Biquaternion Involution Lattice

Introduction

The biquaternion algebra carries four distinguished antilinear maps, the conjugations

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

and the six subspaces of the two preceding articles Biquaternion Relations Between Subspaces and its companions are precisely the fixed and anti-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 the first three, together with the identity, 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 six subspace articles.

The Four Conjugations

Definitions

On $\tilde{Q} = Q_0e_0 + Q_1e_1 + Q_2e_2 + Q_3e_3$ with $Q_\mu = q_\mu + iq'_\mu$:

name	notation	formula	real coordinates
quaternion conjugation	${}^{\natural}$	$\tilde{Q}^{\natural} = Q_0e_0 - Q_1e_1 - Q_2e_2 - Q_3e_3$	$(q_0,-q_1,-q_2,-q_3,q'_0,-q'_1,-q'_2,-q'_3)$
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	${}^{*}$	$\tilde{Q}^{*} = \overline{\tilde{Q}^{\natural}}$	$(q_0,-q_1,-q_2,-q_3,-q'_0,q'_1,q'_2,q'_3)$
reversal	$\flat$	$\tilde{Q}^{\flat} = -\tilde{Q}^{*}$	$(-q_0,q_1,q_2,q_3,q'_0,-q'_1,-q'_2,-q'_3)$

The first two are the conjugations of the quaternion and of the complex structure; the third is their composite; the fourth is the negative of the third. The names are conventional and are those used throughout the series; the notations ${}^{\natural}$ and $\bar{\cdot}$ are not to be confused with the complex conjugate of a coefficient, which is always written with a star on the coefficient alone.

Linearity and Antilinearity

Definition. An antilinear involution of $\mathbb{B}$ is a real-linear map $\sigma$ with $\sigma^2 = \mathrm{id}$ and $\sigma(\lambda \tilde{Q}) = \bar{\lambda}\sigma(\tilde{Q})$ for $\lambda \in \mathbb{C}$.

Proposition. The four conjugations are antilinear involutions, and the first and the third are antiautomorphisms of the algebra, while the second is an automorphism, reversing nothing, and the fourth is an antiautomorphism up to the central sign: for $\tilde{Q}, \tilde{R} \in \mathbb{B}$,

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

Proof. Each formula is a direct computation from the definitions; the sign in the last follows from $\flat = -{}^{*}$ and ${}^{*}$ being an antiautomorphism, since $-\mathrm{id}$ is central. The second formula is the one that does not reverse: coefficient-wise conjugation is multiplicative, because it is a ring homomorphism of $\mathbb{C}$ applied to the coefficients.

The distinction between the second conjugation and the others is the reason the four maps are not interchangeable: $\bar{\cdot}$ reverses nothing, being the coefficient-wise conjugation, while ${}^{\natural}$, ${}^{*}$ and $\flat$ reverse the order of a product.

The Group They Generate

Two Composition Rules

Theorem. The four conjugations are related by

$$ {}^{*} = \bar{\cdot} \circ {}^{\natural} , \qquad \flat = -{}^{*} = {}^{*} \circ (-\mathrm{id}) , $$

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 ${}^{\natural}, \bar{\cdot}, {}^{*}$ are pairwise commuting involutions. Each of the four is its own inverse.

Proof. The first rule is the definition of ${}^{*}$; the second is its definition as well. For the commutativity, ${}^{\natural}$ and $\bar{\cdot}$ are generated by the quaternion and complex structures, which are independent and commute; their composite is ${}^{*}$, and a product of two commuting involutions is an involution.

The set $\{ \mathrm{id}, {}^{\natural}, \bar{\cdot}, {}^{*} \}$ is therefore the involution group of the algebra; the set of four conjugations with $\flat$ in place of one of them is not a group under composition, since $\flat \circ \flat = \mathrm{id}$ but $\flat \circ {}^{\natural}$ is not among the four. The lattice of the next section is generated by the group and by the central sign.

Composition Table

Composing the four conjugations, with the convention that the entry is 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: the composite of two maps of the group is the third, while a composite involving $\flat$ may be the negative of a member of the group — as in the two corners $-\bar{\cdot}$ — or the negative of the identity, as in the pairs $({}^{*},\flat)$ and $(\flat,{}^{*})$.

How the Conjugations Permute Each Other

Theorem. Conjugation by an involution $\sigma$, that is the operation $\tau \mapsto \sigma \circ \tau \circ \sigma^{-1}$, permutes the four conjugations among themselves:

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

and more generally every $\sigma$ in the group fixes every $\tau$ in the group, so the conjugation action of the Klein group on itself is trivial; the reversal is fixed by the action as well. The maps are mutually compatible in the sense that $\sigma \circ \tau = \tau \circ \sigma$ for all pairs of the four.

