Quaternion Automorphisms and Derivations

Introduction

Two invariants of an algebra are its group of automorphisms and its Lie algebra of derivations, and for the quaternion algebra both can be computed completely: the automorphism group is the rotation group of the imaginary subspace, and the derivation algebra is the imaginary subspace itself under the commutator bracket. This article establishes those identifications, computes the dimensions, and links the two by the exponential. It is the quaternion member of the family's automorphism-and-derivation pair; its counterpart is the biquaternion case, where the same questions are answered by the projective Lorentz group and by $\mathrm{SO}(1,3)$, and where the distinction between the complex and the real ground field is decisive. Here the ground field is $\mathbb{R}$ throughout, the algebra is central simple over it, and there is no conjugate-linear coset to add.

The article uses Quaternion Algebra for the algebra and its centre, Quaternion Ideals and Simplicity for the central-simplicity of $\mathbb{H}$ over $\mathbb{R}$, Quaternion Norm and Invertibility for the unit group, and Quaternion Rotations and Reflections for the identification of the unit sphere with the rotation group.

The results are the case of the standard structure theory of a central simple algebra: over a field $k$, every $k$-algebra automorphism of a finite-dimensional central simple algebra is inner, and every derivation is inner. Those two theorems are cited as standard; the specific computations for $\mathbb{H}$ are shown.

Throughout, $\mathbb{H}$ is the quaternion algebra over $\mathbb{R}$ with basis $e_0 = 1, e_1, e_2, e_3$, $e_k^2 = -e_0$; a quaternion is $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ with centre $\mathbb{R}_{\mathbb{H}} = \mathbb{R}e_0$, conjugate $\tilde{q}^{\natural}$, and norm $N(\tilde q) = \tilde q\tilde{q}^{\natural}$; the unit group is $\mathbb{H}^{\times} = \mathbb{H}\setminus\{0\}$ and the unit sphere is $Sp(1) = \{\tilde q : N(\tilde q) = 1\}$. The scalar part of a commutator gives $\mathrm{Sc}([\tilde q,\tilde p]) = 0$, so $[\tilde q,\tilde p]\in\operatorname{Im}\mathbb{H}$ always.

Automorphisms of the Quaternion Algebra

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

Definition. For a unit $u\in\mathbb{H}$ the map $\iota_u(\tilde q) = u\tilde qu^{-1}$ is the inner automorphism determined by $u$. Since the centre is $\mathbb{R}$, one has $\iota_u = \iota_{\lambda u}$ for every non-zero real $\lambda$, so $\iota_u$ depends only on the class of $u$ in $Sp(1)/\{\pm1\}$.

Proposition. $\iota_u$ is an automorphism for every unit $u$, the assignment $u\mapsto\iota_u$ is a group homomorphism $Sp(1)\to\operatorname{Aut}_{\mathbb{R}}(\mathbb{H})$, and its kernel is $\{\pm1\}$.

Proof. For units $u,v$, $\iota_u\iota_v = \iota_{uv}$, and $\iota_u^{-1} = \iota_{u^{-1}}$, so the assignment is a homomorphism into bijective $\mathbb{R}$-algebra maps. Its kernel consists of the units with $u\tilde qu^{-1} = \tilde q$ for all $\tilde q$, namely the units of the centre $\mathbb{R}$, that is $\{\pm1\}$.

Theorem (Skolem–Noether for $\mathbb{H}$). Every $\mathbb{R}$-algebra automorphism of $\mathbb{H}$ is inner. Consequently the homomorphism $Sp(1)\to\operatorname{Aut}_{\mathbb{R}}(\mathbb{H})$ is surjective and

$$ \operatorname{Aut}_{\mathbb{R}}(\mathbb{H})\cong Sp(1)/\{\pm1\} . $$

Proof. The algebra $\mathbb{H}$ is central simple over $\mathbb{R}$ by Quaternion Ideals and Simplicity, and the Skolem–Noether theorem states that every $k$-algebra automorphism of a finite-dimensional central simple $k$-algebra is inner; quoting it as standard gives surjectivity of $u\mapsto\iota_u$ onto the automorphism group. The kernel is $\{\pm1\}$ by the proposition, so the first isomorphism theorem gives the displayed isomorphism.

