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}.