Proof. The group is abelian by the theorem above, and in an abelian group conjugation is the identity.

The practical content of the theorem is that the four conjugations can be applied in any order without changing the result, and that the elementary operator classes — the fourteen products of one of the four conjugations with an element of the algebra — are consistent: the group of operators acts on the algebra, and its action commutes with the action of the four conjugations on operators.

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	generated by
${}^{\natural}$	$\mathbb{C}_{\mathbb{B}}$	$\mathrm{Vect}(\mathbb{B})$	$\mathbb{C}$ and the vector structure
$\bar{\cdot}$	$\mathbb{H}_{\mathbb{B}}$	$i\mathbb{H}_{\mathbb{B}}$	the real structure
${}^{*}$	$\mathbb{M}_+$	$\mathbb{M}_-$	the Hermitian structure
$\flat$	$\mathbb{M}_-$	$\mathbb{M}_+$	the reversal

The eight listed spaces are six, since the Hermitian pair is counted twice: the reversal has the same eigenspaces as Hermitian conjugation with the signs exchanged, and consequently $$ \left\{ \tilde{Q} : \tilde{Q}^{\flat} = \tilde{Q} \right\} = \mathbb{M}_- , \qquad \left\{ \tilde{Q} : \tilde{Q}^{\flat} = -\tilde{Q} \right\} = \mathbb{M}_+ . $$

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

The Lattice

The fixed spaces are related by inclusion with the coordinate blocks of the algebra, 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
$\mathbb{B}$	the whole algebra, dimension $8$
the sums	$\mathbb{C}_{\mathbb{B}} + \mathrm{Vect}(\mathbb{B})$, $\mathbb{H}_{\mathbb{B}} + i\mathbb{H}_{\mathbb{B}}$, $\mathbb{M}_+ + \mathbb{M}_-$ — all equal to $\mathbb{B}$
the four-dimensional spaces	$\mathbb{H}_{\mathbb{B}}, i\mathbb{H}_{\mathbb{B}}, \mathbb{M}_+, \mathbb{M}_-$
the three-dimensional triples	$\langle e_1,e_2,e_3 \rangle = \mathrm{Vect} \cap \mathbb{H}_{\mathbb{B}}$, $\langle ie_1,ie_2,ie_3\rangle = \mathrm{Vect} \cap \mathbb{M}_+$
the two-dimensional centre	$\mathbb{C}_{\mathbb{B}} = \langle e_0 \rangle \oplus \langle ie_0\rangle$
the one-dimensional lines	$\langle e_0\rangle$, $\langle ie_0\rangle$
the origin	$0$

The lattice is not a chain, and it is not closed under sum, being the union of the three decompositions of $\mathbb{B}$ rather than a single distributive lattice — the intersections of the previous article are its meets, and the sums its joins where the sum does not reach the top. Its skeleton is the three decompositions of the first table of Biquaternion Relations Between Subspaces, and the diagram above records how the six subspaces sit over the four coordinate blocks.

Involutions on the Involution Group

The four involutions act on the whole set of operator classes by composition, and their action is best recorded on the generators. For an element $\tilde{Q}$ of the algebra, the four images $\tilde{Q}, \tilde{Q}^{\natural}, \bar{\tilde{Q}}, \tilde{Q}^{*}$ are the four points of an orbit of the involution group, and the orbit has size

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

the size being $1$ for the real centre $\langle e_0\rangle$, $2$ for the elements lying in $\mathbb{C}_{\mathbb{B}}$, in $\mathbb{H}_{\mathbb{B}}$ or in $\mathbb{M}_+$ but not in $\langle e_0\rangle$, and $4$ otherwise. The statement is the orbit-stabilizer theorem applied to the action of the Klein group on the algebra, the stabilizer being the subgroup of the maps fixing the element.

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 $\mathbb{C}_{\mathbb{B}} \cap \mathbb{H}_{\mathbb{B}} = \langle e_0\rangle$, which gives 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 of them — fixed by two it is fixed by the third and falls back in $\langle e_0\rangle$ — and the three fixed spaces of those maps are $\mathbb{C}_{\mathbb{B}}$, $\mathbb{H}_{\mathbb{B}}$ 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

Take $\tilde{Q} = e_0 + 2e_1 + 3ie_1 + 4ie_2$.

$\tilde{Q}^{\natural} = e_0 - 2e_1 - 3ie_1 - 4ie_2$;
$\bar{\tilde{Q}} = e_0 + 2e_1 - 3ie_1 - 4ie_2$;
$\tilde{Q}^{*} = e_0 - 2e_1 + 3ie_1 + 4ie_2$;
$\tilde{Q}^{\flat} = -e_0 + 2e_1 - 3ie_1 - 4ie_2$.

