# Lie algebra

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Infinite simple Lie groups: An, Bn, Cn, Dn,
Exceptional simple Lie groups: G2 F4 E6 E7 E8

In mathematics, a Lie algebra (pronounced /ˈliː/ ("lee"), not /ˈlaɪ/ ("lye")) is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" (after Sophus Lie) was introduced by Hermann Weyl in the 1930s. In older texts, the name "infinitesimal group" is used.

## Contents

### Definition and first properties

A Lie algebra is a vector space $\,\mathfrak{g}$ over some field F together with a binary operation [·, ·]

called the Lie bracket, which satisfies the following axioms:

• Alternating on $\,\mathfrak{g}$: