等質ループの左移動について

その他

The notion of homogeneous systems inntroduced in §1 of this paper is an abstraction of homogeneity
of homogeneous loops and groups. In consideration of the geometric approach tried in [1], the left translatious of homogeneous Lie loops lead us to the notion of (parallel) displacements of homogeneous systems. This paper is aimed at the axiomatic construction of homogeneous systems which will be useful to the study of a certain class of locally reductive spaces including homogemeous Lie loops and Lie groups. In §2 the homogeneous systems of homogeneous loops are found. In §3 it is shown that any autornorphism of a homogeneous systems is an automorphism fixing the origin followed by a displacement from the origin. As the isotropy subgroup of the group of displacements the notion of holonomy groups is obtained which is closely related with (non) associativity of the binary system induced from the homogeneous system. Symmetric homogueous systems are introduced in §4 which should be combined with symmetric homogeneous spaces.