The Euclidean Topology of the Biquaternion Algebra

Introduction

The Hermitian form of the biquaternion algebra $\mathbb{B}$ — the sesquilinear inner product $\langle\tilde{P},\tilde{Q}\rangle=\sum_\mu P_{\bar\mu}Q_\mu$ built on the antilinear conjugation ${}^{*}$ — makes $\mathbb{B}$ into a complex Hilbert space whose real part is a Euclidean inner product, and the topology it defines is the topology in which every topological statement of the corpus is made. This article reads the topology of $\mathbb{B}$ from the Hermitian form: the Euclidean norm, the linear isometry onto $\mathbb{R}^{8}$, the Hilbert-space structure and the Riemannian metric; the normed-algebra inequality with its sharp constant; the contractibility of the algebra and of its six distinguished subspaces; and the Euclidean unit sphere $S^{7}_{E}$, which — unlike the level sets of the biquaternion norm — is a genuine sphere but is not a group and contains zero divisors.

The readings collected here are the Hermitian half of the former joint treatment of the ambient topology. The bilinear half — the null cone, its link and the projective geometry of the norm — is Biquaternion Topology, which quotes this article for the Euclidean structure and the contractibility. The two forms are compared in The Hermitian Form on the Biquaternion Algebra and The Bilinear Form on the Biquaternion Algebra; the matrix reading of the Euclidean norm is The Unit Group and the Frobenius Norm in the Matrix Representation.

Conventions. The algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ with units $e_0=1,e_1,e_2,e_3$, $e_k^2=-e_0$, central scalar imaginary $i$, and a general element $\tilde{Q}=\sum_{\mu=0}^{3}Q_\mu e_\mu$ with $Q_\mu=q_\mu+iq'_\mu\in\mathbb{C}$. The Hermitian conjugation is ${}^{*}={}^{\natural}\circ\bar{\cdot}$, the inner product is $\langle\tilde{P},\tilde{Q}\rangle=\sum_\mu P_{\bar\mu}Q_\mu$, the Euclidean norm is $\|\tilde{Q}\|_E=\bigl(\sum_\mu|Q_\mu|^{2}\bigr)^{1/2}=\bigl(\mathrm{Sc}(\tilde{Q}\tilde{Q}^{*})\bigr)^{1/2}$, and the biquaternion norm is $N(\tilde{Q})=\sum_\mu Q_\mu^{2}$.

The Euclidean Norm and the Isometry onto $\mathbb{R}^{8}$

The real part of the inner product is a genuine inner product. Write $\tilde{Q}=\sum_\mu(q_\mu+iq'_\mu)e_\mu$ and introduce the real inner product

$$ (\tilde{P},\tilde{Q})_{\mathbb{R}}=\mathrm{Re}\,\langle\tilde{P},\tilde{Q}\rangle=\sum_{\mu=0}^{3}\bigl(p_\mu q_\mu+p'_\mu q'_\mu\bigr). $$

Proposition. $(\cdot,\cdot)_{\mathbb{R}}$ is a positive definite real inner product on $\mathbb{B}$, of the form $\mathrm{Re}\langle\tilde{P},\tilde{Q}\rangle$, and its associated norm is $\|\tilde{Q}\|_E$.

Proof. Bilinearity over $\mathbb{R}$ is immediate from the definition; symmetry follows from $\langle\tilde{P},\tilde{Q}\rangle^{*}=\langle\tilde{Q},\tilde{P}\rangle$; and $(\tilde{Q},\tilde{Q})_{\mathbb{R}}=\sum_\mu(q_\mu^{2}+q'_\mu{}^{2})$ is positive off zero, since a vanishing sum of squares forces $q_\mu=q'_\mu=0$. The associated norm is the square root of $(\tilde{Q},\tilde{Q})_{\mathbb{R}}=\sum_\mu|Q_\mu|^{2}=\|\tilde{Q}\|_E^{2}$.

Theorem (the isometry). The coefficient map

$$ \iota:\mathbb{B}\longrightarrow\mathbb{R}^{8},\qquad \tilde{Q}=\sum_\mu(q_\mu+iq'_\mu)e_\mu\longmapsto(q_0,q_1,q_2,q_3,q'_0,q'_1,q'_2,q'_3), $$

is a linear isometry of $(\mathbb{B},(\cdot,\cdot)_{\mathbb{R}})$ onto $\mathbb{R}^{8}$ with the standard inner product. Equivalently, $\mathbb{B}\cong\mathbb{C}^{4}$ as a complex Hilbert space with orthonormal basis $e_0,e_1,e_2,e_3$.

