Quaternion Exponential and Lie Group Structure

Introduction

The quaternion algebra is at once an associative algebra, a Lie algebra under the commutator bracket, and the tangent space at the identity of its group of units; the unit sphere is a compact Lie group, and the exponential map connects the three descriptions. This article develops that structure: the Lie algebra of $\mathbb{H}$ and its decomposition into a centre and an orthogonal part, the exponential map and its closed form, the surjectivity of the exponential onto the group of units, the polar split of the units into a positive scale and the unit sphere, the compact subgroup $Sp(1)$ and its parametrisation, and the relation to the rotations and reflections of the geometry slot. It is the quaternion member of the family's exponential-and-Lie-group pair; its counterpart is the biquaternion case, where the unit group is $GL_2(\mathbb{C})$ and the exponential is complex.

The article uses Quaternion Algebra and Quaternion Norm and Invertibility for the algebra and the units, The Scalar and Vector Subspaces of $\mathbb{H}$ for the decomposition, Quaternion Rotations and Reflections for the orthogonal actions, and Quaternion Automorphisms and Derivations for the derivation algebra, which is the Lie algebra computed here independently. The special functions are in Quaternion Special Functions.

The corpus's default base is a commutative ring with identity, but the exponential series, the closed form and the Lie group statements require a complete ordered field, so everything below is stated over $\mathbb{R}$; the algebraic identities remain meaningful for complex coefficients and are used in the comparison.

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 conjugate $\tilde{q}^{\natural}$, norm $N(\tilde q) = \tilde q\tilde{q}^{\natural}$ and modulus $|\tilde q| = \sqrt{N(\tilde q)}$; the unit group is $\mathbb{H}^{\times} = \mathbb{H}\setminus\{0\}$ and the unit sphere is $Sp(1) = \{\tilde q : N(\tilde q) = 1\}\cong S^3$.

The Lie Algebra Structure

Definition. The commutator bracket on $\mathbb{H}$ is $[\tilde q,\tilde p] = \tilde q\tilde p-\tilde p\tilde q$; the algebra $\mathbb{H}$ with this bracket is the Lie algebra of the quaternion algebra.

Proposition. The bracket is bilinear, alternating and satisfies the Jacobi identity, so $\mathbb{H}$ is a Lie algebra; the centre of the algebra is the centre of the Lie algebra, and the derived subalgebra is the vector subspace,

$$ [\mathbb{H},\mathbb{H}] = \operatorname{Im}\mathbb{H}, \qquad Z(\mathbb{H}) = \mathbb{R}_{\mathbb{H}} . $$

Proof. The bracket of an associative algebra is a Lie bracket, the Jacobi identity being the associativity of multiplication. The commutator of any two quaternions is pure, since $\mathrm{Sc}([\tilde q,\tilde p]) = \mathrm{Sc}(\tilde q\tilde p)-\mathrm{Sc}(\tilde p\tilde q) = 0$, and the pure quaternions are all obtained as commutators, for instance $e_1 = \tfrac12[e_2,e_3]$; hence $[\mathbb{H},\mathbb{H}] = \operatorname{Im}\mathbb{H}$. The centre of the bracket is the set of elements commuting with every quaternion, namely $\mathbb{R}_{\mathbb{H}}$.

Theorem. As a Lie algebra the quaternion algebra splits as the direct sum of its centre and a simple ideal,

$$ \mathbb{H} = \mathbb{R}_{\mathbb{H}}\oplus\operatorname{Im}\mathbb{H}\cong\mathbb{R}\oplus\mathrm{SO}(3), $$

and $\operatorname{Im}\mathbb{H}$ is isomorphic to the cross-product Lie algebra $\mathrm{SO}(3)\cong\mathrm{SU}(2)$, with $[\mathbf{p},\mathbf{q}] = 2(\mathbf{p}\times\mathbf{q})$.

Proof. The centre is an abelian ideal, and it commutes with the vector subspace, so the sum is direct; $\operatorname{Im}\mathbb{H}$ is a subalgebra and an ideal because its bracket lands in itself and the centre commutes with it. The bracket on the vector subspace is $[\mathbf{p},\mathbf{q}] = \mathbf{p}\mathbf{q}-\mathbf{q}\mathbf{p} = 2(\mathbf{p}\times\mathbf{q})$, which is the cross-product Lie algebra, isomorphic to $\mathrm{SO}(3)$ and to $\mathrm{SU}(2)$.

