Split-Biquaternion Automorphisms and Derivations

Introduction

The split biquaternion algebra is $\mathbb{H}_{\mathbb{D}} = \mathbb{D} \otimes_{\mathbb{R}} \mathbb{H}$, an eight-dimensional real algebra isomorphic to $\mathbb{H} \oplus \mathbb{H}$. It carries two structures distinguished by the ground ring: over the split complex algebra $\mathbb{D}$ it is a free $\mathbb{D}$-algebra of rank $4$, and over $\mathbb{R}$ the same set is an eight-dimensional real algebra. This article describes its algebra automorphisms and its derivations, over each ground structure, and compares them with the biquaternion case. The algebra and its decomposition are used from Split-Biquaternion Algebra and Split-Biquaternion Ideals and Peirce Decomposition; the split complex unit and the idempotents are those of Split-Complex Algebra.

We use the conventions of the split biquaternion algebra: the basis $\{e_0,e_1,e_2,e_3\}$ with $e_k^2 = -e_0$; the central split complex unit $j$ with $j^2 = +1$; the idempotents $\tilde\Pi_\pm = \tfrac{1}{2}(1 \pm j)$; the conjugations ${}^{\natural}$ (quaternion), $\bar{\cdot}$ (split complex), ${}^{*} = \bar{\cdot}\circ{}^{\natural}$ (Hermitian) and ${}^{\flat} = -{}^{*}$ (anti-Hermitian); and the four distinguished subspaces $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{H}_{\mathbb{H}_{\mathbb{D}}}$, $\mathbb{M}_+$, $\mathbb{M}_-$. A general element is $\tilde{Q} = A + jB$ with $A, B \in \mathbb{H}$.

No physics is invoked and no new results are claimed. Everything below is the standard structure theory of a product of two quaternion algebras.

Standing Facts: the Centre and the Two Halves

Two structural facts govern the whole article.

The centre. The centre of $\mathbb{H}_{\mathbb{D}}$ is $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} = \mathbb{D} e_0 = \mathrm{span}_{\mathbb{R}}\{e_0, je_0\}$, a real two-dimensional algebra isomorphic to $\mathbb{D}$, with the idempotents $\tilde\Pi_\pm$ as its two nontrivial idempotents. Hence $\mathbb{H}_{\mathbb{D}}$ is not central over $\mathbb{R}$: its centre is strictly larger than $\mathbb{R}e_0$.

The two halves. The algebra is a product of two copies of the quaternion division algebra, $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H} \oplus \mathbb{H}$, the isomorphism sending $\tilde{Q}$ to the pair $(\tilde{Q}_+, \tilde{Q}_-)$ of its idempotent components. It is semisimple but not simple: the two factors are its minimal two-sided ideals. Every automorphism and every derivation must respect this splitting, up to the one permutation that interchanges the two factors.

Automorphisms over the Split Complex Numbers

Throughout this section the ground ring is $\mathbb{D}$, and an automorphism is $\mathbb{D}$-linear.

Definition. A $\mathbb{D}$-algebra automorphism of $\mathbb{H}_{\mathbb{D}}$ is a bijective $\mathbb{D}$-linear map $\sigma$ with $\sigma(\tilde Q\tilde P) = \sigma(\tilde Q)\sigma(\tilde P)$ and $\sigma(1) = 1$. These maps form a group $\mathrm{Aut}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}})$.

Theorem. $\mathrm{Aut}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}}) \cong SO(3) \times SO(3)$.

Proof. A $\mathbb{D}$-linear automorphism fixes the centre $\mathbb{D}$ pointwise, hence fixes the idempotents $\tilde\Pi_\pm$ and preserves each of the two factors $\mathbb{H} \tilde\Pi_+$ and $\mathbb{H} \tilde\Pi_-$. On each factor it is an $\mathbb{R}$-algebra automorphism of $\mathbb{H}$, and every such automorphism is inner, by the Skolem–Noether theorem for the division algebra $\mathbb{H}$: it is $\iota_{u}(\tilde q) = u \tilde q u^{-1}$ for some $u \in \mathbb{H}^\times$, depending only on $u$ modulo the central scalars $\mathbb{R}^\times$. Thus the automorphisms of the first factor form $\mathbb{H}^\times / \mathbb{R}^\times \cong S^3/\{\pm 1\} = SO(3)$, and likewise for the second, giving the product.

