Biquaternion Topology

Introduction

This article collects the topology of the null cone of the biquaternion algebra $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ and its projective geometry: the null cone and its link, the Segre embedding and the two rulings of null planes, the projective null quadric $Q^2$ with its tangency and its polarity, and the Klein–Plücker geometry of the lines of $\mathbb{P}^3$. The Euclidean structure of the ambient space, its contractibility and the Euclidean unit sphere read the Hermitian form and belong to The Euclidean Topology of the Biquaternion Algebra; they are quoted from there wherever the geometry of the null cone needs them.

The norm $N(\tilde{Q})=\tilde{Q}\tilde{Q}^{\natural}=\sum_\mu Q_\mu^2$ decides invertibility and vanishes exactly on the zero divisors together with the origin (Biquaternion Norm and Invertibility, Biquaternion Zero Divisors); both treatments are algebraic, and the projective sections below treat the same form geometrically, as the equation of a quadric. The article uses the algebra and fixed-point subspaces of Biquaternion Algebra, the zero divisor set of Biquaternion Zero Divisors, and Lie Groups. The polarisation of the norm, its real forms and its associated Clifford algebra are in Biquaternion Norm and Invertibility; the Lorentzian and conformal reading of the real slices is in Biquaternion Lorentzian and Conformal Geometry. No physics is invoked and no new result is claimed.

Scope. The topology of the group of units $\mathbb{B}^\times$ — the polar decomposition, the retractions onto its compact subgroups, the homotopy groups and the universal cover — is read on the Hermitian form and is treated in The Unitary Group of the Biquaternion Algebra, the algebraic group itself being The Biquaternion Unit Group as a Topological Group. This article owns the null cone, its link and their projective geometry, quoting the ambient Euclidean structure from The Euclidean Topology of the Biquaternion Algebra and the homotopy type of $\mathbb{B}^\times$ from those articles when a comparison is needed.

Conventions. The units are $e_0=1,e_1,e_2,e_3$ with $e_k^2=-e_0$, the central scalar imaginary is $i$, and $\tilde{Q}=\sum_{\mu=0}^{3}Q_\mu e_\mu$ with $Q_\mu=q_\mu+iq'_\mu\in\mathbb{C}$. The biquaternion norm is $N(\tilde{Q})=\sum_\mu Q_\mu^2$ and the Euclidean norm is $\|\tilde{Q}\|_E=(\sum_\mu|Q_\mu|^2)^{1/2}$.

As a real vector space the algebra is $\mathbb{R}^8$, hence contractible, and the Euclidean structure that exhibits this, the Euclidean unit sphere and the contractibility of the six distinguished subspaces are The Euclidean Topology of the Biquaternion Algebra; only the contractibility of the ambient space is quoted here. The distinguished subset studied in this article is the null cone $\{N(\tilde{Q})=0\}$ together with its link.

The Ambient Space

The algebra is $\mathbb{R}^8$ with the Euclidean topology of The Euclidean Topology of the Biquaternion Algebra, and it is contractible: the straight-line homotopy $H(s,\tilde{Q})=(1-s)\tilde{Q}$ contracts it to the origin, so $\pi_n(\mathbb{B})=0$ for every $n\geq1$ and every map into $\mathbb{B}$ is null-homotopic. The homotopy preserves every linear subspace, hence also the six distinguished subspaces and the null cone, so the null cone is homotopy trivial as well, like the algebra around it; but it is a singular subset of it, and its smooth structure away from the apex is what the rest of the article uses. Nothing in the projective sections rests on the metric; the Euclidean input is used only through the metric topology it induces.

The null cone

The null cone, or singular set, is

$$ \mathcal{N}=\{\tilde{Q}\in\mathbb{B}:N(\tilde{Q})=0\}=\{0\}\cup\mathcal{Z}, $$

where $\mathcal{Z}$ is the zero-divisor set.

