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) |
[2a] | J.A. Dieudonné, "Invariant theory: old and new" , Acad. Press (1971) |
[2b] | D. Mumford, "Geometric invariant theory" , Springer (1965) |
[2c] | D. Mumford, "Projective invariants of projective structures and applications" , Proc. Internat. Congress mathematicians (Stockholm, 1962) , Inst. Mittag-Leffler (1963) pp. 526–530 |
[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 |
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=13276