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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.

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