differential geometry, a Lie group action is a group action adapted to the smooth setting: G {\displaystyle G} is a Lie group, M {\displaystyle M} is a...

In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...

an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations...

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...

a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of...

representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered...

mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points...

transformations of a local Lie group. Conversely, in the presence of a Lie algebra of vector fields, integration gives a local Lie group action. The result now known...

A Lie group integrator is a numerical integration method for differential equations built from coordinate-independent operations such as Lie group actions...

algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent...

the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids. Lie groupoids were introduced by...

behaviour of a smooth Lie group action on a smooth manifold. Such vector fields find important applications in the study of Lie theory, symplectic geometry...

In topology, a continuous group action on a topological space X is a group action of a topological group G that is continuous: i.e., G × X → X , ( g ,...

unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. The matrices of the more general unitary group may...

matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn,...

is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for...

defined as the unit group of the matrix ring M(n, R). The general linear group GL(n, R) over the field of real numbers is a real Lie group of dimension n2...

Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify...

universal complexification of a real Lie group is given by a continuous homomorphism of the group into a complex Lie group with the universal property that...

Lie to Me (stylized as Lie to me*) is an American crime drama television series. It originally ran on the Fox network from January 21, 2009, to January...

971,200. This is the only simple group 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...

Lie groups form a class of topological groups, and the compact Lie groups have a particularly well-developed theory. Basic examples of compact Lie groups...

multiplicative character of a group. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to...

geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way G × G...

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any...

transformations Mathematically, the Lorentz group may be described as the indefinite orthogonal group O(1, 3), the matrix Lie group that preserves the quadratic form...

as abelian groups and semisimple Lie groups.) A basic example is the Fourier transform, which decomposes the action of the additive group R {\displaystyle...

non-abelian Lie group that is of importance as a model in our understanding of the most basic fundamentals of physics. The Poincaré group consists of...

with structure group H. Let G be a Lie group acting smoothly on a manifold M. If the action is a locally free action or free action, then the orbits...

equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact. The orthogonal group in dimension n has two connected components...