In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.
Together with the commutative Lie group of the real numbers, , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1.
The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification.
Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether is a simple Lie group.
The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closed connected normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial center, but is not simple.
In this article the connected simple Lie groups with trivial center are listed. Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a universal cover, whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.
An equivalent definition of a simple Lie group follows from the Lie correspondence: A connected Lie group is simple if its Lie algebra is simple.
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vignette|Le polytope de Gosset : les 240 vecteurs du système de racines En mathématiques, est le plus grand groupe de Lie complexe de type exceptionnel. Son algèbre de Lie est notée . E est de rang 8 et de dimension 248. Il est simplement connexe et son centre est trivial. La structure E a été découverte en 1887 par le mathématicien norvégien Sophus Lie pour étudier la symétrie et jusqu’ici personne ne pensait que cet objet mathématique pourrait être compris, considère , responsable de l’équipe qui réunit 18 mathématiciens et programmeurs dans le monde, dont Fokko du Cloux et .
In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension.
En mathématiques, E6 est le nom d'un groupe de Lie ; son algèbre de Lie est notée . Il s'agit de l'un des cinq groupes de Lie complexes de type exceptionnel. E6 est de rang 6 et de dimension 78. Le groupe fondamental de sa forme compacte est le groupe cyclique Z3 et son groupe d'automorphismes est le groupe cyclique Z2. Sa représentation fondamentale est de dimension complexe 27. Sa représentation duale est également de dimension 27. Une certaine forme non compacte réelle de E6 est le groupe des collinéations du plan projectif octonionique OP2, ou plan de Cayley.
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Lie groups are manifolds with a group structure. The interaction between the geometric and the algebraic structure of these objects gives rise to a rich and beautiful subject with various applications
We will establish the major results in the representation theory of semisimple Lie algebras over the field of complex numbers, and that of the related algebraic groups.
Let Y be a simply connected simple algebraic group over an algebraically closed field k of characteristic p and let X be a maximal closed connected simple subgroup of Y.
Excluding some smal
We consider a natural subclass of harmonic maps from a surface into G/T, namely cyclic primitive maps. Here G is any simple real Lie group (not necessarily compact), T is a Cartan subgroup and both ar
Elsevier2017
Let be a simple exceptional algebraic group of adjoint type over an algebraically closed field of characteristic and let be a subgroup of containing a regular unipotent element of . By a theorem of Te