The geometry of sedenion zero divisors

The Cayley-Dickson algebras form a sequence of real algebras $\mathbb{A}_{n}$, defined recursively beginning with $\mathbb{R}$ and doubling in dimension with each iteration. The first members of this family are the familiar real division algebras: $\mathbb{R}=\mathbb{A}_{0}$, $\mathbb{C}=\mathbb{A}_{1}$, $\mathbb{H}=\mathbb{A}_{2}$ and $\mathbb{O}=\mathbb{A}_{3}$. The next algebra in this sequence is the so-called sedenion algebra $\mathbb{S}=\mathbb{A}_{4}$, which is often overlooked in comparison to its lower-dimensional relatives due to its lack of certain desirable algebraic properties. Nonetheless, this somewhat enigmatic algebra has long intrigued mathematicians and has recently found applications in fields such as theoretical physics [GG19] and machine learning [SA20].

Since $\mathbb{S}$ is not a division algebra, it is interesting to understand the structure of its zero divisors. The topology of the sedenion zero divisors is described by the principal bundle

where $G_{2}$ is the excepcional compact Lie group of rank $2$ and $V_{2}(\mathbb{R}^{7})$ is the Stiefel manifold of orthonormal $2$-frames in $\mathbb{R}^{7}$. Specifically, $G_{2}$ is homeomorphic to the submanifold $\mathcal{Z}(\mathbb{S})\subset\mathbb{S}\times\mathbb{S}$ of normalized sedenion pairs that multiply to zero; $V_{2}(\mathbb{R}^{7})$ is homeomorphic to the submanifold $\operatorname{\mathit{ZD}}(\mathbb{S})\subset\mathbb{S}$ of sedenions with norm $\sqrt{2}$ that have non-trivial annihilators; and, for each $u\in\operatorname{\mathit{ZD}}(\mathbb{S})$, the fiber $\operatorname{SU}(2)$ corresponds to the sphere of the annihilator subspace of $u$ (see [Mor98, BDI08]). However, little is known regarding the geometry of the sedenion zero divisors.

Both $\mathcal{Z}(\mathbb{S})$ and $\operatorname{\mathit{ZD}}(\mathbb{S})$ carry a natural geometry as submanifolds of $\mathbb{R}^{32}$ and $\mathbb{R}^{16}$, respectively. Furthermore, since the zero divisors of $\mathbb{S}$ are invariant under $\operatorname{Aut}(\mathbb{S})$, whose connected component is isomorphic to $G_{2}$, it follows that $\mathcal{Z}(\mathbb{S})$ and $\operatorname{\mathit{ZD}}(\mathbb{S})$ are homogeneous submanifolds. In this article, we study the intrinsic geometry of the zero divisors of $\mathbb{S}$. First, we prove that $\mathcal{Z}(\mathbb{S})$ is isometric to $G_{2}$ with a naturally reductive left-invariant metric, forming the total space of a Riemannian submersion over the exceptional symmetric space $G_{2}/{\operatorname{SO}(4)}$, with fibers locally isometric to a product of two round $3$-spheres with different radii.

Next, we analyze the geometry of $\operatorname{\mathit{ZD}}(\mathbb{S})$, which is isometric to $V_{2}(\mathbb{R}^{7})=G_{2}/{\operatorname{SU}(2)}$ with a particular $G_{2}$-invariant metric. At first glance, the geometry of $\operatorname{\mathit{ZD}}(\mathbb{S})$ does not seem very interesting; however, by shrinking the metric along a certain circle fibration, we obtain a family $g_{r}$ ($r>0$) of $G_{2}$-invariant metrics on $V_{2}(\mathbb{R}^{7})$, where $g_{\frac{2}{3}}$ represents the original metric. This process reveals several distinguished examples. Specifically, we prove, among other things, that $(V_{2}(\mathbb{R}^{7}),g_{r})$:

• •

  has positive scalar curvature if and only of $r<\frac{20}{3}$;

• •

  is an Einstein manifold if and only if $r=\frac{5}{9}$;

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  has non-negative sectional curvature if and only if $0<r\leq\frac{4}{9}$.

These results are quite remarkable, as Einstein metrics and metrics with non-negative curvature are very rare. To the best of our knowledge, the examples presented in this article are new. The known homogeneous Einstein metrics on $V_{2}(\mathbb{R}^{7})=G_{2}/{\operatorname{SU}(2)}=\operatorname{SO}(7)/{\operatorname{SO}(5)}$ are limited to the unique $\operatorname{SO}(7)$-invariant Einstein metric discovered by Sagle and the so-called Jensen metrics (see [Sag70, Jen73, BH87, Ker98]). It is worth noticing that the metric $g_{\frac{5}{9}}$ is neither $\operatorname{SO}(7)$-invariant nor a Jensen metric. Regarding metrics with non-negative sectional curvature, we refer to the survey [Zil07]. Typically, examples of homogeneous metrics with non-negative curvature appear as normal homogeneous metrics or are constructed through a Cheeger deformation of a metric already known to have non-negative curvature. Recall that none of the metrics $g_{r}$ is normal homogeneous (nor even naturally reductive) and that the initial metric $g_{\frac{2}{3}}$ does not possess non-negative sectional curvature.

