nLab nearby homomorphisms from compact Lie groups are conjugate

Idea

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

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

(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 $Hom(G,\Gamma) \,\subset\, Maps(G,\Gamma)$ under the adjoint action by $\Gamma$ is discrete:

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

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

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

Examples

Example

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

In this case, since $S^1$ is abelian so that the conjugation action on $Hom(G,S^1)$ 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

$$\alpha \;\colon\; G \xrightarrow{\;} Aut_{Grp}(\Gamma)$$

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

$$\alpha_{\vert C(G)} \;=\; id_{\Gamma} \,.$$

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 $\phi \;\colon\; G \xrightarrow{\;} \Gamma$ is equivalently a plain group homomorphism $(\phi(-),\,(-)) \,\colon\, G \xrightarrow{\;} \Gamma \rtimes G$ to the semidirect product group, subject to the constraint that its projection to $G$ is the identity morphism. Now say that another crossed homomorphism $\phi'$ is nearby if it is so as a plain homomorphism $(\phi'(-),(-))$ 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 $Hom(G,\Gamma \rtimes G))$.

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

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

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

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

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

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

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

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

exhibits the required crossed conjugation:

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

For non-Lie groups

• While the projective unitary group $\Gamma \,\coloneqq\,$ PU(ℋ) on a separable Hilbert space $\mathcal{H}$ is not a (finite-dimensional) Lie group and hence outside the scope of assumption for the above general theorem, the conclusion still holds, at least for $G$ discrete, hence finite:

  The PU(ℋ)-space of homomorphisms $Hom\big(G,\, PU(\mathcal{H})\big)$ is a disjoint union (3) of orbits of the conjugation action.

  This is established in Uribe & Lück 2013, Sec. 15, p. 38.

Related concepts

• Hilbert's fifth problem

References

• Pierre Conner, Edwin Floyd, Ch. III, Lem. 38.1 in: Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete 33, Springer 1964 (doi:10.1007/978-3-662-41633-4)

• Charles Rezk, Nearby homomorphisms from compact Lie groups are conjugate, 2013 (MO:q/123624)

• Charles Rezk, Rem. 2.2.1 in: Global Homotopy Theory and Cohesion, 2014 (pdf, pdf)

See also:

• Alejandro Adem, Frederick R. Cohen, Lemma 2.5 in: Commuting Elements and Spaces of Homomorphisms, Math. Ann. 338 (2007) 587–626 (arXiv:math/0603197, doi:10.1007/s00208-007-0089-z)

• Charles Rezk, Classifying spaces for 1-truncated compact Lie groups, Algebr. Geom. Topol. 18 (2018) 525-546 (arXiv:1608.02999, doi:10.2140/agt.2018.18.525)