Smooth Bruck loops, symmetric spaces, and nonassociative vector spaces
Our 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 (= K-loops = gyrocommutative gyrogroups). (3) Propose a loop-theoretic analog of topological vector spaces. (4) Derive algebraic consequences and equivalences of smoothness notions, particularly the notion of parallel transport. (5) Illustrate the effective interaction of the algebraic operations of reflection, Bruck addition, and coaddition in the test case of parallelograms in symmetric spaces.
Publication Source (Journal or Book title)
Kim, S., & Lawson, J. (2011). Smooth Bruck loops, symmetric spaces, and nonassociative vector spaces. Demonstratio Mathematica, 44 (4), 755-779. https://doi.org/10.1515/dema-2013-0343