Difference between revisions of "Homogeneous space of an algebraic group"
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− | An [[Algebraic variety|algebraic variety]] | + | {{TEX|done}} |
+ | An [[Algebraic variety|algebraic variety]] $ M $ | ||
+ | together with a regular transitive action of an [[Algebraic group|algebraic group]] $ G $ | ||
+ | given on it. If $ x \in M $ , | ||
+ | then the [[Isotropy group|isotropy group]] $ G _{x} $ | ||
+ | is closed in $ G $ . | ||
+ | Conversely, if $ H $ | ||
+ | is a closed subgroup of an algebraic group $ G $ , | ||
+ | then the set of left cosets $ G /H $ | ||
+ | has the structure of an algebraic variety, making it into a homogeneous space of the algebraic group $ G $ , | ||
+ | where the natural mapping $ \pi : \ G \rightarrow G / H $ | ||
+ | is regular, separable and has the following universal property: For any morphism $ \phi : \ G \rightarrow X $ | ||
+ | constant on cosets, there is a morphism $ \psi : \ G / H \rightarrow X $ | ||
+ | such that $ \psi \pi = \phi $ . | ||
+ | If $ M $ | ||
+ | is any homogeneous space of the algebraic group $ G $ | ||
+ | and $ H = G _{x} $ | ||
+ | for some $ x \in M $ , | ||
+ | then the natural bijection $ \psi : \ G / H \rightarrow M $ | ||
+ | is regular, and if the ground field $ K $ | ||
+ | has characteristic 0, then $ \psi $ | ||
+ | is biregular (see [[#References|[1]]], [[#References|[3]]]). | ||
− | Suppose that the connected group | + | Suppose that the connected group $ G $ , |
+ | the homogeneous space $ M $ | ||
+ | and the action of $ G $ | ||
+ | on $ M $ | ||
+ | are defined over some subfield $ k \subset K $ . | ||
+ | Then the group of $ k $ - | ||
+ | rational points $ G (k) $ | ||
+ | takes $ M (k) $ | ||
+ | into itself and $ G (k) _{x} = G _{x} (k) $ | ||
+ | for $ x \in M (k) $ . | ||
+ | If $ k $ | ||
+ | is finite, then $ M (k) \neq \emptyset $ , | ||
+ | and if moreover the isotropy group $ G _{x} $ | ||
+ | is connected, then $ G (k) $ | ||
+ | acts transitively on $ M (k) $ . | ||
+ | In the general case, the study of the $ k $ - | ||
+ | rational points in $ M $ | ||
+ | reduces to problems in the theory of [[Galois cohomology|Galois cohomology]] (see [[#References|[2]]]). | ||
− | A homogeneous space of an algebraic group | + | A homogeneous space of an algebraic group $ G $ |
+ | is always a smooth quasi-projective variety (see [[#References|[5]]]). If $ G $ | ||
+ | is an affine algebraic group, then the variety $ G / H $ | ||
+ | is projective if and only if $ H $ | ||
+ | is a [[Parabolic subgroup|parabolic subgroup]] in $ G $ ( | ||
+ | see [[#References|[3]]]). If $ G $ | ||
+ | is reductive, then the variety $ G / H $ | ||
+ | is affine if and only if the subgroup $ H $ | ||
+ | is reductive (see [[Matsushima criterion|Matsushima criterion]]). A description is also known of the closed subgroups $ H $ | ||
+ | of a linear algebraic group $ G $ | ||
+ | over an algebraically closed field of characteristic 0 for which $ G / H $ | ||
+ | is quasi-affine (see [[#References|[4]]], [[#References|[6]]]). | ||
====References==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.A. Sukhanov, "A description of observable subgroups of linear algebraic groups" ''Math. USSR-Sb.'' , '''68''' (Forthcoming) ''Mat. Sb.'' , '''137''' : 1 (1988) pp. 90–102</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W. Chow, "On the projective embedding of homogeneous varieties" , ''Algebraic topology; symposium in honour of S. Lefschetz'' , Princeton Univ. Press (1957) pp. 122–128 {{MR|0084851}} {{ZBL|0091.33302}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) {{MR|0620024}} {{ZBL|0589.20025}} </TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) {{MR|0180551}} {{ZBL|0128.26303}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> A.A. Sukhanov, "A description of observable subgroups of linear algebraic groups" ''Math. USSR-Sb.'' , '''68''' (Forthcoming) ''Mat. Sb.'' , '''137''' : 1 (1988) pp. 90–102</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> W. Chow, "On the projective embedding of homogeneous varieties" , ''Algebraic topology; symposium in honour of S. Lefschetz'' , Princeton Univ. Press (1957) pp. 122–128 {{MR|0084851}} {{ZBL|0091.33302}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) {{MR|0620024}} {{ZBL|0589.20025}} </TD></TR></table> |
Latest revision as of 08:31, 16 December 2019
An algebraic variety $ M $ together with a regular transitive action of an algebraic group $ G $ given on it. If $ x \in M $ , then the isotropy group $ G _{x} $ is closed in $ G $ . Conversely, if $ H $ is a closed subgroup of an algebraic group $ G $ , then the set of left cosets $ G /H $ has the structure of an algebraic variety, making it into a homogeneous space of the algebraic group $ G $ , where the natural mapping $ \pi : \ G \rightarrow G / H $ is regular, separable and has the following universal property: For any morphism $ \phi : \ G \rightarrow X $ constant on cosets, there is a morphism $ \psi : \ G / H \rightarrow X $ such that $ \psi \pi = \phi $ . If $ M $ is any homogeneous space of the algebraic group $ G $ and $ H = G _{x} $ for some $ x \in M $ , then the natural bijection $ \psi : \ G / H \rightarrow M $ is regular, and if the ground field $ K $ has characteristic 0, then $ \psi $ is biregular (see [1], [3]).
Suppose that the connected group $ G $ , the homogeneous space $ M $ and the action of $ G $ on $ M $ are defined over some subfield $ k \subset K $ . Then the group of $ k $ - rational points $ G (k) $ takes $ M (k) $ into itself and $ G (k) _{x} = G _{x} (k) $ for $ x \in M (k) $ . If $ k $ is finite, then $ M (k) \neq \emptyset $ , and if moreover the isotropy group $ G _{x} $ is connected, then $ G (k) $ acts transitively on $ M (k) $ . In the general case, the study of the $ k $ - rational points in $ M $ reduces to problems in the theory of Galois cohomology (see [2]).
A homogeneous space of an algebraic group $ G $ is always a smooth quasi-projective variety (see [5]). If $ G $ is an affine algebraic group, then the variety $ G / H $ is projective if and only if $ H $ is a parabolic subgroup in $ G $ ( see [3]). If $ G $ is reductive, then the variety $ G / H $ is affine if and only if the subgroup $ H $ is reductive (see Matsushima criterion). A description is also known of the closed subgroups $ H $ of a linear algebraic group $ G $ over an algebraically closed field of characteristic 0 for which $ G / H $ is quasi-affine (see [4], [6]).
References
[1] | A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201 |
[2] | J.-P. Serre, "Cohomologie Galoisienne" , Springer (1964) MR0180551 Zbl 0128.26303 |
[3] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) MR0396773 Zbl 0325.20039 |
[4] | A.A. Sukhanov, "A description of observable subgroups of linear algebraic groups" Math. USSR-Sb. , 68 (Forthcoming) Mat. Sb. , 137 : 1 (1988) pp. 90–102 |
[5] | W. Chow, "On the projective embedding of homogeneous varieties" , Algebraic topology; symposium in honour of S. Lefschetz , Princeton Univ. Press (1957) pp. 122–128 MR0084851 Zbl 0091.33302 |
[6] | G.P. Hochschild, "Basic theory of algebraic groups and Lie algebras" , Springer (1981) MR0620024 Zbl 0589.20025 |
Homogeneous space of an algebraic group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Homogeneous_space_of_an_algebraic_group&oldid=44250