It is closed, with empty interior, so $\mathbb{B}^\times$ is dense; it is a real algebraic cone with apex $0$, since $N$ is homogeneous of degree $2$; and its real dimension is $6$, as the complex hypersurface $N=0$ in $\mathbb{B}\cong\mathbb{C}^4$. The polynomial $N$ is irreducible and its complex gradient $2(Q_0,Q_1,Q_2,Q_3)$ vanishes only at the origin, so $0$ is the only singular point and the punctured cone $\mathcal{N}\setminus\{0\}=\mathcal{Z}$ is a smooth complex $3$-manifold, exactly the zero-divisor set. Away from the origin it is a real $6$-manifold, and at the origin it is not a manifold; the smooth structure, which is analytic, is in Biquaternion Analysis, and the local picture at the apex in §The link of the null cone.

Contractibility. The homotopy $K(s,\tilde{Q})=(1-s)\tilde{Q}$ maps $[0,1]\times\mathcal{N}$ into $\mathcal{N}$, since scaling a null element by a complex number preserves nullity, and contracts $\mathcal{N}$ to the apex. The null cone is thus homotopy trivial, like the algebra, but is a singular subset of it.

The Segre embedding

Choose on $\mathbb{B}$ the linear coordinates $$ Z_0=Q_0-iQ_3,\qquad Z_1=-iQ_1-Q_2,\qquad Z_2=-iQ_1+Q_2,\qquad Z_3=Q_0+iQ_3, $$ an invertible $\mathbb{C}$-linear change of the coordinates $Q_0,\dots,Q_3$. In them the norm is a split form, $$ N(\tilde{Q})=\sum_{\mu=0}^{3}Q_\mu^2=Z_0Z_3-Z_1Z_2, $$ since $Z_0Z_3=(Q_0-iQ_3)(Q_0+iQ_3)=Q_0^2+Q_3^2$ and $Z_1Z_2=(-iQ_1-Q_2)(-iQ_1+Q_2)=-Q_1^2-Q_2^2$. The null cone is therefore the affine hypersurface $Z_0Z_3=Z_1Z_2$, and each of its nonzero points is a pair of one-dimensional subspaces of $\mathbb{C}^2$. Indeed, for nonzero $u=(\alpha,\beta)$ and $v=(\gamma,\delta)$ the point $$ (Z_0,Z_1,Z_2,Z_3)=(\alpha\gamma,\alpha\delta,\beta\gamma,\beta\delta) $$ lies on the null cone, since $\alpha\gamma\cdot\beta\delta=\alpha\delta\cdot\beta\gamma$, and conversely every null point is of this form: if $Z_0\neq0$ the pair $u=(1,Z_2/Z_0)$, $v=(Z_0,Z_1)$ reproduces it, and the other cases are the same argument applied to a coordinate that is nonzero. The pair is determined only up to $(u,v)\mapsto(\lambda u,\lambda^{-1}v)$. Projectivising gives the Segre embedding $$ s:\mathbb{P}^1\times\mathbb{P}^1\longrightarrow\mathbb{P}^3,\qquad ([u],[v])\mapsto[\alpha\gamma:\alpha\delta:\beta\gamma:\beta\delta], $$ whose image is exactly the projective null quadric $$ \mathbb{P}(\mathcal{N})=\{[\tilde{Q}]\in\mathbb{P}^3:N(\tilde{Q})=0\}. $$ The affine null cone is the cone over the Segre variety $\mathbb{P}^1\times\mathbb{P}^1$, and the zero-divisor set is that cone with its apex removed.

