On Observable Subgroups of Complex Analytic Groups and Algebraic Structures on Analytic Homogeneous Spaces
Journal of Lie Theory, Volume 12 (2002) no. 2, pp. 495-502
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.
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.
DOI:
10.5802/jolt.276
Classification:
22E10, 22E45, 22F30, 20G20, 14L15
Keywords: observable subgroups, faithfully representable complex analytic group, homogeneous space
Keywords: observable subgroups, faithfully representable complex analytic group, homogeneous space
@article{JOLT_2002_12_2_a10,
author = {N. Nahlus},
title = {On {Observable} {Subgroups} of {Complex} {Analytic} {Groups} and {Algebraic} {Structures} on {Analytic} {Homogeneous} {Spaces}},
journal = {Journal of Lie Theory},
pages = {495--502},
year = {2002},
volume = {12},
number = {2},
doi = {10.5802/jolt.276},
zbl = {1007.22015},
url = {https://jolt.centre-mersenne.org/articles/10.5802/jolt.276/}
}
TY - JOUR AU - N. Nahlus TI - On Observable Subgroups of Complex Analytic Groups and Algebraic Structures on Analytic Homogeneous Spaces JO - Journal of Lie Theory PY - 2002 SP - 495 EP - 502 VL - 12 IS - 2 UR - https://jolt.centre-mersenne.org/articles/10.5802/jolt.276/ DO - 10.5802/jolt.276 ID - JOLT_2002_12_2_a10 ER -
%0 Journal Article %A N. Nahlus %T On Observable Subgroups of Complex Analytic Groups and Algebraic Structures on Analytic Homogeneous Spaces %J Journal of Lie Theory %D 2002 %P 495-502 %V 12 %N 2 %U https://jolt.centre-mersenne.org/articles/10.5802/jolt.276/ %R 10.5802/jolt.276 %F JOLT_2002_12_2_a10
N. Nahlus. On Observable Subgroups of Complex Analytic Groups and Algebraic Structures on Analytic Homogeneous Spaces. Journal of Lie Theory, Volume 12 (2002) no. 2, pp. 495-502. doi: 10.5802/jolt.276
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