Split Quaternions: Theory & Applications

Updated 2 August 2025

Split quaternions are a four-dimensional real algebra with an indefinite (2,2) quadratic form, featuring unique zero divisors and non-trivial norm properties.
They underpin advanced function theory and representation theory by enabling spectral decompositions on non-compact Lie groups and facilitating minimal representations in mathematical physics.
Their structure supports innovative computational methods in polynomial factorization, geometric constructions, and signal processing, with applications in robotics and quantum information.

Split quaternions are a four-dimensional real associative algebra that generalizes the classical Hamilton quaternions by replacing the positive-definite norm with an indefinite quadratic form of signature (2,2). This algebraic structure plays a central role in several domains, including harmonic analysis on non-compact Lie groups, representation theory, mathematical physics (notably the Dirac equation and quantum field theory), computational algorithms for polynomials, and geometric constructions related to Clifford algebras and projective geometries.

1. Algebraic Structure and Fundamental Properties

Split quaternions, typically denoted as $\mathbb{H}_{\mathbb{R}}$ or $\mathbb{H}'$ , form a real vector space with basis $\{1, i, j, k\}$ , with multiplication rules distinguishing them from Hamilton’s quaternions:

$i^2 = 1$ , $j^2 = 1$ , $k^2 = -1$ , $ij = k = -ji$ .
The conjugation is defined by $q^* = q_0 - q_1i - q_2j - q_3k$ for $q = q_0 + q_1i + q_2j + q_3k$ .
The quadratic norm is $N(q) = q q^* = q_0^2 - q_1^2 - q_2^2 + q_3^2$ , yielding a signature $(2,2)$ . Unlike the positive-definite Hamiltonian norm, $N(q)$ may vanish for nonzero $q$ (causing the presence of zero divisors, idempotents, and nilpotents).

A canonical isomorphism identifies $\mathbb{H}_{\mathbb{R}}$ with $M_2(\mathbb{R})$ , the $2\times2$ real matrices, with unit norm elements corresponding to $\mathrm{SL}(2,\mathbb{R})$ . This algebra can also be realized as a commutative spin factor or as a subalgebra isomorphic to the Jordan algebra of $2\times2$ real symmetric matrices (1009.2540, Berthier et al., 21 Apr 2025).

2. Quaternionic and Split Quaternionic Function Theory

Split quaternionic analysis extends holomorphic function theory from complex and classical quaternionic variables to the split setting. The basic constructs parallel the Fueter theory and Clifford analysis, but leverage the signature $(2,2)$ :

Regular (Dirac-null) functions: For $f:\mathbb{H}_{\mathbb{R}} \to \mathbb{H}_{\mathbb{R}}$ or $\mathbb{H}_{\mathbb{C}}$ , left-regularity is defined as $\nabla_R^+ f(X) = 0$ for

$\nabla_R^+ = e_0\partial_{x^0} - \tilde{e}_1\partial_{x^1} - \tilde{e}_2\partial_{x^2} + e_3\partial_{x^3}$

where $e_0$ , $\tilde{e}_1$ , $\tilde{e}_2$ , $e_3$ are suitable basis elements (1009.2540).

Integral formulas: The split-quaternionic analogue of the Cauchy–Fueter formula expresses boundary values using the kernel $(Z - W)^{-1} N(Z - W)$ and a regularized Poisson kernel $N(X - W)^{-1} \pm i\varepsilon$ for harmonic projections (1009.2532, 1009.2540).
Cayley transform: An essential tool maps analysis on $\mathrm{SL}(2,\mathbb{R})$ (the “unit sphere” in $\mathbb{H}_{\mathbb{R}}$ ) to analysis on the one-sheeted hyperboloid in Minkowski space, i.e., the imaginary Lobachevski space $\mathrm{SL}(2,\mathbb{C})/\mathrm{SL}(2,\mathbb{R})$ (1009.2532).
Spectral separation: Utilizing these integral formulas and cycle decompositions in the complexified algebra, split quaternionic analysis separates $L^2$ -spaces into discrete and continuous series, with explicit projection operators arising from rational expansion in matrix-coefficient functions $t^{\,l}_{\,mn}(Z)$ (1009.2532, 1009.2540, Libine, 2014).

