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A [[Linear algebraic group|linear algebraic group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804401.png" /> (over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804402.png" />) that satisfies one of the following equivalent conditions: 1) the radical of the connected component <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804403.png" /> of the unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804404.png" /> is an [[Algebraic torus|algebraic torus]]; 2) the unipotent radical of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804405.png" /> is trivial; or 3) the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804406.png" /> is a product of closed normal subgroups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804408.png" /> that are a [[Semi-simple algebraic group|semi-simple algebraic group]] and an algebraic torus, respectively. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r0804409.png" /> is the commutator subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044011.png" /> coincides with the radical of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044012.png" /> as well as with the connected component of the unit element of its centre; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044013.png" /> is finite, and any semi-simple or unipotent subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044014.png" /> is contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044015.png" />.
+
{{MSC|20G}}
 +
{{TEX|done}}
  
A linear algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044016.png" /> is called linearly reductive if either of the two following equivalent conditions is fulfilled: a) each rational linear representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044017.png" /> is completely reducible (cf. [[Reducible representation|Reducible representation]]); or b) for each rational linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044018.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044019.png" />-invariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044020.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044021.png" />-invariant linear function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044022.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044023.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044024.png" />. Any linearly reductive group is reductive. If the characteristic of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044025.png" /> is 0, the converse is true. This is not the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044026.png" />: 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044027.png" /> is called geometrically reductive (or semi-reductive) if for each rational linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044028.png" /> and any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044029.png" />-invariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044030.png" /> there is a non-constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044031.png" />-invariant polynomial function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044032.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044033.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044034.png" />. A linear algebraic group is reductive if and only if it is geometrically reductive (see [[Mumford hypothesis|Mumford hypothesis]]).
+
A ''reductive group'' is a
 +
[[Linear algebraic group|linear algebraic group]] $G$ (over an
 +
algebraically closed field $K$) that satisfies one of the following
 +
equivalent conditions:  
  
The generalized [[Hilbert theorem|Hilbert theorem]] on invariants is true for reductive groups. The converse is also true: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044035.png" /> is a linear algebraic group over an algebraically closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044036.png" /> and if for any locally finite-dimensional rational representation by automorphisms of a finitely-generated associative commutative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044037.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044038.png" /> with identity the algebra of invariants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044039.png" /> is finitely generated, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044040.png" /> is reductive (see [[#References|[4]]]).
+
1) the [[Radical of a group|radical]] of the
 +
[[Connected component of the identity|connected component]] $G^0$ of the unit element of $G$ is an
 +
[[Algebraic torus|algebraic torus]];
  
Any finite linear group is reductive and if its order is not divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044041.png" />, 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|root system]]; [[Weyl group|Weyl group]], etc., see [[#References|[2]]]). This theory extends to groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044042.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044043.png" /> is a connected reductive group defined over a subfield <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044044.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044045.png" /> is the group of its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044046.png" />-rational points (see [[#References|[3]]]). In this case the role of Borel subgroups (cf. [[Borel subgroup|Borel subgroup]]), maximal tori (cf. [[Maximal torus|Maximal torus]]) and Weyl groups is played by minimal parabolic subgroups (cf. [[Parabolic subgroup|Parabolic subgroup]]) defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044047.png" />, maximal tori split over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044048.png" />, and relative Weyl groups (see [[Weyl group|Weyl group]]), respectively. Any two minimal parabolic subgroups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044049.png" /> that are defined over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044050.png" /> are conjugate by an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044051.png" />; this is also true for any two maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044052.png" />-split tori of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044053.png" />.
+
2) the unipotent radical of the group $G^0$ is trivial; or
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044054.png" /> is a connected reductive group defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044055.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044056.png" /> is a split group over a separable extension of finite degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044057.png" />; if, in addition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044058.png" /> is an infinite field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044059.png" /> is dense in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044060.png" /> in the Zariski topology. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044061.png" /> is a reductive group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044062.png" /> is a closed subgroup of it, then the quotient space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044063.png" /> is affine if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044064.png" /> 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|Lie algebra, reductive]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044065.png" />, this is also equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r080/r080440/r08044066.png" /> being the complexification of a compact Lie group (see [[Complexification of a Lie group|Complexification of a Lie group]]).
+
3) the group $G^0$ is a product of closed normal subgroups $S$ and $T$ that are a
 +
[[Semi-simple algebraic group|semi-simple algebraic group]] and an
 +
algebraic torus, respectively.
 +
 
