Split-Quaternion Zero Divisors

Introduction

This article studies the zero divisors of the split-quaternion algebra. It defines them, proves the criterion $N(\tilde q) = 0$ that identifies them with the null cone, exhibits the two families of minimal one-sided ideals into which the zero divisor set splits, proves the existence of nonzero nilpotents, and describes the distribution of the zero divisors among the distinguished subspaces.

The split-quaternion algebra, its central product $N(\tilde q) = \tilde q\tilde{q}^{\natural}$ and its idempotents $\tilde\pi_\pm$ and subspaces $S$, $V$, $\mathbb{D}_2$, $\mathbb{D}_3$ are assumed from Split-Quaternion Algebra, and the metrical reading of $N$ from Split-Quaternion Norm and Invertibility. The invertibility criterion is assumed from Split-Quaternion Norm and Invertibility, §The Invertibility Criterion; it is not re-proved here. The ideal theory of $\mathbb{H}_{\mathrm{s}} \cong M_2(\mathbb{R})$ is assumed from Split-Quaternion Ideals and Peirce Decomposition. Nothing physical is invoked.

Definition and Criterion

Definition. A nonzero element $\tilde q \in \mathbb{H}_{\mathrm{s}}$ is a zero divisor if there exists a nonzero $\tilde p \in \mathbb{H}_{\mathrm{s}}$ with $\tilde q \tilde p = 0$ or a nonzero $\tilde r \in \mathbb{H}_{\mathrm{s}}$ with $\tilde r\tilde q = 0$. The zero divisor set is

$$ Z = \{\tilde q \in \mathbb{H}_{\mathrm{s}} : \tilde q \neq 0 \text{ and } \tilde q \text{ is a zero divisor}\}. $$

Theorem (The Criterion). Let $\tilde q$ be nonzero. Then $\tilde q$ is a zero divisor if and only if $N(\tilde q) = 0$. Equivalently, the zero divisor set is the null cone with the origin removed:

$$ Z = \{\tilde q \neq 0 : N(\tilde q) = 0\}. $$

Proof. The criterion is the corollary of the invertibility criterion proved in Split-Quaternion Norm and Invertibility, §The Invertibility Criterion: a nonzero element is a zero divisor exactly when it is not invertible, and it is invertible exactly when $N(\tilde q) \neq 0$.

The set

$$ \mathcal{N} = \{\tilde q : N(\tilde q) = 0\} = Z \cup \{0\} $$

is the null cone of the product $N$. It is a cone: if $N(\tilde q) = 0$ then $N(\lambda \tilde q) = \lambda^2 N(\tilde q) = 0$ for every real $\lambda$. It has real dimension $3$, the expected dimension of one equation in four real variables. That it is closed, that its only singular point is the origin, and that the complement of $Z$ in $\mathbb{H}_{\mathrm{s}} \setminus \{0\}$ is open, dense and of full measure are topological statements and belong to Split-Quaternion Topology and Split-Quaternion Norm and Invertibility, §Distribution of the Invertible Elements.

That the zero divisor set is connected is a topological statement and belongs to Split-Quaternion Topology. What is used here is the algebraic parametrisation of Split-Quaternion Norm and Invertibility, §Isotropy: every zero divisor $\tilde q$ is a positive multiple of a vector $(\cos\alpha, \sin\alpha, \cos\beta, \sin\beta)$.

The Annihilator of a Zero Divisor

The zero divisors are exactly the elements with a nonzero annihilator: $\tilde q \neq 0$ is a zero divisor if and only if $\{\tilde a : \tilde a\tilde q = 0\}$ or $\{\tilde a : \tilde q\tilde a = 0\}$ is nonzero.

Theorem (The Left Annihilator Is a Minimal Left Ideal). Let $\tilde q \neq 0$ be a zero divisor. Then the left annihilator $P_{\tilde q} = \{\tilde a : \tilde a\tilde q = 0\}$ is a two-dimensional minimal left ideal, and the right annihilator is a two-dimensional minimal right ideal. Consequently the zero divisor set is the union of the nonzero elements of the minimal one-sided ideals, with the origin removed.