3. Representation Theory, Harmonic Analysis, and Minimal Representations

The interplay between split quaternions and non-compact Lie groups is particularly rich:

Unit norm identification: The set of split quaternions $X$ with $N(X) = 1$ is isomorphic to $\mathrm{SL}(2,\mathbb{R})$ , offering a group-theoretic realization central to harmonic analysis (1009.2532).
Series separation: Function spaces on $\mathbb{H}_{\mathbb{R}}$ or $\mathrm{SL}(2,\mathbb{R})$ decompose into discrete and continuous series of the group, with matrix coefficients indexed by $(l, m, n)$ and projections given by integral operators with regularized Poisson or Cauchy–Fueter kernels (1009.2532, 1009.2540, Libine, 2014).
Minimal and conformal representations: The continuous series forms a minimal representation of $\mathrm{SL}(4,\mathbb{R}) \cong \mathrm{SO}(3,3)$ , with $K$ -types

$\mathcal{V} \cong \bigoplus_{l=0}^{\infty} V_l \otimes V_l$

where $V_l$ denotes irreducible $\mathrm{SO}(3)$ modules (1009.2532).

Explicit Plancherel measure: Analysis of boundary values and spectral resolutions yields explicit quaternionic formulas for the Plancherel measure of $\mathrm{SL}(2,\mathbb{R})$ in terms of Poisson integrals with kernels $N(X - W)^{-1} \pm i\varepsilon$ (1009.2532).

4. Connections to Mathematical Physics and Geometry

Split quaternionic structures are encountered throughout mathematical physics and projective geometry:

Dirac spinors and the Dirac equation: The Dirac 4-spinor formalism can be equivalently reformulated as a system of 2-spinors with split-quaternionic components. Lorentz transformations correspond to $2\times 2$ unitary matrices over split quaternions, making the $SO(3,2;\mathbb{R})$ symmetry of the Lorentz-invariant scalar manifest (Antonuccio, 2014). The gamma-matrix algebra is realized within split-quaternionic matrix algebra, and the inner product $\Phi^* \cdot \Phi$ is automatically invariant.
Interpretation of the norm as Minkowski interval: For $s = X + x i + y j + ct(ij)$ ,

$N^2 = X^2 - x^2 - y^2 + c^2 t^2$

with $X$ interpreted as wavelength, $x,y$ as spatial coordinates, $t$ as time. Imposing $N^2 \geq 0$ recovers both relativistic interval invariance and quantum uncertainty, e.g., $\Delta X \Delta p_x \gtrsim \hbar$ (Gogberashvili, 2014).

Analyticity and the Dirac equation: The split quaternionic Cauchy–Riemann (analyticity) condition

$\frac{\partial \Phi}{\partial s^*} = 0$

encodes the $(2+1)$ -dimensional Dirac equation in this framework (Gogberashvili, 2014).

Quantum field theory correspondence: Kernels in split quaternionic integral formulas coincide with massless singular functions, such as $N(Z-W)^{-1} \pm i\varepsilon$ , which are projectors in 4D QFT (1009.2532).
Clifford algebra models: Replacing complex entries in $2\times2$ Pauli matrices by split quaternions provides economical models for large Clifford algebras, with the corresponding Weyl spinors represented by (complexified) split quaternion columns. For example, Certain spinor spaces in $\mathrm{Spin}(4,4)$ and Clifford algebras $\mathrm{Cl}(3,2)$ , $\mathrm{Cl}(2,2)$ , etc., are built in this manner (Bhoja et al., 2022).

5. Polynomial Theory, Zero Divisors, and Factorization

Polynomial equations in split quaternion algebras illustrate the effect of zero divisors and the associated geometric richness:

Quadratic and higher-degree equations: Finding roots requires refined formulas. When coefficients are real, solutions are given by

$x = -\frac{1}{2}b \pm \frac{1}{2}\sqrt{b^2 - 4c}$

but the square root must be interpreted with respect to the indefinite split-quaternion norm, and invertibility constraints for elements like $2x_0 a + b$ must be checked. When noninvertible, root formulae and solution existence depend on nonlinear real systems and additional algebraic or geometric compatibility conditions (Cao, 2019, Cao, 2022).

