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nearby homomorphisms from compact Lie groups are conjugate

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Context

Lie theory

∞-Lie theory (higher geometry)

Background

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Contents

Idea

For GG a compact Lie group and Γ\Gamma any Lie group, every group homomorphisms ϕ:GΓ\phi \,\colon\, G \xrightarrow{\;} \Gamma (as topological groups, hence as Lie groups) has a neighbourhood U ϕU_\phi (in the homomorphism-subspace of the compact open topology) all whose elements ϕ\phi' are conjugate to ϕ\phi, in that

(1)ϕU ϕγΓϕ()=γ 1ϕ()γ. \underset{ \phi' \in U_{\phi} }{\forall} \;\;\; \underset{ \gamma \in \Gamma }{\exists} \;\;\;\;\;\;\; \phi'(-) \,=\, \gamma^{-1} \cdot \phi(-) \cdot \gamma \,.

(Conner & Floyd 1964, Lem. 31.8, Rezk et al. 2013).

It follows (highlighted in Rezk 2014, Rem. 2.2.1) that the quotient space of the homomorphism space Hom(G,Γ)Maps(G,Γ)Hom(G,\Gamma) \,\subset\, Maps(G,\Gamma) under the adjoint action by Γ\Gamma is discrete:

(2)Hom(G,Γ)/ adΓSetTopSp, Hom(G,\Gamma)/_{\!\!ad} \Gamma \;\;\; \in Set \xhookrightarrow{\;} TopSp \,,

and that the homomorphism space itself decomposes, as a Γ \Gamma -space via the adjoint action, into the disjoint union, indexed by (2), of the coset spaces of Γ\Gamma by the corresponding stabilizer subgroups (“centralizer subgroups”) C Γ(ϕ)ΓC_\Gamma(\phi) \subset \Gamma:

(3)Hom(G,Γ)[ϕ]Hom(G,Γ)/ΓΓ/C Γ(ϕ)ΓAct(TopSp). Hom(G,\Gamma) \;\simeq\; \underset{ {[\phi] \, \in } \atop \mathclap{ Hom(G,\Gamma)/\Gamma } }{\bigsqcup} \Gamma/C_{{}_{\Gamma}}(\phi) \;\;\; \in \; \Gamma Act(TopSp) \,.

Examples

Example

(Pontrjagin duality)
In the special case that Γ=S 1\Gamma = S^1 is the circle group and GG is an abelian group, the homomorphism space G^Hom(G,S 1)\widehat G \,\coloneqq\, Hom(G,S^1) is the Pontrjagin dual of GG.

In this case, since S 1S^1 is abelian so that the conjugation action on Hom(G,S 1)Hom(G,S^1) is trivial, the statement (2) is a special case of the classical fact that the Pontrjagin dual of a compact topological group is discrete (see there).

Variants

For crossed homomorphisms

Let

α:GAut Grp(Γ) \alpha \;\colon\; G \xrightarrow{\;} Aut_{Grp}(\Gamma)

be a group action by group automorphisms, with the special property that its restriction to the center C(G)GC(G) \subset G is trivial (for example when GG has trivial center to start with):

α |C(G)=id Γ. \alpha_{\vert C(G)} \;=\; id_{\Gamma} \,.

In that case the above statement (1) generalizes to say that nearby crossed homomorphisms are crossed conjugate.

More precisely, notice (this Prop.) that a crossed homomorphism ϕ:GΓ\phi \;\colon\; G \xrightarrow{\;} \Gamma is equivalently a plain group homomorphism (ϕ(),()):GΓG(\phi(-),\,(-)) \,\colon\, G \xrightarrow{\;} \Gamma \rtimes G to the semidirect product group, subject to the constraint that its projection to GG is the identity morphism. Now say that another crossed homomorphism ϕ\phi' is nearby if it is so as a plain homomorphism (ϕ(),())(\phi'(-),(-)) to the semidirect product group (i.e. we consider neighbourhoods of crossed homomorphism in the sense of neighbourhoods of the points which they represent in Hom(G,ΓG))Hom(G,\Gamma \rtimes G)).

Then the above statement (1) says that there is an element (γ,h)ΓG(\gamma,\,h) \,\in\, \Gamma \rtimes G such that

(4)gG(ϕ(g),g)=(γ,h) 1(ϕ(g),g)(γ,h). \underset{g \in G}{\forall} \;\;\;\;\; \big( \phi'(g),\,g \big) \;\; = \;\; \big( \gamma, \, h \big)^{-1} \cdot \big( \phi(g),\,g \big) \cdot \big( \gamma, \, h \big) \,.

In order for such a conjugation to be a crossed conjugation of ϕ\phi to ϕ\phi', we need its GG-component to be the neutral element: h=eGh = \mathrm{e} \,\in\, G.

Now, the further conjugation of (4) by (e,h 1)\big(\mathrm{e},\,h^{-1}\big) yields:

(α(h 1)(ϕ(g)),g)=(e,h)(γ,h) 1(ϕ(g),g)(γ,h)(e,h 1). \Big( \alpha(h^{-1}) \big( \phi'(g) \big) ,\, g \Big) \;\; = \;\; \big( \mathrm{e},\, h \big) \cdot \big( \gamma, \, h \big)^{-1} \cdot \big( \phi(g),\,g \big) \cdot \big( \gamma, \, h \big) \cdot \big( \mathrm{e},\, h^{-1} \big) \,.

But we know that hC(G)h \in C(G) is in the center of GG, since the projection of both sides of (4) to GG must be the identity, by construction of crossed homomorphisms.

Therefore, by the assumption that the action of GG on Γ\Gamma restricts along the inclusion C(G)GC(G) \xhookrightarrow{\;} G to the trivial action, we have α(h 1)(ϕ(g))=ϕ(g)\alpha(h^{-1})\big( \phi(g) \big) \,=\, \phi(g) and it follows that

(γ,h)(e,h 1)=(γ,e) \big( \gamma, \, h \big) \cdot \big( \mathrm{e},\, h^{-1} \big) \;\; = \;\; \big( \gamma, \, \mathrm{e} \big)

exhibits the required crossed conjugation:

ϕ()=γ 1ϕ()α()(γ). \phi'(-) \;=\; \gamma^{-1} \cdot \phi(-) \cdot \alpha(-)(\gamma) \,.

For non-Lie groups

References

Last revised on September 15, 2021 at 11:15:11. See the history of this page for a list of all contributions to it.