Corollary. As a Lie group, $\mathrm{Aut}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}})$ has dimension $6$ and is connected and compact.

Example. Conjugation by a unit of the first factor, $\sigma(\tilde{Q}_+,\tilde{Q}_-) = (u\tilde{Q}_+ u^{-1}, \tilde{Q}_-)$ with $|u| = 1$, is a $\mathbb{D}$-algebra automorphism; conjugation by a unit of the second factor is the companion. Every $\mathbb{D}$-automorphism is a pair of such conjugations, one per factor.

Automorphisms over $\mathbb{R}$

Now the ground ring is $\mathbb{R}$, so automorphisms need only be $\mathbb{R}$-linear, and the group is strictly larger.

Automorphisms preserve the centre. If $\sigma$ is an $\mathbb{R}$-algebra automorphism and $A$ is central then $\sigma(A)$ is central, so $\sigma$ restricts to an $\mathbb{R}$-algebra automorphism of $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} \cong \mathbb{D}$. Since $\mathbb{D} \cong \mathbb{R} \oplus \mathbb{R}$ has exactly two $\mathbb{R}$-algebra automorphisms, the identity and the swap $\tilde\Pi_+ \leftrightarrow \tilde\Pi_-$, restriction gives a homomorphism

$$ \rho : \mathrm{Aut}_{\mathbb{R}}(\mathbb{H}_{\mathbb{D}}) \longrightarrow \mathrm{Aut}_{\mathbb{R}}(\mathbb{D}) \cong \mathbb{Z}/2 . $$

The kernel is the $\mathbb{D}$-linear part. An automorphism lies in $\ker\rho$ exactly when it fixes the centre pointwise, equivalently when it fixes $\tilde\Pi_+$ and $\tilde\Pi_-$ and so preserves each factor. Hence $\ker\rho = \mathrm{Aut}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}}) \cong SO(3)\times SO(3)$, the group of the previous section, consisting of the inner automorphisms.

The swap coset. The map

$$ c(\tilde{Q}) = \bar{\tilde{Q}} , \qquad c(j) = -j , $$

is an $\mathbb{R}$-algebra automorphism, because $\bar{\cdot}$ is a ring automorphism of the commutative coefficient algebra $\mathbb{D}$ and fixes the quaternion units $e_k$. It induces the nontrivial element of $\mathbb{Z}/2$ on the centre, it is not $\mathbb{D}$-linear since $c(j) = -j \neq j$, and it is not inner because inner automorphisms fix the centre pointwise. So $\rho$ is surjective and the extension is nontrivial. Since $c^{2} = \mathrm{id}$, the sequence splits:

Theorem. $\mathrm{Aut}_{\mathbb{R}}(\mathbb{H}_{\mathbb{D}}) \cong \left(SO(3) \times SO(3)\right) \rtimes \mathbb{Z}/2$, the generator of $\mathbb{Z}/2$ acting by interchanging the two factors of $SO(3)\times SO(3)$.

Proof. There is a short exact sequence $1 \to SO(3)\times SO(3) \to \mathrm{Aut}_{\mathbb{R}}(\mathbb{H}_{\mathbb{D}}) \xrightarrow{\rho} \mathbb{Z}/2 \to 1$, split by $c$ because $c^2 = \mathrm{id}$. The action of $c$ on the kernel conjugates an automorphism of the first factor into one of the second and conversely, that is, swaps the factors.

Concretely, every real automorphism of $\mathbb{H}_{\mathbb{D}}$ has exactly one of the two forms

$$ \sigma(\tilde Q_+, \tilde Q_-) = (u_+ \tilde Q_+ u_+^{-1},\, u_- \tilde Q_- u_-^{-1}) \quad\text{or}\quad \sigma(\tilde Q_+, \tilde Q_-) = \left(u_+ \tilde Q_- u_+^{-1},\, u_- \tilde Q_+ u_-^{-1}\right), $$

with $u_\pm$ determined up to a real scalar; the first family is the identity coset, the second the coset of $c$.