The two rulings of null planes

In the coordinates above the polar form of $N$ is $$ B(\tilde R,\tilde S)=\tfrac12\bigl(Z_0Z'_3+Z_3Z'_0-Z_1Z'_2-Z_2Z'_1\bigr), $$ the polarisation of $Z_0Z_3-Z_1Z_2$, with $B(\tilde R,\tilde R)=N(\tilde R)$. For each $[u]=[\alpha:\beta]\in\mathbb{P}^1$ the two-dimensional subspace $$ W_{[u]}=\operatorname{span}\{(\alpha,0,\beta,0),\,(0,\alpha,0,\beta)\}=\{(\alpha\sigma,\alpha\tau,\beta\sigma,\beta\tau):\sigma,\tau\in\mathbb{C}\} $$ is totally isotropic: for $\tilde R$ built from $\sigma,\tau$ and $\tilde S$ from $\sigma',\tau'$ the form above gives $B(\tilde R,\tilde S)=\tfrac12\alpha\beta(\sigma\tau'+\tau\sigma'-\tau\sigma'-\sigma\tau')=0$. Since $\dim W_{[u]}=2$ is the maximal isotropic dimension for a non-degenerate form in dimension $4$, $W_{[u]}$ is a null plane, and the same holds for $$ W^{[v]}=\operatorname{span}\{(\gamma,\delta,0,0),\,(0,0,\gamma,\delta)\},\qquad [v]=[\gamma:\delta]. $$ Their projectivisations $\ell_{[u]}=\mathbb{P}(W_{[u]})$ and $m_{[v]}=\mathbb{P}(W^{[v]})$ are the two rulings: each family is a $\mathbb{P}^1$ of lines; every quadric point lies on exactly one line of each family; lines of the same family are disjoint, while lines of different families meet in exactly one point, the point $[(\alpha\gamma,\alpha\delta,\beta\gamma,\beta\delta)]$ that the pair $([u],[v])$ determines.

The projective null quadric $Q^2$

The projectivised null cone $$ Q^2=\{[\tilde{Q}]\in\mathbb{P}^3:N(\tilde{Q})=0\} $$ is a smooth irreducible quadric surface, isomorphic to $\mathbb{P}^1\times\mathbb{P}^1$; it is the classical Segre quadric. Non-degeneracy of $B$ gives smoothness, and over $\mathbb{C}$ all smooth quadric surfaces in $\mathbb{P}^3$ are projectively equivalent.

Its real points depend on the real form of Biquaternion Norm and Invertibility, §The Real Forms and Their Signatures: empty for the definite form on $\mathbb{H}_{\mathbb{B}}$; the sphere $S^2$ for the Lorentzian form on $\mathbb{M}_+$ (or $\mathbb{M}_-$); the torus $S^1\times S^1$ for the split form of signature $(2,2)$. Only in the split case does the real quadric contain real lines.

Lines in $\mathbb{P}^3$, the Klein quadric, and the Plücker embedding

A line in $\mathbb{P}^3$ is $\mathbb{P}(U)$ for a two-dimensional subspace $U\subset\mathbb{C}^4$. With a basis $\tilde Q,\tilde P$ of $U$, the Plücker coordinates are the six minors $$ p_{ij}=Q_iP_j-Q_jP_i,\qquad 0\le i < j\le3, $$ the coordinates of the decomposable bivector $\tilde Q\wedge\tilde P\in\Lambda^2\mathbb{C}^4$, defined up to an overall scalar. This gives the Plücker embedding $$ \mathrm{Gr}(2,4)\hookrightarrow\mathbb{P}(\Lambda^2\mathbb{C}^4)=\mathbb{P}^5, $$ whose image is the Klein quadric, the quadric hypersurface of dimension $4$ cut out by the Plücker relation $$ p_{01}p_{23}-p_{02}p_{13}+p_{03}p_{12}=0 . $$ Over $\mathbb{R}$ the Plücker form has signature $(3,3)$. This Klein quadric in $\mathbb{P}^5$ is distinct from the null quadric $Q^2$ in $\mathbb{P}^3$: it is the variety of lines of the projective space in which $Q^2$ sits.

The rulings of $Q^2$ appear in this picture as follows. A line of $Q^2$ is a maximal isotropic two-plane, and its Plücker point lies on the Klein quadric. The lines on $Q^2$ form two components, the two rulings, each a $\mathbb{P}^1$; their Plücker images are two conics on the Klein quadric. (The maximal isotropic subspaces of the Klein quadric itself are two families of projective planes $\mathbb{P}^2$: the stars of lines through a fixed point and the plane fields of lines in a fixed plane.)

