Split-Quaternion Automorphisms and Derivations

Introduction

The split-quaternion algebra is the four-dimensional real algebra

$$ \mathbb{H}_{\mathrm{s}} = \mathrm{Cl}_{1,1}, $$

with basis $1, e_1, e_2, e_3$, the relations $e_1^2 = -1$, $e_2^2 = e_3^2 = +1$, $e_3 = e_1 e_2$. This article describes two standard invariants: the group of algebra automorphisms and the Lie algebra of derivations. Because the algebra is central simple over $\mathbb{R}$, both invariants are as small as the structure permits: every automorphism is inner and every derivation is inner, so there are no outer automorphisms and no exotic derivations.

We use the conventions of the algebra article: the conjugation ${}^{\natural}$, the principal involution $\alpha$ and the reversal $\rho$; the split-quaternion norm $N(\tilde q) = q_0^2 + q_1^2 - q_2^2 - q_3^2$; the vector subspace $V = \operatorname{span}\{e_1,e_2,e_3\}$, with bracket algebra $\mathrm{SL}_2(\mathbb{R})$. A general element is $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$. No physics is invoked, and everything below is standard structure theory of a central simple real algebra.

Standing Facts: Simplicity and the Centre

The automorphism group and the derivation algebra are governed by two structural facts.

Simplicity. The only two-sided ideals of $\mathbb{H}_{\mathrm{s}}$ are $0$ and $\mathbb{H}_{\mathrm{s}}$.

The centre. The centre is $Z(\mathbb{H}_{\mathrm{s}}) = S = \mathbb{R}\cdot 1$, a one-dimensional real space. Hence $\mathbb{H}_{\mathrm{s}}$ is central simple over $\mathbb{R}$: simple, of real dimension $4$, with centre exactly $\mathbb{R}$. This is the point of departure from the biquaternion algebra $\mathbb{B}$, which over $\mathbb{R}$ is simple but not central, its centre being the two-dimensional subspace $\mathbb{C}_{\mathbb{B}}$. Because the centre here is one-dimensional, an automorphism can only fix it, and a derivation can only vanish on it; there is no conjugate-linear coset and no étale direction to allow extra structure.

Automorphisms over $\mathbb{R}$

Definition. An $\mathbb{R}$-algebra automorphism of $\mathbb{H}_{\mathrm{s}}$ is a bijective $\mathbb{R}$-linear map $\sigma : \mathbb{H}_{\mathrm{s}} \to \mathbb{H}_{\mathrm{s}}$ with $\sigma(\tilde q \tilde p) = \sigma(\tilde q)\sigma(\tilde p)$ and $\sigma(1) = 1$. They form a group under composition, written $\mathrm{Aut}(\mathbb{H}_{\mathrm{s}})$.

Inner automorphisms. For an invertible $g \in \mathbb{H}_{\mathrm{s}}$ the map $\iota_g(\tilde q) = g\tilde q g^{-1}$ is an automorphism, the inner automorphism determined by $g$. Since the centre is $\mathbb{R}$, the automorphism $\iota_g$ depends only on the class of $g$ modulo nonzero scalars, so $\iota_g = \iota_{\lambda g}$ for $\lambda \in \mathbb{R}^\times$.

Theorem (Skolem–Noether). For any field $k$ and $n \geq 1$, every $k$-algebra automorphism of $M_n(k)$ is inner: for each $\sigma$ there exists $g \in GL_n(k)$ with $\sigma(\tilde q) = g\tilde q g^{-1}$ for all $\tilde q$.

Theorem. Every $\mathbb{R}$-algebra automorphism of $\mathbb{H}_{\mathrm{s}}$ is inner, and the map

$$ \mathbb{H}_{\mathrm{s}}^\times \longrightarrow \mathrm{Aut}(\mathbb{H}_{\mathrm{s}}), \qquad g \longmapsto \iota_g, $$

is surjective with kernel the centre $\mathbb{R}^\times$. Hence

$$ \mathrm{Aut}(\mathbb{H}_{\mathrm{s}}) \cong \mathbb{H}_{\mathrm{s}}^{\times} / \mathbb{R}^\times = \mathrm{PGL}_2(\mathbb{R}) \cong SO(2,1). $$

Proof. The Skolem–Noether theorem for a central simple algebra applies: every automorphism is inner, which gives the surjectivity and the identification of the automorphism group with the quotient. The kernel of $g \mapsto \iota_g$ is the set of units $g$ with $g\tilde q = \tilde q g$ for all $\tilde q$, i.e. the units of the centre, $\mathbb{R}^\times$. The last isomorphism is the adjoint action on $V$ of Split-Quaternion Rotations and the Lorentz Group, which identifies $\mathrm{PGL}_2(\mathbb{R})$ with $SO(2,1)$.

