Lie group

Lie groupsLie subgroupmatrix Lie groupinfinitesimal generatorreal Lie groupLiep-adic Lie groupp''-adic Lie groupgroup manifoldgroup
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are smooth.wikipedia
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Group (mathematics)

groupgroupsgroup operation
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are smooth.
Lie groups are the symmetry groups used in the Standard Model of particle physics; Poincaré groups, which are also Lie groups, can express the physical symmetry underlying special relativity; and point groups are used to help understand symmetry phenomena in molecular chemistry.

Conformal group

conformal group of spacetimeconformal invariance
Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R 3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.
All conformal groups are Lie groups.

Particle physics and representation theory

is used extensively in particle physicstheory of quarks
Representation theory is used extensively in particle physics.
It links the properties of elementary particles to the structure of Lie groups and Lie algebras.

Topological group

continuous groupclosed subgrouptopological groups
Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups.
The groups mentioned so far are all Lie groups, meaning that they are smooth manifolds in such a way that the group operations are smooth, not just continuous.

Lie group action

actsquotient manifoldacted
The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold.
In differential geometry, a Lie group action on a manifold M is a group action by a Lie group G on M that is a differentiable map; in particular, it is a continuous group action.

Representation theory of the Poincaré group

representations of the Poincaré groupPoincaré groupirreducible unitary representations of the Poincaré group
Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2)), the special unitary group SU(3) and the Poincaré group.
In mathematics, the representation theory of the Poincaré group is an example of the representation theory of a Lie group that is neither a compact group nor a semisimple group.

Armand Borel

BorelBorel, ArmandA. Borel
In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field.
He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups.

Lie algebra representation

representationrepresentation theory of Lie algebrasrepresentations
Here, the representations of the Lie group (or of its Lie algebra) are especially important.
If \phi: G → H is a homomorphism of (real or complex) Lie groups, and \mathfrak g and \mathfrak h are the Lie algebras of G and H respectively, then the differential on tangent spaces at the identities is a Lie algebra homomorphism.

Erlangen program

Erlangen programmeErlanger programnotion of geometry
Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant.
In today's language, the groups concerned in classical geometry are all very well known as Lie groups: the classical groups.

Rotation (mathematics)

rotationrotationsrotate
The rotation group is a Lie group of rotations about a fixed point.

Euclidean group

direct isometriesEuclidean motionrigid motion
Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R 3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.
The Euclidean groups are not only topological groups, they are Lie groups, so that calculus notions can be adapted immediately to this setting.

Orthogonal group

special orthogonal grouprotation grouporthogonal
Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups.
The orthogonal group is an algebraic group and a Lie group.

Affine group

Affine general linear groupgeneral affinegeneral affine group
It is a Lie group if K is the real or complex field or quaternions.

Closed-subgroup theorem

Cartan's theoremclosed subgroupClosed subgroup theorem
According to Cartan's theorem, a closed subgroup of G admits a unique smooth structure which makes it an embedded Lie subgroup of G—i.e. a Lie subgroup such that the inclusion map is a smooth embedding.
is a closed subgroup of a Lie group

Hilbert's fifth problem

Fifth Problem5thcontinuous group
Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples.
Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups.

Lie group–Lie algebra correspondence

Lie correspondencecorrespondenceLie correspondence between Lie groups and Lie algebras
Lie's third theorem says that every finite-dimensional real Lie algebra is the Lie algebra of a Lie group.
In mathematics, Lie group–Lie algebra correspondence allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects.

Sophus Lie

Marius Sophus LieLieLie, Sophus
Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.
Lie's principal tool, and one of his greatest achievements, was the discovery that continuous transformation groups (now called, after him, Lie groups) could be better understood by "linearizing" them, and studying the corresponding generating vector fields (the so-called infinitesimal generators).

Conformal geometry

conformal structureconformal manifoldconformal
Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space R 3, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group.
The conformal group for the Minkowski quadratic form q(x, y) = 2xy in the plane is the abelian Lie group

Automorphism group

transformation grouptransformation groupsgroup of automorphisms
Lie groups are named after Norwegian mathematician Sophus Lie, who laid the foundations of the theory of continuous transformation groups.

Classical group

classical groupsclassical Lie groupclassical Lie algebra
Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups.

Algebraic group

algebraic groupsalgebraicalgebraic subgroup
In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups.
Note that this means that algebraic group is narrower than Lie group, when working over the field of real numbers: there are examples such as the universal cover of the 2×2 special linear group that are Lie groups, but have no faithful linear representation.

Unitary group

U(1)unitaryU(''n'')
The unitary group U(n) is a real Lie group of dimension n 2.

Group of Lie type

Chevalley groupgroups of Lie typefinite groups of Lie type
It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups.
The name "groups of Lie type" is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers.

Representation theory

linear representationrepresentationsrepresentation
Linear actions of Lie groups are especially important, and are studied in representation theory.
The representation theory of linear algebraic groups and Lie groups extends these examples to infinite-dimensional groups, the latter being intimately related to Lie algebra representations.

Claude Chevalley

ChevalleyChevalley, ClaudeClaude Charles Chevalley
In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field.
Around 1950, Chevalley wrote a three-volume treatment of Lie groups.