Reductive 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). Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group. 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 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.

If G ≤ GL_n is a smooth closed -subgroup that acts irreducibly on affine -space over, then G is reductive. It follows that GL_n and SL_n are reductive (the latter being even semisimple).