Dimension and components. Since the unit group has real dimension $4$ and the centre $\mathbb{R}^\times$ has dimension $1$, the group has real dimension $3$. The sign of the split-quaternion norm splits the unit group into the two components $\{N>0\}$ and $\{N<0\}$, and since $N(\lambda g) = \lambda^2 N(g)$ the quotient keeps the sign of the split-quaternion norm, so $\mathrm{Aut}(\mathbb{H}_{\mathrm{s}}) \cong PGL_2(\mathbb{R})$ has two components, matching the two components of $SO(2,1)$. The identity component is

$$ \mathrm{Aut}^0(\mathbb{H}_{\mathrm{s}}) \cong PSL_2(\mathbb{R}) \cong \mathrm{SO}^{+}(2,1), $$

which is also the image of the norm-one group $U = \{N = 1\}$ under $g \mapsto \iota_g$.

Invariants. Every automorphism preserves the split-quaternion norm, $N(\sigma(\tilde q)) = N(\tilde q)$, because $\iota_g$ is inner and $N$ is multiplicative; hence it preserves the group of units, the zero-divisor set, the rank-one idempotents and the subspaces defined by $N$. It also preserves the centre pointwise.

Example. For $g = e_3$, with $e_3^{-1} = e_3$ (since $N(e_3) = -1$ and $e_3^{\natural} = -e_3$, so $e_3^{-1} = -e_3/(-1) = e_3$), the inner automorphism acts on the vector subspace by $\iota_{e_3}(e_1) = -e_1$, $\iota_{e_3}(e_2) = -e_2$, $\iota_{e_3}(e_3) = e_3$, the matrix $\operatorname{diag}(-1,-1,1)$ in the basis $e_1,e_2,e_3$. This transformation is in $SO(2,1)$ but not in its identity component: it fixes the spacelike direction $e_3$ and reverses the sheet of the timelike hyperboloid.

Derivations

Definition. An $\mathbb{R}$-linear derivation of $\mathbb{H}_{\mathrm{s}}$ is an $\mathbb{R}$-linear map $D : \mathbb{H}_{\mathrm{s}} \to \mathbb{H}_{\mathrm{s}}$ satisfying the Leibniz rule

$$ D(\tilde q \tilde p) = D(\tilde q)\,\tilde p + \tilde q\,D(\tilde p) . $$

The derivations form a real Lie algebra under the commutator, written $\mathrm{Der}(\mathbb{H}_{\mathrm{s}})$.

Inner derivations. For $a \in \mathbb{H}_{\mathrm{s}}$ the map

$$ \mathrm{ad}_a : \tilde q \mapsto [a,\tilde q] = a\tilde q - \tilde q a $$

is a derivation, the inner derivation by $a$, and $\mathrm{ad}_a$ depends only on the class of $a$ modulo the centre: $\mathrm{ad}_a = \mathrm{ad}_{a + c}$ for $c \in \mathbb{R}$.

Theorem. Every derivation of $\mathbb{H}_{\mathrm{s}}$ is inner, and the map

$$ \mathbb{H}_{\mathrm{s}} \longrightarrow \mathrm{Der}(\mathbb{H}_{\mathrm{s}}), \qquad a \longmapsto \mathrm{ad}_a, $$

has kernel the centre $\mathbb{R}$. Hence

$$ \mathrm{Der}(\mathbb{H}_{\mathrm{s}}) = \mathrm{ad}(\mathbb{H}_{\mathrm{s}}) \cong \mathbb{H}_{\mathrm{s}}/\mathbb{R} \cong V \cong \mathrm{SL}_2(\mathbb{R}), $$

the vector subspace $V$ under the commutator bracket, of real dimension $3$.

