• 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...
    8 KB (1,332 words) - 23:54, 15 January 2024
  • Thumbnail for Lie group
    In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses...
    64 KB (9,427 words) - 05:48, 28 May 2024
  • Thumbnail for Group action
    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...
    46 KB (5,637 words) - 12:41, 27 June 2024
  • Thumbnail for Adjoint representation
    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...
    21 KB (3,516 words) - 07:00, 12 July 2024
  • Thumbnail for Representation of a Lie group
    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...
    34 KB (5,242 words) - 12:15, 30 June 2024
  • Thumbnail for Simple Lie group
    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...
    34 KB (2,262 words) - 15:54, 14 May 2024
  • Thumbnail for Lie algebra
    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...
    61 KB (10,457 words) - 02:56, 14 July 2024
  • Thumbnail for Group of Lie type
    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...
    22 KB (2,985 words) - 10:42, 28 March 2023
  • A Lie group integrator is a numerical integration method for differential equations built from coordinate-independent operations such as Lie group actions...
    2 KB (175 words) - 17:19, 16 August 2023
  • the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids. Lie groupoids were introduced by...
    44 KB (7,436 words) - 12:16, 15 May 2024
  • Fundamental vector field (category Lie groups)
    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...
    7 KB (1,033 words) - 02:23, 16 September 2023
  • 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...
    6 KB (720 words) - 12:15, 4 January 2024
  • Thumbnail for General linear group
    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...
    23 KB (2,964 words) - 15:21, 15 April 2024
  • Thumbnail for Special unitary group
    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...
    33 KB (5,453 words) - 23:03, 26 July 2024
  • Thumbnail for E8 (mathematics)
    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...
    46 KB (6,107 words) - 02:27, 20 June 2024
  • 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 ,...
    2 KB (355 words) - 06:14, 22 October 2022
  • Thumbnail for Symplectic group
    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,...
    22 KB (3,076 words) - 13:01, 4 July 2024
  • Thumbnail for Reductive group
    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...
    55 KB (7,845 words) - 18:28, 24 April 2024
  • Thumbnail for Semisimple Lie algebra
    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...
    41 KB (5,731 words) - 17:17, 10 April 2024
  • Lie to Me (stylized as Lie to me*) is an American crime drama television series. It originally ran on Fox from January 21, 2009, to January 31, 2011. In...
    38 KB (3,201 words) - 15:04, 24 July 2024
  • Thumbnail for Lorentz group
    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...
    65 KB (9,740 words) - 16:15, 2 February 2024
  • closed manifold of (real) dimension 4n. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line H P 1 {\displaystyle...
    7 KB (1,245 words) - 17:30, 5 June 2023
  • Thumbnail for Exponential map (Lie theory)
    of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to the group, which...
    13 KB (2,207 words) - 21:53, 8 July 2024
  • Thumbnail for Foliation
    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...
    70 KB (8,140 words) - 01:31, 24 April 2024
  • Thumbnail for Topological group
    as abelian groups and semisimple Lie groups.) A basic example is the Fourier transform, which decomposes the action of the additive group R {\displaystyle...
    50 KB (7,492 words) - 16:51, 25 May 2024
  • Thumbnail for Poincaré group
    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...
    15 KB (2,172 words) - 23:34, 20 June 2024
  • Thumbnail for Spin group
    mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R)...
    27 KB (4,183 words) - 01:55, 27 July 2024
  • Thumbnail for Complexification (Lie group)
    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...
    52 KB (7,216 words) - 14:30, 2 December 2022
  • Thumbnail for Orthogonal group
    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...
    56 KB (7,844 words) - 08:45, 30 June 2024
  • 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...
    4 KB (640 words) - 06:58, 5 June 2024