Reductive group
A linear algebraic group (over an algebraically closed field
) that satisfies one of the following equivalent conditions: 1) the radical of the connected component
of the unit element of
is an algebraic torus; 2) the unipotent radical of the group
is trivial; or 3) the group
is a product of closed normal subgroups
and
that are a semi-simple algebraic group and an algebraic torus, respectively. In this case
is the commutator subgroup of
and
coincides with the radical of
as well as with the connected component of the unit element of its centre;
is finite, and any semi-simple or unipotent subgroup of the group
is contained in
.
A linear algebraic group is called linearly reductive if either of the two following equivalent conditions is fulfilled: a) each rational linear representation of
is completely reducible (cf. Reducible representation); or b) for each rational linear representation
and any
-invariant vector
there is a
-invariant linear function
on
such that
. Any linearly reductive group is reductive. If the characteristic of the field
is 0, the converse is true. This is not the case when
: A connected linearly reductive group is an algebraic torus. However, even in the general case, a reductive group can be described in terms of its representation theory. A linear algebraic group
is called geometrically reductive (or semi-reductive) if for each rational linear representation
and any
-invariant vector
there is a non-constant
-invariant polynomial function
on
such that
. A linear algebraic group is reductive if and only if it is geometrically reductive (see Mumford hypothesis).
The generalized Hilbert theorem on invariants is true for reductive groups. The converse is also true: If is a linear algebraic group over an algebraically closed field
and if for any locally finite-dimensional rational representation by automorphisms of a finitely-generated associative commutative
-algebra
with identity the algebra of invariants
is finitely generated, then
is reductive (see [4]).
Any finite linear group is reductive and if its order is not divisible by , then it is also linearly reductive. Connected reductive groups have a structure theory that is largely similar to the structure theory of reductive Lie algebras (root system; Weyl group, etc., see [2]). This theory extends to groups
where
is a connected reductive group defined over a subfield
and
is the group of its
-rational points (see [3]). In this case the role of Borel subgroups (cf. Borel subgroup), maximal tori (cf. Maximal torus) and Weyl groups is played by minimal parabolic subgroups (cf. Parabolic subgroup) defined over
, maximal tori split over
, and relative Weyl groups (see Weyl group), respectively. Any two minimal parabolic subgroups of
that are defined over
are conjugate by an element of
; this is also true for any two maximal
-split tori of
.
If is a connected reductive group defined over a field
, then
is a split group over a separable extension of finite degree of
; if, in addition,
is an infinite field, then
is dense in
in the Zariski topology. If
is a reductive group and
is a closed subgroup of it, then the quotient space
is affine if and only if
is reductive. A linear algebraic group over a field of characteristic 0 is reductive if and only if its Lie algebra is a reductive Lie algebra (cf. Lie algebra, reductive). If
, this is also equivalent to
being the complexification of a compact Lie group (see Complexification of a Lie group).
References
[1] | T.A. Springer, "Invariant theory" , Lect. notes in math. , 585 , Springer (1977) |
[2] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) |
[3] | A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 |
[4] | V.L. Popov, "Hilbert's theorem on invariants" Soviet Math. Dokl. , 20 : 6 (1979) pp. 1318–1322 Dokl. Akad. Nauk SSSR , 249 : 3 (1979) pp. 551–555 |
Reductive group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Reductive_group&oldid=21921