Smooth Bruck loops, symmetric spaces, and nonassociative vector spaces

Sejong Kim,Jimmie Lawson

doi:10.1515/dema-2013-0343

Abstract

AbstractOur purposes in this work include the following: (1) Extend and expand earlier work on symmetric spaces, particularly that done from a nonassociative algebra point of view, from the finite-dimensional setting to the Banach space setting. (2) Take a careful look at the equivalence of the categories of smooth pointed reflection quasigroups (a special class of symmetric spaces) and uniquely 2-divisible Bruck loops (=

Highlights

René Descartes and Pierre de Fermat revolutionized the study of geometry with their introduction of coordinate systems and algebraic methods
Significant outcomes were the rise of analytic geometry and the introduction of Euclidean vector spaces as an appropriate framework for Euclidean geometry
Our goal is to find an algebraic model that is an enriched Bruck loop and that, as closely as possible, resembles a non-associative Banach space, with the smooth structure and the tangent bundle replaced by algebraic notions and operations internal to the loop

Summary

Introduction

René Descartes and Pierre de Fermat revolutionized the study of geometry with their introduction of coordinate systems and algebraic methods. Our goal is to find an algebraic model that is an enriched Bruck loop and that, as closely as possible, resembles a non-associative Banach space, with the smooth structure and the tangent bundle replaced by algebraic notions and operations internal to the loop. These structures may be viewed as variants of the gyrovector space structures of Ungar [16], gyrocommutative gyrogroups with a type of scalar multiplication, or of the odules of Sabanin [13], [14]. In the latter part of the paper we adopt primarily the terminology and notation of Ungar, since we find this the most suitable for highlighting analogies with the classical theory of vector spaces

Smooth symmetric spaces

Smooth gyrocommutative gyrogroups

Parallelogram vector addition