Corollary. $\mathrm{Aut}_{\mathbb{R}}(\mathbb{H}_{\mathbb{D}})$ has real dimension $6$ and exactly two connected components, each a copy of $SO(3)\times SO(3)$.

Remark. Quaternion conjugation ${}^{\natural}$ is an anti-automorphism, not an automorphism, so it is not in either group. The split complex conjugation $\bar{\cdot}$ is an automorphism and appears above. The Hermitian conjugation ${}^{*}$ and the anti-Hermitian conjugation ${}^{\flat}$ are anti-automorphisms and do not appear.

Derivations

Definition. An $\mathbb{R}$-linear derivation of $\mathbb{H}_{\mathbb{D}}$ is an $\mathbb{R}$-linear map $D$ with $D(\tilde Q\tilde P) = D(\tilde Q)\tilde P + \tilde Q D(\tilde P)$. The set is a real vector space $\mathrm{Der}(\mathbb{H}_{\mathbb{D}})$, a Lie algebra under $[D_1,D_2] = D_1D_2 - D_2D_1$. A $\mathbb{D}$-linear derivation is additionally $\mathbb{D}$-linear; the set is $\mathrm{Der}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}})$.

Every real derivation is $\mathbb{D}$-linear. A derivation maps the centre to the centre, since for central $A$ and every $\tilde Q$, $D(A)\tilde Q = D(A\tilde Q) - A D(\tilde Q) = D(\tilde Q A) - D(\tilde Q)A = \tilde Q D(A)$. A derivation of $\mathbb{D}$ over $\mathbb{R}$ vanishes, because $0 = D(1) = D(j^2) = 2jD(j)$ forces $D(j) = 0$ and $j$ generates $\mathbb{D}$ over $\mathbb{R}$. Hence $D$ vanishes on the centre and, since $j$ is central, $D(j\tilde Q) = jD(\tilde Q)$, so $D$ is $\mathbb{D}$-linear. Therefore

$$ \mathrm{Der}(\mathbb{H}_{\mathbb{D}}) = \mathrm{Der}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}}) . $$

Theorem. $\mathrm{Der}(\mathbb{H}_{\mathbb{D}}) \cong \mathrm{SO}(3) \oplus \mathrm{SO}(3) \cong \mathrm{SO}(4)$, a real Lie algebra of dimension $6$, and every derivation is inner.

Proof. A derivation $D$ fixes each central idempotent: from $D(\tilde\Pi_+) = D(\tilde\Pi_+^2) = \tilde\Pi_+D(\tilde\Pi_+) + D(\tilde\Pi_+)\tilde\Pi_+$ and the centrality of $D(\tilde\Pi_+)$ one obtains $D(\tilde\Pi_+) = 0$, and similarly $D(\tilde\Pi_-) = 0$. Hence $D$ kills no element of the form $\tilde Q_+ \tilde\Pi_+$ into the other factor and, by the Leibniz rule, preserves the splitting; so $D$ acts as a derivation of $\mathbb{H}$ on the first factor and one on the second, and

$$ \mathrm{Der}(\mathbb{H}_{\mathbb{D}}) \cong \mathrm{Der}(\mathbb{H}) \oplus \mathrm{Der}(\mathbb{H}) . $$

For the division algebra $\mathbb{H}$, every derivation is inner: $\mathrm{Der}(\mathbb{H}) = \mathrm{ad}(\mathbb{H})$ with kernel the centre $\mathbb{R}$, so $\mathrm{Der}(\mathbb{H}) \cong \mathbb{H}/\mathbb{R} \cong \mathrm{SO}(3)$, of dimension $3$. Summing the two factors gives $\mathrm{SO}(3) \oplus \mathrm{SO}(3) \cong \mathrm{SO}(4)$.

Inner derivations. For $a \in \mathbb{H}_{\mathbb{D}}$ the map $\mathrm{ad}_a(\tilde Q) = a\tilde Q - \tilde Q a$ is a derivation, and $\mathrm{ad}$ has kernel the centre $Z(\mathbb{H}_{\mathbb{D}}) = \mathbb{D}_{\mathbb{H}_{\mathbb{D}}}$ and image all derivations, so

$$ \mathrm{Der}(\mathbb{H}_{\mathbb{D}}) \cong \mathbb{H}_{\mathbb{D}} / \mathbb{D}_{\mathbb{H}_{\mathbb{D}}} , $$

an isomorphism of Lie algebras because $[\mathrm{ad}_a,\mathrm{ad}_b] = \mathrm{ad}_{[a,b]}$.