Remark. The anti-automorphism $\tilde q\mapsto\tilde q^{\natural}$ is not an automorphism, since it reverses products, and it is not the identity $\iota_u$ for any unit; it is excluded from the automorphism group. There is no second coset of automorphisms here, because the centre of $\mathbb{H}$ is $\mathbb{R}$, which has no non-trivial field automorphism.

The Automorphism Group as the Rotation Group

Theorem. The inner automorphisms are exactly the automorphisms that act on the imaginary subspace as rotations, and there is a canonical isomorphism

$$ \operatorname{Aut}_{\mathbb{R}}(\mathbb{H})\cong SO(3), $$

the rotation group of the three-dimensional space $\operatorname{Im}\mathbb{H}$.

Proof. An automorphism $\sigma$ fixes the centre $\mathbb{R}e_0$ and preserves squares, so it preserves the set $\operatorname{Im}\mathbb{H} = \{\mathbf q : \mathbf q^2\in\mathbb{R}e_0,\ \mathbf q^2\leq0\}$: a quaternion has real square exactly when it is real or pure, and the square is non-positive exactly when it is pure (or zero). On that set the quaternion norm is $N(\mathbf q) = -\mathbf q^2$, whence $\sigma$ preserves the quaternion norm. Thus every automorphism restricts to an orthogonal map of $\operatorname{Im}\mathbb{H}$; since by Skolem–Noether every automorphism is inner and every inner automorphism is the adjoint action of a unit, the restriction has determinant $+1$ by Quaternion Rotations and Reflections. Conversely every rotation of $\operatorname{Im}\mathbb{H}$ extends to an automorphism — it is the adjoint action of a unit quaternion — and the extension is unique because $\mathbb{H}$ is generated by $\operatorname{Im}\mathbb{H}$. Identifying the cover $Sp(1)\to SO(3)$ with $u\mapsto\iota_u$ gives the isomorphism.

Corollary. As a real manifold, $\operatorname{Aut}_{\mathbb{R}}(\mathbb{H})$ is diffeomorphic to the projective space $\mathbb{RP}^3$ and has dimension $3$; it is connected and compact, and it is isomorphic to the group of orientation-preserving isometries of $\operatorname{Im}\mathbb{H}$ fixing the origin.

Proof. $Sp(1)/\{\pm1\}\cong S^3/\{\pm1\}\cong\mathbb{RP}^3$, of dimension three; $SO(3)$ is connected and compact.

Example. For $u = e_1$ the inner automorphism $\iota_{e_1}$ fixes $e_1$ and the centre and reverses the signs of $e_2$ and $e_3$: indeed $e_1e_2e_1^{-1} = -(e_1e_2)e_1 = -e_3e_1 = -e_2$, using $e_1e_2 = e_3$ and $e_3e_1 = e_2$. It is the half-turn about the axis $e_1$.

Remark. The automorphisms of order two are exactly the halfturns about the axes. An inner automorphism $\iota_u$ satisfies $\iota_u^2 = \iota_{u^2} = \mathrm{id}$ exactly when $u^2$ is real, that is when $u$ is real or pure imaginary; a real $u$ gives the identity, and a pure $u$ is $u = t\nu$ with $t\neq0$ real and $\nu$ a unit vector, giving $\iota_u = \iota_\nu$, which fixes the plane $\operatorname{span}(e_0,\nu)$ and is the half-turn about the axis line $\mathbb{R}\nu$. There is thus one non-trivial automorphism of order two for each axis line of $\operatorname{Im}\mathbb{H}$, and they are the involutions of Quaternion Involutions and Projections.

Derivations of the Quaternion Algebra

Definition. A derivation of $\mathbb{H}$ is an $\mathbb{R}$-linear map $D : \mathbb{H}\to\mathbb{H}$ satisfying the Leibniz rule $D(\tilde q\tilde p) = D(\tilde q)\tilde p+\tilde q D(\tilde p)$ for all $\tilde q,\tilde p$. The derivations form a real vector space $\operatorname{Der}_{\mathbb{R}}(\mathbb{H})$, a Lie algebra under the commutator bracket $[D_1,D_2] = D_1D_2-D_2D_1$.

