Concept
Groupe réductif
Concepts associés (37)
Geometric invariant theory
In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory. Geometric invariant theory studies an action of a group G on an algebraic variety (or scheme) X and provides techniques for forming the 'quotient' of X by G as a scheme with reasonable properties.
Loi de réciprocité d'Artin
En mathématiques, la 'loi de réciprocité d'Artin' est un résultat important de théorie des nombres établi par Emil Artin dans une série d'articles publiés entre 1924 et 1930. Au cœur de la théorie du corps de classe, la réciprocité d'Artin tire son nom d'une parenté avec la réciprocité quadratique introduite par Gauss, et d'autres lois d'expression similaire, la réciprocité d'Eisenstein, de Kummer, ou de Hilbert. Une des motivations initiales derrière ce résultat était le neuvième problème de Hilbert, auquel la réciprocité d'Artin apporte une réponse partielle.
Semi-simplicity
In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, , and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of simple objects, and simple objects are those that do not contain non-trivial proper sub-objects. The precise definitions of these words depends on the context. For example, if G is a finite group, then a nontrivial finite-dimensional representation V over a field is said to be simple if the only subrepresentations it contains are either {0} or V (these are also called irreducible representations).
Représentation unitaire
En mathématiques, une représentation unitaire d'un groupe G est une représentation linéaire π de G sur un espace de Hilbert complexe V telle que π(g) est un opérateur unitaire pour tout g ∈ G. La théorie générale est bien développée dans le cas où G est un groupe topologique localement compact (séparé) et les représentations sont fortement continues. La théorie a été largement appliquée en mécanique quantique depuis les années 1920, particulièrement sous l'influence par le livre de 1928 de Hermann Weyl, Gruppentheorie und Quantenmechanik.
Parabolic induction
In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups. If G is a reductive algebraic group and is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of taking a representation of , extending it to P by letting N act trivially, and inducing the result from P to G. There are some generalizations of parabolic induction using cohomology, such as cohomological parabolic induction and Deligne–Lusztig theory.
Identity component
In mathematics, specifically group theory, the identity component of a group G refers to several closely related notions of the largest connected subgroup of G containing the identity element. In point set topology, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group. The identity path component of a topological group G is the path component of G that contains the identity element of the group.
Group scheme
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance.