The minimal Lie groupoid and infinity algebroid of the singular octonionic Hopf foliation

Abstract

The famous singular leaf decomposition ${\mathcal{L}}_{OH}$ of ${\mathbb{R}}^{16}\cong {\mathbb{O}}^2$ induced by the Hopf construction for octonions $\mathbb{O}$ has no known Lie group action generating it. In this article we construct a ${\mathrm{G}}_2$-equivariant Lie groupoid $\mathcal{G}\Rightarrow {\mathbb{O}}^2$ whose orbits coincide with ${\mathcal{L}}_{OH}$. Its Lie algebroid $E=\mathrm{Lie}(\mathcal{G})$ is of the form ${\mathbb{O}}^4\to {\mathbb{O}}^2$ with polynomial structure functions. Its sheaf of sections induces a singular foliation ${\mathcal{F}}_{OH}:=\rho (\Gamma (E))$ on ${\mathbb{O}}^2$, which we call the singular octonionic Hopf foliation (SOHF). ${\mathcal{F}}_{OH}$ is shown to be maximal among all singular foliations $\mathcal{F}$ generating ${\mathcal{L}}_{OH}$ -- in the polynomial, the real analytic, as well as in the smooth setting. We extend $E$ to a Lie $3$-algebroid, which is a minimal length representative of the universal Lie $\infty -$algebroid of the SOHF. This permits to prove that $E$ is the minimal rank Lie algebroid and that $\mathcal{G}$ the lowest dimensional Lie groupoid which generate the SOHF. The leaf decomposition ${\mathcal{L}}_{OH}$ is one of the few known examples of a singular Riemannian foliation in the sense of Molino which cannot be generated by local isometries (local non-homogeneity). We improve this result by showing that any smooth singular foliation $\mathcal{F}$ inducing ${\mathcal{L}}_{OH}$ cannot be even Hausdorff Morita equivalent to a singular foliation ${\mathcal{F}}_M$ on a Riemannian manifold $(M,g)$ generated by local isometries. Furthermore, we show that there is no real analytic singular foliation $\mathcal{F}$ generating ${\mathcal{L}}_{OH}$ which turns $({\mathbb{R}}^{16},g_{st},\mathcal{F})$ into a module singular Riemannian foliation as defined in \cite{NS24}.

38 pages