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- Linear_algebraic_group abstract "In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation MTM = I where MT is the transpose of M.The main examples of linear algebraic groups are certain Lie groups, where the underlying field is the real or complex field. (For example, every compact Lie group can be regarded as the group of points of a real linear algebraic group, essentially by the Peter–Weyl theorem.)These were the first algebraic groups to be extensively studied. Such groups were known for a long time before their abstract algebraic theory was developed according to the needs of major applications. Compact Lie groups were considered by Élie Cartan, Ludwig Maurer, Wilhelm Killing, and Sophus Lie in the 1880s and 1890s in the context of differential equations and Galois theory. However, a purely algebraic theory was first developed by Kolchin (1948), with Armand Borel as one of its pioneers. The Picard–Vessiot theory did lead to algebraic groups.The first basic theorem of the subject is that any affine algebraic group is a linear algebraic group: that is, any affine variety V that has an algebraic group law has a faithful linear representation, over the same field, which is also a morphism of varieties. For example the additive group of an n-dimensional vector space has a faithful representation as (n+1)×(n+1) matrices.One can define the Lie algebra of an algebraic group purely algebraically (it consists of the dual number points based at the identity element); and this theorem shows that we get a matrix Lie algebra. A linear algebraic group G consists of a finite number of irreducible components, that are in fact also the connected components: the one Go containing the identity will be a normal subgroup of G.One of the first uses for the theory was to define the Chevalley groups.".
- Linear_algebraic_group wikiPageExternalLink luminysga3.pdf.
- Linear_algebraic_group wikiPageID "395885".
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- Linear_algebraic_group wikiPageWikiLink Affine_variety.
- Linear_algebraic_group wikiPageWikiLink Algebraic_closure.
- Linear_algebraic_group wikiPageWikiLink Algebraic_group.
- Linear_algebraic_group wikiPageWikiLink Algebraic_number_theory.
- Linear_algebraic_group wikiPageWikiLink Algebraic_torus.
- Linear_algebraic_group wikiPageWikiLink Annals_of_Mathematics.
- Linear_algebraic_group wikiPageWikiLink Armand_Borel.
- Linear_algebraic_group wikiPageWikiLink Borel_fixed-point_theorem.
- Linear_algebraic_group wikiPageWikiLink Borel_subgroup.
- Linear_algebraic_group wikiPageWikiLink Brian_Conrad.
- Linear_algebraic_group wikiPageWikiLink Category:Algebraic_groups.
- Linear_algebraic_group wikiPageWikiLink Chevalley_group.
- Linear_algebraic_group wikiPageWikiLink Complete_variety.
- Linear_algebraic_group wikiPageWikiLink Complex_number.
- Linear_algebraic_group wikiPageWikiLink Composition_series.
- Linear_algebraic_group wikiPageWikiLink Covering_space.
- Linear_algebraic_group wikiPageWikiLink Cyclic_group.
- Linear_algebraic_group wikiPageWikiLink Differential_Galois_theory.
- Linear_algebraic_group wikiPageWikiLink Differential_equation.
- Linear_algebraic_group wikiPageWikiLink Dual_number.
- Linear_algebraic_group wikiPageWikiLink Field_(mathematics).
- Linear_algebraic_group wikiPageWikiLink Fundamental_group.
- Linear_algebraic_group wikiPageWikiLink Galois_theory.
- Linear_algebraic_group wikiPageWikiLink Group_of_Lie_type.
- Linear_algebraic_group wikiPageWikiLink Identity_component.
- Linear_algebraic_group wikiPageWikiLink Infinite_cyclic.
- Linear_algebraic_group wikiPageWikiLink Lie_algebra.
- Linear_algebraic_group wikiPageWikiLink Lie_group.
- Linear_algebraic_group wikiPageWikiLink Lie–Kolchin_theorem.
- Linear_algebraic_group wikiPageWikiLink Linear_representation.
- Linear_algebraic_group wikiPageWikiLink Ludwig_Maurer.
- Linear_algebraic_group wikiPageWikiLink Mathematics.
