LIE GROUP Meaning and
Definition
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A Lie group is a mathematical concept that combines the notions of a group and a smooth manifold. A group is a set equipped with a binary operation that satisfies certain properties, such as associativity, existence of an identity element, and existence of inverses for every element. A smooth manifold is a topological space that locally resembles Euclidean space, and is equipped with a smooth structure that allows the formulation of calculus-like operations.
In the case of a Lie group, these two structures are merged together. It is a group wherein the group operations exhibit smoothness with respect to the underlying topology. Specifically, a Lie group possesses a smooth manifold structure that is compatible with the group operations. This means that multiplication and inversion are smooth functions on the manifold, enabling the application of differential and integral calculus techniques.
Lie groups are named after the Norwegian mathematician Sophus Lie, who extensively studied their properties and applications. These groups have wide-ranging applications in various branches of mathematics and physics. They serve as a powerful tool for understanding the symmetries present in mathematical and physical systems. Additionally, Lie groups are an essential ingredient in the formulation of Lie algebras, which provide a linearized description of the group's structure.
Overall, a Lie group can be understood as a smooth manifold equipped with a group structure, where the group operations are compatible with the smooth structure. This mathematical concept finds extensive utility in diverse areas of research, including physics, geometry, and number theory.
Common Misspellings for LIE GROUP
- kie group
- pie group
- oie group
- lue group
- lje group
- lke group
- loe group
- l9e group
- l8e group
- liw group
- lis group
- lid group
- lir group
- li4 group
- li3 group
- lie froup
- lie vroup
- lie broup
- lie hroup
- lie yroup
Etymology of LIE GROUP
The term "lie group" originates from the name of the Norwegian mathematician Marius Sophus Lie (1842-1899). Lie made significant contributions to the theory of continuous transformation groups, which are now known as "Lie groups" in his honor. The word "lie" is pronounced as "lee" to match the pronunciation of Marius Lie's last name.
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