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Coadjoint representation

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In mathematics, the coadjoint representation K {\displaystyle K} of a Lie group G {\displaystyle G} is the dual of the adjoint representation. If g {\displaystyle {\mathfrak {g}}} denotes the Lie algebra of G {\displaystyle G} , the corresponding action of G {\displaystyle G} on g ∗ {\displaystyle {\mathfrak {g}}^{*}} , the dual space to g {\displaystyle {\mathfrak {g}}} , is called the coadjoint action. A geometrical interpretation is as the action by left-translation on the space of right-invariant 1-forms on G {\displaystyle G} .

The importance of the coadjoint representation was emphasised by work of Alexandre Kirillov, who showed that for nilpotent Lie groups G {\displaystyle G} a basic role in their representation theory is played by coadjoint orbits. In the Kirillov method of orbits, representations of G {\displaystyle G} are constructed geometrically starting from the coadjoint orbits. In some sense those play a substitute role for the conjugacy classes of G {\displaystyle G} , which again may be complicated, while the orbits are relatively tractable.

Formal definition

Let G {\displaystyle G} be a Lie group and g {\displaystyle {\mathfrak {g}}} be its Lie algebra. Let A d : G → A u t ( g ) {\displaystyle \mathrm {Ad} :G\rightarrow \mathrm {Aut} ({\mathfrak {g}})} denote the adjoint representation of G {\displaystyle G} . Then the coadjoint representation A d ∗ : G → G L ( g ∗ ) {\displaystyle \mathrm {Ad} ^{*}:G\rightarrow \mathrm {GL} ({\mathfrak {g}}^{*})} is defined by

⟨ A d g ∗ μ , Y ⟩ = ⟨ μ , A d g − 1 Y ⟩ = ⟨ μ , A d g − 1 Y ⟩ {\displaystyle \langle \mathrm {Ad} _{g}^{*}\,\mu ,Y\rangle =\langle \mu ,\mathrm {Ad} _{g}^{-1}Y\rangle =\langle \mu ,\mathrm {Ad} _{g^{-1}}Y\rangle } for g ∈ G , Y ∈ g , μ ∈ g ∗ , {\displaystyle g\in G,Y\in {\mathfrak {g}},\mu \in {\mathfrak {g}}^{*},}

where ⟨ μ , Y ⟩ {\displaystyle \langle \mu ,Y\rangle } denotes the value of the linear functional μ {\displaystyle \mu } on the vector Y {\displaystyle Y} .

Let a d ∗ {\displaystyle \mathrm {ad} ^{*}} denote the representation of the Lie algebra g {\displaystyle {\mathfrak {g}}} on g ∗ {\displaystyle {\mathfrak {g}}^{*}} induced by the coadjoint representation of the Lie group G {\displaystyle G} . Then the infinitesimal version of the defining equation for A d ∗ {\displaystyle \mathrm {Ad} ^{*}} reads:

⟨ a d X ∗ μ , Y ⟩ = ⟨ μ , − a d X Y ⟩ = − ⟨ μ , [ X , Y ] ⟩ {\displaystyle \langle \mathrm {ad} _{X}^{*}\mu ,Y\rangle =\langle \mu ,-\mathrm {ad} _{X}Y\rangle =-\langle \mu ,\rangle } for X , Y ∈ g , μ ∈ g ∗ {\displaystyle X,Y\in {\mathfrak {g}},\mu \in {\mathfrak {g}}^{*}}

where a d {\displaystyle \mathrm {ad} } is the adjoint representation of the Lie algebra g {\displaystyle {\mathfrak {g}}} .

Coadjoint orbit

A coadjoint orbit O μ {\displaystyle {\mathcal {O}}_{\mu }} for μ {\displaystyle \mu } in the dual space g ∗ {\displaystyle {\mathfrak {g}}^{*}} of g {\displaystyle {\mathfrak {g}}} may be defined either extrinsically, as the actual orbit A d G ∗ μ {\displaystyle \mathrm {Ad} _{G}^{*}\mu } inside g ∗ {\displaystyle {\mathfrak {g}}^{*}} , or intrinsically as the homogeneous space G / G μ {\displaystyle G/G_{\mu }} where G μ {\displaystyle G_{\mu }} is the stabilizer of μ {\displaystyle \mu } with respect to the coadjoint action; this distinction is worth making since the embedding of the orbit may be complicated.

The coadjoint orbits are submanifolds of g ∗ {\displaystyle {\mathfrak {g}}^{*}} and carry a natural symplectic structure. On each orbit O μ {\displaystyle {\mathcal {O}}_{\mu }} , there is a closed non-degenerate G {\displaystyle G} -invariant 2-form ω ∈ Ω 2 ( O μ ) {\displaystyle \omega \in \Omega ^{2}({\mathcal {O}}_{\mu })} inherited from g {\displaystyle {\mathfrak {g}}} in the following manner:

ω ν ( a d X ∗ ν , a d Y ∗ ν ) := ⟨ ν , [ X , Y ] ⟩ , ν ∈ O μ , X , Y ∈ g {\displaystyle \omega _{\nu }(\mathrm {ad} _{X}^{*}\nu ,\mathrm {ad} _{Y}^{*}\nu ):=\langle \nu ,\rangle ,\nu \in {\mathcal {O}}_{\mu },X,Y\in {\mathfrak {g}}} .

The well-definedness, non-degeneracy, and G {\displaystyle G} -invariance of ω {\displaystyle \omega } follow from the following facts:

(i) The tangent space T ν O μ = { − a d X ∗ ν : X ∈ g } {\displaystyle \mathrm {T} _{\nu }{\mathcal {O}}_{\mu }=\{-\mathrm {ad} _{X}^{*}\nu :X\in {\mathfrak {g}}\}} may be identified with g / g ν {\displaystyle {\mathfrak {g}}/{\mathfrak {g}}_{\nu }} , where g ν {\displaystyle {\mathfrak {g}}_{\nu }} is the Lie algebra of G ν {\displaystyle G_{\nu }} .

(ii) The kernel of the map X ↦ ⟨ ν , [ X , ⋅ ] ⟩ {\displaystyle X\mapsto \langle \nu ,\rangle } is exactly g ν {\displaystyle {\mathfrak {g}}_{\nu }} .

(iii) The bilinear form ⟨ ν , [ ⋅ , ⋅ ] ⟩ {\displaystyle \langle \nu ,\rangle } on g {\displaystyle {\mathfrak {g}}} is invariant under G ν {\displaystyle G_{\nu }} .

ω {\displaystyle \omega } is also closed. The canonical 2-form ω {\displaystyle \omega } is sometimes referred to as the Kirillov-Kostant-Souriau symplectic form or KKS form on the coadjoint orbit.

Properties of coadjoint orbits

The coadjoint action on a coadjoint orbit ( O μ , ω ) {\displaystyle ({\mathcal {O}}_{\mu },\omega )} is a Hamiltonian G {\displaystyle G} -action with momentum map given by the inclusion O μ ↪ g ∗ {\displaystyle {\mathcal {O}}_{\mu }\hookrightarrow {\mathfrak {g}}^{*}} .

Examples

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See also

References

External links

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