Introduction to the Six Subspaces

Introduction

The biquaternion algebra $\mathbb{B}$ carries four conjugations, and each of them is an involution. Each one splits $\mathbb{B}$ into a fixed space and an anti-fixed space, and the eight spaces so obtained reduce to six distinct subspaces, the six distinguished subspaces of $\mathbb{B}$. The conjugations themselves are not the subject of this article: their definitions, the group they form and the two spaces each defines are in Biquaternion Algebra and Biquaternion Involution Lattice.

This article treats the six subspaces themselves, one to a section. Each section gives the defining condition of the subspace, the coordinate condition it amounts to, the real basis that exhibits it and its real dimension, and closes with the part it plays. The table is the summary; the sections are the detail.

subspace	defining condition	real basis	$\dim_{\mathbb{R}}$	role
the centre $\mathbb{C}_{\mathbb{B}}$	$\tilde{Q}^{\natural} = \tilde{Q}$	$e_0, ie_0$	$2$	the centre of the algebra, a field
the vector subspace $\mathrm{Vect}(\mathbb{B})$	$\tilde{Q}^{\natural} = -\tilde{Q}$	$e_1, e_2, e_3, ie_1, ie_2, ie_3$	$6$	the kernel of the scalar part, a Lie algebra
the quaternion subspace $\mathbb{H}_{\mathbb{B}}$	$\bar{\tilde{Q}} = \tilde{Q}$	$e_0, e_1, e_2, e_3$	$4$	a division algebra
the anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$	$\bar{\tilde{Q}} = -\tilde{Q}$	$ie_0, ie_1, ie_2, ie_3$	$4$	a module over $\mathbb{H}_{\mathbb{B}}$
the Hermitian subspace $\mathbb{M}_+$	$\tilde{Q}^{*} = \tilde{Q}$	$e_0, ie_1, ie_2, ie_3$	$4$	a Jordan algebra
the anti-Hermitian subspace $\mathbb{M}_-$	$\tilde{Q}^{\flat} = \tilde{Q}$	$ie_0, e_1, e_2, e_3$	$4$	a Lie algebra

The six are pairwise distinct. Their relations — the coordinate blocks out of which they are built, their pairwise intersections, their sums, the action of the four conjugations and the action of the central imaginary unit — are the subject of Comparison of the Six Subspaces, and are not treated here. The algebra that each subspace carries is not treated here either: the product and its two halves, the Jordan algebra, the Lie algebra, the units, the zero divisors, the idempotents and projections, the ideals and the Peirce decomposition, the roots of minus one, the two matrix representations, the forms and the analysis are the business of the thematic articles of the group, each stated once for the six.

Nothing below is a physical statement. The elements are written $\tilde{Q}, \tilde{R}, \tilde{P}, \dots$, their complex coefficients $Q_0, Q_1, Q_2, Q_3$, and the real and imaginary parts of a coefficient $Q_\mu = q_\mu + i q'_\mu$; no other coordinates are used.

The Centre Subspace

Definition. The centre subspace $\mathbb{C}_{\mathbb{B}}$ is the fixed space of quaternion conjugation,

$$ \mathbb{C}_{\mathbb{B}} = \left\{ \tilde{Q} \in \mathbb{B} : \tilde{Q}^{\natural} = \tilde{Q} \right\} . $$

Coordinate condition. Writing $\tilde{Q} = Q_0e_0 + Q_1e_1 + Q_2e_2 + Q_3e_3$ and comparing the two sides of $\tilde{Q}^{\natural} = \tilde{Q}$ coefficient by coefficient, the coefficient of $e_0$ gives no condition while the coefficient of $e_k$ gives $-Q_k = Q_k$, that is $Q_k = 0$, for $k = 1, 2, 3$. The subspace is therefore the set of elements with vanishing vector part,

$$ \tilde{Q} = Q_0e_0 , \qquad Q_0 \in \mathbb{C} . $$

Basis and dimension. The real basis is $e_0, ie_0$ and $\dim_{\mathbb{R}}\mathbb{C}_{\mathbb{B}} = 2$. In the eight real coordinates it is the coordinate plane $q_1 = q_2 = q_3 = q'_1 = q'_2 = q'_3 = 0$.

Role. It is the centre of the algebra, the set of elements that commute with every element, and it is a commutative subalgebra isomorphic to $\mathbb{C}$; it is one of the two of the six that are closed under the product. Its elements are the central elements, written $Ae_0$ with $A \in \mathbb{C}$.