Proof. Every derivation of $\mathbb{H}_{\mathrm{s}}$ is inner: the standard computation, valid for any central simple $k$-algebra, shows that for a derivation $D$ the element $a = \sum_i D(u_i)v_i$ — where $\sum_i u_i v_i = 1$ is a separating idempotent decomposition — satisfies $D = \mathrm{ad}_a$. The kernel of $\mathrm{ad}$ is the centre: $[a,\tilde q] = 0$ for all $\tilde q$ iff $a \in Z = \mathbb{R}$. The image is $\mathrm{ad}(V) \cong \mathrm{SL}_2(\mathbb{R})$, since the scalar part of $a$ contributes nothing.

Structure. The bracket of inner derivations is again inner and satisfies $[\mathrm{ad}_a, \mathrm{ad}_b] = \mathrm{ad}_{[a,b]}$, so the map $a \mapsto \mathrm{ad}_a$ is a Lie algebra homomorphism onto $\mathrm{Der}(\mathbb{H}_{\mathrm{s}})$ with kernel the centre. Under it the bracket of $V$ computed in Split-Quaternion Scalar and Vector Subspaces, $[e_1,e_2] = 2e_3$, $[e_2,e_3] = -2e_1$, $[e_3,e_1] = 2e_2$, becomes the bracket of $\mathrm{SL}_2(\mathbb{R})$. In particular $\mathrm{Der}(\mathbb{H}_{\mathrm{s}}) \cong \mathrm{SL}_2(\mathbb{R}) \cong \mathrm{SO}(2,1)$, the Lie algebra of the Lorentz group of the vector subspace.

Every derivation preserves the split-quaternion norm. Because $\mathrm{Der}$ is spanned by the inner derivations $\mathrm{ad}_a$, and each generates the one-parameter group of automorphisms $t \mapsto \exp(t\,\mathrm{ad}_a) = \mathrm{Ad}_{e^{ta}}$, which preserves $N$, the infinitesimal statement $D(N(\tilde q)) = 0$ holds for every derivation and every $\tilde q$. Every derivation therefore vanishes on the centre and annihilates the split-quaternion norm.

Outer Automorphisms

Proposition. The algebra $\mathbb{H}_{\mathrm{s}} \cong M_2(\mathbb{R})$ has no outer automorphisms: the group $\mathrm{Out}(\mathbb{H}_{\mathrm{s}}) = \mathrm{Aut}(\mathbb{H}_{\mathrm{s}})/\mathrm{Inn}(\mathbb{H}_{\mathrm{s}})$ is trivial, and every automorphism is inner.

Proof. By Skolem–Noether every automorphism is inner, so $\mathrm{Aut} = \mathrm{Inn}$ and the quotient is trivial.

The exceptional phenomena that produce outer automorphisms elsewhere — the outer automorphism of $M_n(\mathbb{D})$ for $n \geq 3$ over a division algebra $\mathbb{D} \neq k$, or the triality of $\mathrm{Cl}_{0,8}$ — do not occur for a $2 \times 2$ matrix algebra over a field. Thus for $\mathbb{H}_{\mathrm{s}}$ the outer automorphism group gives no new symmetry, and the whole automorphism group is the inner one, of two components.

Anti-Automorphisms

The conjugations ${}^{\natural}$ and $\rho$ are anti-automorphisms, satisfying $(\tilde q \tilde p)^{\natural} = \tilde p^{\natural}\,\tilde{q}^{\natural}$, and are not automorphisms, so they do not appear in $\mathrm{Aut}(\mathbb{H}_{\mathrm{s}})$. Together with the automorphism $\alpha$ they make up the involution group of the algebra, treated in Split-Quaternion Involution Lattice. In the Clifford description of Split-Quaternion Other Algebraic Element Representations, the two are the reversal and the Clifford conjugation; the existence of anti-automorphisms outside $\mathrm{Aut}$ is the standard asymmetry between an algebra and its opposite, and it is not an outer automorphism phenomenon.

