Hurwitz quaternion

In mathematics, a Hurwitz quaternion is a quaternion a + bi + cj + dk where the coefficients a, b, c, d are either all integers or all half-integers (i.e., integers plus \frac{1}{2}).^[1] The set of all such quaternions, often denoted \mathbb{H}, forms a subring of the quaternion algebra over the rational numbers \mathbb{Q}.^[1] Hurwitz quaternions were introduced by the mathematician Adolf Hurwitz in his 1919 textbook Vorlesungen über die Zahlentheorie der Quaternions, which developed the number-theoretic aspects of quaternions originally discovered by William Rowan Hamilton in 1843 and further explored by Rudolf Lipschitz in the 1880s.^[2] Building on Hurwitz's earlier 1896 research, this work established \mathbb{H} as a foundational structure in non-commutative arithmetic, analogous to the Gaussian integers in the complex domain.^[2] As a ring, \mathbb{H} is a maximal order in the quaternion algebra \left(\frac{-1,-1}{\mathbb{Q}}\right), containing the Lipschitz quaternions with index 2, and it admits a Euclidean algorithm based on the reduced norm, making it a right Euclidean domain.^[1] Every right ideal in \mathbb{H} is principal, and nonzero elements factor uniquely into irreducibles up to units, with the unit group \mathbb{H}^\times isomorphic to the binary tetrahedral group of order 24.^[1] These properties enable powerful applications, including Hurwitz's quaternionic proof of Lagrange's four-square theorem, which shows that every natural number can be expressed as the sum of four integer squares.^[3]

Introduction

Definition

The Hurwitz quaternions form a subring of the quaternion algebra \mathbb{H} over the real numbers \mathbb{R}, defined as the set

H = \left\{ a + bi + cj + dk \in \mathbb{H} \;\middle|\; a, b, c, d \in \mathbb{Z} \;\text{or}\; a, b, c, d \in \mathbb{Z} + \frac{1}{2} \right\}.

^[4] This specification ensures that either all coefficients are integers or all are half-integers, distinguishing H from other integral structures in \mathbb{H}. Representative elements include the integer-coefficient quaternion $1 + i + j + k and the half-integer example \frac{1}{2} + \frac{1}{2}i + \frac{1}{2}j + \frac{1}{2}k.^[4] The underlying quaternion algebra \mathbb{H} is spanned by the standard basis \{1, i, j, [k](/page/K)\} over \mathbb{R}, with the defining relations i^2 = j^2 = [k](/page/K)^2 = ijk = -1.^[4] The set H is closed under both addition and the non-commutative multiplication inherited from \mathbb{H}, thereby constituting a subring.^[4] The subset of Hurwitz quaternions with purely integer coefficients corresponds to the Lipschitz quaternions.^[4]

Historical Context

The discovery of quaternions is credited to William Rowan Hamilton, who introduced them in 1843 as a non-commutative extension of complex numbers to handle three-dimensional rotations.^[5] In 1886, Rudolf Lipschitz advanced the number theory of quaternions by defining a ring of integer quaternions with components in the integers, enabling initial studies of factorization and sums of squares, though this ring lacked unique factorization and a full division algorithm. Adolf Hurwitz built upon this foundation in his 1896 paper "Ueber die Zahlentheorie der Quaternionen," presented to the Royal Society of Sciences in Göttingen, where he motivated the extension of commutative number theory principles—such as unique factorization and the division algorithm—to the non-commutative setting of rational quaternions, the simplest such algebra.^[6] To achieve better arithmetic properties, including the existence of greatest common divisors for any two elements, Hurwitz redefined the integral quaternions to include those with all coefficients either integers or half-integers, addressing the limitations of Lipschitz's stricter integer-based ring.^[6] Hurwitz expanded this work in his 1919 book Vorlesungen über die Zahlentheorie der Quaternionen, published posthumously by Springer in Berlin, which provided a detailed exposition with proofs of the 1896 results and applications to problems like generalizations of Lagrange's four-square theorem.^[7] A key contribution was his proof that this ring, denoted H, is a maximal order in the rational quaternion algebra, meaning no larger subring of quaternions with integer coefficients exists while preserving the algebraic structure.^[6]

