# Definition:Lie Group

## Definition

A Lie group is a group which is also a smooth manifold.

## Source of Name

This entry was named for Marius Sophus Lie.

## Historical Note

When the concept of a Lie group was originally researched, it did not appear to have any immediate practical application.

In more recently times, they have been used:

to describe pseudo-Riemannian locally homogeneous symmetric spaces, to be used in geometric theory of gravitation
to model phenomena in quantum mechanics, from $1925$ to $1926$, via Hermitian matrix representations of observations