 +
In this case $S$ is the commutator subgroup of $G^0$ and $T$ coincides with the radical of $G^0$ as well as with the connected component of the unit element of its centre; $S\cap T$ is finite, and any semi-simple or unipotent subgroup of the group $G^0$ is contained in $S$.
 +
 
 +
A linear algebraic group $G$ is called linearly reductive if either of
 +
the two following equivalent conditions is fulfilled:
 +
 
 +
a) each rational linear representation of $G$ is completely reducible (cf.
 +
[[Reducible representation|Reducible representation]]); or
 +
 
 +
b) for each
 +
rational linear representation $\rho: G\to \def\GL{ {\rm GL}}\GL(W)$
 +
and any $\rho(G)$-invariant vector $w\in W\setminus\{0\}$ there is a
 +
$\rho(G)$-invariant linear function $f$ on $W$ such that $f(w)\ne
 +
0$.
 +
 
 +
Any linearly reductive group is reductive. If the characteristic
 +
of the field $K$ is 0, the converse is true. This is not the case when
 +
$\def\char{ {\rm char}\;}\char K > 0$: 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 $G$ is called geometrically reductive (or semi-reductive) if for each rational linear representation $\rho: G\to \GL(W)$ and any $\rho(G)$-invariant vector $w\in W\setminus\{0\}$ there is a non-constant $\rho(G)$-invariant polynomial function $f$ on $W$ such that $f(w)\ne 0$. A linear algebraic group is reductive if and only if it is geometrically reductive (see
 +
[[Mumford hypothesis|Mumford hypothesis]]).
 +
 
 +
The generalized
 +
[[Hilbert theorem|Hilbert theorem]] on invariants is true for reductive groups. The converse is also true: If $G$ is a linear algebraic group over an algebraically closed field $K$ and if for any locally finite-dimensional rational representation by automorphisms of a finitely-generated associative commutative $K$-algebra $A$ with identity the algebra of invariants $A^G$ is finitely generated, then $G$ is reductive (see
 +
{{Cite|Po}}).
 +
 
 +
Any finite linear group is reductive and if its order is not divisible by $\char K$, 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|root system]];
 +
[[Weyl group|Weyl group]], etc., see
 +
{{Cite|Hu}}). This theory extends to groups $G_k$ where $G$ is a connected reductive group defined over a subfield $k\subset K$ and $G_k$ is the group of its $k$-rational points (see
 +
{{Cite|BoTi}}). In this case the role of Borel subgroups (cf.
 +
[[Borel subgroup|Borel subgroup]]), maximal tori (cf.
 +
[[Maximal torus|Maximal torus]]) and Weyl groups is played by minimal parabolic subgroups (cf.
 +
[[Parabolic subgroup|Parabolic subgroup]]) defined over $k$, maximal tori split over $k$, and relative Weyl groups (see
 +
[[Weyl group|Weyl group]]), respectively. Any two minimal parabolic subgroups of $G$ that are defined over $k$ are conjugate by an element of $G_k$; this is also true for any two maximal $k$-split tori of $G$.
 +
 
 +
If $G$ is a connected reductive group defined over a field $k$, then $G$ is a split group over a separable extension of finite degree of $k$; if, in addition, $k$ is an infinite field, then $G_k$ is dense in $G$ in the Zariski topology. If $G$ is a reductive group and $H$ is a closed subgroup of it, then the quotient space $G/H$ is affine if and only if $H$ 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|Lie algebra, reductive]]). If $K=\C$, this is also equivalent to $G$ being the complexification of a compact Lie group (see
 +
[[Complexification of a Lie group|Complexification of a Lie group]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> T.A. Springer, "Invariant theory" , ''Lect. notes in math.'' , '''585''' , Springer (1977) {{MR|0447428}} {{ZBL|0346.20020}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> 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 {{MR|}} {{ZBL|}} </TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|BoTi}}||valign="top"| A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'', '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}}
 +
|-
 +
|valign="top"|{{Ref|Hu}}||valign="top"| J.E. Humphreys, "Linear algebraic groups", Springer (1975) {{MR|0396773}} {{ZBL|0325.20039}}
 +
|-
 +
|valign="top"|{{Ref|Po}}||valign="top"| 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 {{MR|}} {{ZBL|}}
 +
|-
 +
|valign="top"|{{Ref|Sp}}||valign="top"| T.A. Springer, "Invariant theory", ''Lect. notes in math.'', '''585''', Springer (1977) {{MR|0447428}} {{ZBL|0346.20020}}
 +
|-
 +
|}

