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1873 – Lie Groups

## 1873 – Lie Groups

Topology is the study of how spaces are shaped and connected without worrying about distances. Some properties of spaces are preserved even when the space is being stretched and deformed. Often, the space that is being studied is called a manifold. A line and a circle are simple one dimensional topological manifolds. A plane and the surface of a ball are two dimensional manifolds.

If it is possible to perform differential calculus (find a derivative of a function) on a manifold, the manifold is called a differentiable manifold. Derivatives evaluate the rate of change in a function and when this is applied to manifolds, they can be used to evaluate the rates of change in curves on the manifold. A particular class of these derivatives (named after Sophus Lie) is called Lie derivatives. (Lie is pronounced ‘lee’)

Under certain conditions, collections of Lie derivatives are considered to be symmetry groups and are called Lie Groups.