Tangency and polarity

The form $B$ defines a polarity, the correlation $$ [\tilde{P}]\longmapsto[\tilde{P}]^{\perp}=\{[\tilde{Q}]:B(\tilde{P},\tilde{Q})=0\}, $$ well defined by bilinearity and bijective by non-degeneracy. The quadric is the locus of self-polar points, $[\tilde{P}]\in Q^2\iff B(\tilde{P},\tilde{P})=0$. For $[\tilde{P}]\in Q^2$, since the differential of $N$ at $\tilde{P}$ is $2B(\tilde{P},\cdot\,)$, the polar hyperplane is the tangent hyperplane, and its intersection with the quadric is the pair of ruling lines through $[\tilde{P}]$, $$ Q^2\cap[\tilde{P}]^{\perp}=\ell_{[u]}\cup m_{[v]}, $$ one line from each family. For two distinct null points $[\tilde{P}],[\tilde{Q}]$, the biquaternion norm on the line $\tilde{P}+t\tilde{Q}$ is $2t\,B(\tilde{P},\tilde{Q})$, so $$ [\tilde{P}][\tilde{Q}]\subset Q^2\iff B(\tilde{P},\tilde{Q})=0, $$ in which case the two points lie on a common ruling line. A line through $[\tilde{P}]\in Q^2$ is therefore tangent exactly when its direction lies in the tangent hyperplane; the two ruling lines are tangent, and every other tangent line meets the quadric only at $[\tilde{P}]$.

Automorphisms of the complex quadric

The projective automorphisms of $Q^2$ are induced by the complex orthogonal group of $N$: $$ \operatorname{Aut}(Q^2)\cong PO_4(\mathbb{C})=O_4(\mathbb{C})/\{\pm I\}, $$ acting on $\mathbb{P}^1\times\mathbb{P}^1$ by $([u],[v])\mapsto([Au],[Bv])$, with the $\mathbb{Z}/2$ exchanging the rulings. Its identity component $PSO_4(\mathbb{C})$ preserves each ruling; the outer component swaps them.

The link of the null cone

The link of the null cone is

$$ L=\{\tilde{Q}\in\mathbb{B}:N(\tilde{Q})=0,\ \|\tilde{Q}\|_E=1\}=\mathcal{N}\cap S^7_E. $$

Every nonzero null element is uniquely $t\,u$ with $t=\|\tilde{Q}\|_E>0$ and $u\in L$, so $\mathcal{N}$ is the cone on $L$ and is contractible (§The null cone). Since $\dim_{\mathbb{R}}\mathcal{N}=6$, the link has real dimension $5$. Under $M=uv^{T}$ with $\|u\|=\|v\|=1$,

$$ L\cong(S^3\times S^3)/U(1), $$

where $U(1)=S^1$ acts by $(u,v)\mapsto(\lambda u,\lambda^{-1}v)$, the relation under which $uv^{T}$ is unchanged. Here $Q^2=\mathbb{P}(\mathcal{N})\cong\mathbb{P}^1\times\mathbb{P}^1$ is the projectivised null cone, the Segre quadric of §The Segre embedding and §The two rulings of null planes. The Hopf maps combine to a bundle projection

$$ q:L\to Q^2\cong S^2\times S^2,\qquad[(u,v)]\mapsto([u],[v]), $$

whose fibre over $([u],[v])$ is the circle of pairs $(e^{i\theta}u_0,e^{-i\theta}v_0)$. Hence $L$ is a closed connected $5$-manifold that is an $S^1$-bundle over $S^2\times S^2$.

The map $S^3\times S^3\to L$ is a principal $S^1$-bundle, and its long exact homotopy sequence, with $\pi_1(S^3\times S^3)=\pi_2(S^3\times S^3)=0$, gives

$$ \pi_1(L)=0,\qquad \pi_2(L)\cong\pi_1(S^1)=\mathbb{Z}. $$

So $L$ is simply connected but not homeomorphic to $S^5$, whose $\pi_2$ vanishes: the null cone is genuinely non-manifold at its apex.

