Dual-Numbers Exponential and Lie Group Structure

Introduction

This article develops the Lie theory of the dual-number algebra: the Lie algebra structure on $\mathbb{D}'$ itself, the exponential map, the group of units as a Lie group, the parabolic one-parameter subgroup, and the quotient of the unit group by the scalars. It follows Dual-Numbers Algebra for the algebra and its abelian Lie bracket, Dual-Numbers Norm and Invertibility for the unit group, Dual-Numbers Automorphisms and Derivations for the derivation algebra, and Shears and Parabolic Rotations for the shear. The structural model is Biquaternion Lie Group and Exponential Structure, where the unit group is $GL_2(\mathbb{C})$; here it is the much smaller group $\mathbb{R}^\times \times \mathbb{R}$.

The treatment is mathematically honest: every claim is either proved or stated as a definition. No physics is invoked. Throughout, $R$ is a commutative ring with identity in which $2$ is invertible; the exponential is a formal series when $R$ is a commutative $\mathbb{Q}$-algebra and an analytic series over the geometric specialisation $R = \mathbb{R}$, and then the algebra is written $\mathbb{D}'$. A general dual number is

$$ A = a + \varepsilon a', \qquad a, a' \in R, $$

with $a = \operatorname{Re} A$, $a' = \operatorname{Inf} A$, norm $N(A) = a^2$, maximal ideal $\mathrm{M} = (\varepsilon) = \varepsilon R_{\mathbb{D}'}$, and unit group $(\mathbb{D}')^\times = \{a \neq 0\}$.

The Lie Algebra Structure

The Lie Bracket

Definition. The Lie bracket on $\mathbb{D}'_R$ is the commutator

$$ [A, B] = AB - BA. $$

Theorem. $\mathbb{D}'_R$ is a commutative algebra, so its Lie bracket vanishes identically:

$$ [A, B] = 0 \qquad \text{for all } A, B \in \mathbb{D}'_R. $$

Hence the Lie algebra $\mathrm{G} = \mathbb{D}'_R$ is abelian.

Proof. $AB = BA$ for all $A, B$ because multiplication of dual numbers is commutative.

Nilpotence

Definition. A Lie algebra $\mathrm{G}$ is nilpotent of class $c$ if the descending central series $\mathrm{G}^1 = \mathrm{G}$, $\mathrm{G}^{k+1} = [\mathrm{G}, \mathrm{G}^k]$ reaches $0$ at step $c + 1$, and $\mathrm{G}$ is abelian when $[\mathrm{G}, \mathrm{G}] = 0$.

Proposition. $\mathrm{G} = \mathbb{D}'_R$ is a nilpotent Lie algebra of class $1$; equivalently, its derived subalgebra is zero, $[\mathrm{G}, \mathrm{G}] = 0$.

Proof. The derived subalgebra is spanned by brackets $[A,B]$, all of which vanish; the series is $\mathrm{G} \supset 0$.

So the dual-number Lie algebra is the abelian two-dimensional real Lie algebra, the simplest nilpotent example. Its enveloping algebra is the polynomial algebra on two commuting generators, and its only Lie structure is the underlying vector space.

The Lie Algebra of a Commutative Associative Algebra

Remark. For any associative algebra $A$ the commutator makes $A$ a Lie algebra, and the Jacobi identity is a consequence of associativity. When $A$ is commutative the bracket vanishes, so the Lie algebra records no more than the $R$-module structure. The interesting Lie theory of $\mathbb{D}'$ therefore lives not in $\mathbb{D}'$ itself but in its derivation algebra $\operatorname{Der}(\mathbb{D}') = \mathbb{R}\partial_\varepsilon$ and in the Lie group structure of the unit group, which are treated below.

The Exponential

Definition and Convergence

Definition. Let $R$ be a commutative $\mathbb{Q}$-algebra, so that $k!$ is invertible for every $k$. The exponential of $A \in \mathbb{D}'_R$ is the formal series

$$ \exp(A) = \sum_{k \geq 0} \frac{A^k}{k!}. $$

Over $R = \mathbb{R}$ the series converges for every $A$, in the topology of $\mathbb{D}' \cong \mathbb{R}^2$.

