Lie algebra representation

Adjoint Lie algebra representation

The adjoint representation of a Lie algebra 𝔤 is a representation carried by 𝔤 itself, given by #m/def/lie

ad𝑋:𝔤→𝔤𝑌↦[𝑋,𝑌]

The Jacobi identity implies ad𝑋 is a derivation for all 𝑋 ∈𝔤 (^P1), and away from 2 these conditions are equivalent.

Properties

  1. ad𝑋⁡[𝑌,𝑍] =[ad𝑋⁡𝑌,𝑍] +[𝑌,ad𝑋⁡𝑍]
  2. ad𝑋 has no nonzero eigenvalues for each 𝑋 ∈𝔤
  3. [ad𝑋,ad𝑌] =ad[𝑋,𝑌]
Proof of 1–3

Let 𝑋,𝑌,𝑍 ∈𝔤. Assuming anticommutativity

ad𝑋⁡[𝑌,𝑍]=[𝑋,[𝑌,𝑍]]=−[𝑌,[𝑍,𝑋]]−[𝑍,[𝑋,𝑌]]=[𝑌,[𝑋,𝑍]]+[[𝑋,𝑌],𝑍]=[𝑌,ad𝑋⁡𝑍]+[ad𝑋⁡𝑌,𝑍]

proving ^P1.

Assume ad𝑋⁡(𝑌) =[𝑋,𝑌] =𝜆𝑌. Then

0=[𝑌,[𝑋,𝑌]]+[𝑋,[𝑌,𝑌]]+[𝑋,[𝑌,𝑋]]=[𝑌,𝜆𝑌]+[𝑋,−𝜆𝑌]=−𝜆[𝑋,𝑌]=−𝜆2𝑌

hence either 𝜆 =0 or 𝑌 =0, proving ^P2.

For any 𝑍 ∈𝔤,

[ad𝑋,ad𝑌](𝑍)=ad𝑋⁡ad𝑌⁡(𝑍)−ad𝑌⁡ad𝑋⁡(𝑍)=[𝑋,[𝑌,𝑍]]−[𝑌,[𝑋,𝑍]]=[𝑋,[𝑌,𝑍]]+[𝑌,[𝑍,𝑋]]=−[𝑍,[𝑋,𝑌]]=[[𝑋,𝑌],𝑍]=ad[𝑋,𝑌]⁡(𝑍)

hence [ad𝑋,ad𝑌] =ad[𝑋,𝑌], proving ^P3.

Further terminology

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