Homogeneous space of an algebraic group
An algebraic variety together with a regular transitive action of an algebraic group given on it. If , then the isotropy group is closed in . Conversely, if is a closed subgroup of an algebraic group , then the set of left cosets has the structure of an algebraic variety, making it into a homogeneous space of the algebraic group , where the natural mapping is regular, separable and has the following universal property: For any morphism constant on cosets, there is a morphism such that . If is any homogeneous space of the algebraic group and for some , then the natural bijection is regular, and if the ground field has characteristic 0, then is biregular (see , ).
Suppose that the connected group , the homogeneous space and the action of on are defined over some subfield . Then the group of -rational points takes into itself and for . If is finite, then , and if moreover the isotropy group is connected, then acts transitively on . In the general case, the study of the -rational points in reduces to problems in the theory of Galois cohomology (see ).
A homogeneous space of an algebraic group is always a smooth quasi-projective variety (see ). If is an affine algebraic group, then the variety is projective if and only if is a parabolic subgroup in (see ). If is reductive, then the variety is affine if and only if the subgroup is reductive (see Matsushima criterion). A description is also known of the closed subgroups of a linear algebraic group over an algebraically closed field of characteristic 0 for which is quasi-affine (see , ).
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Homogeneous space of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homogeneous_space_of_an_algebraic_group&oldid=14365