Corollary. The Lie algebra $\mathbb{H}$ is reductive but not semisimple: its centre is one-dimensional, and its derived algebra $\operatorname{Im}\mathbb{H}\cong\mathrm{SO}(3)$ is simple. The Killing form is negative definite on the derived ideal and zero on the centre.

Proof. $\mathrm{SO}(3)$ is simple, being the Lie algebra of the simple compact group $SO(3)$; the centre contributes a one-dimensional abelian factor, so the sum is reductive and not semisimple. The Killing form of a compact simple Lie algebra is negative definite, and it vanishes on the centre, which is abelian.

The Exponential Map

Definition. The exponential of a quaternion is the sum of the series

$$ \exp(\tilde q) = \sum_{n=0}^{\infty}\frac{\tilde q^n}{n!}, $$

which converges for every $\tilde q$ because the quaternion norm is multiplicative and $\mathbb{H}$ is complete as a finite-dimensional real space.

Proposition. For every $\tilde q$ the exponential satisfies $N(\exp(\tilde q)) = e^{2q_0}$, so $\exp(\tilde q)$ is a unit with $|\exp(\tilde q)| = e^{q_0}$; the exponential maps the pure imaginary subspace into the unit sphere.

Proof. By continuity of $\exp$ relative to the rational exponential and the identity $N(\exp(\tilde q)) = \exp(2\mathrm{Sc}\tilde q)$, which holds for rational multiples and extends by continuity; for pure $\tilde q$, $q_0 = 0$ and the quaternion norm is one.

Closed Form

Theorem (closed form). For $\tilde q = q_0 e_0 + q_1 e_1 + q_2 e_2 + q_3 e_3$ with $\mathbf{q}\neq0$,

$$ \exp(\tilde q) = e^{q_0}\Bigl(\cos|\mathbf{q}| + \frac{\mathbf{q}}{|\mathbf{q}|}\sin|\mathbf{q}|\Bigr), $$

and for $\mathbf{q} = 0$ it reduces to the real exponential $\exp(q_0) = e^{q_0}$.

Proof. Write $\mathbf{q} = \mu\rho$ with $\mu$ a pure unit and $\rho = |\mathbf{q}|\geq0$, so $\mu^2 = -1$. Then $(q_0+\mu\rho)^n$ may be expanded, and separating the terms by the parity of $n$ and summing the real exponential gives the displayed expression, exactly as for the complex exponential.

Corollary. The exponential commutes with neither addition nor multiplication of quaternions; it satisfies $N(\exp \tilde q) = e^{2\mathrm{Sc}(\tilde q)}$ and maps the imaginary subspace onto the unit sphere, where it is the parametrisation $\exp(\mu\theta) = \cos\theta+\mu\sin\theta$ for a pure unit $\mu$.

Proof. The failure of additivity and multiplicativity is the non-commutativity of the algebra; the quaternion norm identity is the previous proposition; the pure case is the closed form with $q_0 = 0$.

Surjectivity

Theorem. The exponential is surjective onto the group of units,

$$ \exp(\mathbb{H}) = \mathbb{H}^{\times} = \mathbb{H}\setminus\{0\}, $$

and it is surjective onto the unit sphere when restricted to the imaginary subspace.

Proof. Every non-zero quaternion has the polar form $\tilde q = |\tilde q|u$ with $|\tilde q| > 0$ and $u$ a unit quaternion; writing $u = \cos\theta+\mu\sin\theta$ with $\mu$ a pure unit and $\theta = \arccos(u_0)\in[0,\pi]$ gives $u = \exp(\mu\theta)$ by the closed form, and $|\tilde q| = e^{\ln|\tilde q|}$. Hence $\tilde q = \exp(\ln|\tilde q|+\mu\theta)$. The unit-sphere statement is the case $\ln|\tilde q| = 0$.

Proposition. The exponential is not injective; its kernel is the union of the spheres of pure quaternions of radius $2\pi k$ together with the origin,

$$ \exp^{-1}(1) = \{0\}\cup\bigcup_{k\geq1}\bigl\{\mathbf{q}\in\operatorname{Im}\mathbb{H} : |\mathbf{q}| = 2\pi k\bigr\}, $$

and its restriction to the open set $\{\tilde q : |\mathbf{q}| < \pi\}$ is a diffeomorphism onto $\mathbb{H}^{\times}\setminus(-\mathbb{R}_{>0})$.

