Split-Biquaternion Involution Lattice

Introduction

The split biquaternion algebra carries four distinguished maps, the conjugations

$$ {}^{\natural} , \quad \bar{\cdot} , \quad {}^{*} , \quad \flat , $$

and the four distinguished subspaces of the two preceding articles Split-Biquaternion Relations Between Subspaces and its companions are precisely the fixed spaces of these four maps. The four are not independent: they are generated by three of them, they satisfy two composition rules, and under composition they form a group of order four. This article treats the maps themselves — their definitions, the group they form, the way they permute each other, and the lattice of fixed spaces they produce — and so supplies the operator background for the four subspace articles.

Throughout, $\tilde{Q} = Q_0 e_0 + Q_1 e_1 + Q_2 e_2 + Q_3 e_3$ with $Q_\mu = q_\mu + j q'_\mu \in \mathbb{D}$.

The Four Conjugations

Definitions

The four maps, with their formulas and their action on the eight real coordinates:

name	notation	formula	real coordinates
quaternion conjugation	${}^{\natural}$	$\tilde{Q}^{\natural} = Q_0e_0 - Q_1e_1 - Q_2e_2 - Q_3e_3$	$(q_0,-q_1,-q_2,-q_3,q'_0,-q'_1,-q'_2,-q'_3)$
split complex conjugation	$\bar{\cdot}$	$\bar{\tilde{Q}} = \bar{Q_0}e_0 + \bar{Q_1}e_1 + \bar{Q_2}e_2 + \bar{Q_3}e_3$	$(q_0,q_1,q_2,q_3,-q'_0,-q'_1,-q'_2,-q'_3)$
Hermitian conjugation	${}^{*}$	$\tilde{Q}^{*} = \overline{\tilde{Q}^{\natural}}$	$(q_0,-q_1,-q_2,-q_3,-q'_0,q'_1,q'_2,q'_3)$
anti-Hermitian conjugation	$\flat$	$\tilde{Q}^{\flat} = -\tilde{Q}^{*}$	$(-q_0,q_1,q_2,q_3,q'_0,-q'_1,-q'_2,-q'_3)$

The first two are the conjugations of the quaternion and of the split complex structure; the third is their composite; the fourth is the negative of the third. The sign patterns are identical to those of the biquaternion conjugations, because the definitions involve only the quaternion units and the coefficient conjugation, not the square of the central scalar.

Linearity and Antilinearity

Definition. A split-antilinear involution of $\mathbb{H}_{\mathbb{D}}$ is a real-linear map $\sigma$ with $\sigma^2 = \mathrm{id}$ and $\sigma(\lambda \tilde{Q}) = \bar{\lambda}\sigma(\tilde{Q})$ for all $\lambda \in \mathbb{D}$, where $\bar{\lambda}$ is the split complex conjugate.

Proposition. The four conjugations are involutions, and quaternion conjugation is $\mathbb{D}$-linear while split complex, Hermitian and anti-Hermitian conjugations are split-antilinear, each being conjugate to the identity on scalar multiples: for $\lambda \in \mathbb{D}$,

$$ (\lambda \tilde{Q})^{\natural} = \lambda \tilde{Q}^{\natural}, \qquad (\lambda \tilde{Q})^{*} = \bar{\lambda} \bar{\tilde{Q}}, \qquad (\lambda \tilde{Q})^{\dagger} = \bar{\lambda} \tilde{Q}^{*}, \qquad (\lambda \tilde{Q})^{\flat} = \bar{\lambda} \tilde{Q}^{\flat}. $$

Proof. Split complex conjugation acts on $j$ by $j^{*} = -j$, while quaternion conjugation fixes $j$ and touches only the quaternion units; hence $(\lambda \tilde{Q})^{\natural} = \lambda \tilde{Q}^{\natural}$, and the other three conjugate the coefficient $\lambda$. The action on the basis and the square of each map give the involution property.

