What does lie group mean?
Definitions for lie group
lie group
This dictionary definitions page includes all the possible meanings, example usage and translations of the word lie group.
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Wiktionary
Lie groupnoun
Any of many analytic groups that are also a smooth manifold; they arise as groups of rotational symmetries
Etymology: For, Norwegian mathematician
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.
Wikidata
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie, who laid the foundations of the theory of continuous transformation groups. Lie groups represent the best-developed theory of continuous symmetry of mathematical objects and structures, which makes them indispensable tools for many parts of contemporary mathematics, as well as for modern theoretical physics. They provide a natural framework for analysing the continuous symmetries of differential equations, in much the same way as permutation groups are used in Galois theory for analysing the discrete symmetries of algebraic equations. An extension of Galois theory to the case of continuous symmetry groups was one of Lie's principal motivations.
Numerology
Chaldean Numerology
The numerical value of lie group in Chaldean Numerology is: 8
Pythagorean Numerology
The numerical value of lie group in Pythagorean Numerology is: 4
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"lie group." Definitions.net. STANDS4 LLC, 2024. Web. 26 Apr. 2024. <https://www.definitions.net/definition/lie+group>.
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