Non-Split Quaternion Algebra Overview

• Non-split quaternion algebras are four-dimensional division algebras with a quadratic norm where every nonzero element is invertible and they cannot be represented as full matrix algebras over their base field.
• Their geometric construction via ordered vector pairs recovers Hamilton’s relations and provides a framework for modeling rotations and spatial transformations.
• These algebras underpin applications in topology, arithmetic, and physics, linking SU(2) representations, Clifford algebras, and p-adic analysis in advanced mathematical research.

A non-split quaternion algebra is a four-dimensional associative division algebra over a field $k$, equipped with a quadratic norm such that every nonzero element is invertible and the algebra cannot be realized as a matrix algebra over $k$ or over any algebraic extension of $k$ (unless specified otherwise, as in certain differential or arithmetic contexts). The canonical example is Hamilton’s quaternions $\mathbb{H}$ over $\mathbb{R}$, with basis $\{1, i, j, k\}$ and multiplication rules $i^2 = j^2 = k^2 = -1$, $ij = k = -ji$, and so on. Non-split quaternion algebras arise as central simple algebras that “do not split,” i.e., they are not isomorphic to $M_2(k)$, and their structure is deeply entwined with the theory of quadratic forms, representations, Clifford algebras, and arithmetic geometry.

1. Algebraic and Geometric Construction

A geometric construction of the quaternion algebra in three dimensions, presented in (Palais, 2010), establishes a bijection between equivalence classes of ordered vector pairs $(v, w) \in \mathbb{R}^3 \times \mathbb{R}^3$ and quaternions, via the map

$$T(v, w) = [v \cdot w,\, v \times w] \in \mathbb{R} \times \mathbb{R}^3$$

where $v \cdot w$ is the dot product (scalar part) and $v \times w$ is the cross product (vector part). Two pairs are equivalent if their dot and cross products agree, and multiplication in the algebra is induced by “merging” the ordered pairs: $$[q,\, \vec{q}] \cdot [q',\, \vec{q'}] = [ qq' - \vec{q} \cdot \vec{q'},\,\; q \vec{q'} + q' \vec{q} + \vec{q'} \times \vec{q} ]$$ This formula recovers Hamilton’s relations and distinguishes quaternion multiplication from complex multiplication by the presence of terms involving the vector and cross product, reflecting the non-commutativity intrinsic to the quaternion algebra. The isomorphism to the quaternion algebra $\mathbb{H}$ is explicit via the identifications $1 = [1,0]$, $i = [0, e_1]$, $j = [0, e_2]$, $k = [0, e_3]$.

2. Structure, Matrix Representations, and Linear Algebra

Quaternions form a skew field: multiplication is non-commutative, but every nonzero element is invertible. Several equivalent presentations are available:

• As quadruples $q = a + b i + c j + d k$ over $\mathbb{R}$, with norm $|q|^2 = a^2 + b^2 + c^2 + d^2$.
• As 2×2 complex matrices, with

$$F(q) = \begin{pmatrix} z & \overline{w} \ -w & \overline{z} \end{pmatrix}$$

where $z = a + i b$, $w = -c + i d$, and where $\det F(q) = |q|^2$ (Krishnaswami et al., 2016). This representation exhibits the link between quaternions and the Pauli matrices, and identifies the group of unit quaternions ($|q|=1$) with $SU(2)$, topologically $S^3$, a double cover of $SO(3)$—the group of rotations in three dimensions.

In linear algebra over non-split quaternion algebras (Schulz et al., 2013), matrix operations must be defined to respect the non-commutativity:

• Matrix multiplication requires explicit “left” or “right” ordering:

$$[A \odot B]_{mn} = \sum_k [A]_{mk} \cdot [B]_{kn} \qquad [A \oplus B]_{mn} = \sum_k [B]_{kn} \cdot [A]_{mk}$$

• Transpose and conjugation interact with order:

$$(A \odot B)^T = B^T \oplus A^T, \qquad (A \odot B)^* = A^* \odot B^*$$

• Eigenvalue problems are bifurcated into left and right eigenvalues, complicated by the non-commutative multiplicative structure:

$$A \odot q_L = \lambda_L \cdot q_L, \qquad A \odot q_R = q_R \cdot \lambda_R$$

This structure is essential when considering the action of quaternions on spaces, notably for implementing rotations and spectral analysis.