Proposition. Quaternion conjugation, split complex conjugation and Hermitian conjugation are antiautomorphisms or automorphisms of the algebra, and the anti-Hermitian conjugation is an antiautomorphism up to the central sign: for $\tilde{Q}, \tilde{R} \in \mathbb{H}_{\mathbb{D}}$,

$$ (\tilde{Q}\tilde{R})^{\natural} = \tilde{R}^{\natural}\,\tilde{Q}^{\natural} , \qquad (\tilde{Q}\tilde{R})^{*} = \bar{\tilde{Q}}\bar{\tilde{R}} , \qquad (\tilde{Q}\tilde{R})^{\dagger} = \tilde{R}^{*}\tilde{Q}^{*} , \qquad (\tilde{Q}\tilde{R})^{\flat} = -\tilde{R}^{\flat}\tilde{Q}^{\flat} . $$

Proof. Each formula is a direct computation. The second follows because $\bar{\cdot}$ is coefficient-wise and the coefficients commute with the quaternion units: it is an automorphism, not an antiautomorphism. The third is the composite of the automorphism $\bar{\cdot}$ with the antiautomorphism ${}^{\natural}$; the sign in the last follows from $\flat = -{}^{*}$ and the centrality of $-1$.

The distinction of the second formula — it alone preserves the order of a product — is the reason the four maps are not interchangeable.

The Group They Generate

Two Composition Rules

Theorem. The four conjugations are related by

$$ {}^{*} = \bar{\cdot} \circ {}^{\natural} , \qquad \flat = -{}^{*} , $$

and any two distinct ones among ${}^{\natural}, \bar{\cdot}, {}^{*}$ generate the third, so that $\{ \mathrm{id}, {}^{\natural}, \bar{\cdot}, {}^{*} \}$ is a group isomorphic to the Klein four-group and those three are pairwise commuting involutions. Each of the four is its own inverse.

Proof. The first rule defines ${}^{*}$; the second defines $\flat$. Quaternion conjugation acts on the quaternion units and split complex conjugation acts on the coefficients, so they commute and their composite ${}^{*}$ is an involution; a product of two commuting involutions is an involution.

The set $\{ \mathrm{id}, {}^{\natural}, \bar{\cdot}, {}^{*} \}$ is the involution group of the algebra. The four conjugations including $\flat$ do not form a group, since $\flat \circ {}^{\natural}$ is $-{}^{\natural}\circ\bar{\cdot} = -{}^{*}$, not among the four.

Composition Table

Composing the four conjugations, the entry being the map applied first in the row and then in the column:

$\circ$	${}^{\natural}$	$\bar{\cdot}$	${}^{*}$	$\flat$
${}^{\natural}$	$\mathrm{id}$	${}^{*}$	$\bar{\cdot}$	$-\bar{\cdot}$
$\bar{\cdot}$	${}^{*}$	$\mathrm{id}$	${}^{\natural}$	$-{}^{\natural}$
${}^{*}$	$\bar{\cdot}$	${}^{\natural}$	$\mathrm{id}$	$-\mathrm{id}$
$\flat$	$-\bar{\cdot}$	$-{}^{\natural}$	$-\mathrm{id}$	$\mathrm{id}$

The entry $-\bar{\cdot}$ denotes the map $\tilde{Q} \mapsto -\bar{\tilde{Q}}$. The table contains the two rules of the theorem and shows where the group structure stops: a composite of two members of the group is the third, while a composite involving $\flat$ may be the negative of a member, as in the corners $-\bar{\cdot}$, or the negative of the identity. The table is formally identical to the composition table of the biquaternion conjugations.