Proposition. Every derivation annihilates $1$ and maps the centre into itself.

Proof. $D(1) = D(1\cdot1) = 2D(1)$, so $D(1) = 0$; if $c$ is central then $D(c)\tilde q = D(c\tilde q)-cD(\tilde q) = D(\tilde q c)-D(\tilde q)c = \tilde q D(c)$, so $D(c)$ is central.

Definition. For $a\in\mathbb{H}$ the inner derivation determined by $a$ is $\operatorname{ad}_a(\tilde q) = a\tilde q-\tilde q a = [a,\tilde q]$.

Proposition. $\operatorname{ad}_a$ is a derivation for every $a$, the map $\operatorname{ad} : \mathbb{H}\to\operatorname{Der}_{\mathbb{R}}(\mathbb{H})$ is $\mathbb{R}$-linear with kernel the centre $\mathbb{R}e_0$, and $[\operatorname{ad}_a,\operatorname{ad}_b] = \operatorname{ad}_{[a,b]}$.

Proof. The Jacobi identity in the form $[a,[\tilde q,\tilde p]] = [[a,\tilde q],\tilde p]+[\tilde q,[a,\tilde p]]$ is exactly the Leibniz rule for $\operatorname{ad}_a$; linearity is clear; $\operatorname{ad}_a = 0$ iff $a$ commutes with every quaternion, iff $a$ is central. The bracket identity is the Jacobi identity again.

Theorem. Every derivation of $\mathbb{H}$ is inner, so there is an isomorphism

$$ \operatorname{Der}_{\mathbb{R}}(\mathbb{H})\cong\mathbb{H}/\mathbb{R}e_0\cong\operatorname{Im}\mathbb{H}, $$

of real vector spaces and of Lie algebras, the bracket on the right being $[\mathbf q,\mathbf p] = 2(\mathbf q\times\mathbf p)$.

Proof. For a finite-dimensional central simple algebra over a field, every derivation is inner; quoting this as standard and applying it to $\mathbb{H}$, the map $\operatorname{ad}$ is surjective, and its kernel is the centre, so it induces the first isomorphism. A commutator is pure, and the space $\mathbb{H}/\mathbb{R}e_0$ is identified with the pure quaternions $\operatorname{Im}\mathbb{H}$; the bracket identity of the proposition becomes $[\mathbf q,\mathbf p] = \mathbf q\mathbf p-\mathbf p\mathbf q = 2(\mathbf q\times\mathbf p)$ for pure quaternions by the product rule $\mathbf{p}\mathbf{q} = -\langle\mathbf{p},\mathbf{q}\rangle+\mathbf{p}\times\mathbf{q}$.

Corollary. $\operatorname{Der}_{\mathbb{R}}(\mathbb{H})$ has dimension $3$, with basis $\{D_1,D_2,D_3\}$, $D_k = \tfrac12\operatorname{ad}_{e_k}$, and the structure constants

$$ [D_1,D_2] = D_3, \qquad [D_2,D_3] = D_1, \qquad [D_3,D_1] = D_2 . $$

Proof. The three inner derivations span $\operatorname{Im}\mathbb{H}/\mathbb{R}e_0$ modulo the centre, which is three-dimensional; the brackets follow from $[e_1,e_2] = 2e_3$, $[e_2,e_3] = 2e_1$, $[e_3,e_1] = 2e_2$ and $D_k = \tfrac12\operatorname{ad}_{e_k}$.

Example. $D_1 = \tfrac12\operatorname{ad}_{e_1}$ vanishes on $e_1$ and acts by $D_1(e_2) = e_3$, $D_1(e_3) = -e_2$: indeed $\tfrac12(e_1e_2-e_2e_1) = \tfrac12(e_3+e_3) = e_3$ and $\tfrac12(e_1e_3-e_3e_1) = \tfrac12(-e_2-e_2) = -e_2$. It is the infinitesimal generator of the half-turn about $e_1$.

