Algebraic group of transformations
An algebraic group 
acting regularly on an algebraic variety 
. More precisely, it is a triplet 
where 
(
) is a morphism of algebraic varieties satisfying the conditions: 
, 
for all 
and 
(where 
is the unit of 
). If 
and 
are defined over a field 
, then 
is called an algebraic group of 
-transformations. For instance, 
, where 
is the adjoint action or an action by shifts, is an algebraic group of transformations. If 
is an algebraic subgroup in 
and 
is its natural action on the affine space 
, then 
is an algebraic group of transformations. For each point 
one denotes by 
the orbit of 
, and by 
the stabilizer of 
. The orbit 
need not necessarily be closed in 
, 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 
which is acting rationally (but not necessarily regularly) on an algebraic variety 
(this means that 
is a rational mapping, and the above properties of 
are valid for ordinary points). It was shown by A. Weil [3] that there always exists a variety 
, birationally isomorphic to 
, and such that the action of 
on 
induced by the rational action of 
on 
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.
Algebraic group of transformations. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebraic_group_of_transformations&oldid=21790