How the Conjugations Permute Each Other

Theorem. Conjugation by any of the four maps, $\tau \mapsto \sigma \circ \tau \circ \sigma^{-1}$, fixes each of the four conjugations, so the conjugation action of the Klein group on itself is trivial and $\sigma \circ \tau = \tau \circ \sigma$ for all pairs:

$$ {}^{\natural} \circ \bar{\cdot} \circ {}^{\natural} = \bar{\cdot} , \qquad {}^{\natural} \circ {}^{*} \circ {}^{\natural} = {}^{*} , \qquad {}^{\natural} \circ \flat \circ {}^{\natural} = \flat . $$

Proof. The group $\{ \mathrm{id}, {}^{\natural}, \bar{\cdot}, {}^{*} \}$ is abelian by the preceding theorem, and in an abelian group conjugation is the identity; the anti-Hermitian conjugation is central because it is $-{}^{*}$ with $-\mathrm{id}$ central.

The practical content is that the four conjugations can be applied in any order without changing the result, and that the orbit of an element under the group does not depend on the order of the operations.

The Lattice of Fixed Spaces

The Fixed and Anti-Fixed Spaces

Theorem. To each of the four conjugations belong two subspaces, its fixed space and its anti-fixed space:

involution	fixed space	anti-fixed space
${}^{\natural}$	$\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$	$\mathrm{Vect}(\mathbb{H}_{\mathbb{D}})$
$\bar{\cdot}$	$\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$	$j\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$
${}^{*}$	$\mathbb{M}_+$	$\mathbb{M}_-$
$\flat$	$\mathbb{M}_-$	$\mathbb{M}_+$

The eight labels stand for six distinct subspaces, because the Hermitian pair is counted twice: the anti-Hermitian conjugation has the same eigenspaces as Hermitian conjugation with the signs exchanged, so

$$ \left\{ \tilde{Q} : \tilde{Q}^{\flat} = \tilde{Q} \right\} = \mathbb{M}_- , \qquad \left\{ \tilde{Q} : \tilde{Q}^{\flat} = -\tilde{Q} \right\} = \mathbb{M}_+ . $$

Proof. Each row is the comparison-of-coefficients computation recorded in the article of the corresponding subspace. For the last row, the equations $-\tilde{Q}^{*} = \tilde{Q}$ and $-\tilde{Q}^{*} = -\tilde{Q}$ are those of ${}^{*}$ read with the opposite labels.

The four subspaces that are named as the distinguished subspaces — the fixed spaces of the four involutions — are $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{M}_+$ and $\mathbb{M}_-$. The two remaining distinct sets, $\mathrm{Vect}(\mathbb{H}_{\mathbb{D}})$ of dimension $6$ and the anti-fixed space $j\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$ of split complex conjugation, are sums of coordinate blocks; the second is isometric to $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$ under multiplication by the central scalar $j$, as explained in Split-Biquaternion Relations Between Subspaces.

The Lattice

The fixed spaces are related by inclusion with the coordinate blocks, in the following lattice of subspaces of the eight-dimensional real space, in which a line joins a subspace to the minimal ones containing it:

level	subspaces
$\mathbb{H}_{\mathbb{D}}$	the whole algebra, dimension $8$
the sums	$\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} + \mathrm{Vect}(\mathbb{H}_{\mathbb{D}})$, $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}} + j\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{M}_+ + \mathbb{M}_-$ — all equal to $\mathbb{H}_{\mathbb{D}}$
the four-dimensional spaces	$\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$, $j\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{M}_+$, $\mathbb{M}_-$
the three-dimensional triples	$\langle e_1,e_2,e_3\rangle = \mathrm{Vect} \cap \mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$, $\langle je_1,je_2,je_3\rangle = \mathrm{Vect} \cap \mathbb{M}_+$
the two-dimensional centre	$\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} = \langle e_0 \rangle \oplus \langle je_0\rangle$
the one-dimensional lines	$\langle e_0\rangle$, $\langle je_0\rangle$
the origin	$0$

The lattice is not a chain, and it is not closed under sum, being the union of the three decompositions of the algebra rather than a single distributive lattice. Its skeleton is the three decompositions of the first table of Split-Biquaternion Relations Between Subspaces, and the diagram records how the six distinct fixed and anti-fixed spaces sit over the four coordinate blocks. The comparison with the biquaternion lattice is the replacement of the complex scalar line $\langle ie_0\rangle$ by the split scalar line $\langle je_0\rangle$, and of the anti-quaternion subspace $i\mathbb{H}_{\mathbb{B}}$ by the isometric copy $j\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$.

