Split-Biquaternion Ideals and Peirce Decomposition

Introduction

The algebra article defined the split biquaternion algebra $\mathbb{H}_{\mathbb{D}} = \mathbb{D} \otimes_{\mathbb{R}} \mathbb{H}$ and established the isomorphism $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H} \oplus \mathbb{H}$. This article treats the ideals of $\mathbb{H}_{\mathbb{D}}$, the decomposition of the algebra into its two quaternion halves, and the Peirce decomposition attached to the idempotents.

The treatment is purely mathematical. Every claim is either proved or stated as a definition. No physics is invoked. The split biquaternion algebra $\mathbb{H}_{\mathbb{D}}$ is assumed from the basic algebra article, its four conjugations and its idempotents $\tilde\Pi_+ = \tfrac{1}{2}(1 + j)$, $\tilde\Pi_- = \tfrac{1}{2}(1 - j)$ are assumed from the article on split complex algebra and the basic algebra article, and the quaternion algebra $\mathbb{H}$ is assumed to be a division algebra.

Throughout, a split biquaternion is written $\tilde{Q} = \sum_{\mu=0}^{3} Q_\mu e_\mu$ with $Q_\mu = q_\mu + j q'_\mu$, and the idempotent components are $\tilde{Q}_\pm = \tilde{Q} \tilde\Pi_\pm \in \mathbb{H}$. The four conjugations are $\bar{\cdot}$, ${}^{*}$, ${}^{\dagger} = {}^{*}\circ\bar{\cdot}$ and ${}^{\flat} = -{}^{\dagger}$.

The classification and the structure of the idempotents themselves — their orthogonality, completeness and primitivity, their role as projections, and the four-point nature of the idempotent set — belong to Split-Biquaternion Idempotents and Projections. This article takes those facts as given and develops the objects that the idempotents generate: the ideals and the Peirce decomposition. The two articles share the idempotents $\tilde\Pi_\pm$ but not their ownership: the idempotents article owns the idempotents, this article owns the ideals and the splitting.

Ideals in an Algebra

Let $A$ be an associative algebra. A left ideal is a subset $I \subseteq A$ closed under addition and under left multiplication by $A$: $AI \subseteq I$. A right ideal is closed under right multiplication, $IA \subseteq I$, and a two-sided ideal is both. An ideal is proper if it is neither $0$ nor $A$, and minimal (among nonzero left ideals) if it contains no nonzero proper left ideal. The algebra is simple if it has no proper nonzero two-sided ideal, and semisimple if it is a direct sum of simple algebras, equivalently if its Jacobson radical is zero.

The basic mechanism connecting ideals to idempotents is that an idempotent $e$ produces the decompositions

$$ A = Ae \oplus A(1 - e) \quad (\text{left}), \qquad A = eA \oplus (1 - e)A \quad (\text{right}), $$

and that $Ae$ is a minimal left ideal exactly when $e$ is primitive.

The Two-Sided Ideals: Simplicity Fails

Theorem. The proper nonzero two-sided ideals of $\mathbb{H}_{\mathbb{D}}$ are exactly $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$. Consequently $\mathbb{H}_{\mathbb{D}}$ is not simple.

Proof. Let $I$ be a two-sided ideal. In the idempotent basis $I = I_+ \tilde\Pi_+ \oplus I_- \tilde\Pi_-$ with $I_\pm \subseteq \mathbb{H}$. Because $\tilde\Pi_\pm$ is central, $I$ being an ideal is equivalent to each $I_\pm$ being an ideal of the quaternion algebra $\mathbb{H}$. But $\mathbb{H}$ is a division algebra and has no proper nonzero ideal: if $\tilde q \neq 0$ lies in an ideal $J$ of $\mathbb{H}$, then $1 = \tilde q^{-1} \tilde q \in J$, so $J = \mathbb{H}$. Hence for each sign either $I_\pm = 0$ or $I_\pm = \mathbb{H}$. The proper nonzero possibilities are therefore $I = \mathbb{H} \tilde\Pi_+$ and $I = \mathbb{H} \tilde\Pi_-$. Since these are proper and nonzero, $\mathbb{H}_{\mathbb{D}}$ is not simple.

Corollary. $\mathbb{H}_{\mathbb{D}} = \mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_-$ is the unique decomposition of $\mathbb{H}_{\mathbb{D}}$ as a direct sum of two proper nonzero two-sided ideals, and the two summands are isomorphic to $\mathbb{H}$.

