Lie algebra representation

Adjoint Lie algebra representation

The adjoint representation of a Lie algebra is a representation carried by itself, given by lie

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

Properties

2. has no nonzero eigenvalues for each

Proof of 1–3

Let . Assuming anticommutativity

proving ^P1.

Assume . Then

hence either or , proving ^P2.

For any ,

hence , proving ^P3.

Further terminology

• We say is -nilpotent iff is nilpotent.

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