Smooth Quasigroups and Loops

1. Front Matter

   Pages i-xvi

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2. Introductory Survey: Quasigroups, Loopuscular Geometry And Nonlinear Geometric Algebra

   1. Introductory Survey: Quasigroups, Loopuscular Geometry and Nonlinear Geometric Algebra

      • Lev V. Sabinin

      Pages 1-20

3. Fundamental Structures of Nonlinear Geometric Algebra

   1. Front Matter

      Pages 21-21

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   2. Basic Algebraic Structures

      • Lev V. Sabinin

      Pages 23-35

   3. Semidirect Products of a Quasigroup by its Transassociants

      • Lev V. Sabinin

      Pages 36-46

   4. Basic Smooth Structures

      • Lev V. Sabinin

      Pages 47-56

4. Smooth Loops and Hyperalgebras

   1. Front Matter

      Pages 57-57

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   2. Infinitesimal Theory of Smooth Loops

      • Lev V. Sabinin

      Pages 59-86

   3. Smooth Bol Loops and Bol Algebras

      • Lev V. Sabinin

      Pages 87-104

   4. Smooth Moufang Loops and Mal’Cev Algebras

      • Lev V. Sabinin

      Pages 105-110

   5. Smooth Hyporeductive and Pseudoreductive Loops

      • Lev V. Sabinin

      Pages 111-128

5. Loopuscular Geometry

   1. Front Matter

      Pages 129-129

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   2. Affine Connections and Loopuscular Structures

      • Lev V. Sabinin

      Pages 131-145

   3. Reductive Geoodular Spaces

      • Lev V. Sabinin

      Pages 146-154

   4. Symmetric Geoodular Spaces

      • Lev V. Sabinin

      Pages 155-165

   5. s-SPACES

      • Lev V. Sabinin

      Pages 166-174

   6. Geometry of Smooth Bol and Moufang Loops

      • Lev V. Sabinin

      Pages 175-182

6. Back Matter

   Pages 183-253

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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 purely geometric structures such as affinely connected spaces. The notion ofodule was introduced as a fundamental algebraic invariant of differential 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 of algebraic 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 space with affine connection (in the form ofgeoodular structure) which can be used in the old problem of discrete space-time in relativity, essential for the quantum space-time theory.

• Algebraic structure
• Group theory
• Volume
• algebra
• differential geometry

• Russian Friendship of Nations University, Moscow, Russia

  Lev V. Sabinin

• Michoacan University, Mexico

  Lev V. Sabinin

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