Synopses & Reviews
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.
Synopsis
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.
Table of Contents
Preface. Introduction. 0. Introductory Survey: Quasigroups, Loopuscular Geometry and Nonlinear Geometric Algebra. Part One: Fundamental Structures of Nonlinear Geometric Algebra. 1. Basic Algebraic Structures. 2. Semidirect Products of a Quasigroup by its Transassociants. 3. Basic Smooth Structures. Part Two: Smooth Loops and Hyperalgebras. 4. Infinitesimal Theory of Smooth Loops. 5. Smooth Bol Loops and Bol Algebras. 6. Smooth Moufang Loops and Mal'Cev Algebras. 7. Smooth Hyporeductive and Pseudoreductive Loops. Part Three: Loopuscular Geometry. 8. Affine Connections and Loopuscular Structures. 9. Reductive Geoodular Spaces. 10. Symmetric Geoodular Spaces. 11. s-Paces. 12. Geometry of Smooth Bol and Moufang Loops. Appendices. Bibliography. Index.