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Sabinin L.V. — Smooth Quasigroups and Loops
Sabinin L.V. — Smooth Quasigroups and Loops

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Название: Smooth Quasigroups and Loops

Автор: Sabinin L.V.

Аннотация:

This monograph presents the complete theory of smooth quasigroups and loops, as well as its geometric and algebraic applications. Based on a generalisation of the Lie-group theory, it establishes new backgrounds for differential geometry in the form of nonlinear geometric algebra and `loopuscular' geometry. It will prove useful in applications in such diverse fields as mathematical physics, relativity, Poisson and symplectic mechanics, quantum gravity, dislocation theory, etc. Audience: This volume will be of interest to researchers, lecturers and postgraduate students whose work involves geometry, group theory, nonassociative rings and algebras, and mathematical and theoretical physics.


Язык: en

Рубрика: Математика/Алгебра/Абстрактная алгебра/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1999

Количество страниц: 249

Добавлена в каталог: 19.03.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\varepsilon$-identity      29 79
Absolute parallelism      26 133
Affinely connected s-manifold      166
Affinizable left loopuscular algebra      132
Algebra of pseudodifferentiations      100
Algebra tangent to antisymmetric space      222
Almost symmetric affinely connected manifold      218
Almost symmetric geoodular manifold      218
Anti-automorphic inverse identity      105
Antiproduct of a loop (with itself)      19 30
Antisymmetric algebra      222
Antisymmetric geoodular manifold      218 221
Antisymmetric space (of antisymmetric algebra)      222
Associant      12 36
Associativity      24
Automorphic inverse identity (property)      7 160 161 162 163 164
Basic fundamental left vector fields      2 55
Basic fundamental right vector fields of a loop      55 112
Binary-binary-ternary algebra tangent to hyporeductive left smooth loop      115
Binary-Lie algebra      109 222
Binary-Lie property of a Mal'cev algebra      109
Bol algebra      15 87 100
Canonical affine connection of a transsymmetric space      206
Canonical loopuscular structure of a left F-quasigroup      10 189
Canonical operations      56 61
Canonical prediodule      56 62
Chern connection      19
Chern loopuscular algebra      30
Correct left F-quasigroup      9 189
Correct ts-structure      9 199 205
correctness      9 199 205
Curvature      16 219
Defining identity of Mal'cev algebra      109
Diassociative loop      53
Diassociative odule      53
Diodular structure (manifold)      4
Elementary holonomy      6 26 78
Enlarged algebra of inner pseudodifferentiations of a (right) Bol algebra      101
Enlarged algebra of pseudodifferentiations of a (right) Bol algebra      101
Enveloping algebra of a (right) Bol algebra      102
Enveloping Lie algebra      186
Exponential function      2 55 61
Exponential map      2 55 61
F-identity      9 199
First geoodular identity      4 28 31
First identity of reductivity      148
First left A-identity (property)      162
First left g-half Bol property (identity)      210 216
First s-identity      156
Flat loopuscular structure (algebra)      6 26
Fundamental left covector fields      60
Fundamental left vector fields      60
Fundamental right vector fields      112
G general s-identity      171
Generalized Bruck identity      173
Generalized symmetric space (s-space)      9
Geodesic odule      6
Geodesic smooth partial left (right) $\mathcal{K}$-odular (diodular) manifold (algebra, structure, space)      26 53
Geodetic algebra      30
Geodetic space      19
Geoodular axiomatics      31
Geoodular space      31
Geoodular structure      4
Geoodular structure associated with a geometric odule      79
Germ of a smooth partial algebra      49
Group of transvections      169
h-identity      29 79
Holonomial geoodule      8
Holonomial loop      8
Holonomial odule      8 29 79
Holonomial prediodule (diodule)      29
Holonomial preodule      29
Homogeneity identity      8 29
Homogeneous Bol algebra      16
Homogeneous space      45 97 203
Hyperalgebra, F-hyperalgebra, $\nu$-hyperalgebra, $\nu$-hyperalgebra tangent to a loop      13 14 70
Hypo-automorphism      111
Hyporeductive algebra      17 122
Hyporeductive decomposition      116
Hyporeductive pair      17 116
Hyporeductive quadruplet      17
Hyporeductive system of vector fields      17
Hyporeductive triple algebra of vector fields      115
Hyporeductive triplet      17 116
Idempotent identity      166
Idempotent left distributive quasigroup      8
Identity      47
Identity of idempotency      155
Identity of left distributivity      155
Inner derivation of a Lie triple algebra      185
Inner pseudodifferentiations of a (right) Bol algebra      101
Isoclinic Bol algebra      88
Isotopy      26
Joint identity      29 78 79
Left (p, k)-canonical subloop      144
Left associant      36
Left basic fundamental vector fields      2 55
Left Bol identity (property)      7 157 160 161 162 163 164
Left canonical (right canonical) partial loop      55
Left canonical partial loopuscular structure      56
Left canonical preodular (predioduilar) algebra      56
Left canonical preodule      2 56
Left canonical unary operations      2
Left companion      123
Left diodule      25
Left distributive identity      9
Left division      1
Left F-identity (property)      9 189 206
Left F-quasigroup      189 206
Left first g-half Bol loop      210
Left geoodiodular structure      4
Left geoodular structure      4
Left geoprediodular (geodiodular) algebra      28
Left geopreodular (geoodular) algebra      28
Left hyporeductive loop      17 112
Left hypospecial loop (left hypo-A-loop)      16 111
Left inverse property      105
Left key identity      155
Left loop      23
Left loopuscular algebra      25
Left M-loop      192
Left modular algebra      26
Left monoalternative identity      3 163 171 174
