
purelygeometric structuressuch as affinely connected spaces. The notionofodule
was introduced as a fundamental algebraic invariant ofdifferential geometry. For
any space with an affine connection loopuscular, odular and geoodular structures
(partial smooth algebras of a special kind) were introduced and studied. As
it happened, the natural geoodular structure of an affinely connected space al
lows us to reconstruct this space in a unique way. Moreover, any smooth ab
stractly given geoodular structure generates in a unique manner an affinely con
nected space with the natural geoodular structure isomorphic to the initial one.
The above said means that any affinely connected (in particular, Riemannian)
space can be treated as a purely algebraic structure equipped with smoothness.
Numerous habitual geometric properties may be expressed in the language of
geoodular structures by means ofalgebraic identities, etc..
Our treatment has led us to the purely algebraic concept of affinely connected
(in particular, Riemannian) spaces; for example, one can consider a discrete, or,
even, finite spacewith affineconnection (in theform ofgeoodular structure) which
can be used in the old problem of discrete space-time in relativity, essential for
the quantum space-time theory.
All the above has given a start to new branches ofmathematics-'loopuscular
and odular geometry' and 'nonlinear geometric algebra'. All this has required, in
particular, thedevelopmentofthe infinitesimaltheory ofsmooth loops, odulesand
quasigroups, analogous to Lie group theory. Such a theory has been developed in
our works in theframe ofour scientificschool. Adequate infinitesimalobjectshave
been discovered (so called v-hyperalgebras and F-hyperalgebras generalizing the
concept of Lie algebras). There have been constructed: smooth Bol lo')ps-Bol
algebras infinitesimal theory (generalizing Lie groups-Lie algebras and smooth
Moufang loops-Mal'cev algebras theories), smooth reductive loops-triple Lie
algebras theory, smooth hyporeductive loops-hyporeductive algebras theory and
so on.
One can be sure that these new theories have a good promise for application
to geometry, algebra, mathematical physics, classical and quantum mechanics,
dislocation theory, general and specialrelativity, etc.. For example, the relativistic
lawofaddition ofthree-dimensional velocities in specialrelativityequips thespace
of all three-dimensional velocities with the structure of a noncommutative loop
ix
x PREFACE
with the left Bol and the left Bruck identities,
x +(y+(x+z)) = (x+(y +x)) +z, x +(y +(y +x)) = (x+y) +(x +y).
Thereisalreadyanumberofpublicationsaboutapplicationsofour newtheories
to the natural sciences.
The first attempt at laying out the mentioned above theories in the form of
a monograph in English was presented in [L.V. Sabinin, P.O. Miheev 90]. That
presentation is no longer up to date, since §§7, 8written jointly with P.O. Miheev
need a new consideration owing to new advances and results in the field. For
example, the treatment ofthe infinitesimal theory ofsmooth loops given there for
theanalyticcaseonlyshould bereplaced byadifferent theory, sincethatdefinition
of tangent hyperalgebra does not work for the nonanalytic case. Moreover, we
need, in fact, a completelynewconstruction and theory asa whole. Thetreatment
ofsmooth Bol loops given there can be modernized, as well. As to §§1--6 written
by myself, they need only small improvements and corrections. Additionally, one
should take into account many new results in the field obtained during the last
ten years, for example, the theory of hyporeductive and pseudoreductive smooth
loops.
Thus it is quite a substantial enterprise to write a treatise on the subject. But
it would take some time. Therefore, in order to inform the world mathematical
community about new results in the field promptly, we have decided to publish
the book 'Smooth quasigroups and loops'which gives an up to date self-contained
presentation of the subject. In fact, this book can be considered a preliminary
version ofpart ofa treatise to be written in the near future.
As to the content, we have included in this book:
Enlargement ofthe Introduction from 'Quasigroups and differential geometry'
[L.V. Sabinin, P.O. Miheev 90] in order to clarify the historicaldevelopmentofthe
area.
Chapter O. 'Introductory survey: Quasigroups, loopuscular geometry and
nonlinear geometric algebra', where the whole area is described in brief.
Chapter 1. 'Basic algebraic structures', where the main algebraic concepts of
nonlinear geometric algebra are considered.
Chapter 2. 'Semidirect productsofa quasigroup by itstransassociants', where
the remarkable construction ofinclusion ofa loop into a group is presented.
Chapter 3. 'Basic smooth structures', where the basic smooth concepts of
nonlinear geometric algebra are treated.
Chapter 4. 'Infinitesimal theory ofsmooth loops', where the generalization of
Lie groups-Lie algebras theory is given.
Chapter 5. 'Smooth Bolloops and Bol algebras', containing a new treatment
of the subject.
Chapter 6. 'Smooth Moufang loops and Mal'cev algebras', where the smooth
Moufangloopstheory iselaboratedasa particularsubcaseofthesmooth Bolloops
theory.
PREFACE xi
Chapter 7. 'Smooth hyporeductive and pseudoreductive loops', where the
remarkable generalization ofboth Bol loops and reductive loops is given, and the
proper infinitesimal theory is suggested.
Chapter 8. 'Affineconnections and loopuscular structures', where it is proved
that an affinely connected manifold and a smooth geoodular manifold are essen
tially the same objects.
Chapter 9. 'Reductive geoodular spaces', where the notion ofreductive space
is formulated and studied, mostly, in a purely algebraic way in the framework of
nonlinear geometric algebra.
Chapter 10. 'Symmetricgeoodularspaces',wheresymmetricspacesaretreated
as smooth universal algebras.
Chapter 11. "s-spaces', dealing with the algebraic theory ofs-spaces (general
ized symmetric spaces).
Chapter 12. 'Geometry of smooth Bol and Moufang loops', where Bol and
Moufang loops are treated as affinely connected manifolds ofzero curvature with
the torsion ofspecial kind.
Appendix 1. 'TripleLiealgebrasand reductivespaces', where the infinitesimal
theory ofreductivespacesand related reductiveloopsaredeveloped in ageometric
way.
Appendix2. 'Left F-quasigroups. Loopuscularapproach', wherethecanonical
loopuscular structure ofa left F-quasigroup is investigated.
Appendix3. 'LeftF-quasigroupsandreductivespaces', wherethepresentation
ofany left F-quasigroup as a reductive spaceofa special kind with multiplication
is given.
Appendix 4. 'Geometry of transsymmetric spaces', outlining the transsym
metric spaces theory (being the generalization of the well known symmetric
spaces theory), as well as the smooth left F-quasigroups theory involved in.
Appendix5. 'HalfBolloops', containingonegeneralizationofBolloopsneeded
for geometrical applications.
Appendix 6. 'Almost symmetric and antisymmetric manifolds', concerning
one more generalization of the concept of a symmetric space. Adequate infini
tesimal objects are introduced and the general theory ofsuch spaces is given. The
antisymmetric spaces are ofspecial interest, since the corresponding infinitesimal
object is a binary algebra (with some characteristic identities close to Mal'cev
algebra identities).
Appendix 7. 'Right alternative local analytic loops', where it is proved that
any analytic right alternative loop is right monoalternative.
There is given a detailed Bibliography, containing a wide range of publica
tions on the subject and related matters. The titles concerning applications to
mathematical physics are included as well.
It is hoped that our efforts in creating this book will be of significance for
mathematics.