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

TL;DR

This paper constructs a minimal Lie groupoid and Lie algebroid for the singular octonionic Hopf foliation in 16-dimensional space, revealing its non-homogeneous and non-analytic nature and extending the structure to a Lie 3-algebroid.

Contribution

It introduces the first Lie groupoid and Lie algebroid models for the singular octonionic Hopf foliation, demonstrating their minimality and non-homogeneity.

Findings

Constructed a $ ext{G}_2$-equivariant Lie groupoid for the foliation.

Proved the foliation is maximal among all singular foliations generating it.

Showed the foliation cannot be generated by local isometries or real analytic structures.

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}.