Lie theory MOC

Lie algebra representation

A representation 𝜋 :𝔤 →𝔤𝔩(𝑉) of a Lie algebra 𝔤 is a Lie algebra homomorphism from 𝔤 to the commutator algebra of the Endomorphism ring of 𝑉. lie Thus for any 𝑋,𝑌 ∈𝔤,

𝜋([𝑋,𝑌])=𝜋(𝑋)𝜋(𝑌)−𝜋(𝑌)𝜋(𝑋)

A space 𝑉 carrying a representation of 𝔤 is sometimes called a 𝔤-module,¹ though this is an abuse of terminology.

Properties

• Every Lie algebra has a natural representation on itself, the adjoint Lie algebra representation.
• Every representation Γ of a Lie group induces an Infinitesimal representation 𝑑Γ of the corresponding Lie algebra.
• Sum of commuting Lie algebra representations
• Tensor product of Lie algebra representations

---

tidy | en | SemBr

Footnotes

1. 1988. Vertex operator algebras and the Monster, p. 5 ↩