Pure biquaternions and reality conditions

A biquaternion is pure if its complex scalar part vanishes,

$$ P=\{\tilde{Q}\in\mathbb{B}:Q_0=0\}=\{Q_1e_1+Q_2e_2+Q_3e_3:Q_k\in\mathbb{C}\}\cong\mathbb{C}^3, $$

a contractible real six-dimensional subspace; its topology is trivial, and the interesting sets are the real forms cut out by a reality condition.

Complex conjugation splits $P$ into two real three-dimensional subspaces. The real condition $\bar{\tilde{Q}}=\tilde{Q}$ gives the pure real quaternions

$$ P\cap\mathbb{H}_{\mathbb{B}}=\operatorname{span}_{\mathbb{R}}\{e_1,e_2,e_3\}\cong\mathbb{R}^3, $$

whose unit sphere consists of the unit imaginary quaternions $\mu$ with $\mu^2=-e_0$, the family of real roots of $-1$ classified in Biquaternion Square Roots of Minus One, Zero and Plus One. The imaginary condition $\bar{\tilde{Q}}=-\tilde{Q}$ gives

$$ P\cap i\mathbb{H}_{\mathbb{B}}=\operatorname{span}_{\mathbb{R}}\{ie_1,ie_2,ie_3\}\cong\mathbb{R}^3, $$

Each is a copy of $\mathbb{R}^3$; for the first the unit sphere is

$$ S^2=\{\mu\in\mathbb{H}_{\mathbb{B}}:\mu^2=-e_0\}, $$

compact, connected and simply connected, with $\pi_2(S^2)\cong\mathbb{Z}$, and not a group, since only $S^0,S^1,S^3$ are groups. It is a homogeneous space of the unit quaternions and the base of the Hopf fibration $S^3\to S^2$; the sphere of roots of $+1$ is its image under multiplication by $i$, homeomorphic to it.

Finally, on either pure reality slice the restricted norm is definite, being $q_1^2+q_2^2+q_3^2$ on $P\cap\mathbb{H}_{\mathbb{B}}$ and $-((q'_1)^2+(q'_2)^2+(q'_3)^2)$ on $P\cap i\mathbb{H}_{\mathbb{B}}$. Hence $N(\tilde{Q})=0$ forces $\tilde{Q}=0$: the only null element of either pure reality slice is the origin, so the light cone meets these three-dimensional spaces only at its apex, unlike the Minkowski slices $\mathbb{M}_\pm$, whose null set is the three-dimensional light cone (Biquaternion Lorentzian and Conformal Geometry, §The Lorentzian Slice and Its Light Cone).

Summary

$\mathbb{B}\cong\mathbb{R}^8$ with the Euclidean topology is contractible, hence path-connected and simply connected with $\pi_n(\mathbb{B})=0$ for all $n\geq1$; the Euclidean structure itself, the six distinguished real subspaces and the Euclidean sphere are The Euclidean Topology of the Biquaternion Algebra, §The Contractibility of the Algebra and §The Euclidean Unit Sphere.
The Euclidean unit sphere $S^7_E$ is a closed, compact, connected $7$-manifold but not a group: $\|\cdot\|_E$ is not multiplicative and $S^7_E\not\subseteq\mathbb{B}^\times$. Its Hermitian topology is The Euclidean Topology of the Biquaternion Algebra; the level set that is a group, $N=1$, is the norm-one group treated in The Biquaternion Unit Group as a Topological Group.
The null cone $\mathcal{N}=\{N(\tilde{Q})=0\}=\{0\}\cup\mathcal{Z}$ is a closed real algebraic cone of real dimension $6$, irreducible, with the origin as its only singular point; it is a manifold away from the apex and non-manifold at the apex, and contractible, being the cone on its link; punctured, it is exactly the zero-divisor set.
The link $L=\mathcal{N}\cap S^7_E$ is a closed connected $5$-manifold, an $S^1$-bundle over $S^2\times S^2$, simply connected with $\pi_2(L)\cong\mathbb{Z}$. Since $\pi_2(S^5)=0$, the apex is genuinely singular.
The projectivised null cone is the smooth quadric $Q^2\cong\mathbb{P}^1\times\mathbb{P}^1$, the Segre quadric; the null cone is the affine cone over it, and its two rulings are the two families of maximal isotropic null planes, each a $\mathbb{P}^1$.
In the coordinates of §The Segre embedding the norm is $Z_0Z_3-Z_1Z_2$ and the polar form is $B$; the quadric is the locus of self-polar points, and the polarity gives the tangency, with $Q^2\cap[\tilde{P}]^\perp=\ell_{[u]}\cup m_{[v]}$ at a point of the quadric.
The quadric sits in the Plücker–Klein geometry of the lines of $\mathbb{P}^3$; its rulings are two conics on the Klein quadric. $\operatorname{Aut}(Q^2)\cong PO_4(\mathbb{C})$, while $SO^+(1,3)$ is the conformal group of the projective null cone $S^2$, not the automorphism group of the complex quadric.
The pure subspace $P\cong\mathbb{C}^3$ is contractible. In its real slice $P\cap\mathbb{H}_{\mathbb{B}}\cong\mathbb{R}^3$ the unit sphere is the sphere of real roots of $-1$, $S^2$, which is compact, connected and simply connected with $\pi_2(S^2)\cong\mathbb{Z}$ and is not a group.
The restricted norm is definite on each pure real slice, so the light cone of the Minkowski slices meets those slices only at the apex.

