Lie Bracket

A Lie bracket on a vector space $\mathfrak{g}$ is a bilinear map

such that, for all $X,Y,Z\in\mathfrak{g}$,

A vector space equipped with a Lie bracket is a Lie algebra .

Intuition and standard source of examples

If an associative product $AB$ is available (e.g. matrices), the commutator

is a Lie bracket. This is why Lie brackets are often thought of as measuring “noncommutativity.”

Another fundamental example is the commutator of vector fields on a manifold.

Adjoint operator

Given a Lie algebra $\mathfrak{g}$, each $X\in\mathfrak{g}$ defines a linear map

which is the adjoint representation at the Lie algebra level.

Brackets from Lie groups

For a Lie group $G$, the canonical Lie bracket on $T_eG$ is defined using left-invariant vector fields ; see Lie algebra of a Lie group .