Connected non-abelian Lie group lacking nontrivial connected normal subgroups
This article is about the Killing-Cartan classification. For a smaller list of groups that commonly occur in theoretical physics, see Table of Lie groups. For groups of dimension at most 3, see Bianchi classification.
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Lie groups and Lie algebras
Classical groups
General linear GL(n)
Special linear SL(n)
Orthogonal O(n)
Special orthogonal SO(n)
Unitary U(n)
Special unitary SU(n)
Symplectic Sp(n)
Simple Lie groups
Classical
An
Bn
Cn
Dn
Exceptional
G2
F4
E6
E7
E8
Other Lie groups
Circle
Lorentz
Poincaré
Conformal group
Diffeomorphism
Loop
Euclidean
Lie algebras
Lie group–Lie algebra correspondence
Exponential map
Adjoint representation
Killing form
Index
Simple Lie algebra
Loop algebra
Affine Lie algebra
Semisimple Lie algebra
Dynkin diagrams
Cartan subalgebra
Root system
Weyl group
Real form
Complexification
Split Lie algebra
Compact Lie algebra
Representation theory
Lie group representation
Lie algebra representation
Representation theory of semisimple Lie algebras
Representations of classical Lie groups
Theorem of the highest weight
Borel–Weil–Bott theorem
Lie groups in physics
Particle physics and representation theory
Lorentz group representations
Poincaré group representations
Galilean group representations
Scientists
Sophus Lie
Henri Poincaré
Wilhelm Killing
Élie Cartan
Hermann Weyl
Claude Chevalley
Harish-Chandra
Armand Borel
Glossary
Table of Lie groups
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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 odd n > 1, when it is isomorphic to the projective special linear group.
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.
In mathematics, a Lie algebra is semisimple if it is a direct sum of simpleLie algebras. (A simpleLie algebra is a non-abelian Lie algebra without any...
direct sum of simpleLie algebras is called a semisimple Lie algebra. A simpleLiegroup is a connected Liegroup whose Lie algebra is simple. A finite-dimensional...
finite simplegroups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simplegroups. The...
group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). For more examples of Lie...
In mathematics, a Liegroup (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...
group E8, corresponding to the three real forms of E8. The groups of Lie type are the finite simplegroups constructed from simple algebraic groups over...
classification of Lie groups in terms of Lie algebras, which are simpler objects of linear algebra. In more detail: for any Liegroup, the multiplication operation...
matrices which represent the groups. In Cartan's classification of the simpleLie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn,...
mathematics, a simplegroup is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken...
mathematics, the classification of finite simplegroups is a result of group theory stating that every finite simplegroup is either cyclic, or alternating, or...
algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simpleLiegroup SL(n,R).) The simpleLiegroups were...
classification of finite simplegroups states that every finite simplegroup is cyclic, or alternating, or in one of 16 families of groups of Lie type, or one of...
finite simplegroups is the Tits group T, which is sometimes considered of Lie type or sporadic — it is almost but not strictly a group of Lie type —...
17,971,200. This is the only simplegroup that is a derivative of a group of Lie type that is not strictly a group of Lie type in any series due to exceptional...
unitary group of degree n, denoted SU(n), is the Liegroup of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may...
representation of a Liegroup. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Liegroups, while the representations...
simplegroups. Inspection of the list of finite simplegroups shows that groups of Lie type over a finite field include all the finite simplegroups other...
Complexification (Liegroup) SimpleLiegroup Compact Liegroup, Compact real form Semisimple Lie algebra Root system Simply laced group ADE classification...
particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system...
connected, simply-connected Liegroup K is a product of finitely many compact, connected, simply-connected simpleLiegroups Ki each of which is isomorphic...
the simpleLiegroup SU(5), was proposed by Howard Georgi and Sheldon Glashow in 1974. The Georgi–Glashow model was preceded by the semisimple Lie algebra...
Jordan algebra Fundamental representation Killing form SimpleLiegroup Georgi, Howard (1999-10-22). Lie Algebras in Particle Physics (2 ed.). Westview Press...
classical Liegroups are four infinite families of Liegroups that together with the exceptional groups exhaust the classification of simpleLiegroups. The...
the mathematical theories of Liegroups and Lie algebras. For the topics in the representation theory of Liegroups and Lie algebras, see Glossary of representation...
this group. The unitary group U(n) is a real Liegroup of dimension n2. The Lie algebra of U(n) consists of n × n skew-Hermitian matrices, with the Lie bracket...