Algebraic Properties

Ring and Order Structure

The Hurwitz quaternion ring, denoted \mathbb{H}, is additively a free abelian group of rank 4 over \mathbb{Z}, generated by the basis \left\{ \frac{1 + i + j + k}{2}, i, j, k \right\}.^[8] This structure arises because elements of \mathbb{H} are quaternions a + bi + cj + dk where a, b, c, d are either all integers or all half-integers, ensuring closure under addition and forming a \mathbb{Z}-lattice of full rank in the 4-dimensional real vector space of quaternions.^[9] Multiplicatively, \mathbb{H} forms a non-commutative ring under the standard quaternion multiplication rules i^2 = j^2 = k^2 = ijk = -1, with the set closed under this operation.^[8] As an order in the division algebra of rational quaternions \mathbb{H}(\mathbb{Q}), \mathbb{H} contains no zero divisors and is thus an integral domain.^[9] The ring \mathbb{H} is a maximal order in \mathbb{H}(\mathbb{Q}), meaning no larger subring of \mathbb{H}(\mathbb{Q}) contains \mathbb{H} while sharing the same field of fractions \mathbb{H}(\mathbb{Q}).^[9] It properly contains the Lipschitz quaternion ring \mathbb{L} = \mathbb{Z} + \mathbb{Z} i + \mathbb{Z} j + \mathbb{Z} k as a subring of index 2.^[8]

Units

In the Hurwitz quaternion ring \mathbb{H}, the units are the invertible elements, which form a finite multiplicative group. The subring of Lipschitz quaternions \mathbb{L}, consisting of quaternions with integer coefficients, has units given by the quaternion group Q_8 of order 8: \{\pm 1, \pm i, \pm j, \pm k\}.^[10] The full group of units in \mathbb{H} is the binary tetrahedral group of order 24, which extends Q_8 by including 16 additional elements of the form \frac{1}{2}(\pm 1 \pm i \pm j \pm k), where all possible combinations of signs are taken.^[10] These half-integer quaternions, together with the elements of Q_8, close under multiplication to form the complete set of units.^[10] Equivalently, as an abstract group, it is isomorphic to \mathrm{SL}(2, \mathbb{F}_3).^[11] All units in \mathbb{H} have norm 1, ensuring their inverses (given by the conjugate) also lie in the ring.^[10]

Geometric Aspects

Lattice Embedding

The Hurwitz quaternions \mathbb{H} can be identified with a lattice in the 4-dimensional Euclidean space \mathbb{R}^4 via the embedding that maps each element a + bi + cj + dk \in \mathbb{H}, where a, b, c, d \in \mathbb{Z} or a, b, c, d \in \mathbb{Z} + \frac{1}{2} with all coefficients of the same type, to the vector (a, b, c, d). This mapping endows \mathbb{H} with the standard Euclidean inner product \langle (a,b,c,d), (a',b',c',d') \rangle = a a' + b b' + c c' + d d', making \mathbb{H} a discrete additive subgroup of \mathbb{R}^4, hence a lattice.^[12] This lattice is isometric to the F_4 lattice, recognized as the root lattice associated with the Lie algebra \mathfrak{f}_4, where a basis can be chosen consisting of vectors corresponding to the simple roots that generate the Lie algebra.^[13] The F_4 structure arises from the geometric arrangement of the lattice points, with the roots embedded within it.^[12] The 24 shortest non-zero vectors in this lattice each have length 1 and form the vertices of a 24-cell in the associated sphere packing. These include the 8 vectors of the form (\pm 1, 0, 0, 0) and permutations, and the 16 vectors of the form (\pm \frac{1}{2}, \pm \frac{1}{2}, \pm \frac{1}{2}, \pm \frac{1}{2}). There are also 24 vectors of length \sqrt{2} that correspond to the short roots of the F_4 root system, given by all permutations and sign changes of ( \pm 1, \pm 1, 0, 0 ).^[12] Under the standard inner product, the Hurwitz quaternion lattice is self-dual, meaning it coincides with its dual lattice \{ y \in \mathbb{R}^4 \mid \langle y, x \rangle \in \mathbb{Z} \ \forall x \in \mathbb{H} \}, a property that underscores its unimodular nature and symmetry in \mathbb{R}^4.^[12]

Norm and Enumeration

The norm of a Hurwitz quaternion q = a + bi + cj + dk, where a, b, c, d are either all integers or all half-integers, is defined by the quadratic form

N(q) = a^2 + b^2 + c^2 + d^2.