Left monoalternative loop      53
Left monoalternative odule      53
Left monoalternative property      3 24
Left neutral      24
Left nucleus      215
Left odular algebra      26
Left odular structure (manifold)      3
Left odule      3 25
Left prediodular algebra      26
Left prediodule      25
Left preodular algebra      26
Left preodule      2 24
Left pseudoassociativity      25
Left pseudoreductive loop      18 123
Left pseudospecial identity (property)      18 123 213
Left pseudospecial loop (left pseudo-A-loop)      18 123
Left quasigroup      23
Left reductive loop      18 112
Left transassociant      37
Lie group      48
Lie triple algebra      185
Lie triple family      179
Lie triple system      15 87
Linear diodular structure      4
Linear diodule      25
Linear left prediodular (diodular) algebra      25
Linear prediodule      25
Locally equivalent smooth algebras      51
Loop      24
Loopuscular (odular) structure      3 4
Loopuscular geometry      1
Loopuscular manifold      3
Loopuscular structure canonical for idempotent quasigroup structure      167
Loopuscular structure with absolute parallelism      6
Loopuscular structure without curvature (of zero curvature)      6
Magma      1
Mal'cev algebra      109
Maximal enveloping algebra      102 186
Middle nucleus      215
Minimal (standard) enveloping algebra      102
Monoassociativity      3 24
Morphism of Lie triple algebras      185
Moufang loop      16 105
N natural F-space      199
Natural geoodular (linear geodiodular) structure of an affinely connected space      4 135
Nonlinear geometric algebra      1
Normal coordinates      2 62
Odular (diodular) s-structure      167
Odular (diodular) structure      4 26
Odule      25
Partial automorphism of a left odular space      146
Partial Lie group      48
Partial loop      1
Perfect s-space      9
Perfect s-structure      172
Plane axiom      88
Polar (normal) coordinates      177
Prediodule      2 25
Preodule      2 24
Prime quasi-reductant (reductant)      42
Principal right isotope of a quasigroup      191
Projectively flat affinely connected space      7
Property of left F-regularity      9
Pseudo-automorphism      123
Pseudoderivative      100
Pseudodifferentiation (pseudoderivation)      88 100
Pseudoequivalent smooth algebras      50
Pseudolinearity property      7
Pseudoreductive smooth loop      18 123
Quasi-flat left geoodular manifold      140
Quasi-flat preodular algebra      26
Quasi-reductant      11 42
Quasigroup      8 23
Reductant      12 42
Reductive left loopuscular (odular) structure (algebra)      7 27 134
Reductive left prediodular (diodular) algebra      27 148
Reductive left preodular (odular) algebra      27 148
Reductive pair      17
Reductive quadruplet      17
Reductive space      147
Reductive triplet      17
Regular ts-structure      206
Riemannian s-manifold      167
Right alternative identity      143 226
Right basic fundamental vector fields      2 112
Right Bol property (identity)      14 88 105
Right diodule      25
Right division      1
Right geometric identity      3 30
Right geometric preodule (odule)      3 30
Right loop      24
Right monoalternative identity (property)      3 25 226
Right monoalternative loop      53 94 143
Right monoalternative odule      53 94 143
Right neutral      23
Right odule      25
Right prediodule      25
Right preodule      24
Right pseudoassociativity      25
Right quasigroup      23
s-automorphic identity      171 174
s-identity      156
Sagle algebra      225
Second 5-identity      156
Second geoodular identity      4 28 31 163
Second identity of reductivity      148
Second left A-property (identity)      153
Second left g-half Bol property (identity)      216
Semicanonical affine connection      83
Semicanonical geoodular structure      83
Semidirect product of a left loop by its transassociant      12 40
Semijreopreodular (semigeoodular) algebra      28
Similarity identity      24
Smooth algebraic system      20
Smooth left odule over a linear $\mathbb{R}$-algebra      51
Smooth local algebra      49
Smooth partial (global) operation      47
Smooth partial idempotent left quasigroup structure      166
Smooth partial left (right) $\mathbb{R}$-odular manifold (algebra structure, space)      53
Smooth partial left loop      48
Smooth partial left quasigroup      47 50
Smooth partial left R-odule      51
Smooth partial loop      48 50
Smooth partial quasigroup      48 50
Smooth partial right loop      48
Smooth partial right quasigroup      48 50
Smooth space with geodesies      19
Smooth subodule      53
Smooth universal algebra      20 47
Special s-half Bol identity      174
Standard enveloping Lie algebra of a Lie triple algebra      186
Standard quasigroup      203
Structure of a canonical preodule of a smooth loop      62
Symmetric left loopuscular algebra      27
Symmetric left odular (diodular) space      155
Symmetric left prediodular (diodular) algebra      27
Symmetric left preodular (odular) algebra      27
Symmetric Lie algebra      87
Symmetric loopuscular (odular) structure      7
Symmetric space      7
Tangent $\nu$-hyperalgebra of a loop      70
Tangent affine connection      5 132
Tangent binary-ternary algebra      88
Tangent Bol algebra of a right Bol loop      15 100
Tangent dibinary-ternary algebra of hyporeductive loop      122
Tangent ft-identity      79
Tangent geometric identity      73
Tangent hyporeductive algebra of a left hyporeductive loop      18
Third geoodular identity      4 28 163
Third identity of reductivity      148
Three-web      19
Torsion tensor      16
Transassociant      12 37
Transformation of an elementary holonomy      6 26 133
Transsymmetric space      10 206
Transsymmetric structure      9
Transsymmetry      9 205
Transversal      11
Triple Lie family of transformations      179
Triple Lie system      87
Triple Lie system of vector fields      97
Trivial curvature      26
ts-structure (manifold)      9 205
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