Topology in the Space of Biquaternions

Introduction

This article builds the topology of the space of biquaternions, and its only input is the linear structure of that space. The space is a real vector space of dimension eight; a norm is defined on it; because the dimension is finite, all its norms are equivalent, so the space carries exactly one topological norm and one topology; that topology is then built explicitly and it is the Euclidean topology of $\mathbb{R}^{8}$; and in it the product of biquaternions is continuous, so that the space is a topological algebra as well as a topological vector space. The article uses the linear structure and the product, and nothing else.

Notation

The algebra is $\mathbb{B}=\mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$, with units $e_t=1,\vec e_x,\vec e_y,\vec e_z$ satisfying $\vec e_k^{2}=-e_t$ and $\vec e_x\vec e_y=\vec e_z$, $\vec e_y\vec e_z=\vec e_x$, $\vec e_z\vec e_x=\vec e_y$. The central imaginary is written $i$, so that it does not collide with the quaternion units. A general element and its real coordinates are

$$ \tilde{Q}=Q_te_t+Q_x\vec e_x+Q_y\vec e_y+Q_z\vec e_z,\qquad Q_\mu=q_\mu+iq'_\mu,\qquad q_\mu,q'_\mu\in\mathbb{R}, $$

so that, as a real vector space,

$$ \mathbb{B}=\mathbb{R}\text{-span}\{e_t,\vec e_x,\vec e_y,\vec e_z,ie_t,i\vec e_x,i\vec e_y,i\vec e_z\},\qquad \dim_{\mathbb{R}}\mathbb{B}=8, $$

with $\mathrm{Sc}(\tilde{Q})=Q_t$ the scalar part and the product of the algebra as in The Realification and the Effect of the Change of Scalars.

1. The Norm

Definition (norm). Let $V$ be a real vector space. A norm on $V$ is a function $\lVert\cdot\rVert:V\to[0,\infty)$ such that

$\lVert x\rVert=0$ if and only if $x=0$ (definiteness);
$\lVert\lambda x\rVert=\lvert\lambda\rvert\,\lVert x\rVert$ for every $\lambda\in\mathbb{R}$ and every $x\in V$ (homogeneity);
$\lVert x+y\rVert\leq\lVert x\rVert+\lVert y\rVert$ for all $x,y\in V$ (triangle inequality).

A pair $(V,\lVert\cdot\rVert)$ is a normed space, the function $d(x,y)=\lVert x-y\rVert$ is a metric on it, and the topology generated by that metric is the norm topology.

Definition (the norm of the space). On $\mathbb{B}$ define

$$ \lVert\tilde{Q}\rVert=\Bigl(\sum_{\mu\in\{t,x,y,z\}}\lvert Q_\mu\rvert^{2}\Bigr)^{1/2} =\Bigl(\sum_{\mu\in\{t,x,y,z\}}\bigl(q_\mu^{2}+(q'_\mu)^{2}\bigr)\Bigr)^{1/2}. $$

Proposition. $\lVert\cdot\rVert$ is a norm on $\mathbb{B}$.

Proof. Definiteness: a sum of squares vanishes only when every $Q_\mu=0$, and then $\tilde{Q}=0$. Homogeneity: $\lvert\lambda Q_\mu\rvert=\lvert\lambda\rvert\,\lvert Q_\mu\rvert$ for real $\lambda$. Triangle inequality: the coordinates $Q_\mu\mapsto(q_\mu,q'_\mu)$ identify $\mathbb{B}$ with $\mathbb{R}^{8}$, the expression is the Euclidean norm of that space, and the triangle inequality there is the Cauchy–Schwarz inequality.

2. The Topological Norm: Equivalence and Uniqueness

Definition (equivalent norms). Two norms $\lVert\cdot\rVert_1$ and $\lVert\cdot\rVert_2$ on $V$ are equivalent when there are constants $c,C>0$ with

$$ c\,\lVert x\rVert_1\leq\lVert x\rVert_2\leq C\,\lVert x\rVert_1\qquad\text{for every }x\in V . $$

Equivalent norms have the same open sets and therefore generate the same topology. The common class of a norm is called here its topological norm.

Theorem (finite dimension). Let $V$ be a finite-dimensional real vector space. Then

every two norms on $V$ are equivalent, and
$V$ carries exactly one Hausdorff topology compatible with its vector-space structure, namely the topology induced by any one of its norms.