Involutions on the Involution Group

The four conjugations act on the algebra, and for an element $\tilde{Q}$ the images under the Klein group $\{ \mathrm{id}, {}^{\natural}, \bar{\cdot}, {}^{*} \}$ are the points of an orbit. By the orbit–stabilizer theorem the orbit has size

$$ 1 \quad \text{or} \quad 2 \quad \text{or} \quad 4 , $$

the size being $1$ for the real line $\langle e_0\rangle$, $2$ for the elements lying in $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$, in $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$ or in $\mathbb{M}_+$ but not in $\langle e_0\rangle$, and $4$ otherwise.

Proof. An element is fixed by the whole group exactly when it is fixed by ${}^{\natural}$ and by $\bar{\cdot}$, since ${}^{*}$ is their composite; that is $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} \cap \mathbb{H}_{\mathbb{H}_{\mathbb{D}}} = \langle e_0\rangle$, giving orbit size $1$. The stabilizer has order two exactly when the element is fixed by one of ${}^{\natural}, \bar{\cdot}, {}^{*}$ and by no second one — fixed by two it is fixed by the third and falls back into $\langle e_0\rangle$ — and the three fixed spaces of those maps are $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$ and $\mathbb{M}_+$, whose pairwise intersections are all $\langle e_0\rangle$; that gives orbit size $2$ exactly on the stated set, and orbit size $4$ on its complement.

Examples

An Element of Each Orbit Size

The element $e_0$ is fixed by all four conjugations: orbit size $1$.
The element $j = je_0$ lies in the centre $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$; quaternion conjugation fixes it and split complex conjugation negates it, so its orbit is $\{j, -j\}$, of size $2$.
The element $e_1$ lies in the quaternion subspace; split complex conjugation fixes it and quaternion conjugation negates it, so its orbit is $\{e_1, -e_1\}$, of size $2$.
The element $\tilde{Q} = e_1 + j$ lies in no one of the three fixed spaces: $\tilde{Q}^{\natural} = -e_1 + j$, $\bar{\tilde{Q}} = e_1 - j$, $\tilde{Q}^{*} = -e_1 - j$, $\tilde{Q}^{\flat} = e_1 - j$. The orbit has four distinct elements, of size $4$.

The Stabilizer of a Vector

For the element $e_1 \in \mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$ the stabilizer is $\{ \mathrm{id}, \bar{\cdot} \}$, of order $2$, and the orbit is $\{e_1, -e_1\}$; the orbit–stabilizer count $|\text{orbit}| \cdot |\text{stabilizer}| = 2 \cdot 2 = 4$ is the order of the group. For $e_0$ the stabilizer is the whole group, of order $4$, and the orbit has size $1$, the count again being $4$.

A Hermitian Element

For the Hermitian element $\tilde{Q} = e_0 + je_1$ the stabilizer contains ${}^{*}$ by definition; moreover $\tilde{Q}^{\natural} = e_0 - je_1 = \bar{\cdot}\tilde{Q}$, so quaternion and split complex conjugations also send it to $e_0 - je_1$, and the stabilizer is $\{ \mathrm{id}, {}^{*} \}$, of order $2$, with orbit $\{e_0 + je_1, e_0 - je_1\}$.

The Analysis and the Geometry of the Lattice

The lattice is algebraic, and the four spaces that it organises carry the four analytical and geometrical readings.

The Analysis on the Labelled Spaces

The fixed spaces of the four conjugations are the four subspaces of Split-Biquaternion Relations Between Subspaces, and the analysis they carry is accordingly the elliptic analysis on the quaternion subspace, the hyperbolic analysis on the centre, and the restricted ambient analysis on the two Hermitian sectors; the group generated by the four conjugations acts on the functions by pull-back, permuting the four readings.

