On Smooth Power-Alternative Loops

Abstract

The aim of this paper is to show that if at every point of some neighborhood of a smooth manifold with an affine connection the geodesic loop is power-alternative and left A-loop, then this loop is a Moufang loop. We give an example of the tangent algebra of such a loop.

Key Words:

• Geodesic loop
• A _ℓ-Loop
• Moufang loop
• Malcev algebra
• Power-alternativity
• Reductive Kikkawa space

2000 Mathematics Subject Classifications:

• Primary: 53B05
• Secondary: 20N05
• 17D10

Acknowledgments

We would like to thank the referee for important remarks. We also are grateful to A.N. Grishkov for useful conversations and finally we are greatly indebted to A. Elduque who pointed out to us the properties of algebras with X ² = 0, J(X, Y, XZ) = J(X, Y, Z)X = 0.

Notes

^#Communicated by I. Shestakov.