Zero divisor structure: The presence of nontrivial zero divisors in $\mathbb{H}_{\mathbb{R}}$ implies certain equations lack unique solutions, unlike the division algebra case of Hamilton's quaternions. Analysis often splits the solution space into branches ( $SI$ , $SZ$ ) according to invertibility properties (Cao, 2019).
Factorization of polynomials: Quadratic polynomials can have up to six factorizations into linear split quaternion factors, in contrast with only two for Hamiltonian quaternions. Factorization is tied to the real roots of the norm polynomial $P(t)\overline{P}(t)$ , with each pair corresponding to geometric structures (e.g., conic intersections, four-bar linkages in hyperbolic geometry) (Li et al., 2018, Scharler et al., 2019). Algorithmic approaches compute all possible factorizations, leveraging geometry of the null quadric and rulings in projective space (Scharler et al., 2020).
Moore–Penrose inverse and matrix equations: By extending the Moore–Penrose inverse to split quaternions, explicit solutions can be obtained for matrix and scalar equations of the form $a x b = d$ , $x a = b x$ , and versions with consimilarity. The theory exploits real matrix representations and scalar invariants for classification (Cao et al., 2019, Kosal et al., 2014).

6. Advanced Applications and Computational Aspects

Plancherel measure and explicit spectral decompositions: Regularized Poisson-type integrals provide direct formulas for the Plancherel measure on $\mathrm{SL}(2,\mathbb{R})$ , with spectral resolution arising naturally from expansions in rational matrix coefficients (1009.2532).
Krawtchouk matrices and quantum information: Krawtchouk matrices, central to quantum probability and coding, can be constructed from split quaternionic algebraic operations. The algebraic identity $F H = H G$ , with $F$ and $G$ split quaternionic units and $H$ the Hadamard matrix, underlies tensor constructions yielding the Krawtchouk matrices and their spectral properties (Kocik, 2016).
Orthogonal Planes Split (OPS) and quaternion Fourier transforms: The OPS divides a quaternion into two orthogonal $2$-planes and generalizes the quaternion Fourier transform structure, allowing more flexible signal representations and efficient FFT computation strategies (Hitzer, 2013, Hitzer et al., 2013).
Computational geometry and robotics: Factorization algorithms for split quaternion polynomials decompose rational motions in the hyperbolic plane into sequences of hyperbolic rotations, with extensions to dual quaternion polynomials and motion synthesis in robotics and kinematics (Scharler et al., 2020).
Perceptual models and color science: Split quaternionic subalgebras yield perceptual chromatic adaptation transforms for perceptual white balance. The algebraic sandwich formula $q' = p_e^{1/2} q_s p_e^{1/2}$ , with $p_e$ encoding the effect (illuminant) and $q_s$ the color state, provides a quantum-like model for color measurement and adaptation, shown experimentally to match or surpass performance of classical von Kries transforms (Berthier et al., 21 Apr 2025).

7. Geometry, Projective Varieties, and Magic Square Constructions

Split quaternions and their degenerations appear in geometric characterizations of subvarieties within the Freudenthal–Tits magic square and related projective geometries:

Veronese varieties and degenerate algebras: Veronese mappings constructed from quadratic alternative (sometimes degenerate) algebras, including the degenerate split quaternions $CD(L',0)$ , yield subvarieties of the standard $E_6$ -variety whose incidence structure is governed by natural point–quadric axioms (Schepper, 2020). These results unify non-degenerate cases (split quaternions, ternions, sextonions) and their geometric interpretation in the broader algebraic context.
Clifford algebra models: Models of Clifford algebras $\mathrm{Cl}(r,s)$ for various signatures are efficiently realized via $2\times2$ matrices with split quaternionic off-diagonal entries, enabling explicit construction of Weyl spinors and the investigation of their orbit spaces under spin groups (Bhoja et al., 2022). This approach reveals invariants and stabilizers of pure and mixed spinor classes, significant for both representation theory and applications in physics.

This synthesis highlights that split quaternions furnish an algebraic and geometric framework with deep consequences for function theory, representation theory, mathematical physics, projective geometry, computational algorithms, and perceptual modeling. Their indefinite quadratic form fundamentally distinguishes them from the classical quaternions, engendering unique spectral properties, polynomial factorization behavior, and connections to non-compact Lie groups and conformal field theories.