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In mathematics, a simple Lie group is a Lie group which is also a simple group. These groups, and groups closely related to them, include many of the so-called classical groups of geometry, which lie behind projective geometry and other geometries derived from it by the Erlangen programme of Felix Klein. They also include some exceptional groups, that were first discovered by those pursuing the classification of simple Lie groups. The exceptional groups account for many special examples and configurations in other branches of mathematics. In particular the classification of finite simple groups depended on a thorough prior knowledge of the 'exceptional' possibilities. Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ...
This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ...
In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ...
Geometry (Greek γεωμετÏία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ...
Projective geometry can be thought of informally as the geometry which arises from placing ones eye at a point. ...
An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ...
Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician. ...
The classification of the finite simple groups is a vast body of work in mathematics, mostly published between around 1955 and 1983, which is thought to classify all of the finite simple groups. ...
The complete list of simple Lie groups is the basis for the theory of the semisimple Lie groups and reductive groups, and their representation theory. This has turned out not only to be a major extension of the theory of compact Lie groups (and their representation theory), but to be of basic significance in particle physics. In mathematics, the simple Lie groups were classified by Élie Cartan. ...
In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...
In mathematics, a reductive group is an algebraic group G such that the unipotent radical of the identity component of G is trivial. ...
In mathematics Representation theory is the name given to the study of standard representations of abstract mathematical structures. ...
In mathematics, a compact (topological, often understood) group is a topological group that is also a compact space. ...
Particles erupt from the collision point of two relativistic (100GeV) gold ions in the STAR detector of the Relativistic Heavy Ion Collider. ...
Method of classification Such groups are classified using the prior classification of the complex simple Lie algebras: for which see the page on root systems. It is shown that a simple Lie group has a simple Lie algebra that will occur on the list given there, once it is complexified (that is, made into a complex vector space rather than a real one). This reduces the classification to two further matters. In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ...
In mathematics, the simple Lie groups were classified by Élie Cartan. ...
Real forms The groups SO(p,q,R) and SO(p+q,R), for example, give rise to different real Lie algebras, but having the same Dynkin diagram. In general there may be different real forms of the same complex Lie algebra. In mathematics, the generalized orthogonal group, O(p, q) is the Lie group of all linear transformations of a p + q = n dimensional real vector space which leave invariant a nondegenerate, symmetric, bilinear form of signature (p, q). ...
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
See also Simple Lie group. ...
Relationship of simple Lie algebras to groups Secondly the Lie algebra only determines uniquely the simply connected (universal) cover G* of the component containing the identity of a Lie group G. It may well happen that G* isn't actually a simple group, for example having a non-trivial center. We have therefore to worry about the global topology, by computing the fundamental group of G (an abelian group: a Lie group is an H-space). This was done by Cartan. A geometrical object is called simply connected if it consists of one piece and doesnt have any circle-shaped holes or handles. Higher-dimensional holes are allowed. ...
In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if...
Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...
In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...
In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ...
In mathematics, an H-space is a topological space X (generally assumed to be connected) together with a continuous map μ : X × X → X with an identity element e so that μ(e, x) = μ(x, e) = x for all x in X. Alternatively, the maps μ(e, x) and...
Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...
For an example, take the special orthogonal groups in even dimension. With −I a scalar matrix in the center, these aren't actually simple groups; and having a two-fold spin cover, they aren't simply-connected either. They lie 'between' G* and G, in the notation above. In linear algebra, a diagonal matrix is a square matrix in which only the entries in the main diagonal are non-zero. ...
In mathematics the spinor group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ...
Classification by Dynkin diagram See main article root system In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. ...
According to Dynkin's classification, we have as possibilities these only, where n is the number of nodes:
 Image File history File links Connected Dynkin diagrams File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...
Infinite series
A series A1, A2, ...
Ar corresponds to the special unitary group, SU(r+1). In mathematics, the special unitary group of degree is the group of by unitary matrices with determinant and entries from the field of complex numbers, with the group operation that of matrix multiplication. ...
In mathematics, the special unitary group of degree is the group of by unitary matrices with determinant and entries from the field of complex numbers, with the group operation that of matrix multiplication. ...
B series B1, B2, ...
Br corresponds to the special orthogonal group, SO(2r+1). In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
C series C1, C2, ...
Cr corresponds to the symplectic group, Sp(2r). In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ...
In mathematics, the name symplectic group can refer to two different, but closely related, types of mathematical groups. ...
D series D2, D3, ...
Dr corresponds to the special orthogonal group, SO(2r). Note that SO(4) is not a simple group, though. The Dynkin diagram has two nodes that are not connected. There is a surjective homomorphism from SO(3)* × SO(3)* to SO(4) given by quaternion multiplication; see quaternions and spatial rotation. Therefore the simple groups here start with D3, which as a diagram straightens out to A3. With D4 there is an 'exotic' symmetry of the diagram, corresponding to so-called triality. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...
In mathematics, the quaternions are a non-commutative extension of the complex numbers. ...
The Wikipedia article on quaternions describes the history and purely mathematical properties of the algebra of quaternions. ...
Categories: Stub | Lie groups ...
Exceptional algebras
G2 See G2. In mathematics, G2 is the name of a Lie group and also its Lie algebra . ...
F4 See F4. In mathematics, F4 is the name of a Lie group and also its Lie algebra . ...
E6 See E6. In mathematics, E6 is the name of a Lie group and also its Lie algebra . ...
E7 See E7. In mathematics, E7 is the name of a Lie group and also its Lie algebra . ...
E8 See E8. In mathematics, E8 is the name of a Lie group and also its Lie algebra . ...
Simply laced groups A simply laced group is a Lie group whose Dynkin diagram only contain simple links, and therefore all the nonzero roots of the corresponding Lie algebra have the same length. The A, D and E series groups are all simply laced. This article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ...
See also Simple Lie group. ...
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