The Geometry on the Labelled Spaces

Geometrically the four conjugations are four linear involutions of $\mathbb{R}^8$, and their fixed and anti-fixed spaces are the four invariant planes of the lattice. The orbit of a point under the Klein four group has size $1$, $2$ or $4$, according to the dimension of the stabiliser, and the geometric content of the lattice is that the four invariant planes are the axes along which the group acts; the reduction to four labelled spaces is the geometric form of the reduction from the six subspaces of $\mathbb{B}$ to the four here.

Summary

The split biquaternion algebra carries four conjugations — quaternion conjugation ${}^{\natural}$, split complex conjugation $\bar{\cdot}$, Hermitian conjugation ${}^{*} = \bar{\cdot}\circ{}^{\natural}$ and the anti-Hermitian conjugation $\flat = -{}^{*}$ — with the same coordinate sign patterns as in the biquaternion algebra. Quaternion conjugation is $\mathbb{D}$-linear and an antiautomorphism; split complex conjugation is split-antilinear and an automorphism; Hermitian conjugation is split-antilinear and an antiautomorphism; the anti-Hermitian conjugation satisfies $(\tilde{Q}\tilde{R})^{\flat} = -\tilde{R}^{\flat}\tilde{Q}^{\flat}$. The maps ${}^{\natural}, \bar{\cdot}, {}^{*}$ generate the Klein four-group $\{ \mathrm{id}, {}^{\natural}, \bar{\cdot}, {}^{*} \}$, whose composition table is identical to that of the biquaternion conjugations; a composite involving $\flat$ may be minus a member of the group. The conjugation action of the group on itself is trivial because the group is abelian. Each involution has a fixed and an anti-fixed space, giving eight labels but six distinct sets: the fixed spaces $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{M}_+$, $\mathbb{M}_-$ and the anti-fixed spaces $\mathrm{Vect}(\mathbb{H}_{\mathbb{D}})$ and $j\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$. The lattice over the four coordinate blocks records how these sit among each other; it is the biquaternion lattice with the complex scalar line replaced by the split scalar line and the anti-quaternion subspace by the isometric copy $j\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$. Under the action of the group an element has orbit size $1$ on the real line, $2$ in $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} \cup \mathbb{H}_{\mathbb{H}_{\mathbb{D}}} \cup \mathbb{M}_+$ away from it, and $4$ otherwise. The family of named subspaces is four, whereas the biquaternion family is six, because the split anti-fixed spaces are a six-dimensional non-fixed space and an isometric copy of the quaternion subspace.

Summary of Notation

Symbol	Meaning
$\mathbb{H}_{\mathbb{D}} = \mathbb{D} \otimes_{\mathbb{R}} \mathbb{H}$	Split biquaternion algebra, real dimension $8$
$\tilde{Q} = \sum_\mu Q_\mu e_\mu$, $Q_\mu = q_\mu + j q'_\mu$	General split biquaternion
${}^{\natural}, \bar{\cdot}, {}^{*}, {}^{\flat}$	The four conjugations
$\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}, \mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$	Centre and quaternion subspace, fixed spaces of ${}^{\natural}$ and $\bar{\cdot}$
$\mathrm{Vect}(\mathbb{H}_{\mathbb{D}}), j\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$	Anti-fixed spaces of ${}^{\natural}$ and $\bar{\cdot}$
$\mathbb{M}_+, \mathbb{M}_-$	Sectors, the fixed and anti-fixed spaces of ${}^{*}$ and $\flat$
$\langle e_0\rangle, \langle je_0\rangle, \langle e_1,e_2,e_3\rangle, \langle je_1,je_2,je_3\rangle$	The four coordinate blocks
$\bar{\lambda}$	Split complex conjugate of a scalar $\lambda \in \mathbb{D}$
orbit size $1, 2, 4$	Sizes under the Klein group, by the orbit–stabilizer theorem