Proof. $\exp(\tilde q) = 1$ forces $e^{q_0} = 1$ and $\sin|\mathbf{q}| = 0$ with $\cos|\mathbf{q}| = 1$, so $q_0 = 0$ and $|\mathbf{q}| = 2\pi k$ for an integer $k\geq0$, the case $k = 0$ giving the origin; conversely any such pure quaternion gives $\exp = 1$ by the closed form. The restriction statement is the classical property of the complex exponential applied to each axis direction $\mu$.

The Group of Units

Theorem. The group of units is the direct product of the positive scalars and the unit sphere,

$$ \mathbb{H}^{\times}\cong\mathbb{R}_{>0}\times Sp(1), \qquad \tilde q = |\tilde q|\,u, \quad u = \frac{\tilde q}{|\tilde q|}, $$

and the two factors commute and are the image under the exponential of the centre and of the imaginary subspace respectively.

Proof. The polar split is Quaternion Norm and Invertibility; the scale factor is $\exp(q_0)$ with $q_0$ real and the unit factor is $\exp(\mathbf{q})$ up to the periodicity of the pure component, so the two exponential images are the two factors.

Theorem. The polar form and the exponential form are the same decomposition: every unit has the parametrisation

$$ u = \exp(\mu\theta) = \cos\theta+\mu\sin\theta, \qquad \mu\in\operatorname{Im}\mathbb{H},\quad N(\mu) = 1,\quad \theta\in[0,\pi], $$

and the map $(\mu,\theta)\mapsto\exp(\mu\theta)$ is a diffeomorphism of the open set $\theta\in(0,\pi)$ onto the unit sphere with the two poles removed.

Proof. The parametrisation is the closed form on the pure subspace, and the failure of injectivity occurs exactly at the poles $u = \pm1$, where the axis $\mu$ is undetermined; away from them the axis and the angle are unique.

The Compact Subgroup

Theorem. The unit sphere $Sp(1)$ is a compact connected Lie group of dimension three, diffeomorphic to $S^3$ and isomorphic to $SU(2)$; it is the maximal compact subgroup of $\mathbb{H}^{\times}$, and the group of units is the product of it with the non-compact factor $\mathbb{R}_{>0}$.

Proof. The unit sphere is the set $N(\tilde q) = 1$, a closed bounded subset of $\mathbb{R}^4$, hence compact; multiplication is smooth and the inverse is smooth, so it is a Lie group of dimension three, and its identification with $SU(2)$ is standard. It is the maximal compact subgroup of the product $\mathbb{R}_{>0}\times Sp(1)$ because the line has no non-trivial compact subgroup.

Corollary. The exponential image of the imaginary subspace is the whole of $Sp(1)$, so $Sp(1)$ is generated by its one-parameter subgroups $\exp(\mathbb{R}\mu)$, one for each pure unit $\mu$; the one-parameter subgroups are the great circles through the identity.

Proof. Surjectivity onto the unit sphere is the theorem above; a one-parameter subgroup is the image of $\mathbb{R}\mu$ under the exponential, which is the circle $t\mapsto\cos t+\mu\sin t$; the geodesic reading of these circles is in Quaternion Rotations and Reflections.

Remark (versors, hyperbolic versors, and one-parameter groups). The unit quaternions are Hamilton's versors, and the subgroup $\exp(\mathbb{R}\mu)$ generated by a unit pure $\mu$ is the family of versors of the fixed axis $\mu$; the naming and the arc picture are in Quaternion Rotations and Reflections. A hyperbolic versor is the same expression $\exp(tr) = \cosh t + r\sinh t$ with $r^2 = +1$ in place of $r^2 = -1$. It cannot be formed in $\mathbb{H}$, whose only elements of square $+1$ are the scalars $\pm1$, and it requires an algebra of split signature, the split-complex numbers or the split-quaternions of Split-Complex Exponential and Lie Group Structure and Split-Quaternion Exponential and Lie Group Structure. Lie's one-parameter group is the notion that subsumes both: for each $r$ with $r^2 = \pm1$ the map $t\mapsto\exp(tr)$ is a one-parameter subgroup, elliptic for $r^2 = -1$ and hyperbolic for $r^2 = +1$. In the definite case the generator directions $r$ and $-r$ trace the same circle in opposite directions, which is the reversal of direction of the corresponding rotation, while in the indefinite case they give the two branches $e^{tr}$ and $e^{-tr}$ of a hyperbola, and the exponential is injective along each line.

