Worked Examples in the Biquaternion Algebra

Introduction

Biquaternion Algebra ($\mathbb{B}$) developed the algebra $\mathbb{B} = \mathbb{C}\otimes_\mathbb{R}\mathbb{H}$, its six distinguished real subspaces, its three decompositions, and the product whose table is written out below is the one of Biquaternion Multiplication. This article works the computations out on explicit elements. The aim is a reference of concrete facts: the multiplication table of the basis, the six subspaces exhibited on one element, the four conjugations applied to that element, and explicit zero-divisor pairs. The idempotents and the minimal left ideals are the subject of Biquaternion Idempotents and Projections, which owns the general construction.

Notation follows Biquaternion Algebra ($\mathbb{B}$). A biquaternion is written in developed form

$$ \tilde{Q} = Q_0e_0 + Q_1e_1 + Q_2e_2 + Q_3e_3, \qquad Q_\mu \in \mathbb{C}, $$

with the quaternion basis $e_0 = 1$, $e_1, e_2, e_3$, and the central complex unit $i$, $i^2 = -1$, commuting with every $e_\mu$. The scalar part of $\tilde{Q}$ is $\mathrm{Sc}\,\tilde{Q} = Q_0$, its vector part is $\mathbf{Q} = Q_1e_1 + Q_2e_2 + Q_3e_3$, and the quaternion conjugate is $\tilde{Q}^{\natural} = Q_0e_0 - \mathbf{Q}$. The fixed element used throughout is

$$ \tilde{Q} = (2+i)e_0 + (1-i)e_1 + 3e_2 + ie_3 . $$

The Multiplication Table

The products of the quaternion basis elements follow the cyclic rule $e_1e_2 = e_3$, $e_2e_3 = e_1$, $e_3e_1 = e_2$, with $e_k^2 = -e_0$ and $e_je_k = -e_ke_j$ for $j\neq k$. Writing the products in a table, with the left factor as the row and the right factor as the column:

The table is the quaternion table, but the coefficients are now complex and the central element $i$ multiplies every entry: $i e_\mu = e_\mu i$. The full multiplication of $\mathbb{B}$ is therefore determined by the table above together with $i^2 = -1$ and centrality of $i$. As a $\mathbb{C}$-algebra the dimension is $4$; as a real algebra the basis may be taken as $e_0, e_1, e_2, e_3, ie_0, ie_1, ie_2, ie_3$ and the dimension is $8$.

Example. The products used below are

$$ e_1^2 = e_2^2 = e_3^2 = -e_0, \qquad e_1e_2 = e_3, \quad e_2e_1 = -e_3, \quad e_1e_3 = -e_2, \quad e_3e_2 = -e_1 . $$

The Six Subspaces on a Concrete Element

The six distinguished real subspaces are the fixed and anti-fixed spaces of the three commuting involutions ${}^{\natural}$, $\bar{\cdot}$ and ${}^{*} = {}^{\natural}\circ\bar{\cdot}$; the fourth conjugation $\flat = -{}^{*}$ has the same two eigenspaces as ${}^{*}$ with the roles of the fixed and anti-fixed space interchanged, so it contributes no further subspaces. On the fixed element the decompositions are as follows.

The decomposition into scalar and vector parts reads $\tilde{Q} = (2+i)e_0 + \bigl((1-i)e_1 + 3e_2 + ie_3\bigr)$, and the decomposition into quaternion and anti-quaternion parts reads

$$ \tilde{Q} = \bigl(2e_0 + e_1 + 3e_2\bigr) + i\bigl(e_0 - e_1 + e_3\bigr) = \tilde{Q}_r + i\tilde{Q}_i, $$

with $\tilde{Q}_r, \tilde{Q}_i \in \mathbb{H}_{\mathbb{B}}$, as the basis of the quaternion subspace consists of the basis elements with real coefficients. The Hermitian decomposition reads

$$ \tilde{Q} = \bigl(2e_0 - ie_1 + ie_3\bigr) + \bigl(ie_0 + e_1 + 3e_2\bigr) = \tilde{Q}_+ + \tilde{Q}_-, $$

and the two pieces are distinguished by the signs of the scalar and vector coefficients: $\tilde{Q}_+$ has real scalar part and purely imaginary vector part, $\tilde{Q}_-$ has purely imaginary scalar part and real vector part. Each of the three decompositions is a direct sum of two of the six subspaces, and the components above are the concrete instances.