The four points are pairwise distinct, so the orbit has size $4$: the element lies in none of the subspaces $\mathbb{C}_{\mathbb{B}}$, $\mathbb{H}_{\mathbb{B}}$, $\mathbb{M}_+$, which is the criterion of the theorem. Two further elements show the smaller orbits. For $\tilde{Q} = (1+3i)e_0$, a central but non-real element, the images are $(1+3i)e_0$ under ${}^{\natural}$ and under $\mathrm{id}$, and $(1-3i)e_0$ under $\bar{\cdot}$ and under ${}^{*}$, so the orbit is $\{(1+3i)e_0, (1-3i)e_0\}$ of size $2$; for $\tilde{Q} = 2e_0$ every image is $2e_0$ and the orbit has size $1$.

The Stabilizer of a Vector

For $\tilde{Q} = e_1$ the two conjugations ${}^{\natural}$ and ${}^{*}$ negate it, while $\bar{\cdot}$ fixes it because the coefficient is real:

$$ {}^{\natural}(e_1) = -e_1 , \qquad \bar{\cdot}(e_1) = e_1 , \qquad {}^{*}(e_1) = -e_1 , \qquad \flat(e_1) = e_1 , $$

so the orbit is $\{e_1,-e_1\}$ of size two and the stabilizer of $e_1$ is $\{ \mathrm{id}, \bar{\cdot} \}$, of order two and equal to the subgroup fixing $\mathbb{H}_{\mathbb{B}}$ pointwise — which is the blockwise reading of the entries of the sign table of Biquaternion Relations Between Subspaces.

A Hermitian Element

For $\tilde{Q} = e_0 + ie_1$:

$$ \tilde{Q}^{\natural} = e_0 - ie_1 , \qquad \bar{\tilde{Q}} = e_0 - ie_1 , \qquad \tilde{Q}^{*} = e_0 + ie_1 = \tilde{Q} , \qquad \tilde{Q}^{\flat} = -e_0 - ie_1 , $$

so the orbit is $\{e_0+ie_1, e_0-ie_1, -e_0-ie_1, -e_0+ie_1\}$ of size four, the element not being central. Its class in the quotient of the algebra by the action of the group is the unordered pair of orthogonal idempotents $\{\tilde\Pi_+,\tilde\Pi_-\}$ of the Hermitian subspace up to sign, which is the operator-theoretic form of the Peirce decomposition.

Summary

The biquaternion algebra carries four antilinear involutions: quaternion conjugation ${}^{\natural}$, complex conjugation $\bar{\cdot}$, their composite Hermitian conjugation ${}^{*}$, and the reversal $\flat = -{}^{*}$. The first three commute and form, together with the identity, a Klein four-group, of which the fourth is the negative of a member; the four are related by ${}^{*} = \bar{\cdot}\circ{}^{\natural}$ and by the sign rule $\flat = -{}^{*}$, and they commute pairwise, so they may be applied in any order. Each has a fixed and an anti-fixed space, and these are the six distinguished subspaces: the centre and the vector subspace for ${}^{\natural}$, the quaternion and anti-quaternion subspaces for $\bar{\cdot}$, the Hermitian and anti-Hermitian subspaces for ${}^{*}$ and, with the signs exchanged, for $\flat$. Only the Hermitian pair is counted twice, so eight labelled spaces reduce to six. The fixed spaces are assembled into a lattice over the four coordinate blocks — the two scalar lines and the two vector triples — and the group acts on the algebra with orbits of size one, two or four according to whether an element lies on the real line $\langle e_0\rangle$, in the centre, the quaternion subspace or the Hermitian subspace but off that line, or is general.

Summary of Notation

symbol	meaning
${}^{\natural}$	quaternion conjugation, the quaternion structure
$\bar{\cdot}$	complex conjugation, the coefficient-wise conjugation
${}^{*}$	Hermitian conjugation, $\bar{\cdot}\circ{}^{\natural}$
$\flat$	reversal, $-{}^{*}$
$-\mathrm{id}$	the central sign, $\tilde{Q} \mapsto -\tilde{Q}$
involution group	$\{ \mathrm{id}, {}^{\natural}, \bar{\cdot}, {}^{*} \}$, a Klein four-group
fixed space	the elements with $\sigma(\tilde{Q}) = \tilde{Q}$
anti-fixed space	the elements with $\sigma(\tilde{Q}) = -\tilde{Q}$
orbit	the set of the four conjugated images of an element
stabilizer	the subgroup fixing an element
$\mathbb{C}_{\mathbb{B}}, \mathrm{Vect}(\mathbb{B}), \mathbb{H}_{\mathbb{B}}, i\mathbb{H}_{\mathbb{B}}, \mathbb{M}_+, \mathbb{M}_-$	the six distinguished subspaces