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Coadjoint representation of a Lie algebra

The dual of the adjoint representation, acting on the dual space g*.

Let g\mathfrak{g} be a and let g∗\mathfrak{g}^* be its dual vector space.

Definition. The coadjoint representation is the linear map

ad∗:g→End(g∗)\mathrm{ad}^*:\mathfrak{g}\to \mathrm{End}(\mathfrak{g}^*)

defined by dualizing the ad:g→End(g)\mathrm{ad}:\mathfrak{g}\to \mathrm{End}(\mathfrak{g}): for X∈gX\in \mathfrak{g} and ℓ∈g∗\ell\in \mathfrak{g}^*,

adX∗(ℓ)=− ℓ∘adX.\mathrm{ad}^*_X(\ell) = -\,\ell\circ \mathrm{ad}_X.
Equivalent characterizations

Equivalently, using the natural pairing ⟨ℓ,Y⟩=ℓ(Y)\langle\ell,Y\rangle=\ell(Y),

⟨adX∗ℓ, Y⟩=−⟨ℓ, [X,Y]⟩for all Y∈g.\langle \mathrm{ad}^*_X\ell,\, Y\rangle = -\langle \ell,\,[X,Y]\rangle \quad \text{for all } Y\in \mathfrak{g}.
Remarks

Key point. The minus sign ensures ad∗\mathrm{ad}^* is a , i.e. [adX∗,adY∗]=ad[X,Y]∗[\mathrm{ad}^*_X,\mathrm{ad}^*_Y]=\mathrm{ad}^*_{[X,Y]}.

Context. in g∗\mathfrak{g}^* (via the integrated ) carry canonical and play a central role in geometric representation theory (Kirillov’s ). In the semisimple case, identifying g≅g∗\mathfrak{g}\cong\mathfrak{g}^* via the relates coadjoint and adjoint pictures.