Lie groups - meaning and definition. What is Lie groups
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What (who) is Lie groups - definition

GROUP THAT IS ALSO A DIFFERENTIABLE MANIFOLD WITH GROUP OPERATIONS THAT ARE SMOOTH
LieGroup; Lie groups; Lie subgroup; Matrix Lie group; Lie group homomorphism; Lie Group; P-adic Lie group; Infinite dimensional Lie group; Matrix lie group; Analytic group; Analytic subgroup; Group manifold; Real Lie group; Lie Groups; Lie group theory; Infinite-dimensional Lie group; Acceptable Lie group
  • The set of all [[complex number]]s with [[absolute value]] 1 (corresponding to points on the [[circle]] of center 0 and radius 1 in the [[complex plane]]) is a Lie group under complex multiplication:  the [[circle group]].
  • center
  • A portion of the group <math>H</math> inside <math>\mathbb T^2</math>. Small neighborhoods of the element <math>h\in H</math> are disconnected in the subset topology on <math>H</math>

Lie group         
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be a group, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction).
Theory of Lie groups         
The theory of Lie groups; Théorie des groupes de Lie; Theorie des groupes de Lie
In mathematics, Theory of Lie groups is a series of books on Lie groups by . The first in the series was one of the earliest books on Lie groups to treat them from the global point of view, and for many years was the standard text on Lie groups.
Simple Lie group         
TYPE OF LIE GROUP
Simply laced group; Exceptional Lie group; Exceptional group; Exceptional groups; Simple Lie groups; List of simple Lie groups; List of simple lie groups; List of Lie groups; Exceptional Group; Exceptional simple Lie group; Exceptional Lie groups; Simply laced lie group; List of symmetric spaces; List of simple Lie algebras; Complex simple Lie algebra; Exceptional lie group; Simply laced groups
In mathematics, a 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 used to read off the list of simple Lie algebras and Riemannian symmetric spaces.

Wikipedia

Lie group

In mathematics, a Lie group (pronounced LEE) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses are continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.

Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the rotational symmetry in three dimensions (given by the special orthogonal group SO ( 3 ) {\displaystyle {\text{SO}}(3)} ). Lie groups are widely used in many parts of modern mathematics and physics.

Lie groups were first found by studying matrix subgroups G {\displaystyle G} contained in GL n ( R ) {\displaystyle {\text{GL}}_{n}(\mathbb {R} )} or GL n ( C ) {\displaystyle {\text{GL}}_{n}(\mathbb {C} )} , the groups of n × n {\displaystyle n\times n} invertible matrices over R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } . These are now called the classical groups, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.