Skip to main content
SpringerLink
Log in

Smooth Quasigroups and Loops

  • Book
  • © 1999

Overview

Authors:
  1. Lev V. Sabinin

    1. Russian Friendship of Nations University, Moscow, Russia
      Michoacan University, Mexico

    View author publications

    You can also search for this author in PubMed   Google Scholar

Part of the book series: Mathematics and Its Applications (MAIA, volume 492)

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (13 chapters)

  1. Front Matter

    Pages i-xvi
    Download chapter PDF

  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
      Download chapter PDF

    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
      Download chapter PDF

    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
      Download chapter PDF

    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
    Download chapter PDF
Back to top

Keywords

About this book

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.

Authors and Affiliations

  • Russian Friendship of Nations University, Moscow, Russia

    Lev V. Sabinin

  • Michoacan University, Mexico

    Lev V. Sabinin

Bibliographic Information

Publish with us

Policies and ethics

Back to top

Search

Navigation