Proof. The map is bijective and $\mathbb{R}$-linear because $e_0,e_1,e_2,e_3,ie_0,ie_1,ie_2,ie_3$ is a real basis; it preserves the inner product by the displayed formula. The complex structure is the central multiplication by $i$, which acts as $i$ on each coefficient, so the $\mathbb{C}$-span of the same four vectors is $\mathbb{B}$ and the complex inner product is $\langle\cdot,\cdot\rangle$.

Two consequences are used throughout. First, $\mathbb{B}$ is complete in $\|\cdot\|_E$: it is finite-dimensional, and a finite-dimensional inner product space is a Hilbert space. Second, $\|\cdot\|_E$ defines the unique Hausdorff vector-space topology on $\mathbb{B}$, the Euclidean topology, and this is the topology of the corpus. The metric is $d(\tilde{P},\tilde{Q})=\|\tilde{P}-\tilde{Q}\|_E$.

Remark (distinct from the bilinear norm). The corpus carries a second quadratic function, the biquaternion norm $N(\tilde{Q})=\sum_\mu Q_\mu^{2}$, complex-valued and indefinite; it is not a norm and defines no topology. The two agree on the quaternion subspace and differ by a sign on the anti-quaternion subspace (Biquaternion Norm and Invertibility, §The Euclidean Norm and the Hermitian Form). Everything topological uses $\|\cdot\|_E$, never $N$.

The Algebra as a Normed Algebra

The multiplication is continuous, and the estimate is sharp.

Theorem (the normed-algebra inequality). For all $\tilde{Q},\tilde{R}\in\mathbb{B}$,

$$ \|\tilde{Q}\tilde{R}\|_E\leq\sqrt{2}\,\|\tilde{Q}\|_E\|\tilde{R}\|_E, $$

and the constant $\sqrt{2}$ cannot be lowered.

Proof. Under the algebra isomorphism $\Phi:\mathbb{B}\to M_2(\mathbb{C})$ of Biquaternion 2×2 Matrix Element Representation, which is a linear isometry up to the factor $\sqrt2$, one has $\|\Phi(\tilde{T})\|_F=\sqrt2\|\tilde{T}\|_E$, where $\|\cdot\|_F$ is the Frobenius norm (The Forms in the Matrix Representation of the Biquaternion Algebra, §The Inner Product as the Hilbert–Schmidt Pairing). The Frobenius norm is submultiplicative, so $$ \sqrt2\,\|\tilde{Q}\tilde{R}\|_E=\|\Phi(\tilde{Q})\Phi(\tilde{R})\|_F\leq\|\Phi(\tilde{Q})\|_F\|\Phi(\tilde{R})\|_F=2\|\tilde{Q}\|_E\|\tilde{R}\|_E, $$ which is the inequality. For sharpness take $\tilde{Q}=\tilde{R}=e_0+ie_1$: then $\tilde{Q}^{2}=2e_0+2ie_1$ and $\|\tilde{Q}\|_E^{2}=2$, $\|\tilde{Q}^{2}\|_E=2\sqrt2$, so $\|\tilde{Q}^{2}\|_E=\sqrt2\,\|\tilde{Q}\|_E^{2}$.

Corollary. Multiplication $\mathbb{B}\times\mathbb{B}\to\mathbb{B}$ is continuous, inversion is continuous on the units, and $\mathbb{B}$ is a topological algebra over $\mathbb{R}$ with $\mathbb{B}^{\times}$ a topological group.

Proof. The inequality bounds the product in terms of the factors; inversion is $N(\tilde{Q})^{-1}\tilde{Q}^{\natural}$-based and $N^{-1}$ is continuous off the closed null cone, so the standard arguments apply; the group axioms with continuous operations give a topological group.

Proposition (the isometries that the algebra supplies). Multiplication by a central element and the conjugations are orthogonal:

$$ \|(A e_0)\tilde{Q}\|_E=|A|\,\|\tilde{Q}\|_E,\qquad \|\tilde{Q}^{\natural}\|_E=\|\bar{\tilde{Q}}\|_E=\|\tilde{Q}^{*}\|_E=\|\tilde{Q}\|_E . $$

Moreover, for every unitary biquaternion $\tilde{U}$ the inner conjugation $\Theta_{\tilde{U}}(\tilde{T})=\tilde{U}\tilde{T}\tilde{U}^{*}$ is a Euclidean isometry.