The Exponential Parametrisation

Theorem. Every non-zero quaternion can be written uniquely as

$$ \tilde q = e^{\alpha}\exp(\mu\theta), \qquad \alpha\in\mathbb{R}, \quad \mu\in\operatorname{Im}\mathbb{H},\ N(\mu) = 1, \quad \theta\in[0,\pi], $$

with the further identification $(\mu,\pi)\sim(-\mu,\pi)$ at the negative real axis.

Proof. Take $\alpha = \ln|\tilde q|$ and $\mu\theta$ the axis-angle form of the unit factor; the closed form gives the product, and the uniqueness away from the poles is that of the polar form and of the axis-angle pair.

Proposition. In this parametrisation the quaternion norm is $N(\tilde q) = e^{2\alpha}$, the modulus is $|\tilde q| = e^{\alpha}$, and the argument is $\theta$ along the direction $\mu$; the logarithm

$$ \log(\tilde q) = \ln|\tilde q| + \mu\theta $$

is a right inverse of the exponential, defined on the complement of the negative real axis, and is multivalued on the whole of $\mathbb{H}^{\times}$.

Proof. The quaternion norm identity is the closed form; the logarithm inverts $\exp$ on the stated set by the surjectivity proof, and the ambiguity is the period $2\pi$ in the pure direction.

The Adjoint Action of the Exponential

Theorem. The adjoint action of the exponential is the conjugation by its value,

$$ \exp(t\,\operatorname{ad}_{\mu})(\tilde q) = e^{t\mu}\,\tilde q\,e^{-t\mu} = \operatorname{Ad}_{e^{t\mu}}(\tilde q), $$

for every $\tilde q\in\mathbb{H}$ and every $\mu\in\operatorname{Im}\mathbb{H}$; equivalently the adjoint representation of the Lie algebra and the adjoint action of the group of units agree under the exponential.

Proof. The identity $\exp(\operatorname{ad}_a) = \operatorname{Ad}_{\exp a}$ is the standard Lie-theoretic identity for an inner derivation, here computed from the series.

The identification of the adjoint action with the rotations of $\operatorname{Im}\mathbb{H}$, the half-angle rule, and the completion of the rotation group to the full orthogonal group by the orientation-reversing maps are the subject of Quaternion Rotations and Reflections; this article supplies the exponential and the adjoint identity that the geometry slot uses.

Comparison with the Biquaternion Case

The biquaternion algebra $\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$ has the same underlying multiplication but complex coefficients, so its Lie algebra is the complexification $\mathbb{H}\otimes_{\mathbb{R}}\mathbb{C}$, and its group of units is $GL_2(\mathbb{C})$.

Feature	$\mathbb{H}$	$\mathbb{B} = \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}$
Lie algebra	$\mathbb{R}\oplus\mathrm{SO}(3)$, real dimension $4$	$\mathrm{GL}_2(\mathbb{C})$, complex dimension $4$
Exponential	surjective onto $\mathbb{H}^{\times}$	surjective onto $GL_2(\mathbb{C})$
Closed form	$e^{q_0}(\cos\lvert\mathbf{q}\rvert+\mu\sin\lvert\mathbf{q}\rvert)$	the same with complex $q_0,\mathbf{q}$
Group of units	$\mathbb{R}_{>0}\times S^3$, compact factor $S^3$	$GL_2(\mathbb{C})$, non-compact, connected
Maximal compact subgroup	$Sp(1)\cong SU(2)$	$U(2)$
Rotation group from the adjoint action	$SO(3)$	$PSL(2,\mathbb{C})$, the projective Lorentz group

The quaternion exponential is real and its unit group is the product of a line and a compact three-sphere; the biquaternion exponential is complex, its unit group is the non-compact $GL_2(\mathbb{C})$, and the adjoint action generates the projective Lorentz group rather than the rotation group. The reason is the same throughout: the quaternion norm is definite, so the unit sphere is compact and the one-parameter groups are circles, while the complex norm is indefinite, so the unit group is non-compact and the one-parameter groups include hyperbolic rotations. The biquaternion account is in Biquaternion Lie Group and Exponential Structure.

