Lie algebras MOC

Lie algebra

A Lie algebra 𝔤 is a vector space over a field 𝕂 with an alternating bilinear map [ −, −] :𝔤 ×𝔤 →𝔤 satisfying the Jacobi identity lie

[𝑋,[𝑌,𝑍]]+[𝑌,[𝑍,𝑋]]+[𝑍,[𝑋,𝑌]]=0

which 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 ∈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 {𝑋𝑗}𝑗∈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

[𝑋𝑗,𝑋𝑘]=∑ℓ∈𝐽𝑐ℓ𝑗𝑘𝑋ℓ

Properties

1. Alternating iff anticommutative away from 2
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

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