The Vector Subspace

Definition. The vector subspace $\mathrm{Vect}(\mathbb{B})$ is the anti-fixed space of quaternion conjugation,

$$ \mathrm{Vect}(\mathbb{B}) = \left\{ \tilde{Q} \in \mathbb{B} : \tilde{Q}^{\natural} = -\tilde{Q} \right\} . $$

Coordinate condition. Comparing coefficients, the condition is $Q_0 = 0$: the scalar part vanishes, and the subspace is the kernel of the scalar-part functional $\operatorname{Sc}(\tilde{Q}) = Q_0 = \tfrac{1}{2}(\tilde{Q} + \tilde{Q}^{\natural})$,

$$ \tilde{Q} = Q_1e_1 + Q_2e_2 + Q_3e_3 , \qquad Q_1, Q_2, Q_3 \in \mathbb{C} . $$

Basis and dimension. The real basis is $e_1, e_2, e_3, ie_1, ie_2, ie_3$ and $\dim_{\mathbb{R}}\mathrm{Vect}(\mathbb{B}) = 6$. It is the largest of the six, and the only one of dimension $6$.

Role. It is not a subalgebra, since the product of two pure vectors has a scalar part; it is closed under the commutator, the bracket of two pure vectors being twice their cross product. It splits into the real vector triple $\operatorname{span}\{e_1,e_2,e_3\}$ and the imaginary vector triple $\operatorname{span}\{ie_1,ie_2,ie_3\}$, and its elements are the pure vectors, written $\mathbf{Q}$.

The Quaternion Subspace

Definition. The quaternion subspace $\mathbb{H}_{\mathbb{B}}$ is the fixed space of complex conjugation,

$$ \mathbb{H}_{\mathbb{B}} = \left\{ \tilde{Q} \in \mathbb{B} : \bar{\tilde{Q}} = \tilde{Q} \right\} . $$

Coordinate condition. The condition is that each coefficient be fixed by complex conjugation, that is, that each be real, $Q_\mu = q_\mu$ for $\mu = 0, 1, 2, 3$.

Basis and dimension. The real basis is $e_0, e_1, e_2, e_3$ and $\dim_{\mathbb{R}}\mathbb{H}_{\mathbb{B}} = 4$. It is the copy of the real quaternion algebra inside $\mathbb{B}$.

Role. It is a subalgebra, the second of the two of the six that are closed under the product, and it is a division algebra: every non-zero element is a unit. It is the only one of the six that is non-commutative, and it is the image of the structure map $\mathbb{H} \to \mathbb{B}$, $h \mapsto he_0$.

The Anti-Quaternion Subspace

Definition. The anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$ is the anti-fixed space of complex conjugation,

$$ i\mathbb{H}_{\mathbb{B}} = \left\{ \tilde{Q} \in \mathbb{B} : \bar{\tilde{Q}} = -\tilde{Q} \right\} . $$

Coordinate condition. The condition is that each coefficient be negated by complex conjugation, that is, that each be purely imaginary, $Q_\mu = iq'_\mu$ for $\mu = 0, 1, 2, 3$.

Basis and dimension. The real basis is $ie_0, ie_1, ie_2, ie_3$ and $\dim_{\mathbb{R}}i\mathbb{H}_{\mathbb{B}} = 4$.

Role. Its elements are exactly the products $i\tilde{P}$ with $\tilde{P} \in \mathbb{H}_{\mathbb{B}}$, so the subspace is the image of $\mathbb{H}_{\mathbb{B}}$ under multiplication by the central imaginary unit. It is not a subalgebra — the product of two of its elements lands in the quaternion subspace — but it is a two-sided module over $\mathbb{H}_{\mathbb{B}}$. It splits into the imaginary scalar line $\mathbb{R}(ie_0)$ and the imaginary vector triple $\operatorname{span}\{ie_1,ie_2,ie_3\}$.

The Hermitian Subspace

Definition. The Hermitian subspace $\mathbb{M}_+$ is the fixed space of Hermitian conjugation,

$$ \mathbb{M}_+ = \left\{ \tilde{Q} \in \mathbb{B} : \tilde{Q}^{*} = \tilde{Q} \right\} , \qquad \tilde{Q}^{*} = \bar{\tilde{Q}}^{\natural} . $$

Coordinate condition. Since ${}^{*} = \bar{\cdot}\circ{}^{\natural}$, the condition is that the scalar coefficient be real while the three vector coefficients be purely imaginary,

$$ \tilde{Q} = q_0e_0 + iq'_1e_1 + iq'_2e_2 + iq'_3e_3 , \qquad q_0, q'_1, q'_2, q'_3 \in \mathbb{R} . $$

Basis and dimension. The real basis is $e_0, ie_1, ie_2, ie_3$ and $\dim_{\mathbb{R}}\mathbb{M}_+ = 4$.

Role. It is not a subalgebra, but it is closed under the symmetrized product $\tilde{P} \bullet \tilde{Q} = \tfrac{1}{2}(\tilde{P}\tilde{Q} + \tilde{Q}\tilde{P})$, which makes it a Jordan algebra. It is the set of elements with real scalar part and purely imaginary vector part.