Derivations and automorphisms. The two structures are linked by the exponential, $\exp(t\,\mathrm{ad}_a)(\tilde Q) = e^{ta}\tilde Q e^{-ta}$, the inner automorphism determined by $e^{ta}$. Hence the Lie algebra of the identity component of $\mathrm{Aut}_{\mathbb{R}}(\mathbb{H}_{\mathbb{D}})$ is the derivation algebra: $\mathrm{Lie}\left(SO(3)\times SO(3)\right) \cong \mathrm{SO}(3)\oplus\mathrm{SO}(3)$.

An explicit basis. The derivations $A_k = \tfrac{1}{2}\mathrm{ad}_{e_k}$, $k = 1,2,3$, act on the two factors simultaneously, each as $\mathrm{ad}_{e_k}$, and satisfy $[A_1,A_2] = A_3$ together with its cyclic permutations; the companion derivations $\tfrac{1}{2}\mathrm{ad}_{je_k}$ act with opposite signs on the two factors. Their combinations

$$ D_k^{+} = \tfrac{1}{2}\left(A_k + \tfrac{1}{2}\mathrm{ad}_{je_k}\right) = \tfrac{1}{2}\mathrm{ad}_{e_k \tilde\Pi_+}, \qquad D_k^{-} = \tfrac{1}{2}\left(A_k - \tfrac{1}{2}\mathrm{ad}_{je_k}\right) = \tfrac{1}{2}\mathrm{ad}_{e_k \tilde\Pi_-} $$

act on the first and on the second factor respectively and vanish on the other, and each triple satisfies $[D_1^{\pm},D_2^{\pm}] = D_3^{\pm}$ together with its cyclic permutations. The six elements $D_k^{\pm}$ are linearly independent and form a real basis of the six-dimensional derivation algebra.

Worked Examples

The factor swap. The map $c(\tilde{Q}) = \tilde{Q}^{*}$ fixes $e_k$ and $je_k$ in the sense $c(e_k) = e_k$, $c(je_k) = -je_k$, and swaps the two halves: $c(\tilde Q_+,\tilde Q_-) = (\tilde Q_-,\tilde Q_+)$ in components. It satisfies $c^2 = \mathrm{id}$, preserves the product because ${}^{*}$ is an automorphism of $\mathbb{D}$, and is not inner, since the inner automorphisms fix the centre pointwise.

An inner automorphism. For $u = e_1$ in the first factor, $\sigma(\tilde Q_+,\tilde Q_-) = (e_1 \tilde Q_+ e_1^{-1}, \tilde Q_-)$ has $e_1^{-1} = -e_1$, so it fixes $e_1$ and reverses the signs of $e_2$ and $e_3$ in the first factor while leaving the second untouched: $\sigma(e_2,0) = (-e_2,0)$, $\sigma(e_3,0) = (-e_3,0)$.

A derivation. Take $D = D_3^{+} = \tfrac{1}{2}\mathrm{ad}_{e_3 \tilde\Pi_+}$, acting on the first factor alone. Its exponential acts by $\exp(tD)(e_1,0) = (\cos t\,e_1 + \sin t\,e_2, 0)$, matching $A(e^{te_3/2},0)$; the second factor is fixed, because $\mathrm{ad}_{e_3\tilde\Pi_+}$ acts as zero there.

A derivation that mixes nothing. Any $a = a_+ \tilde\Pi_+ + a_- \tilde\Pi_-$ gives the pair $\mathrm{ad}_a = (\mathrm{ad}_{a_+}, \mathrm{ad}_{a_-})$, and $\mathrm{ad}_a$ depends only on $a$ modulo the centre: adding a central element changes neither component.

