Lie group
Quick Summary
A Lie group is a topological group where the operations of multiplication and inversion are smooth functions. Essentially, it's a group that also behaves like a smooth manifold, meaning it has a continuous, differentiable structure.12
Lie groups are closely related to Lie algebras, which are algebraic structures named after mathematician Sophus Lie. A Lie algebra can be thought of as the abstract setting for matrix algebra and describes the "infinitesimal" structure of a Lie group. Lie groups themselves can represent geometric concepts like rotations in space, where the set of all rotations forms a Lie group. This theory has broad applications in geometry and physics.32
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mathematicsRiemann’s influence…to be the theory of Lie groups. The Hungarian mathematician Alfréd Haar showed how to define the concept of measure so that functions defined on Lie groups could be integrated. This became a crucial part of Hermann Weyl’s way of representing a Lie group as acting linearly on the space…
Read More…to be the theory of Lie groups. The Hungarian mathematician Alfréd Haar showed how to define the concept of measure so that functions defined on Lie groups could be integrated. This became a crucial part of Hermann Weyl’s way of representing a Lie group as acting linearly on the space…
Read More- mathematicsMathematical physics and the theory of groups
…were a small number of Lie groups, and there was a canonical simplest one to choose in each case. The groups had an even simpler geometric interpretation than the corresponding algebras, for they turned out to describe motions that leave certain properties of figures unaltered. For example, in Euclidean three-dimensional…
Read More…were a small number of Lie groups, and there was a canonical simplest one to choose in each case. The groups had an even simpler geometric interpretation than the corresponding algebras, for they turned out to describe motions that leave certain properties of figures unaltered. For example, in Euclidean three-dimensional…
Read More