Summary

The quaternion algebra is a Lie algebra under the commutator bracket, with derived subalgebra the imaginary subspace and centre the real line, splitting as $\mathbb{R}_{\mathbb{H}}\oplus\operatorname{Im}\mathbb{H}\cong\mathbb{R}\oplus\mathrm{SO}(3)$; it is reductive and not semisimple. The exponential has the closed form $\exp(q_0+\mathbf{q}) = e^{q_0}(\cos|\mathbf{q}|+\frac{\mathbf{q}}{|\mathbf{q}|}\sin|\mathbf{q}|)$, it satisfies $N(\exp \tilde q) = e^{2q_0}$, and it is surjective onto the group of units but not injective, with kernel the spheres of pure quaternions of radius $2\pi k$.

The group of units splits as $\mathbb{H}^{\times}\cong\mathbb{R}_{>0}\times Sp(1)$, and the unit sphere is the compact connected three-dimensional Lie group $Sp(1)\cong S^3\cong SU(2)$, the maximal compact subgroup; the polar and exponential forms of an element are the same decomposition, $\tilde q = e^{\alpha}\exp(\mu\theta)$ with $\mu$ a pure unit and $\theta\in[0,\pi]$, and the logarithm $\log \tilde q = \ln|\tilde q|+\mu\theta$ inverts the exponential off the negative real axis. The adjoint action of the exponential is the conjugation by its value, $\exp(t\operatorname{ad}_\mu) = \operatorname{Ad}_{e^{t\mu}}$, and the reading of this adjoint action as the rotations of the imaginary subspace is in Quaternion Rotations and Reflections.

The biquaternion case replaces the definite norm by the indefinite complex one: the unit group becomes $GL_2(\mathbb{C})$, the maximal compact subgroup becomes $U(2)$, and the adjoint action generates the projective Lorentz group. The compactness of $Sp(1)$ and the circularity of its one-parameter subgroups are exactly what the definite quaternion norm buys.

Summary of Notation

Symbol	Meaning
$\mathbb{H}$	The quaternion algebra, as algebra and as Lie algebra
$[\tilde q,\tilde p] = \tilde q\tilde p-\tilde p\tilde q$	Commutator bracket
$\mathbb{R}_{\mathbb{H}}, \operatorname{Im}\mathbb{H}$	Centre and derived subalgebra
$\mathbb{H}\cong\mathbb{R}\oplus\mathrm{SO}(3)$	Lie algebra decomposition
$\exp(\tilde q) = \sum_n \tilde q^n/n!$	Exponential map
$\exp(q_0+\mathbf{q}) = e^{q_0}(\cos\lvert\mathbf{q}\rvert+\frac{\mathbf{q}}{\lvert\mathbf{q}\rvert}\sin\lvert\mathbf{q}\rvert)$	Closed form
$\exp(\mu\theta) = \cos\theta+\mu\sin\theta$	Unit-sphere parametrisation, $\mu$ a pure unit
$\exp^{-1}(1) = \{2\pi k\mu\}$	Kernel of the exponential
$\mathbb{H}^{\times}\cong\mathbb{R}_{>0}\times Sp(1)$	Polar split of the units
$Sp(1)\cong S^3\cong SU(2)$	Compact subgroup, maximal compact
$\log \tilde q = \ln\lvert \tilde q\rvert+\mu\theta$	Logarithm, inverse off the negative axis
$\operatorname{Ad}_u(\tilde q) = u\tilde q u^{-1}$	Adjoint action; $\exp(\operatorname{ad}_a) = \operatorname{Ad}_{\exp a}$
$O(3)$	Orthogonal group of $\operatorname{Im}\mathbb{H}$, from Quaternion Rotations and Reflections, where $O(3) = \{\pm\operatorname{Ad}_u : u\in Sp(1)\}$
$GL_2(\mathbb{C}), U(2), PSL(2,\mathbb{C})$	Biquaternion unit group, maximal compact, adjoint group