Closed Form

Theorem. For every $A = a + \varepsilon a'$,

$$ \exp(a + \varepsilon a') = e^{a}\bigl(1 + \varepsilon a'\bigr) = e^{a} + e^{a}\varepsilon a'. $$

Proof. By the binomial expansion and $\varepsilon^2 = 0$, $(a + \varepsilon a')^k = a^k + k a^{k-1}\varepsilon a'$ for $k \geq 1$. Summing,

$$ \exp(a + \varepsilon a') = \sum_{k \geq 0}\frac{a^k}{k!} + \varepsilon \sum_{k \geq 1}\frac{k a^{k-1}a'}{k!} = e^{a} + \varepsilon\, a' \sum_{k \geq 1}\frac{a^{k-1}}{(k-1)!} = e^{a} + e^{a}\varepsilon a', $$

the two series being the exponential of $a$.

Corollary. $\exp(\varepsilon) = 1 + \varepsilon$ and $\exp(s\varepsilon) = 1 + s\varepsilon$ for every $s$; the exponential of a purely infinitesimal element is a shear.

Worked Values

Example. $\exp(2 + 3\varepsilon) = e^{2}(1 + 3\varepsilon)$; numerically $e^{2} = 7.389056\ldots$, so the real part is $7.389056\ldots$ and the infinitesimal part is $3e^{2} = 22.167168\ldots$.

Example. The dual logarithm inverts the exponential on the shear group $1 + \mathrm{M}$, where it is $\log(1 + s\varepsilon) = s\varepsilon$; thus $\log(1 + 5\varepsilon) = 5\varepsilon$ and $\log(1 - \varepsilon) = -\varepsilon$.

The Group Homomorphism Property

Theorem. The exponential is a homomorphism from the additive group $(\mathbb{D}'_R, +)$ to the multiplicative group $(\mathbb{D}'_R)^\times$:

$$ \exp(A + B) = \exp(A)\exp(B). $$

Proof. The exponential is the usual exponential of the commutative associative algebra $\mathbb{D}'_R$, and for commuting $A, B$ the identity $\exp(A+B) = \exp(A)\exp(B)$ follows by the binomial theorem. Alternatively, $\exp(a + \varepsilon a')\exp(b + \varepsilon b') = e^a(1 + \varepsilon a')e^b(1 + \varepsilon b') = e^{a+b}(1 + (a'+b')\varepsilon) = \exp((a+b) + \varepsilon(a'+b'))$, matching $A + B$.

So $\exp$ is a homomorphism of groups, and the image is a subgroup of the unit group.

The Group of Units

The Lie Group

Theorem. The group of units

$$ (\mathbb{D}'_R)^\times = \{a + \varepsilon a' : a \in R^\times\} $$

is a group under multiplication, with $(a + \varepsilon a')^{-1} = a^{-1} - a^{-2}\varepsilon a'$. Over $R = \mathbb{R}$ it is a two-dimensional abelian Lie group with the following structure:

$$ (\mathbb{D}')^\times \cong \mathbb{R}^\times \times (\mathbb{R}, +), $$

the isomorphism being $a + \varepsilon a' \leftrightarrow (a,\, a'/a)$.

Proof. The group axioms and the inverse formula are from Dual-Numbers Norm and Invertibility; the isomorphism is the direct-product decomposition of Shears and Parabolic Rotations, where the second factor is the shear group $1 + \mathrm{M}$.

Connected Components

Proposition. Over $R = \mathbb{R}$ the unit group has exactly two connected components, $\{a > 0\}$ and $\{a < 0\}$, each contractible; the identity component is

$$ (\mathbb{D}')^\times_0 = \{a > 0\} \cong \mathbb{R}_{>0} \times \mathbb{R}. $$

Proof. The sign of $a$ is a continuous surjective homomorphism $(\mathbb{D}')^\times \to \{\pm 1\}$; the two fibres are half-planes, hence connected and contractible.

Surjectivity of the Exponential onto the Identity Component

Theorem. Over $R = \mathbb{R}$ the exponential

$$ \exp : \mathrm{G} = \mathbb{D}' \longrightarrow (\mathbb{D}')^\times $$

is injective, with image exactly the identity component $(\mathbb{D}')^\times_0 = \{a > 0\}$. It is therefore a bijection, and in fact a diffeomorphism, onto the identity component.

Proof. $\exp(a + \varepsilon a') = e^{a}(1 + \varepsilon a')$ has real part $e^{a} > 0$, so the image is contained in $\{a > 0\}$. Conversely, for $b + \varepsilon b'$ with $b > 0$ set $a = \log b$ and $a' = b'/b$; then $\exp(a + \varepsilon a') = b(1 + \varepsilon b'/b) = b + \varepsilon b'$, so the image is $\{a > 0\}$. For injectivity, $\exp(a + \varepsilon a') = 1$ forces $e^{a} = 1$, hence $a = 0$, and then $e^{a}a' = a' = 0$. The map is smooth with the explicit smooth inverse $b + \varepsilon b' \mapsto \log b + (b'/b)\varepsilon$ on $\{b > 0\}$, so it is a diffeomorphism.

