In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial (i.e., the group of unipotent elements of the radical of G). More generally, over fields that are not necessarily algebraically closed, a reductive group is a smooth, affine algebraic group such that the unipotent radical of G over the algebraic closure is trivial. The intervention of an algebraic closure in this definition is necessary to include the case of imperfect ground fields, such as local and global function fields over finite fields. Algebraic groups over (possibly imperfect) fields k such that the k-unipotent radical is trivial are called pseudo-reductive groups.
The name comes from the complete reducibility of linear representations of such a group, which is a property in fact holding only for representations of the algebraic group over fields of characteristic zero. (This only applies to representations of the algebraic group: finite-dimensional representations of the underlying discrete group need not be completely reducible even in characteristic 0.) Haboush's theorem shows that a certain rather weaker property called geometric reductivity holds for reductive groups in the positive characteristic case.
- Any semisimple algebraic group is reductive.
- Every algebraic torus is a reductive group.
- Any smooth subgroup of which acts irreducibly on is a reductive algebraic group.
- In particular, and are reductive groups (the latter being even semisimple).
Over all algebraically closed field, Chevalley classified the reductive algebraic groups, obtaining a classification similar to that of compact Lie groups, with simple groups of types An, Bn, Cn, Dn, E6, E7, E8, F4, G2. At the time it was considered surprising that this classification still worked in non-zero characteristic, because in characteristic 0 the classification of simple Lie groups depended on the classification of simple Lie algebras, and in non-zero characteristic there are many simple Lie algebras with no analogs in characteristic 0. Chevalley's classification shows that these extra Lie algebras in nonzero characteristic do not correspond to algebraic groups.
Lie group case
More generally, in the case of Lie groups, a reductive Lie group G can be defined in terms of its Lie algebra, namely a reductive Lie group is one whose Lie algebra g is a reductive Lie algebra; concretely, a Lie algebra that is the sum of an abelian and a semisimple Lie algebra. Sometimes the condition that the identity component G0 of G is of finite index is added.
A Lie algebra is reductive if and only if its adjoint representation is completely reducible, but this does not imply that all of its finite-dimensional representations are completely reducible. The concept of reductive is not quite the same for Lie groups as it is for algebraic groups because a reductive Lie group can be the group of real points of a unipotent algebraic group.
For example, the one-dimensional, abelian Lie algebra R is reductive, and is the Lie algebra of both a reductive algebraic group Gm (the multiplicative group of nonzero real numbers) and also a unipotent (non-reductive) algebraic group Ga (the additive group of real numbers). These are not isomorphic as algebraic groups; at the Lie algebra level we see the same structure, but this is not enough to make any stronger assertion (essentially because the exponential map is not an algebraic function).
- See Springer 1998, exercise 2.4.15
- Borel, Armand (1991), Linear Algebraic Groups, Graduate Texts in Mathematics, 126 (2nd ed.), New York: Springer-Verlag, ISBN 978-0-387-97370-8.
- A. Borel, J. Tits, Groupes réductifs Publ. Math. IHES, 27 (1965) pp. 55–150; Compléments à l'article «Groupes réductifs». Publications Mathématiques de l'IHÉS, 41 (1972), p. 253–276
- Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local : I. Données radicielles valuées. Publications Mathématiques de l'IHÉS, 41 (1972), p. 5–251 II. Schémas en groupes. Existence d'une donnée radicielle valuée. Publications Mathématiques de l'IHÉS, 60 (1984), p. 5–184
- V.L. Popov (2001), "Reductive group", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- A.L. Onishchik (2001), "Lie algebra, reductive", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Springer, Tonny A. (1979), "Reductive groups" (PDF), Automorphic forms, representations, and L-functions, 1, pp. 3–27, ISBN 0-8218-3347-2
- Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713