# Lie, Marius Sophus (1842–1899)

Sophus Lie was a Norwegian mathematician who, along with his close friend Felix Klein,
introduced group theory into geometry and
used it to classify geometries. Lie discovered the **contact transformation**,
which maps curves into surfaces (1870), and Lie groups, which use continuous
or infinitesimal transformations. He used these groups to classify partial
differential equations, making the traditional methods of solution all
reduce to a single principle. Lie groups also provided a basis for the growth
of modern topology.

## Lie algebra

Lie algebra is an algebra in which multiplication satisfies
properties similar to the so-called **bracket operation** on matrices given by [*A*, *B*] = *AB* - *BA*, where the operation on the right-hand side involves
ordinary multiplication and subtraction of matrices. The operation is not associative.