ABSTRACT
Abstract We discuss some properties of Kikkawa spaces (families of geodesic rightmonoalternative loops) in the context of the theory of smooth loops and smooth odules.
Key words: Geodesic loop; Al-loop; Moufang loop; Preparallel relation; Kikkawa space
2000 MSC: Primary: 53B05; Secondary: 20N05
Right-monoalternativity has long been recognized as one of the key properties of smooth loops. Kikkawa in his paper [5] introduced the concept of a geodesic loop and studied the case of a right-monoalternative geodesic loop without torsion. He established that for such a loop the curvature of the associated connection vanishes. Mikheev and Sabinin (see [12]) obtained a differential equation defining right-monoalternative smooth loops. In particular, they showed that for every right-monoalternative smooth loop there exists a smooth manifold with an affine connection (which can be chosen to be flat) with the property that the geodesic loop of such a manifold is isomorphic to the initial loop. Using the results on right-monoalternative smooth loops they established the identities for the tangent algebra of a smooth Bol loop (called a Bol algebra), and proved that the correspondence between smooth Bol loops and their Bol algebras is analogous to the correspondence between Lie groups and Lie algebras. Moreover, their study of smooth right-monoalternative loops permitted them to describe the tangent algebra of a general smooth loop. They called this object a hyperalgebra, now known as a Sabinin algebra. The theory of smooth Bol loops (in particular, smooth Bruck loops) and Bol algebras is in the intersection of several areas: differential geometry, algebra, and physics, as these structures describe symmetric spaces and their generalizations in an algebraic way (first suggested by O. Loos). Shestakov and Umirbaev [15] discovered the fact that Sabinin algebras appear as generalizations of Lie algebras in the context of the theory of bialgebras; this stimulated further research on the of
geodesic loops, we described the class of Malcev algebras that appear as tangent algebras to geodesic Moufang loops of reductive homogeneous spaces. In this chapter we present some geometric properties of right-monoalternativity of geodesic loops and compare the results from [3] with other known results.
