Reductive algebraic group
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.
A simple non-example of a reductive group is an abelian variety (hence an elliptic curve).
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.
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