The Lie Algebra Statement

The three statements fit together as follows. The automorphism group is the Lie group $\mathrm{Aut}(\mathbb{H}_{\mathrm{s}}) \cong \mathrm{PGL}_2(\mathbb{R}) \cong SO(2,1)$, of dimension $3$ with two components. Its identity component is $\mathrm{PSL}_2(\mathbb{R}) \cong \mathrm{SO}^{+}(2,1)$, with Lie algebra $\mathrm{SO}(2,1)$. The derivation algebra is $\mathrm{Der}(\mathbb{H}_{\mathrm{s}}) \cong \mathrm{SL}_2(\mathbb{R}) \cong \mathrm{SO}(2,1)$, and it is exactly the Lie algebra of $\mathrm{Aut}$:

$$ \operatorname{Lie}\bigl(\mathrm{Aut}(\mathbb{H}_{\mathrm{s}})\bigr) \;=\; \mathrm{Der}(\mathbb{H}_{\mathrm{s}}) \;\cong\; \mathrm{SL}_2(\mathbb{R}) \;\cong\; \mathrm{SO}(2,1). $$

The correspondence is the exponential of the inner derivations, $\exp(t\,\mathrm{ad}_a) = \iota_{e^{ta}}$, which is the usual Lie-theoretic identification of the inner derivations with the Lie algebra of the inner automorphism group. Since $\mathrm{Aut}$ has two components, its Lie algebra is generated by the identity component alone; the two components are distinguished by the determinant, and any automorphism of negative determinant, such as $\iota_{e_3}$ of the example above, lies in the non-identity component.

Worked Examples

Example (the elliptic one-parameter group). Take $a = e_1$, so that $\mathrm{ad}_{e_1}(e_2) = [e_1,e_2] = 2e_3$ and $\mathrm{ad}_{e_1}(e_3) = [e_1,e_3] = -2e_2$, with $\mathrm{ad}_{e_1}(e_1) = 0$. The exponential $t \mapsto \exp(t\,\mathrm{ad}_{e_1})$ is the inner automorphism group generated by $e^{t e_1} = \cos t + \sin t\, e_1$ (a unit of norm $1$), which acts on $V$ as the rotation of the plane $\operatorname{span}\{e_2,e_3\}$ by the doubled angle $2t$, the elliptic subgroup of Split-Quaternion Rotations and the Lorentz Group.

Example (the hyperbolic one-parameter group). Take $a = e_2$, so that $\mathrm{ad}_{e_2}(e_1) = [e_2,e_1] = -2e_3$ and $\mathrm{ad}_{e_2}(e_3) = [e_2,e_3] = -2e_1$, with $\mathrm{ad}_{e_2}(e_2) = 0$. The exponential is generated by $e^{t e_2/2} = \cosh(t/2) + \sinh(t/2)\, e_2$ (a unit of norm $1$), acting on $V$ as a boost of the plane $\operatorname{span}\{e_1,e_3\}$ of signature $(1,1)$ fixing $e_2$.

Example (a derivation that is not a scalar multiple of a basis one). For $a = e_1 + e_2$ the derivation $\mathrm{ad}_a$ has $\mathrm{ad}_a(e_3) = [e_1 + e_2, e_3] = -2e_1 - 2e_2$ and mixes the two involutions' eigenspaces, so it is not proportional to any of $\mathrm{ad}_{e_1}, \mathrm{ad}_{e_2}, \mathrm{ad}_{e_3}$.

Example (an automorphism of the outer component). The inner automorphism $\iota_{e_3}$ of the example above has matrix $\operatorname{diag}(-1,-1,1)$ on $V$, of determinant $+1$, in the component of $SO(2,1)$ away from the identity. Its square is the identity.

The Contrast with the Biquaternion Case

The biquaternion article Biquaternion Automorphisms and Derivations separates the automorphisms over $\mathbb{C}$ from those over $\mathbb{R}$, because over $\mathbb{R}$ the centre of $\mathbb{B}$ is the two-dimensional $\mathbb{C}_{\mathbb{B}} \cong \mathbb{C}$ and complex conjugation adds a second, conjugate-linear coset, so that $\mathrm{Aut}_{\mathbb{R}}(\mathbb{B}) \cong PGL(2,\mathbb{C}) \rtimes \mathbb{Z}/2$ is strictly larger than $\mathrm{Aut}_{\mathbb{C}}(\mathbb{B}) \cong PGL(2,\mathbb{C})$. For $\mathbb{H}_{\mathrm{s}}$ the centre is one-dimensional over $\mathbb{R}$ and there is no conjugation available, so the group is a single inner group, $\mathrm{Aut}(\mathbb{H}_{\mathrm{s}}) \cong PGL_2(\mathbb{R}) \cong SO(2,1)$, with no outer automorphisms and no semilinear coset; the outer automorphism group, which over $\mathbb{R}$ is nontrivial for $\mathbb{B}$, is trivial here. For derivations the two cases agree in shape: all derivations are inner, and the derivation algebra is the vector subspace under the commutator, $\mathrm{SL}_2$ in both, purely real here, realified from $\mathbb{C}$ there. The complexification of $\mathbb{H}_{\mathrm{s}}$ is $\mathbb{C} \otimes_{\mathbb{R}} \mathbb{H}_{\mathrm{s}} = M_2(\mathbb{C}) \cong \mathbb{B}$, so the two categories describe the same complex algebra over different real forms; the involution and automorphism picture is the part where the real forms differ.

