Is there a non-abelian analogue of quantum flips in algebraic spectral chromatography?
For example, if $P\to P^+$ is the shifting operator on a normal Siegel graph $\mathcal{G}$, can we construct the associated operator on its $\mathrm{SU}(2)$-bundle in a canonical way?
More precisely, the shifts would be compatible with flips on the total space, and descend to the augmentation map $\mathcal{G}\to \mathcal{G}^+$ when passing to $\mathfrak{su}(2)$-quivers through the generalized $\mathrm{Diff}(\mathcal{G})$-operator. At least, this is possible in the trivial case $\mathcal{G}=\mathbb{CP}^1\rtimes_\theta\mathrm{SU}(2)$, where $\mathrm{SU}(2)$ acts via Mobius transforms and $\theta$ is the Euler twist. Simply take the direct sum with the standard representation of $\mathrm{Sp}(1)$ on $\mathbb{H}$ with the flat metric. This gives (via Eskin's theorem on meromorphic singularities) an holomorphic shifting map with the desired properties.