Summary of Notation

Symbol	Meaning
$\mathbb{B} \cong \mathbb{R}^8$	Biquaternion algebra as a real topological space; contractible
$\\|\tilde{Q}\\|_E = \big(\sum_{\mu=0}^{3} \\|Q_\mu\\|^2\big)^{1/2}$	Euclidean norm; makes $\mathbb{B}$ a topological algebra
$S^7_E = \{\\|\tilde{Q}\\|_E = 1\}$	Euclidean unit sphere; a $7$-manifold, not a group
$N(\tilde{Q}) = \sum_\mu Q_\mu^2$	Biquaternion norm
$\mathcal{N} = \{N(\tilde{Q}) = 0\}$	Null cone; closed, real dimension $6$, contractible, singular at $0$
$\mathcal{Z}$	Zero-divisor set; $\mathcal{N} = \{0\} \cup \mathcal{Z}$
$L = \mathcal{N} \cap S^7_E$	Link of the null cone; $S^1$-bundle over $S^2 \times S^2$, $\pi_1 = 0$, $\pi_2 \cong \mathbb{Z}$
$Q^2 = \mathbb{P}(\mathcal{N}) \cong \mathbb{P}^1 \times \mathbb{P}^1$	Projectivised null cone, the Segre quadric
$B(\tilde{P},\tilde{Q}) = \sum_\mu P_\mu Q_\mu$	Polar form of $N$, the complex bilinear dot product; $B(e_\mu,e_\nu) = \delta_{\mu\nu}$
$Z_0,\dots,Z_3$	Linear coordinates on $\mathbb{B}$ in which $N = Z_0Z_3 - Z_1Z_2$; see §The Segre embedding
$s : \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^3$	Segre embedding, $([u],[v]) \mapsto [\alpha\gamma:\alpha\delta:\beta\gamma:\beta\delta]$; its image is $Q^2$
$[u] = [\alpha:\beta]$, $[v] = [\gamma:\delta]$	The two factors of a null point, determined up to $(u,v)\mapsto(\lambda u,\lambda^{-1}v)$
$\ell_{[u]}, m_{[v]}$	The two rulings of $Q^2$; the lines through $[\tilde{P}] \in Q^2$
$p_{ij} = Q_i P_j - Q_j P_i$	Plücker coordinates; the Klein quadric in $\mathbb{P}^5$ is their Plücker locus
$\operatorname{Aut}(Q^2) \cong PO_4(\mathbb{C})$	Projective automorphisms of the quadric
$P = \{Q_0 = 0\} \cong \mathbb{C}^3$	Pure biquaternions; contractible
$S^2 = \{\mu \in \mathbb{H}_{\mathbb{B}} : \mu^2 = -e_0\}$	Sphere of real roots of $-1$
$\mathbb{B}^\times$	Group of units; its topology is in The Biquaternion Unit Group as a Topological Group