Journal of Lie Theory, Vol. 10, No. 2, pp. 455-461 (2000)

Geodesic loops

Ágota Figula

Mathematisches Institut
der Universität Erlangen-Nürnberg
Bismarckstr. 1$1\over2$,
D-91054 Erlangen, Germany

Abstract: In this paper we obtain for geodesic loops $(L,\nabla ,e)$ analogous statements to Lie's theorems using new tangential objects, which we call $\Lambda $-algebras. We prove that any geodesic loop $L$ defines in the tangential space $T_eL$ a unique $\Lambda $-algebra, and that to any finite dimensional real $\Lambda $-algebra $F$ there exists a geodesic loop $L$, the $\Lambda $-algebra of which is isomorphic to $F$. Geodesic loops which have isomorphic $\Lambda $-algebras, are locally isomorphic. If the local loop $L$ is diassociative then to the $\Lambda $-algebra of $L$ there corresponds a subseries of the Hausdorff-Campbell formula with respect to the binary Lie algebra belonging to $L$.

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