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Journal of Lie Theory 12 (2002), No. 2, 495--502
Copyright Heldermann Verlag 2002

On Observable Subgroups of Complex Analytic Groups and Algebraic Structures on Analytic Homogeneous Spaces

Nazih Nahlus
Dept. of Mathematics, American University, Beirut, Lebanon, New York, NY 10022-6297, U.S.A.

Let L be a closed analytic subgroup of a faithfully representable complex analytic group G, let R(G) be the algebra of complex analytic representative functions on G, and let G0 be the universal algebraic subgroup (or algebraic kernel) of G.
In this paper, we show many characterizations of the property that the homogenous space G/L is (representationally) separable, i.e, R(G)L separates the points of G/L. This yield new characterizations for the observability of L in G and new characterizations for the existence of a quasi-affine structure on G/L. For example, G/L is separable if and only if the intersection of G0 and L is an observable algebraic subgroup of G0. Moreover, L is observable in G if and only if G/L is separable and L0 is equal to the intersection of G0 and L.
Similarly, we discuss a weaker separability of G/L and the existence of a representative algebraic structure on it.

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