Let us comment briefly on the proof our main results. In order to study the geometry of $\mathcal{Z}(\mathbb{S})$, it is necessary to “fix an origin” so that the metric can be identified with a left-invariant metric on $G_{2}$. Any choice of such an origin for $\mathcal{Z}(\mathbb{S})$ leads to isometric metrics on $G_{2}$, but a well-chosen origin can greatly simplify computations. We select the origin from among the so-called 84 standard zero divisors of $\mathbb{S}$. Then, using the results in [DZ79], we show that $\mathcal{Z}(\mathbb{S})$ is a naturally reductive space. A similar approach applies to the study of $\operatorname{\mathit{ZD}}(\mathbb{S})$ with the metric $g_{r}$. Here, we select another standard zero divisor (different from the previous one) so that the isotropy subgroup of $G_{2}$ acts trivially on the usual subalgebra $\mathbb{H}\subset\mathbb{S}$. This choice allows the metric $g_{r}$ to be expressed in diagonal form with respect to the normal homogeneous metric, making it possible to derive a nice expression for the Ricci tensor of $g_{r}$.

The most challenging part is to determine the sign of the sectional curvatures of $g_{r}$. Since there is no manageable expression for the curvature (as there is for naturally reductive spaces), using algebraic manipulation proves to be nearly impossible. Indeed, the sectional curvature function $F_{r}$ of $g_{r}$ can be interpreted as a homogeneous polynomial of degree $4$ in $22$ real variables, which, for a generic $r$, has 285 non-trivial coefficients. To show that $F_{r}$ is non-negative for $0<r\leq\frac{4}{9}$, we reduce the problem to proving that both $F_{0}$ (the formal extension of $F_{r}$ at $r=0$) and $F_{\frac{4}{9}}$ are non-negative. By using convex optimization techniques, we are able to prove the stronger result that $F_{0}$ and $F_{\frac{4}{9}}$ are polynomial sums of squares.

Finally we want to mention that the computations required in the proof of some of our results are often cumbersome and were computer checked using the software SageMath. The code used to verify our results is available at [Reg24].

We believe this work shows that the study of the geometry of Cayley-Dickson algebras, particularly regarding their zero divisors, deserves further attention, as it may have interesting implications in differential geometry of compact homogeneous spaces.

The main references for this section are [Mor98, BDI08] on Cayley-Dickson algebras and their zero divisors, [Arv03] on the geometry of homogeneous spaces and [DZ79] on naturally reductive left-invariant metrics on compact Lie groups. Observe that in this section, as well as throughout the rest of the article, we start counting indices from 0.

Consider $\mathcal{Z}(\mathbb{S})$ as a submanifold of $\mathbb{S}\times\mathbb{S}\simeq\mathbb{R}^{32}$ with the induced metric. Although this reduction is not necessary here, one could lower the codimension of $\mathcal{Z}(\mathbb{S})$. In fact, by Proposition 2.1, $\mathcal{Z}(\mathbb{S})$ is a submanifold of $S^{6}\times S^{6}\times S^{6}\times S^{6}$. Since $G_{2}=\operatorname{Aut}(\mathbb{O})\subset\operatorname{Aut}(\mathbb{S})$ acts isometrically on $\mathbb{S}\times\mathbb{S}$, we have that $\mathcal{Z}(\mathbb{S})$ is a homogeneous submanifold. Furthermore, by Theorem 2.2, the diffeomorphism $\mathcal{Z}(\mathbb{S})\simeq G_{2}$ induces a left-invariant metric $g$ on $G_{2}$.

Now we direct our attention to the geometry of $\operatorname{\mathit{ZD}}(\mathbb{S})$ with the metric induced from the ambient space $\mathbb{S}\simeq\mathbb{R}^{16}$. Since $G_{2}$ acts isometrically and transitively on $\operatorname{\mathit{ZD}}(\mathbb{S})$, we have that $\operatorname{\mathit{ZD}}(\mathbb{S})$ is isometric to the Stiefel manifold $G_{2}\cdot u_{0}=G_{2}/{\operatorname{SU}(2)}=V_{2}(\mathbb{R}^{7})$, equipped with a certain $G_{2}$-invariant metric, where $\operatorname{SU}(2)$ is the isotropy subgroup of $u_{0}\in\operatorname{\mathit{ZD}}(\mathbb{S})$. We again denote by $g$ such a metric, which is defined by

Similarly to the case of $\mathcal{Z}(\mathbb{S})$, we can choose $u_{0}$ appropriately so that the Lie algebra of $\operatorname{SU}(2)$ is $\mathfrak{k}_{0}$. Taking $u_{0}=e_{1}+e_{10}$, we obtain that

is a reductive decomposition for $G_{2}/{\operatorname{SU}(2)}$. The corresponding $\operatorname{Ad}(\operatorname{SU}(2))$-invariant inner product on $\mathfrak{m}$ is given by

A detailed study of the isometry group and the curvature of $g$ is given in the next section. Before proceeding, we note a simple fact about the sectional curvatures of $g$.

For each $r>0$, we consider on $V_{2}(\mathbb{R}^{7})$ the family of $G_{2}$-invariant metrics given by

Indeed, $g_{r}$ gives an $\operatorname{Ad}(\operatorname{SU}(2))$-invariant inner product on $\mathfrak{m}$ since $\mathfrak{m}_{0}$ is the subspace of fixed points of the isotropy representation of $G_{2}/{\operatorname{SU}(2)}$ and

Next, we compute the connected component of the full isometry group of $g_{r}$.

Now, we compute the Ricci and scalar curvature of $g_{r}$. In particular, we obtain the following result.

In order to prove our theorem, we will need the following result, which is a particular case of Theorem 1 in [PW98] (see also [CLR95]).

Let us mention that the vector $\boldsymbol{x}$ has $\frac{(n+2)(n+1)}{2}$ coordinates and the subspace of, not necessarily positive semi-definite, matrices $H$ satisfying (4.3) has dimension $\frac{(n+2)(n+1)^{2}n}{12}$. Thus, finding an exact (positive semi-definite) solution $H$ for equation (4.3) can be quite difficult, even for relatively small values of $n$.