The Lie Algebra Statement

Theorem. The derivation Lie algebra is the orthogonal Lie algebra of the vector subspace,

$$ \operatorname{Der}_{\mathbb{R}}(\mathbb{H})\cong\operatorname{Im}\mathbb{H}\cong\mathrm{SO}(3)\cong\mathrm{SU}(2), $$

and it is the Lie algebra of the automorphism group: $\operatorname{Lie}\operatorname{Aut}_{\mathbb{R}}(\mathbb{H}) = \operatorname{Der}_{\mathbb{R}}(\mathbb{H})$.

Proof. With the bracket $[\mathbf q,\mathbf p] = 2(\mathbf q\times\mathbf p)$ the vector subspace is the cross-product Lie algebra, isomorphic to $\mathrm{SO}(3)$, equivalently to $\mathrm{SU}(2)$; the two are isomorphic as real Lie algebras. The exponential identity $\exp(t\operatorname{ad}_a)(\tilde q) = e^{ta}\,\tilde q\,e^{-ta} = \iota_{e^{ta}}(\tilde q)$ holds by the standard series computation, so every inner derivation integrates to an inner automorphism and the Lie functor applied to $\operatorname{Aut}_{\mathbb{R}}(\mathbb{H})\cong SO(3)$ recovers the derivation algebra.

Proposition. Every element of $\operatorname{Im}\mathbb{H}$ gives an inner derivation, and $\operatorname{ad}_a$ depends only on the class of $a$ modulo the centre; the exponential of a derivation acts on the imaginary subspace as the rotation generated by the cross-product field of the corresponding vector.

Proof. The dependence only on the class modulo the centre is the kernel of $\operatorname{ad}$; the exponential statement is the identity quoted in the theorem.

Contrast with the Biquaternion Case

The biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ is central simple over $\mathbb{C}$ but only simple, not central, over $\mathbb{R}$, and the difference changes both invariants.

Over $\mathbb{C}$. Every $\mathbb{C}$-algebra automorphism of $\mathbb{B}\cong M_2(\mathbb{C})$ is inner, and $\operatorname{Aut}_{\mathbb{C}}(\mathbb{B})\cong GL_2(\mathbb{C})/\mathbb{C}^{*} = PGL(2,\mathbb{C}) = PSL(2,\mathbb{C})$, a complex group of dimension $3$, the projective Lorentz group; its derivation algebra is $\mathrm{SL}(2,\mathbb{C})$, of complex dimension $3$.

Over $\mathbb{R}$. The centre is $\mathbb{C}_{\mathbb{B}}\cong\mathbb{C}$, which has a non-trivial real automorphism, complex conjugation; consequently $\operatorname{Aut}_{\mathbb{R}}(\mathbb{B})\cong PGL(2,\mathbb{C})\rtimes\mathbb{Z}/2$, with two connected components and real dimension $6$, the second coset consisting of the conjugate-linear automorphisms.

Feature	$\mathbb{H}$ over $\mathbb{R}$	$\mathbb{B}$ over $\mathbb{C}$	$\mathbb{B}$ over $\mathbb{R}$
Centre	$\mathbb{R}$	$\mathbb{C}$	$\mathbb{C}$
Central simple?	yes	yes	no
Automorphism group	$SO(3)$, dimension $3$	$PGL(2,\mathbb{C}) = PSL(2,\mathbb{C})$, dimension $3$ over $\mathbb{C}$	$PGL(2,\mathbb{C})\rtimes\mathbb{Z}/2$, dimension $6$
Derivation algebra	$\mathrm{SO}(3)\cong\mathrm{SU}(2)$, dimension $3$	$\mathrm{SL}(2,\mathbb{C})$, dimension $3$ over $\mathbb{C}$	$\mathrm{SL}(2,\mathbb{C})_{\mathbb{R}}\cong\mathrm{SO}(1,3)$
Conjugate-linear coset	none	none	present

The compact group $SO(3)$ here is replaced there by the non-compact projective Lorentz group; the reason is the change of the norm from definite to indefinite under complexification, which turns the rotation group into the Lorentz group while leaving both groups of dimension three over their respective fields. The conjugate-linear coset appears only over $\mathbb{R}$, where the centre $\mathbb{C}$ is larger than the base field. No such coset exists here, because the centre of $\mathbb{H}$ is $\mathbb{R}$ itself. The biquaternion account is in Biquaternion Automorphisms and Derivations.