Latest revision as of 13:15, 23 November 2013

2010 Mathematics Subject Classification: Primary: 20G [MSN][ZBL]

A reductive group is a linear algebraic group $G$ (over an algebraically closed field $K$) that satisfies one of the following equivalent conditions:

1) the radical of the connected component $G^0$ of the unit element of $G$ is an algebraic torus;

2) the unipotent radical of the group $G^0$ is trivial; or

3) the group $G^0$ is a product of closed normal subgroups $S$ and $T$ that are a semi-simple algebraic group and an algebraic torus, respectively.

In this case $S$ is the commutator subgroup of $G^0$ and $T$ coincides with the radical of $G^0$ as well as with the connected component of the unit element of its centre; $S\cap T$ is finite, and any semi-simple or unipotent subgroup of the group $G^0$ is contained in $S$.

A linear algebraic group $G$ is called linearly reductive if either of the two following equivalent conditions is fulfilled:

a) each rational linear representation of $G$ is completely reducible (cf. Reducible representation); or

b) for each rational linear representation $\rho: G\to \def\GL{ {\rm GL}}\GL(W)$ and any $\rho(G)$-invariant vector $w\in W\setminus\{0\}$ there is a $\rho(G)$-invariant linear function $f$ on $W$ such that $f(w)\ne 0$.

Any linearly reductive group is reductive. If the characteristic of the field $K$ is 0, the converse is true. This is not the case when $\def\char{ {\rm char}\;}\char K > 0$: 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 $G$ is called geometrically reductive (or semi-reductive) if for each rational linear representation $\rho: G\to \GL(W)$ and any $\rho(G)$-invariant vector $w\in W\setminus\{0\}$ there is a non-constant $\rho(G)$-invariant polynomial function $f$ on $W$ such that $f(w)\ne 0$. 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 $G$ is a linear algebraic group over an algebraically closed field $K$ and if for any locally finite-dimensional rational representation by automorphisms of a finitely-generated associative commutative $K$-algebra $A$ with identity the algebra of invariants $A^G$ is finitely generated, then $G$ is reductive (see [Po]).

Any finite linear group is reductive and if its order is not divisible by $\char K$, 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 [Hu]). This theory extends to groups $G_k$ where $G$ is a connected reductive group defined over a subfield $k\subset K$ and $G_k$ is the group of its $k$-rational points (see [BoTi]). 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 $k$, maximal tori split over $k$, and relative Weyl groups (see Weyl group), respectively. Any two minimal parabolic subgroups of $G$ that are defined over $k$ are conjugate by an element of $G_k$; this is also true for any two maximal $k$-split tori of $G$.

If $G$ is a connected reductive group defined over a field $k$, then $G$ is a split group over a separable extension of finite degree of $k$; if, in addition, $k$ is an infinite field, then $G_k$ is dense in $G$ in the Zariski topology. If $G$ is a reductive group and $H$ is a closed subgroup of it, then the quotient space $G/H$ is affine if and only if $H$ 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 $K=\C$, this is also equivalent to $G$ being the complexification of a compact Lie group (see Complexification of a Lie group).

References

[BoTi] A. Borel, J. Tits, "Groupes réductifs" Publ. Math. IHES, 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402
[Hu] J.E. Humphreys, "Linear algebraic groups", Springer (1975) MR0396773 Zbl 0325.20039
[Po] 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
[Sp] T.A. Springer, "Invariant theory", Lect. notes in math., 585, Springer (1977) MR0447428 Zbl 0346.20020
How to Cite This Entry:
Reductive group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Reductive_group&oldid=21994
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article