# Lie algebra of a real Lie group

## Contents

This article gives a basic definition in the following area: Lie theory
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## Definition

Suppose $G$ is a real Lie group, i.e., a Lie group over the field of real numbers $\R$. The Lie algebra of $G$ is a Lie algebra over $\R$ defined as follows: it is the collection of all left-$G$-invariant derivations on $G$ (viewed as a differential manifold). The structures are described in the table below:

Item Description
underlying set All left-$G$-invariant derivations on $G$, or equivalently, all left-$G$-invariant vector fields. Here, a derivation on $G$ means a derivation of its sheaf of infinitely differentiable functions. Further, because of left-$G$-invariance, this set can be identified canonically with the tangent space at the identity.
addition and scalar multiplication pointwise addition and scalar multiplication of derivations or vector fields.
Lie bracket $[X,Y] \mapsto XY - YX$, where the multiplication here is obtained by composing derivations. Note that the composite of two derivations is a second-order differential operator and usually not a derivation, because it fails the Leibniz rule, but the Lie bracket as defined here is a derivation.

Although the definition here uses left invariance, an equivalent definition using right invariance gives the same Lie algebra because every Lie group is naturally isomorphic to its opposite Lie group via the inverse map (a Lie group version of the fact that every group is naturally isomorphic to its opposite group via the inverse map).