Proof. The left annihilator is a left ideal, since $\tilde a\tilde q = 0$ implies $\tilde b\tilde a\tilde q = 0$. It is nonzero: the central product is symmetric, $\tilde q\tilde{q}^{\natural} = \tilde{q}^{\natural}\tilde q = N(\tilde q)$, and $N(\tilde q) = 0$ by the criterion, so the conjugate $\tilde{q}^{\natural}$ is nonzero and lies in it. It is proper, because $1 \cdot \tilde q = \tilde q \neq 0$ excludes $\tilde a = 1$. In $\mathbb{H}_{\mathrm{s}} \cong M_2(\mathbb{R})$ every nonzero proper left ideal is a minimal left ideal and every minimal left ideal is two-dimensional, by Split-Quaternion Ideals and Peirce Decomposition; hence $P_{\tilde q}$ is two-dimensional and minimal. The right annihilator is the image of a left annihilator under the conjugation, which reverses the order of multiplication, so it is a two-dimensional minimal right ideal. Every nonzero element of a left annihilator $P_{\tilde q}$ satisfies $\tilde a\tilde q = 0$ with $\tilde q \neq 0$, hence is a zero divisor; conversely every zero divisor lies in its own annihilator. Since the annihilator depends only on the line $\mathbb{R}\tilde q$, the zero divisor set is the union of the nonzero elements of the minimal left and right ideals.

The Two Families

Definition. For a zero-divisor line $L = \mathbb{R}\tilde q$, the left annihilator and the right annihilator are

$$ \mathcal{K}_L = \{\tilde a : \tilde a\tilde q = 0\}, \qquad \mathcal{R}_L = \{\tilde a : \tilde q\tilde a = 0\}, $$

and each depends only on the line $L$ and not on the chosen representative $\tilde q$.

Theorem (The Two Families). For every zero-divisor line $L$ the subspaces $\mathcal{K}_L$ and $\mathcal{R}_L$ are two-dimensional minimal one-sided ideals. Every minimal left ideal of $\mathbb{H}_{\mathrm{s}}$ is a $\mathcal{K}_L$ and every minimal right ideal is an $\mathcal{R}_L$. The collections

$$ \mathcal{K} = \{\mathcal{K}_L\}, \qquad \mathcal{R} = \{\mathcal{R}_L\}, $$

each parametrised by the projective line, are the two families; the two families are disjoint.

Proof. The left annihilator is the minimal left ideal $P_{\tilde q}$ of the preceding theorem, hence two-dimensional. The right annihilator is the image of the left annihilator of $\tilde{q}^{\natural}$ under the conjugation, which reverses the order of multiplication and preserves $N$, so it has the same properties. The condition $\tilde a(\lambda\tilde q) = 0$ for $\lambda \neq 0$ is the condition $\tilde a\tilde q = 0$, so the annihilator depends only on the zero-divisor line. The algebra $\mathbb{H}_{\mathrm{s}} \cong M_2(\mathbb{R})$ has two families of minimal one-sided ideals, the left and the right, each parametrised by the projective line, by Split-Quaternion Ideals and Peirce Decomposition. The two annihilator constructions realise the two families.

Corollary (Each Zero-Divisor Line Lies in One Ideal of Each Family). Through each zero-divisor line pass exactly two minimal one-sided ideals, one from each family; the zero divisor set is the union of their nonzero elements. Two ideals of opposite families meet in a line, and two distinct ideals of the same family meet only at the origin.

Proof. Every zero-divisor line consists of the multiples of one zero divisor, which by the preceding theorem has a two-dimensional annihilator in each family; the incidence is that of the one-sided ideals of $M_2(\mathbb{R})$ (Split-Quaternion Ideals and Peirce Decomposition).

Corollary (The Minimal Ideals Are Members of the Families). Both minimal left ideals $\mathbb{H}_{\mathrm{s}}\tilde\pi_\pm$ are members of the family $\mathcal{K}$, and both minimal right ideals $\tilde\pi_\pm\mathbb{H}_{\mathrm{s}}$ are members of the family $\mathcal{R}$.

Proof. A left ideal is closed under left multiplication, so $\mathbb{H}_{\mathrm{s}}\tilde\pi_\pm$ is the left annihilator of a zero-divisor line; it is a two-dimensional minimal left ideal by Split-Quaternion Ideals and Peirce Decomposition. The right ideals are the images of left ideals under the conjugation and lie in $\mathcal{R}$.

Corollary (The Anti-Automorphism $\tau$ Swaps the Families). The assignment on the generators

$$ \tau(e_1) = -e_1, \qquad \tau(e_2) = e_2, \qquad \tau(e_3) = e_3 $$

extends to an involutive algebra anti-automorphism that preserves $N$ and interchanges the two families: $\tau(\mathcal{K}_L) \in \mathcal{R}$ and $\tau(\mathcal{R}_L) \in \mathcal{K}$ for every zero-divisor line $L$.

Proof. As for the conjugation the relations are checked on the generators, giving an anti-automorphism with $\tau^2 = \mathrm{id}$; and $N(\tau(\tilde q)) = q_0^2 + q_1^2 - q_2^2 - q_3^2 = N(\tilde q)$ for $\tilde q = q_0 + q_1e_1 + q_2e_2 + q_3e_3$, so $\tau$ preserves $N$. An anti-automorphism carries a left ideal to a right ideal and preserves minimality, so it carries a member of $\mathcal{K}$ to a member of $\mathcal{R}$; being an involution, it interchanges the families.

Nonzero Nilpotents

Definition. A nonzero element $\tilde q$ is nilpotent if $\tilde q^2 = 0$.

Theorem (The Nilpotents). A nonzero element $\tilde q$ is nilpotent if and only if

$$ \tilde q \in V \quad \text{and} \quad N(\tilde q) = 0, $$

that is, if and only if $\tilde q$ is a nonzero vector of the vector subspace lying on the level set $N = 0$, $q_1^2 = q_2^2 + q_3^2$. The set of nilpotents is therefore a two-dimensional cone, and in particular

$$ (e_1 + e_3)^2 = e_1^2 + e_1 e_3 + e_3 e_1 + e_3^2 = -1 + 0 + 1 = 0 . $$

Every nilpotent is a zero divisor, and the nilpotents form a proper subset of the zero divisor set.

Proof. Write $\tilde q = q_0 + \mathbf v$ with $q_0 \in S$ and $\mathbf v \in V$. Since $e_1, e_2, e_3$ are traceless and the product of two distinct generators is the third with a sign, the square is

$$ \tilde q^2 = q_0^2 + 2q_0\mathbf v + \mathbf v^2 = q_0^2 + 2q_0\mathbf v - N(\mathbf v), $$

where $\mathbf v^2 = -N(\mathbf v)$ is the identity for pure vectors recorded in (Split-Quaternion Norm and Invertibility, §The Split-Quaternion Norm). If $\tilde q^2 = 0$, then comparing the components in $S$ and in $V$ gives $2q_0\mathbf v = 0$, so $q_0 = 0$ or $\mathbf v = 0$. If $\mathbf v = 0$ then $q_0^2 = 0$, so $q_0 = 0$ and $\tilde q = 0$, excluded by hypothesis; hence $q_0 = 0$ and $\tilde q = \mathbf v \in V$. Then $\tilde q^2 = -N(\tilde q)$, so $\tilde q^2 = 0$ exactly when $N(\tilde q) = 0$. Conversely every such $\tilde q$ has $\tilde q^2 = 0$. The computation for $e_1 + e_3$ uses $e_1 e_3 = -e_3 e_1$ and $e_1^2 = -1$, $e_3^2 = +1$. Every nilpotent satisfies $N(\tilde q)^2 = N(\tilde q^2) = 0$, hence $N(\tilde q) = 0$, so it is a zero divisor; the element $1 + e_2$ is a zero divisor with $N(1+e_2) = 0$ but $(1+e_2)^2 = 2(1+e_2) \neq 0$, so the inclusion is proper.

The nilpotent set is the level set $N = 0$ in $V$; by Split-Quaternion Norm and Invertibility, §Isotropy, its lines are the curve $\mathbb{R}(e_1 + \cos\theta\, e_2 + \sin\theta\, e_3)$.

The Contrast with the Division Algebras

The existence of nilpotents is the sharpest structural contrast the category has.

Theorem (No Nilpotents in the Division Algebras). The quaternion algebra $\mathbb{H}$ has no nonzero nilpotent, and the eight-dimensional $\mathbb{H}_{\mathbb{D}} = \mathbb{D} \otimes_{\mathbb{R}} \mathbb{H}$ has no nonzero nilpotent.

Proof. The quaternion algebra is a division algebra by Quaternion Algebra, so $\tilde q \neq 0$ implies $\tilde q$ invertible, and $\tilde q^2 = 0$ would give $\tilde q = 0$ after multiplying by $\tilde q^{-1}$. The eight-dimensional algebra is a product of two copies of $\mathbb{H}$ by the dictionary of The Number Systems as Clifford Algebras, and a nilpotent in a product of algebras would have a nilpotent component in one of the factors; a division algebra has none, so the product has none.

Remark. The comparison isolates the phenomenon. A simple real algebra with nilpotents, such as $\mathbb{H}_{\mathrm{s}}$, and a semisimple, non-simple product of division algebras without nilpotents, such as $\mathbb{H}_{\mathbb{D}}$, lie on opposite sides of the line that the nilpotent draws. The eight-dimensional algebra is treated later in Part V, under Split-Biquaternions; nothing of it is used here beyond the identification already stated in The Number Systems as Clifford Algebras.

Distribution of the Zero Divisors

The zero divisors are distributed over the distinguished subspaces as follows. The subspaces are those of Split-Quaternion Algebra.

