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- Reductive_group abstract "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 ≤ GLn is a smooth closed -subgroup that acts irreducibly on affine -space over , then G is reductive. It follows that GLn and SLn are reductive (the latter being even semisimple).".
- Reductive_group wikiPageExternalLink pspum331.
- Reductive_group wikiPageExternalLink pspum331-ptI-1.pdf.
- Reductive_group wikiPageExternalLink item?id=PMIHES_1965__27__55_0.
- Reductive_group wikiPageExternalLink item?id=PMIHES_1972__41__253_0.
- Reductive_group wikiPageExternalLink item?id=PMIHES_1972__41__5_0.
- Reductive_group wikiPageExternalLink item?id=PMIHES_1984__60__5_0.
- Reductive_group wikiPageExternalLink 978-0-387-97370-8.
- Reductive_group wikiPageID "1088386".
- Reductive_group wikiPageRevisionID "602394655".
- Reductive_group author Vladimir_Leonidovich_Popov.
- Reductive_group author "A.L. Onishchik".
- Reductive_group hasPhotoCollection Reductive_group.
- Reductive_group id "R/r080440".
- Reductive_group id "l/l058500".
- Reductive_group title "Lie algebra, reductive".
- Reductive_group title "Reductive group".
- Reductive_group subject Category:Lie_groups.
- Reductive_group subject Category:Representation_theory_of_algebraic_groups.
- Reductive_group type Abstraction100002137.
- Reductive_group type Group100031264.
- Reductive_group type LieGroups.
- Reductive_group comment "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.".
- Reductive_group label "Groupe réductif".
- Reductive_group label "Grupo reductivo".
- Reductive_group label "Reductive group".
- Reductive_group label "Редуктивная группа".
- Reductive_group label "約化群".
- Reductive_group sameAs Grupo_reductivo.
- Reductive_group sameAs Groupe_réductif.
- Reductive_group sameAs m.044yqm.
- Reductive_group sameAs Q1006450.
- Reductive_group sameAs Q1006450.
- Reductive_group sameAs Reductive_group.
- Reductive_group wasDerivedFrom Reductive_group?oldid=602394655.
- Reductive_group isPrimaryTopicOf Reductive_group.