- Linear_algebraic_group wikiPageWikiLink Matrix_(mathematics).
- Linear_algebraic_group wikiPageWikiLink Matrix_multiplication.
- Linear_algebraic_group wikiPageWikiLink Maximal_torus.
- Linear_algebraic_group wikiPageWikiLink Normal_subgroup.
- Linear_algebraic_group wikiPageWikiLink Orthogonal_group.
- Linear_algebraic_group wikiPageWikiLink Peter–Weyl_theorem.
- Linear_algebraic_group wikiPageWikiLink Picard–Vessiot_theory.
- Linear_algebraic_group wikiPageWikiLink Polynomial.
- Linear_algebraic_group wikiPageWikiLink Projective_variety.
- Linear_algebraic_group wikiPageWikiLink Real_number.
- Linear_algebraic_group wikiPageWikiLink Reductive_group.
- Linear_algebraic_group wikiPageWikiLink Representation_theory.
- Linear_algebraic_group wikiPageWikiLink Root_system.
- Linear_algebraic_group wikiPageWikiLink SL2(R).
- Linear_algebraic_group wikiPageWikiLink Solvable_Lie_algebra.
- Linear_algebraic_group wikiPageWikiLink Solvable_Lie_group.
- Linear_algebraic_group wikiPageWikiLink Sophus_Lie.
- Linear_algebraic_group wikiPageWikiLink Subgroup.
- Linear_algebraic_group wikiPageWikiLink Transpose.
- Linear_algebraic_group wikiPageWikiLink Unipotent.
- Linear_algebraic_group wikiPageWikiLink Unipotent_group.
- Linear_algebraic_group wikiPageWikiLink Universal_cover.
- Linear_algebraic_group wikiPageWikiLink Vector_space.
- Linear_algebraic_group wikiPageWikiLink Wilhelm_Killing.
- Linear_algebraic_group wikiPageWikiLink Élie_Cartan.
- Linear_algebraic_group wikiPageWikiLinkText "Additive group".
- Linear_algebraic_group wikiPageWikiLinkText "Linear Algebraic Groups".
- Linear_algebraic_group wikiPageWikiLinkText "Linear algebraic group".
- Linear_algebraic_group wikiPageWikiLinkText "algebraic group".
- Linear_algebraic_group wikiPageWikiLinkText "algebraic".
- Linear_algebraic_group wikiPageWikiLinkText "anisotropic".
- Linear_algebraic_group wikiPageWikiLinkText "isotropic algebraic group".
- Linear_algebraic_group wikiPageWikiLinkText "linear algebraic group".
- Linear_algebraic_group hasPhotoCollection Linear_algebraic_group.
- Linear_algebraic_group id "p/l059070".
- Linear_algebraic_group title "Linear algebraic group".
- Linear_algebraic_group wikiPageUsesTemplate Template:Citation.
- Linear_algebraic_group wikiPageUsesTemplate Template:Group_theory_sidebar.
- Linear_algebraic_group wikiPageUsesTemplate Template:Harvs.
- Linear_algebraic_group wikiPageUsesTemplate Template:Springer.
- Linear_algebraic_group subject Category:Algebraic_groups.
- Linear_algebraic_group hypernym Subgroup.
- Linear_algebraic_group type EthnicGroup.
- Linear_algebraic_group type Group.
- Linear_algebraic_group type Group.
- Linear_algebraic_group type Variety.
- Linear_algebraic_group comment "In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation MTM = I where MT is the transpose of M.The main examples of linear algebraic groups are certain Lie groups, where the underlying field is the real or complex field.".
- Linear_algebraic_group label "Linear algebraic group".
- Linear_algebraic_group sameAs Strop_algébrich_linear.
- Linear_algebraic_group sameAs m.0238_j.
- Linear_algebraic_group sameAs Q6553407.
- Linear_algebraic_group sameAs Q6553407.
- Linear_algebraic_group wasDerivedFrom Linear_algebraic_group?oldid=673431682.
- Linear_algebraic_group isPrimaryTopicOf Linear_algebraic_group.