Smooth quasigroups and loops

This monograph written by a leading specialist in this new rapidly developing field of mathematics presents the complete up-to-date theory of smooth quasigroups and loops, as well as its geometric and algebraic applications. Based on a generalization of the Lie group theory, it establishes a new background for differential geometry in the form of the nonlinear geometric algebra and "loopuscular" geometry. The book could be useful in applications in such diverse fields as mathematical physics, relativity, Poisson and symplectic mechanics, quantum gravity, dislocation theory, etc. par We will list the chapter titles with a short description of their contents. par Chapter 0. {it Introductory survey: quasigroups, loopuscular geometry and nonlinear geometric algebra} describes briefly the whole area in its historical perspective. The text in this chapter is an enlargement of the introduction from the paper of L. V. Sabinin and P. O. Miheev published as Chapter XII of the book [{it Quasigroups and loops: theory and applications}, Heldermann, Berlin, 1990; [msn] MR1125806 (93g:20133) [/msn]], the paper [L. V. Sabinin, Eesti NSV Tead. Akad. Füüs. Inst. Uurim. No.~66 (1990), 24--53; [msn] MR1158692 (93d:20125) [/msn]], and the author's lectures delivered at the international conferences [in {it Universal algebra, quasigroups and related systems (Jadwisin, 1989}), Tech. Univ. Warsaw, Warsaw, 1991; per revr.; in {it Classical and quantum theory of homogeneous spaces (Moscow, 1994}); per revr.]. par Chapter 1. {it Basic algebraic structures} presents the main algebraic concepts of nonlinear geometric algebra. Chapter 2. {it Semidirect products of a quasigroup by its transassociants} gives the construction of inclusion of a loop into a group. Chapter 3. {it Basic smooth structures} treats the basic smooth concepts of nonlinear geometric algebra. Chapter 4. {it Infinitesimal theory of smooth loops} gives a generalization of the Lie groups-Lie algebras theory. Chapter 5. {it Smooth Bol loops and Bol algebras} contains a new treatment of the subject. Chapter 6. {it Smooth Moufang loops and Malʹtsev algebras} elaborates smooth Moufang loop theory as a particular case of smooth Bol loop theory. par Chapter 7. {it Smooth hyporeductive and pseudoreductive loops} suggests a generalization of both Bol loops and reductive loops and the proper infinitesimal theory. Chapter 8. {it Affine connections and loopuscular structures} proves that an affinely connected manifold and a smooth geoodular manifold are essentially the same objects. Chapter 9. {it Reductive geoodular spaces} formulates the notion of reductive space in a purely algebraic way in the framework of nonlinear geometric algebra. Chapter 10. {it Symmetric geoodular spaces} treats the symmetric spaces as smooth universal algebras. Chapter 11. {it s-spaces} deals with the algebraic theory of s-spaces (generalized symmetric spaces). par Chapter 12. {it Geometry of smooth Bol and Moufang loops} describes the Bol and Moufang loops as affinely connected manifolds of zero curvature with torsion of a special kind. par The book has seven appendices. Appendix 1. {it Triple Lie algebras and reductive spaces} develops the infinitesimal theory of reductive spaces and related reductive loops in a geometric way. Appendix 2. {it Left F-quasigroups. Loopuscular approach} investigates the canonical loopuscular structure of a left F-quasigroup. Appendix 3. {it Left F-quasigroups and reductive spaces} gives the presentation of any left F-quasigroup as a reductive space of a special kind with multiplication. Appendix 4. {it Geometry of transsymmetric spaces} outlines the transsymmetric space theory (a generalization of the well-known symmetric space theory) as well as smooth left F-quasigroups theory. par Appendix 5. {it Half-Bol loops} contains a generalization of Bol loops needed for geometrical applications. Appendix 6. {it Almost symmetric and antisymmetric manifolds} concerns one more generalization of the concept of a symmetric space. In this appendix adequate infinitesimal objects are introduced, and the general theory of such spaces is given. The antisymmetric spaces are of special interest, since the corresponding infinitesimal object is a binary algebra (with some characteristic identities close to Malʹtsev algebra identities). Appendix 7. {it Right alternative local analytic loops} proves that any analytic right alternative loop is right monoalternative. par A detailed bibliography containing a wide range of publications on the subject and related matters is given at the end of the book. It contains the titles concerning applications to mathematical physics. par This book 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.