Lie algebra

A Lie algebra $𝔤$ is a vector space over a commutative ring $K$ with an alternating bilinear map $[-, -] : 𝔤 \times 𝔤 \to 𝔤$ satisfying the Jacobi identity #m/def/lie

[𝑋, [𝑌, 𝑍]] + [𝑌, [𝑍, 𝑋]] + [𝑍, [𝑋, 𝑌]] = 0

which for $K$ a field away from 2 is equivalent to demanding the Lie bracket is a derivation on itself (see ^P1). A Lie algebra is one of the simplest kinds of non-associative, non-commutative algebras (in fact it is anticommutative).

Lie algebras were first encountered as tangent spaces of Lie groups. They naturally arise as the commutator algebra of a unital associative algebra, and the existence of the universal enveloping algebra gives a sense in which all Lie algebras are of this form.

Further terminology

An Abelian Lie algebra has $[𝑋, 𝑌] = 0$ for all $𝑋, 𝑌 \in 𝔤$
A Lie subalgebra is a linear subspace closed under the bracket.
A Lie algebra ideal is a linear subspace which “absorbs all elements“.
Lie algebra extension

Basis

Since $𝔤$ is a vector space we can find a basis ${𝑋 𝑗} 𝑗 \in 𝐽$ . The behaviour of the Lie bracket on all elements is completely determined by the basis generators due to linearity. We describe this using the so-called structure constants

[𝑋 𝑗, 𝑋 𝑘] = \sum ℓ \in 𝐽 𝑐 ℓ 𝑗 𝑘 𝑋 ℓ

Properties

Alternating iff anticommutative away from 2
Every Lie algebra may be constructed as a subalgebra of the commutator of its Universal enveloping algebra (see Poincaré-Birkhoff-Witt theorem)

Examples

Commutator of any associative algebra

#state/tidy | #lang/en | #SemBr