Subspace	Zero divisors	Description
$S = \mathbb{R}\cdot 1$	none	every nonzero scalar is a unit
$V$	the nonzero vectors with $q_1^2 = q_2^2 + q_3^2$	the level set $N = 0$, a two-dimensional cone, every nonzero point of which is nilpotent
$\mathbb{D}_2 = \operatorname{span}\{1,e_2\}$	the nonzero multiples of $1 \pm e_2$	two zero-divisor lines
$\mathbb{D}_3 = \operatorname{span}\{1,e_3\}$	the nonzero multiples of $1 \pm e_3$	two zero-divisor lines
$\mathbb{H}_{\mathrm{s}} \tilde\pi_\pm$, $\tilde\pi_\pm \mathbb{H}_{\mathrm{s}}$	the whole subspace minus the origin	four minimal one-sided ideals
$\tilde\pi_\pm$ themselves	$\tilde\pi_+$ and $\tilde\pi_-$	the two non-central idempotents

The table is completed by the following observations.

The vector subspace. On $V$ the zero divisors are exactly the lightlike vectors, and by Nonzero Nilpotents they are exactly the nonzero nilpotents. Every zero divisor of $V$ has square zero; this is peculiar to the traceless part and does not hold in the whole algebra.

The idempotents. The idempotents $\tilde\pi_\pm = \tfrac12(1 \pm e_2)$ are zero divisors with $\tilde\pi_+ \tilde\pi_- = 0$; they are not nilpotent, since $\tilde\pi_\pm^2 = \tilde\pi_\pm \neq 0$. Together with $0$ and $1$ they are two of the idempotents of the algebra: the general non-scalar idempotent is $\tfrac12(1 \pm \eta)$ for a root $\eta$ of $+1$ in the vector subspace, a one-sheeted hyperboloid's worth of idempotents, as recorded in Split-Quaternion Roots of Minus One.

The splitting. The zero divisor set is the union of the planes of the two families of The Two Families. The nilpotent set is the two-dimensional subcone of $V$, and the non-scalar idempotents form a further two-dimensional set of points of the zero divisor set lying outside that subcone.

Measure. That the zero divisor set is closed, that it has measure zero in $\mathbb{R}^4$, and that the units are open and dense are topological and measure-theoretic statements and belong to Split-Quaternion Topology; what is used here is only that the zero divisor set is the vanishing set of the single polynomial $N$.

Summary

A nonzero split-quaternion is a zero divisor exactly when $N(\tilde q) = 0$, that is, when $\tilde q\tilde{q}^{\natural} = 0$, and the zero divisor set is the null cone with the origin removed, the union of the minimal one-sided ideals with the origin removed.

The zero divisor set splits into the two families of minimal one-sided ideals, each parametrised by the projective line; each ideal is two-dimensional and minimal, and the two families are the two rulings of the projective zero set. Every zero-divisor line lies in exactly one member of each family, the two families are exchanged by the anti-automorphism $\tau$, and the four minimal ideals of the algebra are members of the families.

The algebra has nonzero nilpotents: a nonzero element is nilpotent exactly when it lies in the vector subspace and on the level set $N = 0$, and $(e_1 + e_3)^2 = 0$. The quaternion algebra and the eight-dimensional $\mathbb{H}_{\mathbb{D}}$ have no nonzero nilpotent, the first because it is a division algebra and the second because it is a product of division algebras. The zero divisor set is a connected three-dimensional cone of measure zero, the nilpotents form its two-dimensional subcone inside $V$, and the two non-central idempotents are two further points of it.

Summary of Notation

Symbol	Meaning	Article
$Z$	the zero divisor set $\{\tilde q \neq 0 : N(\tilde q) = 0\}$	this article
$\mathcal{N}$	the null cone $\{\tilde q : N(\tilde q) = 0\}$	this article
$\mathbb{R}\tilde q$, $[\tilde q]$	a zero-divisor line of the projective zero set $Q$	this article
$\mathcal{R}$, $\mathcal{K}$	the two families of minimal one-sided ideals	this article
$\mathcal{K}_L$, $\mathcal{R}_L$	the left and right annihilators of a zero-divisor line $L$	this article
$\tau$	the anti-automorphism $e_1 \mapsto -e_1$, $e_2, e_3 \mapsto e_2, e_3$	this article
nilpotent	nonzero $\tilde q$ with $\tilde q^2 = 0$	this article
$\tilde\pi_\pm = \tfrac12(1 \pm e_2)$	the non-central idempotents	Split-Quaternion Algebra
$N(\tilde q) = \tilde q\tilde{q}^{\natural}$	the central product, formed algebraically; the metrical reading is in Split-Quaternion Norm and Invertibility	Split-Quaternion Algebra