Summary

The split biquaternion algebra is the product $\mathbb{H} \oplus \mathbb{H}$ of two quaternion division algebras, with centre $\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} \cong \mathbb{D}$ and two minimal two-sided ideals. Over the split complex algebra every automorphism is $\mathbb{D}$-linear and consists of an inner automorphism of each factor, so $\mathrm{Aut}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}}) \cong SO(3) \times SO(3)$, connected, of dimension $6$. Over $\mathbb{R}$ an automorphism may also interchange the two factors, and the split complex conjugation $c(\tilde{Q}) = \bar{\tilde{Q}}$ realises that swap, an outer automorphism; hence $\mathrm{Aut}_{\mathbb{R}}(\mathbb{H}_{\mathbb{D}}) \cong (SO(3)\times SO(3)) \rtimes \mathbb{Z}/2$, of dimension $6$ with two connected components. Every real derivation is automatically $\mathbb{D}$-linear, because the centre $\mathbb{D}$ admits no nonzero derivation; every derivation preserves the splitting and is inner, so $\mathrm{Der}(\mathbb{H}_{\mathbb{D}}) \cong \mathbb{H}_{\mathbb{D}}/\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} \cong \mathrm{SO}(3)\oplus\mathrm{SO}(3) \cong \mathrm{SO}(4)$, of dimension $6$, the Lie algebra of the identity component of the automorphism group. The contrast with the biquaternion case is exact: there the algebra is simple and central over $\mathbb{C}$, every automorphism is inner, $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) = PGL(2,\mathbb{C}) = PSL(2,\mathbb{C})$ and $\mathrm{Aut}_{\mathbb{R}}(\mathbb{B}) \cong PGL(2,\mathbb{C}) \rtimes \mathbb{Z}/2$, with derivation algebra $\mathrm{SL}(2,\mathbb{C})_{\mathbb{R}} \cong \mathrm{SO}(1,3)$; here the semisimple product structure replaces the simple algebra, the compact group $SO(3)\times SO(3)$ replaces the projective Lorentz group, and the derivation algebra $\mathrm{SO}(4)$ replaces $\mathrm{SO}(1,3)$.

Summary of Notation

Symbol	Meaning
$\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H} \cong \mathbb{H}\oplus\mathbb{H}$	Split biquaternion algebra; product of two quaternion algebras
$e_0,e_1,e_2,e_3$	Algebra basis, $e_0 = 1$, $e_k^2 = -e_0$
$j$	Central split complex unit, $j^2 = +1$
$\tilde\Pi_\pm = \tfrac{1}{2}(1\pm j)$	Central idempotents, the two factors
$\mathbb{D}_{\mathbb{H}_{\mathbb{D}}} = \mathrm{span}_{\mathbb{R}}\{e_0, je_0\}$	Centre of the algebra
$\tilde{Q} \mapsto (\tilde{Q}_+,\tilde{Q}_-)$	Isomorphism to $\mathbb{H}\oplus\mathbb{H}$
${}^{\natural}, \bar\cdot, {}^{*}, {}^{\flat}$	Quaternion, split complex, Hermitian, anti-Hermitian conjugations
$\mathrm{Aut}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}}) \cong SO(3)\times SO(3)$	$\mathbb{D}$-linear automorphisms
$\mathrm{Aut}_{\mathbb{R}}(\mathbb{H}_{\mathbb{D}}) \cong (SO(3)\times SO(3))\rtimes\mathbb{Z}/2$	Real automorphisms
$c(\tilde{Q}) = \bar{\tilde{Q}}$	Factor swap, the outer real automorphism
$\mathrm{Der}(\mathbb{H}_{\mathbb{D}}) = \mathrm{Der}_{\mathbb{D}}(\mathbb{H}_{\mathbb{D}}) \cong \mathrm{SO}(3)\oplus\mathrm{SO}(3)\cong\mathrm{SO}(4)$	Derivation algebra
$\mathrm{ad}_a(\tilde Q) = a\tilde Q - \tilde Q a$	Inner derivation
$D_k^{\pm} = \tfrac{1}{2}\mathrm{ad}_{e_k \tilde\Pi_\pm}$	Derivation basis: $D_k^{+}$ acts on the first factor, $D_k^{-}$ on the second, $[D_1^{\pm},D_2^{\pm}]=D_3^{\pm}$