Proof. The decomposition follows from $\tilde\Pi_+ + \tilde\Pi_- = 1$; uniqueness from the theorem, since any decomposition into two nonzero two-sided ideals must consist of the two listed ideals. Each summand $\mathbb{H} \tilde\Pi_\pm$ is identified with $\mathbb{H}$ by the projection $\tilde{Q} \mapsto \tilde{Q}_\pm$.

The Algebra Is Semisimple

Proposition. The Jacobson radical of $\mathbb{H}_{\mathbb{D}}$ is zero, so $\mathbb{H}_{\mathbb{D}}$ is semisimple.

Proof. By the theorem, the only proper nonzero two-sided ideals are $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$; their intersection is $\mathbb{H} \tilde\Pi_+ \cap \mathbb{H} \tilde\Pi_- = 0$, so the radical, being the intersection of the maximal two-sided ideals, is zero.

The algebra is thus semisimple but not simple, and its two simple components are both the same division algebra $\mathbb{H}$. This is the algebraic content of the isomorphism $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H} \oplus \mathbb{H}$: the right-hand side is a product of two division algebras.

The Idempotents Revisited

The four idempotents of $\mathbb{H}_{\mathbb{D}}$ are $0$, $\tilde\Pi_+$, $\tilde\Pi_-$ and $1$, and all of them are central; this is proved in Split-Biquaternion Idempotents and Projections. Here only the following facts are used: $\{\tilde\Pi_+, \tilde\Pi_-\}$ is a complete family of orthogonal, primitive, central idempotents, with

$$ \tilde\Pi_+^2 = \tilde\Pi_+, \qquad \tilde\Pi_-^2 = \tilde\Pi_-, \qquad \tilde\Pi_+ \tilde\Pi_- = \tilde\Pi_- \tilde\Pi_+ = 0, \qquad \tilde\Pi_+ + \tilde\Pi_- = 1. $$

Because they are central, the left, right and two-sided ideals they generate coincide, and the primitivity criterion gives that each of $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$ is a minimal ideal.

The Peirce Decomposition

Let $e$ be an idempotent of an algebra $A$. Relative to $e$ and its complement $1 - e$, the algebra decomposes as a direct sum of four Peirce spaces:

$$ A = eAe \oplus eA(1 - e) \oplus (1 - e)Ae \oplus (1 - e)A(1 - e). $$

Theorem. For $e = \tilde\Pi_+$ in $\mathbb{H}_{\mathbb{D}}$ the Peirce decomposition is

$$ \mathbb{H}_{\mathbb{D}} = \mathbb{H} \tilde\Pi_+ \oplus 0 \oplus 0 \oplus \mathbb{H} \tilde\Pi_-, $$

the two off-diagonal Peirce spaces $\tilde\Pi_+ \mathbb{H}_{\mathbb{D}} \tilde\Pi_-$ and $\tilde\Pi_- \mathbb{H}_{\mathbb{D}} \tilde\Pi_+$ both vanishing.

Proof. Since $\tilde\Pi_+$ is central,

$$ \tilde\Pi_+ \mathbb{H}_{\mathbb{D}} \tilde\Pi_+ = \mathbb{H}_{\mathbb{D}} \tilde\Pi_+ = \mathbb{H} \tilde\Pi_+, \qquad \tilde\Pi_+ \mathbb{H}_{\mathbb{D}} \tilde\Pi_- = \mathbb{H}_{\mathbb{D}} \tilde\Pi_+ \tilde\Pi_- = 0, $$

and similarly $(1 - \tilde\Pi_+) \mathbb{H}_{\mathbb{D}} (1 - \tilde\Pi_+) = \mathbb{H} \tilde\Pi_-$ and the other off-diagonal space vanishes. The direct sum is the whole algebra because $\tilde\Pi_+ + \tilde\Pi_- = 1$.

The vanishing of the off-diagonal Peirce spaces is the algebraic statement that the two halves of $\mathbb{H}_{\mathbb{D}}$ do not mix: no element of one half acts on the other. This is the sharpest contrast with the biquaternion algebra, where the Peirce spaces relative to a noncentral primitive idempotent have nonzero off-diagonal parts and carry off-diagonal elements $E_{12}$, $E_{21}$.

The Decomposition into Minimal Ideals

The Peirce decomposition can be read as the sum of the two minimal ideals. For the complete family $\{\tilde\Pi_+, \tilde\Pi_-\}$,

$$ \mathbb{H}_{\mathbb{D}} = \mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_-, $$

and each summand is a minimal two-sided ideal, equivalently a minimal left ideal and a minimal right ideal, isomorphic to $\mathbb{H}$. Summing over the family, the algebra is the direct sum of its minimal ideals, one for each idempotent of the complete family:

Proposition. $\mathbb{H}_{\mathbb{D}}$ is the direct sum of its two minimal ideals $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$, and every minimal left ideal of $\mathbb{H}_{\mathbb{D}}$ is one of these two.