3. Rotations, Topology, and Fiber Bundles

Non-split quaternion algebras encode rotations in three dimensions via the action $v' = q v q^{-1}$, where $q$ is a unit quaternion and $v$ is a pure imaginary quaternion (Krishnaswami et al., 2016). This gives rise to the Rodrigues formula: $$v' = \cos\theta\, v + \sin\theta\, (n \times v) + (1-\cos\theta)(n \cdot v) n$$ for rotation by angle $\theta$ about axis $n$. The identification of unit quaternions $|q|=1$ with $SU(2)$ and the mapping $SU(2) \to SO(3)$ provides a topological double-cover, explaining phenomena such as spin-½ objects requiring 720° rotation for return to initial state.

These topological properties extend to the Hopf fibrations:

• Complex numbers: $S^3 \to S^2$ with $S^1$ fibers,
• Quaternions: $S^7 \to S^4$ with $S^3$ fibers (Krishnaswami et al., 2016), relating the non-split quaternion structure with higher-dimensional geometric and topological phenomena. Applications appear in quantum information theory and gauge theory.

4. Extensions, Clifford Algebras, and Typification

Non-split quaternion algebra structure is also present, abstractly, in certain Clifford algebras (Shirokov, 2011). A “basis-free” quaternion type classification is made using two canonical involutions—the grade involution $\alpha$ and the reversion $t$: $$\overline{k}(V,Q) = \{ U \in \mathrm{Cl}(V,Q) \mid U^{\mathcal{c}} = (-1)^k U,\, U^{\sim} = (-1)^{k(k-1)/2} U \},\quad k=0,1,2,3$$ This partitions algebra elements into four principal types that reflect quaternionic symmetries. These algebraic properties generalize the multiplication and commutator structures of quaternions, establishing an internal “quaternion-type” grading useful for decomposing Clifford algebras and identifying hidden quaternionic (non-split) substructures within larger non-commutative algebras.

5. Differential and Arithmetic Aspects

In the context of splitting fields and differential algebra, non-split quaternion algebras display markedly different behavior from their split counterparts (Gupta et al., 2022). While every classical quaternion algebra splits over a quadratic extension, differential quaternion algebras (i.e., quaternion algebras equipped with a derivation) may require transcendental extensions for splitting. Key techniques involve solving Riccati differential equations

$X' = \alpha_0 + \alpha_1 X + \alpha_2 X^2$

whose solutions govern the existence and transcendence degree of splitting fields. Non-split differential quaternion algebras are distinguished by the absence of algebraic solutions to these Riccati equations, which necessitates transcendental, rather than algebraic, splitting.

6. Arithmetic Representations and p-adic Theory

Non-split quaternion algebras over local fields (notably $\mathbb{Q}_p$) play a pivotal role in $p$-adic representation theory and the local/global Jacquet–Langlands correspondences (Liu et al., 22 Oct 2025, Hu et al., 2023). For instance, Scholze’s functor $S^1$ attaches admissible unitary Banach space representations of $D^\times$ (the multiplicative group of a non-split quaternion algebra $D$ over $\mathbb{Q}_p$) to Galois representations, with the important property:

• Such representations, though infinite-dimensional as Banach spaces, are topologically of finite length; i.e., they possess only finitely many irreducible subquotients in their composition series.
• The functor respects intricate multiplicity and cohomological invariants, and is compatible with derived and patched module constructions. Multiplicity criteria (e.g., $\mu \leq 4$ for finite length) and filtration analyses using Iwasawa algebra techniques are central in characterizing the representations’ structure.

These results underpin significant progress in understanding the $p$-adic Langlands program for $GL_2$ and its inner forms, using non-split quaternion algebras as a testing ground for arithmetic and geometric phenomena.

7. Applications and Broader Significance

The algebraic, differential, and arithmetic properties of non-split quaternion algebras have direct applications in implementing and interpolating rotations (via quaternionic SLERP), encoding physical symmetries (such as spin and helicity in quantum mechanics (Sanctuary, 27 Mar 2025)), decomposing high-dimensional rotations (Arizmendi et al., 2019), and exploring deep connections between topology, fiber bundles, and representation theory. In quantum variance and automorphic forms, non-split quaternion algebras provide compact quotients where spectral and analytic techniques—such as theta correspondence and metaplectic Rankin–Selberg convolutions—yield fine-grained results (e.g., optimal error terms, explicit limiting variance matrices (Nelson, 2016)).

In summary, non-split quaternion algebras constitute a class of four-dimensional division algebras whose rich structural properties—rooted in their algebraic non-commutativity, geometric realization, topological behavior, typification within Clifford algebras, differential splitting theory, and arithmetic representation theory—underlie their central importance in advanced mathematics and theoretical physics. Their paper enables a unified framework for analyzing symmetry, rotation, and representation phenomena across diverse mathematical domains.