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An algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115201.png" /> acting regularly on an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115202.png" />. More precisely, it is a triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115203.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115204.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115205.png" />) is a morphism of algebraic varieties satisfying the conditions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115206.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115207.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115208.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a0115209.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152010.png" /> is the unit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152011.png" />). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152013.png" /> are defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152014.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152015.png" /> is called an algebraic group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152017.png" />-transformations. For instance, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152019.png" /> is the adjoint action or an action by shifts, is an algebraic group of transformations. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152020.png" /> is an algebraic subgroup in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152022.png" /> is its natural action on the affine space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152023.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152024.png" /> is an algebraic group of transformations. For each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152025.png" /> one denotes by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152026.png" /> the orbit of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152027.png" />, and by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152028.png" /> the stabilizer of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152029.png" />. The orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152030.png" /> need not necessarily be closed in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152031.png" />, but closed orbits exist always, e.g. orbits of minimal dimension are closed. An algebraic group of transformations is sometimes understood to mean a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152032.png" /> which is acting rationally (but not necessarily regularly) on an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152033.png" /> (this means that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152034.png" /> is a rational mapping, and the above properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152035.png" /> are valid for ordinary points). It was shown by A. Weil [[#References|[3]]] that there always exists a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152036.png" />, birationally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152037.png" />, and such that the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152038.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152039.png" /> induced by the rational action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a011/a011520/a01152041.png" /> is regular. The problem of describing the orbits, stabilizers, fields of invariant rational functions (cf. [[Invariants, theory of|Invariants, theory of]]), and of constructing quotient varieties are fundamental in the theory of algebraic groups of transformations and have numerous applications.
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An algebraic group $  G $  acting regularly on an algebraic variety $  V $ . More precisely, it is a triplet $  (G,\  V,\  \tau ) $  where $  \tau : \  G \times V \rightarrow V $  ( $  \tau (g,\  x ) = gx $ ) is a morphism of algebraic varieties satisfying the conditions: $  ex = x $ , $  g(hx) = (gh)x $  for all $  x \in V $  and $  g,\  h \in G $  (where $  e $  is the unit of $  G $ ). If $  G,\  V $  and $  \tau $  are defined over a field $  k $ , then $  ( G,\  V,\  \tau ) $  is called an algebraic group of $  k $ -transformations. For instance, $  ( G,\  G,\  \tau ) $ , where $  \tau $  is the adjoint action or an action by shifts, is an algebraic group of transformations. If $  G $  is an algebraic subgroup in $  \mathop{\rm GL}\nolimits (n) $  and $  \tau $  is its natural action on the affine space $  V = k ^{n} $ , then $  (G,\  V,\  \tau ) $  is an algebraic group of transformations. For each point $  x \in V $  one denotes by $  G(x) = \{ {gx} : {g \in G} \} $  the orbit of $  x $ , and by $  G _{x} = \{ {g \in G} : {gx = x} \} $  the stabilizer of $  x $ . The orbit $  G(x) $  need not necessarily be closed in $  V $ , but closed orbits exist always, e.g. orbits of minimal dimension are closed. An algebraic group of transformations is sometimes understood to mean a group $  G $  which is acting rationally (but not necessarily regularly) on an algebraic variety $  V $  (this means that $  \tau : \  G \times V \rightarrow V $  is a rational mapping, and the above properties of $  \tau $  are valid for ordinary points). It was shown by A. Weil [[#References|[3]]] that there always exists a variety $  V ^{1} $ , birationally isomorphic to $  V $ , and such that the action of $  G $  on $  V ^{1} $  induced by the rational action of $  G $  on $  V $  is regular. The problem of describing the orbits, stabilizers, fields of invariant rational functions (cf. [[Invariants, theory of|Invariants, theory of]]), and of constructing quotient varieties are fundamental in the theory of algebraic groups of transformations and have numerous applications.
  
 
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Latest revision as of 20:27, 12 December 2019

An algebraic group $ G $ acting regularly on an algebraic variety $ V $ . More precisely, it is a triplet $ (G,\ V,\ \tau ) $ where $ \tau : \ G \times V \rightarrow V $ ( $ \tau (g,\ x ) = gx $ ) is a morphism of algebraic varieties satisfying the conditions: $ ex = x $ , $ g(hx) = (gh)x $ for all $ x \in V $ and $ g,\ h \in G $ (where $ e $ is the unit of $ G $ ). If $ G,\ V $ and $ \tau $ are defined over a field $ k $ , then $ ( G,\ V,\ \tau ) $ is called an algebraic group of $ k $ -transformations. For instance, $ ( G,\ G,\ \tau ) $ , where $ \tau $ is the adjoint action or an action by shifts, is an algebraic group of transformations. If $ G $ is an algebraic subgroup in $ \mathop{\rm GL}\nolimits (n) $ and $ \tau $ is its natural action on the affine space $ V = k ^{n} $ , then $ (G,\ V,\ \tau ) $ is an algebraic group of transformations. For each point $ x \in V $ one denotes by $ G(x) = \{ {gx} : {g \in G} \} $ the orbit of $ x $ , and by $ G _{x} = \{ {g \in G} : {gx = x} \} $ the stabilizer of $ x $ . The orbit $ G(x) $ need not necessarily be closed in $ V $ , but closed orbits exist always, e.g. orbits of minimal dimension are closed. An algebraic group of transformations is sometimes understood to mean a group $ G $ which is acting rationally (but not necessarily regularly) on an algebraic variety $ V $ (this means that $ \tau : \ G \times V \rightarrow V $ is a rational mapping, and the above properties of $ \tau $ are valid for ordinary points). It was shown by A. Weil [3] that there always exists a variety $ V ^{1} $ , birationally isomorphic to $ V $ , and such that the action of $ G $ on $ V ^{1} $ induced by the rational action of $ G $ on $ V $ is regular. The problem of describing the orbits, stabilizers, fields of invariant rational functions (cf. Invariants, theory of), and of constructing quotient varieties are fundamental in the theory of algebraic groups of transformations and have numerous applications.

References

[1] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[2a] J.A. Dieudonné, "Invariant theory: old and new" , Acad. Press (1971) MR0279102 Zbl 0258.14011
[2b] D. Mumford, "Geometric invariant theory" , Springer (1965) MR0214602 Zbl 0147.39304
[2c] D. Mumford, "Projective invariants of projective structures and applications" , Proc. Internat. Congress mathematicians (Stockholm, 1962) , Inst. Mittag-Leffler (1963) pp. 526–530 MR0175899 Zbl 0154.20702
[2d] C.S. Seshadri, "Quotient spaces modulo reductive groupes and applications to moduli of vector bundles on algebraic curves" , Proc. Internat. Congress Mathematicians (Nice, 1970) , 1 , Gauthier-Villars (1971) pp. 479–482
[3] A. Weil, "On algebraic groups and homogeneous spaces" Amer. J. Math. , 77 : 2 (1955) pp. 355–391 MR0074084 Zbl 0065.14202


Comments

The notion in question is also called an algebraic transformation space.

How to Cite This Entry:
Algebraic group of transformations. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Algebraic_group_of_transformations&oldid=21847
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article