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

Nazih Nahlus

Abstract: Let $\srm L$ be a closed analytic subgroup of a faithfully representable complex analytic group $\srm G$, let $\srm R(G)$ be the algebra of complex analytic representative functions on $\srm G$, and let $\srm G_0$ be the universal algebraic subgroup (or algebraic kernel) of $\srm G$. In this paper, we show many characterizations of the property that the homogenous space $\srm G/L$ is (representationally) {\it separable}, i.e, $\srm R(G)^L$ separates the points of $\srm G/L$. This yield new characterizations for the observability of $\srm L$ in $\srm G$ and new characterizations for the existence of a quasi-affine structure on $\srm G/L$. For example, $\srm G/L$ is separable if and only if $\srm G_0 \cap\ L$ is an observable algebraic subgroup of $\srm G_0$. Moreover, $\srm L$ is observable in $\srm G$ if and only if $\srm G/L$ is separable and $\srm L_0 = G_0 \cap\ L$. Similarly, we discuss a weaker separability of $\srm G/L$ and the existence of a representative algebraic structure on it.

Classification (MSC2000): 22E10, 22E45, 22F30, 20G20, 14L15

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