Statement 1 is the theorem of the equivalence of norms, and statement 2 the theorem that every finite-dimensional Hausdorff topological vector space is linearly homeomorphic to $\mathbb{R}^{n}$. Both are in Normed and Banach Spaces and Comparison of Norms and Invertibility, and in the standard references (Rudin, Functional Analysis, theorems 1.21 and 1.22; Bourbaki, Topological Vector Spaces, chapter I).

Corollary. The space of biquaternions carries exactly one topological norm and exactly one topology. The choice of a norm is a matter of convenience and never a topological choice: the Euclidean norm of §1, the maximum norm $\max_\mu\lvert Q_\mu\rvert$, the sum norm $\sum_\mu(\lvert q_\mu\rvert+\lvert q'_\mu\rvert)$ and every other norm of $\mathbb{R}^{8}$ generate that one topology.

3. The Topology

Definition (the topology of $\mathbb{B}$). On $\mathbb{B}$ take the norm of §1, the metric $d(\tilde{Q}_1,\tilde{Q}_2)=\lVert\tilde{Q}_1-\tilde{Q}_2\rVert$, and the topology generated by the open balls

$$ B(\tilde{Q}_1,r)=\bigl\{\tilde{Q}_2\in\mathbb{B}:\lVert\tilde{Q}_2-\tilde{Q}_1\rVert0 . $$

Theorem. The topology of $\mathbb{B}$ is the Euclidean topology of $\mathbb{R}^{8}$ under the coordinate isomorphism $\tilde{Q}\mapsto(q_t,q'_t,q_x,q'_x,q_y,q'_y,q_z,q'_z)$, and by §2 it is the only topology the space carries.

It is Euclidean in the precise sense that this coordinate isomorphism is a linear bijection carrying the norm of §1 to the Euclidean norm of $\mathbb{R}^{8}$ and the balls $B(\tilde{Q}_1,r)$ to the Euclidean balls, so the open sets are those of the usual metric topology of $\mathbb{R}^{8}$ and of nothing else.

The topology is the usual one of a finite-dimensional real space: Hausdorff, second countable, locally compact, complete, path connected, and with the Heine–Borel property that a subset is compact exactly when it is closed and bounded (Metric, Uniform and Complete Spaces). It is also the topology inherited from any faithful finite-dimensional real representation: the regular representation gives a linear injection $\mathbb{B}\hookrightarrow M_4(\mathbb{R})$ (The Realification and the Effect of the Change of Scalars), and the topology defined above is the subspace topology of the operator norm there, by §2.

The norm of §1 is the topological norm: it is the norm that defines the topology of §3, and by §2 every other norm of the space gives the same topology.

4. The Algebra Structure

Proposition (the product is bounded). For all $\tilde{Q}_1,\tilde{Q}_2\in\mathbb{B}$,

$$ \boxed{\; \lVert\tilde{Q}_1\tilde{Q}_2\rVert\leq\sqrt{2}\,\lVert\tilde{Q}_1\rVert\,\lVert\tilde{Q}_2\rVert \;}. $$

Proof. The estimate is the sharp normed-algebra inequality of The Euclidean Topology of the Biquaternion Algebra, §The Algebra as a Normed Algebra, where it is proved from $\lVert\mathsf{M}_2(\tilde{Q})\rVert_F=\sqrt{2}\,\lVert\tilde{Q}\rVert$ and the submultiplicativity of the Frobenius norm.

Corollary. The product is continuous, and addition and scalar multiplication are continuous, so $\mathbb{B}$ is a topological algebra. The rescaled norm $\sqrt2\,\lVert\cdot\rVert$ is submultiplicative, so with that norm $\mathbb{B}$ is a normed algebra, and being finite-dimensional it is complete, hence a Banach algebra (Topological Algebras and Banach Algebras). The group $\mathbb{B}^{\times}$ of invertible elements is open in the topology and inversion is continuous on it.

5. Summary

The space of biquaternions is a real vector space of dimension eight. It carries a norm (§1), all its norms are equivalent and it carries exactly one topological norm and one topology (§2), that topology is built in §3 and is the Euclidean topology of $\mathbb{R}^{8}$, and in it the product is continuous, so the space is a topological, normed and Banach algebra with an open group of units (§4). One norm, one topological norm, one topology, and it is the Euclidean one.