Summary

The split-quaternion algebra is central simple over $\mathbb{R}$, with centre the scalar line $S = \mathbb{R}\cdot1$. Every $\mathbb{R}$-algebra automorphism is inner, by Skolem–Noether, and

$$ \mathrm{Aut}(\mathbb{H}_{\mathrm{s}}) \cong \mathrm{PGL}_2(\mathbb{R}) \cong SO(2,1), $$

of dimension $3$, with identity component $\mathrm{Aut}^0 \cong \mathrm{PSL}_2(\mathbb{R}) \cong \mathrm{SO}^{+}(2,1)$ and a unique nontrivial component; every automorphism preserves the split-quaternion norm and the centre. There are no outer automorphisms, $\mathrm{Out}(\mathbb{H}_{\mathrm{s}}) = 1$; the anti-automorphisms ${}^{\natural}$ and $\rho$ are not automorphisms and belong to the involution group instead.

Every derivation is inner, so $\mathrm{Der}(\mathbb{H}_{\mathrm{s}}) = \mathrm{ad}(\mathbb{H}_{\mathrm{s}}) \cong \mathbb{H}_{\mathrm{s}}/\mathbb{R} \cong V \cong \mathrm{SL}_2(\mathbb{R}) \cong \mathrm{SO}(2,1)$, the traceless matrices under the commutator bracket, of real dimension $3$; every derivation vanishes on the centre and annihilates the split-quaternion norm, and the map $a \mapsto \mathrm{ad}_a$ has kernel the centre. The derivation algebra is the Lie algebra of the automorphism group. The contrast with the biquaternion case is exact: there the real automorphism group is strictly larger than the complex one and has a nontrivial outer part, here the group is a single inner group of dimension $3$ with two components; the two systems nevertheless share the same derivation shape, the traceless matrices under the bracket.

Summary of Notation

Symbol	Meaning	Article
$\mathbb{H}_{\mathrm{s}}$	the split-quaternion algebra, $\mathrm{Cl}_{1,1}$	Split-Quaternion Algebra
$Z(\mathbb{H}_{\mathrm{s}}) = S = \mathbb{R}\cdot 1$	the centre, one-dimensional over $\mathbb{R}$	Split-Quaternion Scalar and Vector Subspaces
$\mathrm{Aut}(\mathbb{H}_{\mathrm{s}})$	the algebra automorphisms, $\cong \mathrm{PGL}_2(\mathbb{R}) \cong SO(2,1)$	this article
$\mathrm{Inn}(\mathbb{H}_{\mathrm{s}})$, $\mathrm{Out}(\mathbb{H}_{\mathrm{s}})$	inner and outer automorphism groups; $\mathrm{Out} = 1$	this article
$\iota_g(\tilde q) = g\tilde q g^{-1}$	the inner automorphism by a unit $g$	this article
$\mathrm{Der}(\mathbb{H}_{\mathrm{s}})$	the derivations, $\cong \mathrm{SL}_2(\mathbb{R}) \cong \mathrm{SO}(2,1)$	this article
$\mathrm{ad}_a(\tilde q) = [a,\tilde q]$	the inner derivation by $a$	this article
$V, \mathrm{SL}_2(\mathbb{R})$	the vector subspace and the traceless matrices	Split-Quaternion Scalar and Vector Subspaces
$N(\tilde q) = q_0^2+q_1^2-q_2^2-q_3^2$	the split-quaternion norm, invariant under automorphisms and derivations	Split-Quaternion Norm and Invertibility
$\alpha, \rho, {}^{\natural}$	the involutions; $\rho, {}^{\natural}$ are anti-automorphisms	Split-Quaternion Subspaces and the Involutions
$U = \{N=1\}$	the norm-one group, mapping onto $\mathrm{SO}^{+}(2,1)$	Split-Quaternion Norm and Invertibility