Proof. The direct-sum statement is the corollary above. For the second statement, let $L$ be a minimal left ideal. If $L$ contains an element with a nonzero $\tilde\Pi_+$-component, then that component lies in $L \cap \mathbb{H} \tilde\Pi_+$, a nonzero left ideal contained in the minimal ideal $\mathbb{H} \tilde\Pi_+$, hence equal to it; otherwise $L \subseteq \mathbb{H} \tilde\Pi_-$. Minimality then forces $L$ to equal $\mathbb{H} \tilde\Pi_+$ or $\mathbb{H} \tilde\Pi_-$.

Minimal Left and Right Ideals

Because the idempotents are central, the left ideal, the right ideal and the two-sided ideal generated by an idempotent coincide. Hence the minimal left ideals, the minimal right ideals and the minimal two-sided ideals of $\mathbb{H}_{\mathbb{D}}$ are the same two objects, $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$, each isomorphic to the division algebra $\mathbb{H}$.

$$ \mathbb{H}_{\mathbb{D}} \tilde\Pi_+ = \tilde\Pi_+ \mathbb{H}_{\mathbb{D}} = \tilde\Pi_+ \mathbb{H}_{\mathbb{D}} \tilde\Pi_+ = \mathbb{H} \tilde\Pi_+ \cong \mathbb{H}, $$

and likewise for $\tilde\Pi_-$. Each summand is annihilated by the complementary idempotent — $\mathbb{H} \tilde\Pi_+$ by $\tilde\Pi_-$ and $\mathbb{H} \tilde\Pi_-$ by $\tilde\Pi_+$ — so the two ideals are disjoint, and together they fill the algebra: $\mathbb{H}_{\mathbb{D}} = \mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_- \cong \mathbb{H} \oplus \mathbb{H}$, both summands isomorphic to the division algebra $\mathbb{H}$ as rings.

In the biquaternion algebra the corresponding statement is different in every respect: the minimal left ideals $\mathbb{B}p$ and $\mathbb{B}q$ are not two-sided, are not right ideals, and are isomorphic to $\mathbb{C}^2$ rather than to a division algebra.

The Lattice of Left Ideals

Proposition. The left ideals of $\mathbb{H}_{\mathbb{D}}$ are exactly

$$ 0, \qquad \mathbb{H} \tilde\Pi_+, \qquad \mathbb{H} \tilde\Pi_-, \qquad \mathbb{H}_{\mathbb{D}}. $$

They form a lattice that is the diamond: $0$ is contained in $\mathbb{H} \tilde\Pi_\pm$, each of these is contained in $\mathbb{H}_{\mathbb{D}}$, and $\mathbb{H} \tilde\Pi_+$, $\mathbb{H} \tilde\Pi_-$ are incomparable. The same description holds for the right ideals and for the two-sided ideals.

Proof. A left ideal $L$ is contained in $\mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_-$. Its intersections $L \cap \mathbb{H} \tilde\Pi_+$ and $L \cap \mathbb{H} \tilde\Pi_-$ are each either $0$ or the whole summand, since each summand is minimal. The four combinations give the four listed ideals, and no second idempotent pair occurs, so there are no further left ideals.

Corollary. Every left ideal of $\mathbb{H}_{\mathbb{D}}$ is two-sided, and $\mathbb{H}_{\mathbb{D}}$ has exactly two maximal left ideals, $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$.

The contrast with $\mathbb{B}$ is extreme. In $\mathbb{B} \cong M_2(\mathbb{C})$ the left ideals form a lattice isomorphic to the projective line $\mathbb{P}^1$, with infinitely many maximal left ideals; in $\mathbb{H}_{\mathbb{D}}$ the lattice has four elements and the two maximal left ideals are the two halves. The reduction is a direct consequence of the vanishing of the off-diagonal Peirce spaces.

The Radical

The Jacobson radical $J(A)$ of a finite-dimensional algebra $A$ is the intersection of its maximal left ideals, equivalently of its maximal right ideals, equivalently the largest nilpotent two-sided ideal; in particular $J(A) = 0$ if and only if $A$ is semisimple.

Proposition. $J(\mathbb{H}_{\mathbb{D}}) = 0$. Moreover $J(\mathbb{H}_{\mathbb{D}}) = \mathbb{H} \tilde\Pi_+ \cap \mathbb{H} \tilde\Pi_-$.

