# nLab nearby homomorphisms from compact Lie groups are conjugate

Contents

## Examples

### $\infty$-Lie algebras

#### Mapping spaces

internal hom/mapping space

# Contents

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

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:

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