Summary

Every $\mathbb{R}$-algebra automorphism of the quaternion algebra is inner, by Skolem–Noether applied to the central simple algebra $\mathbb{H}$, and the inner automorphisms form the group $Sp(1)/\{\pm1\}\cong SO(3)$ of rotations of the imaginary subspace; the group is connected and compact, diffeomorphic to $\mathbb{RP}^3$, of dimension three, and the anti-automorphism of quaternion conjugation is not among its elements. There is no conjugate-linear coset, because the centre $\mathbb{R}$ has no non-trivial automorphism.

Every derivation of $\mathbb{H}$ is inner, so $\operatorname{Der}_{\mathbb{R}}(\mathbb{H})\cong\mathbb{H}/\mathbb{R}e_0\cong\operatorname{Im}\mathbb{H}$, a three-dimensional real Lie algebra with basis $D_k = \tfrac12\operatorname{ad}_{e_k}$ and brackets $[D_1,D_2] = D_3$ and cyclic; under the bracket $[\mathbf q,\mathbf p] = 2(\mathbf q\times\mathbf p)$ it is $\mathrm{SO}(3)\cong\mathrm{SU}(2)$, and $\exp(t\operatorname{ad}_a) = \iota_{e^{ta}}$ identifies it as the Lie algebra of the automorphism group.

The biquaternion algebra, being central simple only over $\mathbb{C}$ and having centre $\mathbb{C}$ over $\mathbb{R}$, has the projective Lorentz group $PGL(2,\mathbb{C}) = PSL(2,\mathbb{C})$ as its complex automorphism group, with a conjugate-linear second coset over $\mathbb{R}$ and the derivation algebra $\mathrm{SL}(2,\mathbb{C})$, realified to $\mathrm{SO}(1,3)$. The definite norm of $\mathbb{H}$ is what keeps its two groups compact, and the absence of a conjugate-linear coset is the smallness of its centre.

Summary of Notation

Symbol	Meaning
$\mathbb{H}$	The quaternion algebra, central simple over $\mathbb{R}$
$e_0 = 1, e_1, e_2, e_3$	Basis, $e_k^2 = -e_0$
${}^{\natural}$	Quaternion conjugation, an anti-automorphism, not in $\operatorname{Aut}$
$\mathbb{R}_{\mathbb{H}} = \mathbb{R}e_0$	Centre; kernel of every derivation
$Sp(1) = \{\tilde q : N(\tilde q) = 1\}$	Unit quaternions
$\iota_u(\tilde q) = u\tilde qu^{-1}$	Inner automorphism determined by the unit $u$
$\operatorname{Aut}_{\mathbb{R}}(\mathbb{H})\cong Sp(1)/\{\pm1\}\cong SO(3)$	Automorphism group, $\cong\mathbb{RP}^3$
$\operatorname{ad}_a(\tilde q) = [a,\tilde q]$	Inner derivation determined by $a$
$\operatorname{Der}_{\mathbb{R}}(\mathbb{H})\cong\operatorname{Im}\mathbb{H}$	Derivation algebra, dimension $3$
$D_k = \tfrac12\operatorname{ad}_{e_k}$	Basis with $[D_1,D_2] = D_3$ and cyclic
$[\mathbf q,\mathbf p] = 2(\mathbf q\times\mathbf p)$	Bracket on $\operatorname{Im}\mathbb{H}$
$\mathrm{SO}(3)\cong\mathrm{SU}(2)$	Orthogonal and special unitary Lie algebras
$PGL(2,\mathbb{C}) = PSL(2,\mathbb{C})$	Biquaternion automorphism group over $\mathbb{C}$
$\mathrm{SL}(2,\mathbb{C})$, $\mathrm{SO}(1,3)$	Biquaternion derivation algebras