Résumé
In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today. One of the first uses for the theory was to define the Chevalley groups. For a positive integer , the general linear group over a field , consisting of all invertible matrices, is a linear algebraic group over . It contains the subgroups consisting of matrices of the form, resp., and . The group is an example of a unipotent linear algebraic group, the group is an example of a solvable algebraic group called the Borel subgroup of . It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of is conjugated into . Any unipotent subgroup can be conjugated into . Another algebraic subgroup of is the special linear group of matrices with determinant 1. The group is called the multiplicative group, usually denoted by . The group of -points is the multiplicative group of nonzero elements of the field . The additive group , whose -points are isomorphic to the additive group of , can also be expressed as a matrix group, for example as the subgroup in : These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups).
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Publications associées (25)
Concepts associés (75)
Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation where is the transpose of . Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).
Variété de drapeaux généralisée
En mathématiques, une variété de drapeaux généralisée ou tordue est un espace homogène d'un groupe (algébrique ou de Lie) qui généralise les espaces projectifs, les grassmanniennes, les quadriques projectives et l'espace de tous les drapeaux de signature donnée d'un espace vectoriel. La plupart des espaces homogènes de points ou de figures de la géométrie classique sont des variétés de drapeaux généralisées ou des espaces symétriques ou des variétés symétriques (analogues en géométrie algébrique des espaces symétriques), ou leur sont liés.
Groupe réductif
En mathématiques, un groupe réductif est un groupe algébrique G sur un corps algébriquement clos tel que le radical unipotent de G (c'est-à-dire le sous-groupe des éléments unipotents de ) soit trivial. Tout est réductif, de même que tout tore algébrique et tout groupe général linéaire. Plus généralement, sur un corps k non nécessairement algébriquement clos, un groupe réductif est un groupe algébrique affine lisse G tel que le radical unipotent de G sur la clôture algébrique de k soit trivial.
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Cours associés (27)
MATH-111(a): Linear Algebra
L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.
MATH-535: Topics in algebraic geometry
This course is aimed to give students an introduction to the theory of algebraic curves and surfaces. In particular, it aims to develop the students' geometric intuition and combined with the basic al
MATH-479: Linear algebraic groups
The aim of the course is to give an introduction to linear algebraic groups and to give an insight into a beautiful subject that combines algebraic geometry with group theory.
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MOOCs associés (9)
Algèbre Linéaire (Partie 1)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Algèbre Linéaire (Partie 1)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
Algèbre Linéaire (Partie 2)
Un MOOC francophone d'algèbre linéaire accessible à tous, enseigné de manière rigoureuse et ne nécessitant aucun prérequis.
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