Remark. The kernel of the exponential is trivial. This is the sharpest contrast with the complex and matrix cases, where the exponential has a nontrivial kernel: on $\mathbb{C}$ the kernel is $2\pi i\mathbb{Z}$, and on $GL_2(\mathbb{C})$ the exponential is surjective but not injective, with kernel the matrices having eigenvalues in $2\pi i\mathbb{Z}$. Here the exponential is a global diffeomorphism from the abelian Lie algebra onto the identity component, the maximal simply connected abelian case.

The Parabolic One-Parameter Subgroup

Definition

Definition. The parabolic one-parameter subgroup of $(\mathbb{D}')^\times$ is

$$ G = \{1 + s\varepsilon : s \in \mathbb{R}\} = 1 + \mathrm{M}. $$

Identification with the Additive Line

Theorem. The maps

$$ (\mathbb{R}, +) \longrightarrow G, \qquad s \longmapsto 1 + s\varepsilon, \qquad G \longrightarrow \mathrm{M}, \qquad 1 + s\varepsilon \longmapsto s\varepsilon $$

are inverse isomorphisms of groups, where $\mathrm{M}$ carries the additive group structure. In particular $G \cong (\mathbb{R},+)$ is a connected, simply connected, one-dimensional abelian Lie group, and it is exactly the image of the exponential of the maximal ideal, $G = \exp(\mathrm{M})$.

Proof. $(1 + s\varepsilon)(1 + t\varepsilon) = 1 + (s+t)\varepsilon$ because $\varepsilon^2 = 0$, so the first map is a homomorphism; it is bijective with the displayed inverse; and $\exp(s\varepsilon) = 1 + s\varepsilon$, so $G = \exp(\mathrm{M})$.

The Unipotent Character

Proposition. Every element of $G$ is unipotent: it differs from the identity by a nilpotent, $(1 + s\varepsilon) - 1 = s\varepsilon$ with $(s\varepsilon)^2 = 0$. The subgroup $G$ is the unique one-parameter subgroup of $(\mathbb{D}')^\times$ generated by a nilpotent element of the algebra, and it is the image of the algebra's own nilpotent direction.

Proof. $(1 + s\varepsilon) - 1 = s\varepsilon$ and $(s\varepsilon)^2 = s^2\varepsilon^2 = 0$, so the displacement from the identity is nilpotent of index two, which is the algebraic form of unipotence; in particular $1 + s\varepsilon$ has no eigenvalue different from $1$ as a linear map of the algebra. The generator is the nilpotent $\varepsilon$, whose exponential is polynomial.

The Group of Units Modulo Scaling

The Quotient

Theorem. The central scalars embed as $\mathbb{R}^\times \hookrightarrow (\mathbb{D}')^\times$, $a \mapsto a$, and

$$ (\mathbb{D}')^\times / \mathbb{R}^\times \;\cong\; 1 + \mathrm{M} \;\cong\; (\mathbb{R}, +), $$

the isomorphism being the class of $a + \varepsilon a'$ mapped to $1 + (a'/a)\varepsilon$, equivalently to the parabolic angle $s = a'/a$.

Proof. Every unit is $a(1 + \varepsilon a'/a)$, and multiplying by the scalar $a$ is exactly the quotient by the subgroup $\mathbb{R}^\times$; two units have the same class exactly when their shear parameters agree.

Corollary. The quotient is connected, contractible and one-dimensional; it is the space of shears, and it is the same object as the parabolic one-parameter subgroup $G$.

Comparison with the Split Complex and Biquaternion Cases

The Split Complex Case

For the split complex algebra $\mathbb{D} = \mathbb{R}[j]$, $j^2 = +1$, the unit group is $\mathbb{D}^\times \cong \mathbb{R}^\times \times \mathbb{R}^\times$, of real dimension two and with four contractible components; the exponential from the abelian Lie algebra $\mathbb{R}^2$ is a bijection onto the identity component, exactly as in the dual case. The quotient is

$$ \mathbb{D}^\times/\mathbb{R}^\times \cong \mathbb{R}^\times, $$

obtained by the ratio of the two idempotent components, and it has two components—unlike the connected quotient $(\mathbb{D}')^\times/\mathbb{R}^\times \cong \mathbb{R}$. The difference is the degeneracy of the norm: the split complex form has two independent directions, so the quotient retains a hyperbola-like two-component structure, while the dual form has one, so the quotient collapses to a line.

