Smooth Quasigroups and Loops

Cover
Title page
PREFACE
INTRODUCTION
CHAPTER 0. INTRODUCTORY SURVEY: QUASIGROUPS, LOOPUSCULAR GEOMETRY AND NONLINEAR GEOMETRIC ALGEBRA
PART ONE. FUNDAMENTAL STRUCTURES OF NONLINEAR GEOMETRIC ALGEBRA
CHAPTER 1. BASIC ALGEBRAIC STRUCTURES
A. Quasigroups, Ioops, odules, diodules
B. Loopuscular, odular, diodular algebras
C. Holonomial and geometric odules
D. Antiproducts, Chern algebras, geodetic spaces
E. Geoodular axiomatics of affine spaces
CHAPTER 2. SEMIDIRECT PRODUCTS OF A QUASIGROUP BY ITS TRANSASSOCIANTS
CHAPTER 3. BASIC SMOOTH STRUCTURES
A. Smooth universai algebras
B. Maximal partial algebras
C. Smooth adules and odular structures
D. Canonicai odules and odular structures
PART TWO. SMOOTH LOOPS AND HYPERALGEBRAS
CHAPTER 4. INFINITESIMAL THEORY OF SMOOTH LOOPS
A. General theory
B. Smooth local geometric odules
C. Smooth holonomial odules
D. Additional differential equations
CHAPTER 5. SMOOTH BOL LOOPS AND BOL ALGEBRAS
A. Bol algebras
B. General theory
C. Smooth Bol loops and homogeneous spaces
D. Triple Lie systems of vector fields
E. Pseudoderivatives of Bol algebras
F. Enveloping Lie algebras of a Bol algebra
G. Infinitesimal theory
H. Final notices
CHAPTER 6. SMOOTH MOUFANG LOOPS AND MAL'CEV ALGEBRAS
CHAPTER 7. SMOOTH HYPOREDUCTIVE AND PSEUDOREDUCTIVE LOOPS
A. Smooth hyporeductive loops
B. Smooth pseudoreductive loops
PART THREE. LOOPUSCULAR GEOMETRY
CHAPTER 8. AFFINE CONNECTIONS AND LOOPUSCULAR STRUCTURES
A. Tangent affine connections of loopuscular structures
B. Natural geoodular (linear geodiodular) structures of an affinely connected manifold
C. Flat geoodular manifolds
D. DifferentiaI geometry of right monoalternative loops
E. Final remarks
CHAPTER 9. REDUCTIVE GEOODULAR SPACES
CHAPTER 10. SYMMETRIC GEOODULAR SPACES
CHAPTER 11. s-SPACES
A. General theory
B. Perfect s-structures
CHAPTER 12. GEOMETRY OF SMOOTH BOL AND MOUFANG LOOPS
A. Differential geometry of smooth Bol loops
B. Main structure theorem of the theory of smooth Bol loops
C. Differential geometry of smooth Moufang loops
APPENDICES
APPENDIX 1. LIE TRIPLE ALGEBRAS AND REDUCTIVE SPACES
APPENDIX 2. LEFT F-QUASIGROUPS. LOOPUSCULAR APPROACH
APPENDIX 3. LEFT F-QUASIGROUPS AND REDUCTIVE SPACES
APPENDIX 4. GEOMETRY OF TRANSSYMMETRIC SPACES
APPENDIX 5. HALF BOL LOOPS
APPENDIX 6. ALMOST SYMMETRIC AND ANTISYMMETRIC MANIFOLDS
APPENDIX 7. RIGHT ALTERNATIVE LOCAL ANALYTIC LOOPS
BIBLIOGRAPHY
INDEX