Proof. A central multiplier scales every coefficient by $A$, and the three conjugations $\natural$, $\bar{\cdot}$, ${}^{*}$ (together with the reversal $\flat=-{}^{*}$) permute the coefficients among $\pm Q_\mu$ and $\pm\bar Q_\mu$, preserving $\sum_\mu|Q_\mu|^{2}$. For the last statement, $\Phi$ carries $\Theta_{\tilde{U}}$ to $M\mapsto M_{\tilde{U}}XM_{\tilde{U}}^{\dagger}$ with $M_{\tilde{U}}$ unitary, and the Frobenius norm is invariant under unitary similarity.

The Contractibility of the Algebra

Theorem. $\mathbb{B}$ is contractible; hence it is path-connected and simply connected, with $\pi_n(\mathbb{B})=0$ for every $n\geq1$.

Proof. The straight-line homotopy $H(t,\tilde{Q})=(1-t)\tilde{Q}$, $t\in[0,1]$, is jointly continuous with $H(0,\cdot)=\mathrm{id}$ and $H(1,\cdot)\equiv0$, so the identity is homotopic to a constant map.

Corollary (the fixed subspaces). The centre, the vector subspace, the Hermitian subspace, the anti-Hermitian subspace, the quaternion subspace and the anti-quaternion subspace are each contractible, the homotopy above preserving every linear subspace, so each is a point as far as homotopy is concerned:

$$ \mathbb{C}_{\mathbb{B}}\cong\mathbb{R}^{2},\qquad \mathbb{V}_{\mathbb{B}}\cong\mathbb{R}^{6},\qquad \mathbb{H}_{\mathbb{B}}\cong i\mathbb{H}_{\mathbb{B}}\cong\mathbb{R}^{4},\qquad \mathbb{M}_{+}\cong\mathbb{M}_{-}\cong\mathbb{R}^{4}. $$

Proof. Each is a linear subspace of $\mathbb{B}$, and the homotopy above preserves it. The dimensions are those of Biquaternion Relations Between Subspaces.

Corollary. Every map into $\mathbb{B}$ is null-homotopic, and $\mathbb{B}$ carries no topological obstruction of its own; the topology of the algebra is entirely the topology of its distinguished subsets — the unit group, the null cone and the spheres.

Proof. Immediate from the theorem.

The complement of the null cone is dense in $\mathbb{B}$, because the null cone is a proper algebraic subset of real codimension two — a complex hypersurface of $\mathbb{B}\cong\mathbb{C}^{4}$ — with empty interior (Biquaternion Topology, §The null cone); this is what makes the group of units, an open dense subset, carry the topology it does.

The Euclidean Unit Sphere

$$ S^{7}_{E}=\{\tilde{Q}\in\mathbb{B}:\|\tilde{Q}\|_E=1\}\cong S^{7} $$

is, by the isometry of §The Euclidean Norm and the Isometry onto $\mathbb{R}^{8}$, the standard unit sphere of $\mathbb{R}^{8}$: closed, compact, connected, and a smooth $7$-manifold. It is nevertheless the wrong sphere for the algebra, and the reason is that $\|\cdot\|_E$ is not multiplicative: for $\tilde{Q}=e_1+ie_2$ one has $\tilde{Q}^{2}=0$ while $\|\tilde{Q}\|_E=\sqrt2$, so

$$ \tilde{Q}_0=\frac{e_1+ie_2}{\sqrt2}\in S^{7}_{E},\qquad N(\tilde{Q}_0)=0, $$

and $\tilde{Q}_0$ is a zero divisor. Hence $S^{7}_{E}\not\subseteq\mathbb{B}^{\times}$ and $S^{7}_{E}$ is not a subgroup of $\mathbb{B}^{\times}$.

Proposition (the sphere meets the null cone in a compact $5$-manifold). The intersection $S^{7}_{E}\cap\mathcal{N}$, with $\mathcal{N}=\{N=0\}$, is a compact real $5$-manifold without boundary, homeomorphic to the link of the null cone.

Proof. The null cone is a real algebraic cone of real dimension $6$ with its only singular point at the origin (Biquaternion Topology, §The null cone); intersecting the smooth part with the transverse unit sphere and removing the origin gives a compact smooth manifold of dimension $6-1=5$, and it is the link by definition (Biquaternion Topology, §The link of the null cone).

Remark (the three spherical level sets). The algebra carries three level sets that look like unit spheres, and they are genuinely different objects; only the group-theoretic ones are developed in The Unitary Group of the Biquaternion Algebra.