The Four Conjugations on the Concrete Element

Applying the four conjugations to $\tilde{Q} = (2+i)e_0 + (1-i)e_1 + 3e_2 + ie_3$ gives

$$ \tilde{Q}^{\natural} = (2+i)e_0 - (1-i)e_1 - 3e_2 - ie_3, $$

$$ \bar{\tilde{Q}} = (2-i)e_0 + (1+i)e_1 + 3e_2 - ie_3, $$

$$ \tilde{Q}^{*} = \overline{\tilde{Q}^{\natural}} = (2-i)e_0 - (1+i)e_1 - 3e_2 + ie_3, $$

$$ \tilde{Q}^{\flat} = -\tilde{Q}^{*} = -(2-i)e_0 + (1+i)e_1 + 3e_2 - ie_3 . $$

Each is an involution, and the Klein group is visible in the relations $\tilde{Q}^{*} = \overline{\tilde{Q}^{\natural}} = \bar{\tilde{Q}}^{\natural}$; the fourth conjugation satisfies $(\tilde{Q}^{*})^{\flat} = -\tilde{Q}$ and $(\tilde{Q}^{\flat})^{\dagger} = -\tilde{Q}$, so it is the composition of ${}^{*}$ with the central sign $-1$. The fixed points of each involution give one of the six subspaces tabulated above: for instance, the Hermitian part of $\tilde{Q}$ is $\tfrac{1}{2}(\tilde{Q} + \tilde{Q}^{*}) = 2e_0 - ie_1 + ie_3$, which agrees with the table.

Idempotents and the Two Minimal Left Ideals

The standard idempotents $\tilde\Pi_1 = \tfrac{1}{2}(e_0 + ie_3)$ and $\tilde\Pi_2 = \tfrac{1}{2}(e_0 - ie_3)$ satisfy $\tilde\Pi_1^2 = \tilde\Pi_1$, $\tilde\Pi_2^2 = \tilde\Pi_2$, $\tilde\Pi_1\tilde\Pi_2 = \tilde\Pi_2\tilde\Pi_1 = 0$ and $\tilde\Pi_1 + \tilde\Pi_2 = e_0$; they are primitive; and they give the decomposition $\mathbb{B} = \mathbb{B}\tilde\Pi_1 \oplus \mathbb{B}\tilde\Pi_2$ into two minimal left ideals of real dimension $4$. The verification is carried out in Biquaternion Idempotents and Projections, which owns the idempotents of the series.

Explicit Zero Divisors

Example (an explicit pair). With the standard idempotents $\tilde\Pi_1, \tilde\Pi_2$ of Biquaternion Idempotents and Projections, $\tilde\Pi_1\tilde\Pi_2 = 0$ and both factors are nonzero: this is a zero-divisor pair. In unnormalised form, set $\tilde A = e_0 + ie_3$ and $\tilde B = e_0 - ie_3$. Then

$$ \tilde A\tilde B = e_0 - (ie_3)^2 = e_0 - 1 = 0, \qquad \tilde A \neq 0, \qquad \tilde B \neq 0, $$

Both $\tilde A$ and $\tilde B$ have two nonzero complex coefficients, $\tilde A = e_0 + ie_3$ and $\tilde B = e_0 - ie_3$.

The computations show all the structural features asserted in Biquaternion Algebra ($\mathbb{B}$) on a single element and on the standard idempotents: the six subspaces split the element into its centre, vector, quaternion, anti-quaternion, Hermitian and anti-Hermitian parts; the four conjugations act as listed; and the two primitive idempotents produce the decomposition $\mathbb{B} = \mathbb{B}\tilde\Pi_1 \oplus \mathbb{B}\tilde\Pi_2$ into minimal left ideals.

Summary

The multiplication of $\mathbb{B}$ is the quaternion table with complex coefficients and central $i$, $i^2 = -1$. On the element $\tilde{Q} = (2+i)e_0 + (1-i)e_1 + 3e_2 + ie_3$ the six subspaces are exhibited by the scalar-vector, quaternion-anti-quaternion and Hermitian-anti-Hermitian decompositions of $\tilde{Q}$, and the four conjugations ${}^{\natural}$, $\bar{\cdot}$, ${}^{*}$ and ${}^{\flat} = -{}^{*}$ act on it as listed.

The idempotents $\tilde\Pi_1, \tilde\Pi_2$ of Biquaternion Idempotents and Projections satisfy $\tilde\Pi_1\tilde\Pi_2 = 0$ and give $\mathbb{B} = \mathbb{B}\tilde\Pi_1 \oplus \mathbb{B}\tilde\Pi_2$ with each summand minimal; the pair $\tilde A = e_0+ie_3$, $\tilde B = e_0-ie_3$ is an explicit zero-divisor pair. As a left module over itself, $\mathbb{B}$ has the decomposition $\mathbb{B} = \mathbb{B}\tilde\Pi_1 \oplus \mathbb{B}\tilde\Pi_2$ into minimal left ideals.

Summary of Notation

Further Reading

• Richard S. Pierce, Associative Algebras (Springer, 1982), for idempotents and minimal left ideals in the structure theory of finite-dimensional algebras.
• Frank W. Anderson and Kent R. Fuller, Rings and Categories of Modules (Springer, 2nd ed. 1992), for idempotents, minimal ideals and the decomposition of a ring into simple modules.
• John H. Conway and Derek A. Smith, On Quaternions and Octonions (A K Peters, 2003), for the quaternion and biquaternion computations.
• Pertti Lounesto, Clifford Algebras and Spinors (Cambridge, 2nd ed. 2001), for the complexification of the quaternions.