Simple Lie group

Lie groups
Classical groups General linear group GL(n) Special linear group SL(n) Orthogonal group O(n) Special orthogonal group SO(n) Unitary group U(n) Special unitary group SU(n) Symplectic group Sp(n)
Simple Lie groups List of simple Lie groups Infinite simple Lie groups: An, Bn, Cn, Dn, Exceptional simple Lie groups: G2 F4 E6 E7 E8
Other Lie groups Circle group Lorentz group Poincaré group Conformal group Diffeomorphism group Loop group
Lie algebras Exponential map Adjoint representation of a Lie group Adjoint representation of a Lie algebra Killing form Lie point symmetry
Structure of semi-simple Lie groups Dynkin diagrams Cartan subalgebra Root system Real form Complexification Split Lie algebra Compact Lie algebra
Representation theory Representation of a Lie group Representation of a Lie algebra
Lie groups in Physics Particle physics and representation theory Representation theory of the Lorentz group Representation theory of the Poincaré group Representation theory of the Galilean group
v · group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. A simple Lie algebra is a non-abelian Lie algebra whose only ideals are 0 and itself. A direct sum of simple Lie algebras is called a semisimple Lie algebra. 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. An important technical point is that a simple Lie group may contain discrete normal subgroups, hence being a simple Lie group is different from being simple as an abstract group. Simple Lie groups include many classical Lie groups, which provide a group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen programme. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics. While the notion of a simple Lie group is satisfying from the axiomatic perspective, in applications of Lie theory, such as the theory of Riemannian symmetric spaces, somewhat more general notions of semisimple and reductive Lie groups proved to be even more useful. In particular, every connected compact Lie group is reductive, and the study of representations of general reductive groups is a major branch of representation theory. Comments on the definition Unfortunately there is no single standard definition of a simple Lie group. The definition given above is sometimes varied in the following ways: Connectedness: Usually simple Lie groups are connected by definition. This excludes discrete simple groups (these are zero-dimensional Lie groups that are simple as abstract groups) as well as disconnected orthogonal groups. Center: Usually simple Lie groups are allowed to have a discrete center; for example, SL2(R) has a center of order 2, but is still counted as a simple Lie group. If the center is non-trivial (and not the whole group) then the simple Lie group is not simple as an abstract group. Some authors require that the center of a simple Lie group be finite (or trivial); the universal cover of SL2(R) is an example of a simple Lie group with infinite center. R: Usually the group R of real numbers under addition (and its quotient R/Z) are not counted as simple Lie groups, even though they are connected and have a Lie algebra with no proper non-zero ideals. Occasionally authors define simple Lie groups in such a way that R is simple, though this sometimes seems to be an accident caused by overlooking this case. Matrix groups: Some authors restrict themselves to Lie groups that can be represented as groups of finite matrices. The metaplectic group is an example of a simple Lie group that cannot be represented in this way. Complex Lie algebras: The definition of a simple Lie algebra is not stable under the extension of scalars. The complexification of a complex simple Lie algebra, such as sln(C) is semisimple, but not simple. The most common definition is the one above: simple Lie groups have to be connected, they are allowed to have non-trivial centers (possibly infinite), they need not be representable by finite matrices, and they must be non-abelian. Method of classification Main article: list of simple Lie groups 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. 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. 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 Élie Cartan. For an example, take the special orthogonal groups in even dimension. With the non-identity matrix −I in the center, these aren't actually simple groups; and having a twofold spin cover, they aren't simply-connected either. They lie 'between' G* and G, in the notation above. Classification by Dynkin diagram Main article: root system According to Dynkin's classification, we have as possibilities these only, where n is the number of nodes: Infinite series A series A1, A2, ... Ar corresponds to the special linear group, SL(r+1). B series B2, B3, ... Br corresponds to the special orthogonal group, SO(2r+1). C series C2, C3, ... Cr corresponds to the symplectic group, Sp(2r). D series D4, D5, ... 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. Exceptional cases For the so-called exceptional cases see G2, F4, E6, E7, and E8. These cases are deemed 'exceptional' because they do not fall into infinite series of groups of increasing dimension. From the point of view of each group taken separately, there is nothing so unusual about them. These exceptional groups were discovered around 1890 in the classification of the simple Lie algebras, over the complex numbers (Wilhelm Killing, re-done by Élie Cartan). For some time it was a research issue to find concrete ways in which they arise, for example as a symmetry group of a differential system. See also E7½ 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, but no group of type B, C, F, or G is simply laced. See also Cartan matrix Coxeter matrix Weyl group Coxeter group Kac–Moody algebra Catastrophe theory References Jacobson, Nathan (1971-06-01). Exceptional Lie Algebras (1 ed.). CRC Press. ISBN 0824713265.  Wikimedia Foundation. 2010. Игры ⚽ Поможем решить контрольную работу Bitstream Vera Volturnalia Look at other dictionaries: Lie group — Lie groups …   Wikipedia List of simple Lie groups — In mathematics, the simple Lie groups were classified by Élie Cartan.The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces. See also the table of Lie groups for a smaller list of… …   Wikipedia List of Lie group topics — This is a list of Lie group topics, by Wikipedia page. Examples See Table of Lie groups for a list *General linear group, special linear group **SL2(R) **SL2(C) *Unitary group, special unitary group **SU(2) **SU(3) *Orthogonal group, special… …   Wikipedia Simple group — In mathematics, a simple group is a group which is not the trivial group and whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the… …   Wikipedia Lie algebra representation — Lie groups …   Wikipedia Group theory — is a mathematical discipline, the part of abstract algebra that studies the algebraic structures known as groups. The development of group theory sprang from three main sources: number theory, theory of algebraic equations, and geometry. The… …   Wikipedia Group (mathematics) — This article covers basic notions. For advanced topics, see Group theory. The possible manipulations of this Rubik s Cube form a group. In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines …   Wikipedia Group of Lie type — In mathematics, a group of Lie type G(k) is a (not necessarily finite) group of rational points of a reductive linear algebraic group G with values in the field k. Finite groups of Lie type form the bulk of nonabelian finite simple groups.… …   Wikipedia Lie algebra — In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term… …   Wikipedia Group extension — In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence:1 ightarrow N… …   Wikipedia 18+ © Academic, 2000-2024 Contact us: Technical Support, Advertising Dictionaries export, created on PHP, Joomla, Drupal, WordPress, MODx. Mark and share Search through all dictionaries Translate… Search Internet Share the article and excerpts Direct link … Do a right-click on the link above and select “Copy Link”