Level set	Geometry	Algebra
$\\|\tilde{Q}\\|_E=1$	$S^{7}$, compact $7$-manifold	no group structure; contains zero divisors
$N(\tilde{Q})=1$	non-compact real $6$-manifold, homotopy equivalent to $S^{3}$	closed subgroup $\mathbb{B}^{\times}_1$
$\tilde{Q}^{*}\tilde{Q}=e_0$	$\cong S^{1}\times S^{3}$	closed subgroup $U(\mathbb{B})\cong U(2)$

The middle row is The Biquaternion Unit Group as a Topological Group; the bottom row is The Unitary Group of the Biquaternion Algebra, where the retraction of the group of units onto it is proved.

The Hilbert Space and the Duality

The inner product $\langle\cdot,\cdot\rangle$ makes $\mathbb{B}$ a Hilbert space of complex dimension four, and the real inner product $(\cdot,\cdot)_{\mathbb{R}}$ a Euclidean space of dimension eight. Two structures follow and are recorded here because later articles use them.

The Riesz duality. Every $\mathbb{C}$-linear functional on $\mathbb{B}$ is $\tilde{Q}\mapsto\langle\tilde{P},\tilde{Q}\rangle$ for a unique $\tilde{P}$; every $\mathbb{R}$-linear functional is $(\tilde{P},\cdot)_{\mathbb{R}}$ for a unique $\tilde{P}$. This is the finite-dimensional Riesz representation theorem, and it is what makes the adjoint operations of the operator articles well defined.

The Riemannian metric. $(\cdot,\cdot)_{\mathbb{R}}$ is the flat Riemannian metric of $\mathbb{B}\cong\mathbb{R}^{8}$; the orthogonal group $O(8)$ is its isometry group, and the $\mathbb{C}$-linear isometries that preserve the algebra structure are exactly the inner automorphisms by the unitary biquaternions, of group $PU(2)\cong SO(3)$ (Biquaternion Automorphisms and Derivations). Over $\mathbb{R}$ the coefficient conjugation adds one more coset, since it is an $\mathbb{R}$-algebra automorphism of unit Euclidean norm, so the real-linear automorphisms that are Euclidean isometries form $PU(2)\rtimes\mathbb{Z}/2$.

Summary

The Hermitian form equips $\mathbb{B}$ with the real inner product $(\tilde{P},\tilde{Q})_{\mathbb{R}}=\mathrm{Re}\sum_\mu P_{\bar\mu}Q_\mu$ and the Euclidean norm $\|\tilde{Q}\|_E=(\sum_\mu|Q_\mu|^{2})^{1/2}$, and the coefficient map is a linear isometry $\mathbb{B}\cong\mathbb{R}^{8}$, equivalently $\mathbb{B}\cong\mathbb{C}^{4}$ as a complex Hilbert space. Multiplication satisfies the sharp inequality $\|\tilde{Q}\tilde{R}\|_E\leq\sqrt2\|\tilde{Q}\|_E\|\tilde{R}\|_E$, and central multipliers, the conjugations and inner conjugations by unitary elements are Euclidean isometries. The algebra is contractible, as is each of its six distinguished subspaces, so every map into $\mathbb{B}$ is null-homotopic. The Euclidean unit sphere $S^{7}_{E}$ is a genuine $S^{7}$ but is not a group and contains zero divisors; its intersection with the null cone is a compact $5$-manifold, the link. The three spherical level sets — Euclidean, norm-one and unitary — are distinct, and only the last two are groups.

Summary of Notation

Symbol	Meaning
$(\tilde{P},\tilde{Q})_{\mathbb{R}}=\mathrm{Re}\,\langle\tilde{P},\tilde{Q}\rangle$	Real inner product; positive definite
$\\|\tilde{Q}\\|_E=(\sum_\mu\lvert Q_\mu\rvert^{2})^{1/2}$	Euclidean norm
$\iota:\mathbb{B}\to\mathbb{R}^{8}$	Linear isometry onto $\mathbb{R}^{8}$
$\\|\tilde{Q}\tilde{R}\\|_E\leq\sqrt2\\|\tilde{Q}\\|_E\\|\tilde{R}\\|_E$	Normed-algebra inequality, sharp
$H(t,\tilde{Q})=(1-t)\tilde{Q}$	Contraction of $\mathbb{B}$ to $0$
$S^{7}_{E}=\{\\|\tilde{Q}\\|_E=1\}$	Euclidean unit sphere; not a group
$S^{7}_{E}\cap\mathcal{N}$	Compact $5$-manifold; the link of the null cone
$O(8)$	Isometry group of the Euclidean structure