Proof. The maximal left ideals are $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$ by the corollary above, and their intersection is $0$ because $\tilde\Pi_+ \tilde\Pi_- = 0$ and the two ideals are complementary summands.

So $\mathbb{H}_{\mathbb{D}}$ is semisimple in the strongest sense: its radical vanishes. This is one of the senses in which $\mathbb{H}_{\mathbb{D}}$ is smaller and better behaved than a general associative algebra — it is a product of two division algebras, and products of division algebras have vanishing radical and only finitely many ideals.

The Real Structure

The algebra $\mathbb{H}_{\mathbb{D}}$ is an eight-dimensional real algebra and a four-dimensional algebra over $\mathbb{D}$. The ideals $\mathbb{H} \tilde\Pi_\pm$ are real vector subspaces of dimension $4$ and are closed under multiplication by $\mathbb{D}$; neither is a free $\mathbb{D}$-span, since $j - 1$ annihilates $\mathbb{H} \tilde\Pi_+$ and $j + 1$ annihilates $\mathbb{H} \tilde\Pi_-$, whereas a free $\mathbb{D}$-span has no such annihilator. The whole algebra $\mathbb{H}_{\mathbb{D}} = \mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_-$ has $\mathbb{D}$-basis $e_0, e_1, e_2, e_3$.

The $\mathbb{R}$-automorphisms and $\mathbb{R}$-derivations of $\mathbb{H}_{\mathbb{D}}$ act on this two-term decomposition. Because the two simple factors are isomorphic, the automorphism group contains the exchange of the factors, and the derivations contain the corresponding off-diagonal maps; the precise statements, together with the comparison with the biquaternion case, are the subject of Split-Biquaternion Automorphisms and Derivations. The splitting $\mathbb{H}_{\mathbb{D}} = \mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_-$ established here is the object on which those automorphisms act.

Summary

The split biquaternion algebra $\mathbb{H}_{\mathbb{D}}$ has exactly two proper nonzero two-sided ideals, $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$; it is therefore not simple, but it is semisimple, with vanishing radical. The two ideals are the two quaternion halves of the decomposition $\mathbb{H}_{\mathbb{D}} = \mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_- \cong \mathbb{H} \oplus \mathbb{H}$, each isomorphic to the division algebra $\mathbb{H}$.

The minimal left ideals, the minimal right ideals and the minimal two-sided ideals coincide and are exactly $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$, because the underlying idempotents are central. The Peirce decomposition relative to $\tilde\Pi_+$ has vanishing off-diagonal spaces,

$$ \mathbb{H}_{\mathbb{D}} = \tilde\Pi_+ \mathbb{H}_{\mathbb{D}} \tilde\Pi_+ \oplus (1 - \tilde\Pi_+) \mathbb{H}_{\mathbb{D}} (1 - \tilde\Pi_+) = \mathbb{H} \tilde\Pi_+ \oplus \mathbb{H} \tilde\Pi_-, $$

so the two halves do not mix. The left ideals are exactly $0$, $\mathbb{H} \tilde\Pi_+$, $\mathbb{H} \tilde\Pi_-$, $\mathbb{H}_{\mathbb{D}}$, forming a diamond lattice with two maximal ideals, in contrast with the projective line of left ideals of the simple biquaternion algebra $\mathbb{B} \cong M_2(\mathbb{C})$. The idempotents themselves, their classification, their role as projections and their relation to the roots of $-1$, are the subject of Split-Biquaternion Idempotents and Projections.

Summary of Notation

Symbol	Meaning
$\mathbb{H}_{\mathbb{D}} = \mathbb{D} \otimes_{\mathbb{R}} \mathbb{H}$	Split biquaternion algebra, $\cong \mathbb{H} \oplus \mathbb{H}$
$\tilde\Pi_\pm = \tfrac{1}{2}(1 \pm j)$	The complete family of primitive central idempotents
$\mathbb{H} \tilde\Pi_+, \mathbb{H} \tilde\Pi_-$	The two minimal (left, right and two-sided) ideals, each $\cong \mathbb{H}$
$\tilde{Q} = \tilde{Q}_+ \tilde\Pi_+ + \tilde{Q}_- \tilde\Pi_-$	Idempotent decomposition, $\tilde{Q}_\pm \in \mathbb{H}$
$eAe$, $eA(1-e)$, $(1-e)Ae$, $(1-e)A(1-e)$	Peirce spaces relative to an idempotent $e$
$J(\mathbb{H}_{\mathbb{D}})$	Jacobson radical, equal to $0$
$\mathbb{B} = \mathbb{C} \otimes_{\mathbb{R}} \mathbb{H}$	Biquaternion algebra, $\cong M_2(\mathbb{C})$, simple