The Anti-Hermitian Subspace

Definition. The anti-Hermitian subspace $\mathbb{M}_-$ is the anti-fixed space of Hermitian conjugation, which is also the fixed space of the reversal $\flat = -{}^{*}$,

$$ \mathbb{M}_- = \left\{ \tilde{Q} \in \mathbb{B} : \tilde{Q}^{*} = -\tilde{Q} \right\} = \left\{ \tilde{Q} \in \mathbb{B} : \tilde{Q}^{\flat} = \tilde{Q} \right\} . $$

Coordinate condition. The condition is the reverse of the Hermitian one: the scalar coefficient is purely imaginary and the three vector coefficients are real,

$$ \tilde{Q} = iq'_0e_0 + q_1e_1 + q_2e_2 + q_3e_3 , \qquad q'_0, q_1, q_2, q_3 \in \mathbb{R} . $$

Basis and dimension. The real basis is $ie_0, e_1, e_2, e_3$ and $\dim_{\mathbb{R}}\mathbb{M}_- = 4$.

Role. It is not a subalgebra and it is not closed under the symmetrized product; it is closed under the commutator, and it is a real Lie algebra of dimension four whose centre is the line $\mathbb{R}(ie_0)$. It is the set of elements with purely imaginary scalar part and real vector part, and it is the anti-fixed space of ${}^{*}$ as well as the fixed space of $\flat$.

Summary

The biquaternion algebra carries four conjugations whose fixed and anti-fixed spaces reduce to six distinct subspaces, the six distinguished subspaces. The centre $\mathbb{C}_{\mathbb{B}}$, the fixed space of quaternion conjugation, is the set of elements with vanishing vector part, of real basis $e_0, ie_0$ and dimension $2$; it is the centre of the algebra and a field. The vector subspace $\mathrm{Vect}(\mathbb{B})$, the anti-fixed space of quaternion conjugation, is the kernel of the scalar part, of real basis $e_1, e_2, e_3, ie_1, ie_2, ie_3$ and dimension $6$; it is a Lie algebra and not a subalgebra. The quaternion subspace $\mathbb{H}_{\mathbb{B}}$, the fixed space of complex conjugation, is the set of elements with real coefficients, of real basis $e_0, e_1, e_2, e_3$ and dimension $4$; it is a division subalgebra. The anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$, the anti-fixed space of complex conjugation, is the set of elements with purely imaginary coefficients, of real basis $ie_0, ie_1, ie_2, ie_3$ and dimension $4$; it is a module over the quaternion subspace and not a subalgebra. The Hermitian subspace $\mathbb{M}_+$, the fixed space of Hermitian conjugation, is the set of elements with real scalar part and purely imaginary vector part, of real basis $e_0, ie_1, ie_2, ie_3$ and dimension $4$; it is a Jordan algebra under the symmetrized product. The anti-Hermitian subspace $\mathbb{M}_-$, the anti-fixed space of Hermitian conjugation and the fixed space of the reversal, is the set of elements with purely imaginary scalar part and real vector part, of real basis $ie_0, e_1, e_2, e_3$ and dimension $4$; it is a real Lie algebra. Exactly two of the six are closed under the product, the centre and the quaternion subspace, and both are division algebras. The six are pairwise distinct; their relations are collected in Comparison of the Six Subspaces, and the algebra each one carries belongs to the thematic articles.

Summary of Notation

symbol	meaning
$\mathbb{B}$	the biquaternion algebra, $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{H}$
$e_0, e_1, e_2, e_3$	the basis, with $e_0$ the unit and $e_k^2 = -e_0$
$i$	the central imaginary unit, $i^2 = -1$, commuting with every element
$\tilde{Q} = \sum_\mu Q_\mu e_\mu$	a general biquaternion
$Q_\mu = q_\mu + i q'_\mu$	the complex coefficient of $e_\mu$, with $q_\mu, q'_\mu \in \mathbb{R}$
${}^{\natural}, \bar{\cdot}, {}^{*}, \flat$	quaternion, complex, Hermitian conjugation and reversal
$\operatorname{Sc}(\tilde{Q})$	the scalar part $Q_0$, equal to $\tfrac{1}{2}(\tilde{Q} + \tilde{Q}^{\natural})$
$\mathbb{C}_{\mathbb{B}}$	the centre subspace, fixed by ${}^{\natural}$
$\mathrm{Vect}(\mathbb{B})$	the vector subspace, anti-fixed by ${}^{\natural}$
$\mathbb{H}_{\mathbb{B}}, i\mathbb{H}_{\mathbb{B}}$	the quaternion and anti-quaternion subspaces
$\mathbb{M}_+, \mathbb{M}_-$	the Hermitian and anti-Hermitian subspaces
$\tilde{P} \bullet \tilde{Q}$	the symmetrized product $\tfrac{1}{2}(\tilde{P}\tilde{Q} + \tilde{Q}\tilde{P})$