View source for Reductive group

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" />.

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]]).

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]]]).

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" />.

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]]).

====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>