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On homogeneous systems. V. (English) Zbl 0538.53053
A homogeneous system $$(G,\eta)$$ is a set G with a ternary operation $$\eta$$ on G satisfying the axioms 1) $$\eta(x,x,y)=\eta(x,y,x)=y$$, 2) $$\eta(x,y,\eta(y,x,z))=z$$, 3) $$\eta(x,y,\eta(u,v,w))=\eta(\eta(x,y,u),\eta(x,y,v),\eta(x,y,w))$$. It was introduced by the author in ibid. 10, 19-25 (1976; Zbl 0223.70014) to generalize the theory of Lie groups and Lie algebras, and it has been investigated in a series of papers [the author, ibid. 11, 9-17 (1977; Zbl 0377.53029); 12, 5-13 (1978; Zbl 0396.53024,; 14, 41-46 (1980; Zbl 0478.53038) and 15, 1-7 (1981; Zbl 0478.53039)]. In this paper, it is shown that any abstract homogeneous system G can be characterized as a subset of a group, satisfying certain conditions. By using this fact the author gives a method of constructing a simply connected analytic homogeneous system whose tangent Lie triple algebra is isomorphic to a given finite dimensional real Lie triple algebra, which is a generalization of the converse to the 3rd fundamental theorem of S. Lie.

##### MSC:
 53C30 Differential geometry of homogeneous manifolds 22E99 Lie groups 20N10 Ternary systems (heaps, semiheaps, heapoids, etc.) 17A40 Ternary compositions