The Biquaternion Case

For the biquaternion algebra $\mathbb{B} \cong M_2(\mathbb{C})$ the unit group is $GL_2(\mathbb{C})$, of real dimension eight, connected. The exponential of $\mathrm{GL}(2,\mathbb{C})$ is surjective onto $GL_2(\mathbb{C})$ but not injective, with kernel the matrices whose eigenvalue differences lie in $2\pi i\mathbb{Z}$; the exponential is not a diffeomorphism, and there is no global logarithm. The quotient by the scalars is

$$ GL_2(\mathbb{C})/\mathbb{C}^\times = PGL(2,\mathbb{C}) = PSL(2,\mathbb{C}), $$

the automorphism group of Biquaternion Automorphisms and Derivations, connected of real dimension six.

The Table

	$\mathbb{D}'$	$\mathbb{D}$	$\mathbb{B}$
Lie algebra	$\mathbb{R}^2$, abelian	$\mathbb{R}^2$, abelian	$\mathrm{GL}(2,\mathbb{C})$
Unit group	$\mathbb{R}^\times\times\mathbb{R}$, two components	$\mathbb{R}^\times\times\mathbb{R}^\times$, four components	$GL_2(\mathbb{C})$, connected
Exponential	bijection onto identity component	bijection onto identity component	surjective, not injective
Kernel of $\exp$	$0$	$0$	eigenvalue gaps in $2\pi i\mathbb{Z}$
Unit group $/$ scalars	$\mathbb{R}$, connected	$\mathbb{R}^\times$, two components	$PGL(2,\mathbb{C})$, connected

The dual row is the two-dimensional row with the collapse of the quotient: the parabolic case replaces the hyperbolic two-component quotient of the split complex numbers by a single contractible line, while sharing with it the property that the exponential is a global diffeomorphism.

Summary

The dual-number algebra $\mathbb{D}'$ is an abelian two-dimensional real Lie algebra, nilpotent of class one, with vanishing bracket. The exponential has the closed form $\exp(a + \varepsilon a') = e^{a}(1 + \varepsilon a')$ and is a group homomorphism from $(\mathbb{D}',+)$ to the unit group. The unit group is $(\mathbb{D}')^\times \cong \mathbb{R}^\times \times \mathbb{R}$: a two-dimensional abelian Lie group with two contractible components, and the exponential is a diffeomorphism from the Lie algebra onto the identity component $\{a > 0\}$, with trivial kernel. The parabolic one-parameter subgroup $G = 1 + \mathrm{M} = \{1 + s\varepsilon\}$ is isomorphic to the additive line $(\mathbb{R},+)$, is the image of the exponential of the maximal ideal, and consists of unipotent elements; the quotient of the unit group by the central scalars is $(\mathbb{D}')^\times/\mathbb{R}^\times \cong 1 + \mathrm{M} \cong \mathbb{R}$. The comparison with the split complex case shows the same exponential behaviour but a two-component quotient $\mathbb{R}^\times$; the comparison with the biquaternion case shows a connected unit group $GL_2(\mathbb{C})$ on which the exponential is surjective but not injective.

Summary of Notation

Symbol	Meaning
$\mathbb{D}'_R = R[\varepsilon]/(\varepsilon^2)$	Dual-number algebra over $R$; $\varepsilon^2 = 0$
$\mathbb{D}' = \mathbb{D}'_{\mathbb{R}}$	Dual-number algebra over $\mathbb{R}$
$\mathbb{D}$	Split-complex algebra, unit $j$, $j^2 = +1$
$A = a + \varepsilon a'$	General dual number
$a = \operatorname{Re} A$, $a' = \operatorname{Inf} A$	Real and infinitesimal parts
$[A,B] = AB - BA$	Lie bracket, identically zero
$\mathrm{G} = \mathbb{D}'$	Abelian Lie algebra of the dual numbers
$\mathrm{M} = (\varepsilon)$	Maximal ideal, the nilpotent direction
$\exp(a + \varepsilon a') = e^{a}(1 + \varepsilon a')$	Exponential
$(\mathbb{D}')^\times = \{a \neq 0\}$	Group of units
$G = 1 + \mathrm{M}$	Parabolic one-parameter subgroup $\cong (\mathbb{R},+)$
$(\mathbb{D}')^\times/\mathbb{R}^\times \cong \mathbb{R}$	Unit group modulo scaling
$\partial_\varepsilon$	Derivation generator, $\operatorname{Der}(\mathbb{D}') = \mathbb{R}\partial_\varepsilon$
$\mathbb{B} \cong M_2(\mathbb{C})$	Biquaternion algebra, the comparison model