This yields a non-negative integer value for every q \in H, reflecting the integral structure of the coefficients.^[14] The norm satisfies the multiplicative property N(q_1 q_2) = N(q_1) N(q_2) for all q_1, q_2 \in [H](/page/H+), which facilitates applications in factorization and divisibility within the ring. Additionally, N(q) = 0 if and only if q = 0, ensuring the norm distinguishes the zero element uniquely. These properties stem from the underlying structure of the quaternion algebra and the Euclidean embedding of the Hurwitz order.^[14] The enumeration of Hurwitz quaternions by norm is captured by the representation function r(n), which counts the number of q \in H such that N(q) = n for n > 0. This function is given explicitly by

r(n) = 24 \sum_{\substack{d \mid n \\ d \ odd}} d,

where the sum runs over the positive odd divisors of n. For n = 0, r(0) = 1 corresponding to the zero quaternion. This formula arises from the geometry of the Hurwitz lattice in \mathbb{R}^4, where the count reflects the symmetry and density of points at each norm level, with r(1) = 24 accounting for the units of norm 1.^[15] The theta series associated with the Hurwitz quaternions provides an analytic generating function for these representation numbers:

\theta_H(\tau) = \sum_{q \in H} e^{2\pi i \tau N(q)} = 1 + \sum_{n=1}^\infty r(n) \, q^n,

where q = e^{2\pi i \tau} and \tau lies in the upper half-plane. Substituting the expression for r(n), this simplifies to

\theta_H(\tau) = 1 + 24 \sum_{n=1}^\infty \left( \sum_{\substack{d \mid n \\ d \ odd}} d \right) q^n.

As a function on the modular surface, \theta_H(\tau) is a modular form of weight 2 for the congruence subgroup \Gamma_0(4). It admits an expression in terms of Eisenstein series of weight 2, specifically \theta_H(\tau) = -\frac{3}{\pi^2} G_{2,2}(\tau), where G_{2,2}(\tau) is the Eisenstein series

G_{2,2}(\tau) = -\frac{\pi^2}{3} \left( 1 + 24 \sum_{n=1}^\infty \left( \sum_{\substack{d \mid n \\ d \ odd}} d \right) q^n \right)

for \Gamma_0(4). This connection highlights the arithmetic depth of the series, linking lattice counts to classical modular form theory.^[16]

Number-Theoretic Properties

Irreducibility and Factorization

In the ring of Hurwitz quaternions, denoted \mathcal{H}, an element q \in \mathcal{H} is irreducible if it is a non-unit and its norm N(q) is a prime number in \mathbb{Z}.^[17] Conversely, if N(q) is composite, then q admits a non-trivial factorization into non-unit elements whose norms multiply to N(q).^[17] This characterization follows from the multiplicativity of the norm and the Euclidean division algorithm in \mathcal{H}, which ensures that factors correspond to the prime factorization of N(q).^[18] Every non-zero, non-unit element in \mathcal{H} factors into a product of irreducible elements, but this factorization is generally not unique.^[17] For instance, the integer 2 admits the factorizations $2 = (1 + i)(1 - i) and $2 = (-i)(1 + i)^2, where the equivalence arises from associating units (here, -i is a unit) to the factors.^[17] Such ambiguities stem from the non-commutativity of quaternion multiplication and the abundance of units (24 in total), allowing for "unit migrations" where units are redistributed among factors, along with reordering and pairings that yield unit recombinations. For primitive Hurwitz quaternions—those that cannot be expressed as an integer m > 1 times another element of \mathcal{H}—factorization into irreducibles is unique up to the order of factors, unit migrations, and recombinations.^[17] If a primitive q has norm N(q) = p_1 \cdots p_n with each p_i prime in \mathbb{Z}, then q = u P_1 \cdots P_n for a unit u and irreducibles P_i with N(P_i) = p_i, and any other such decomposition differs only by permuting the P_i, inserting units between them, or recombining pairs into units.^[17] This partial uniqueness highlights the arithmetic structure of \mathcal{H} despite its failure to be a unique factorization domain.^[18] The norm plays a pivotal role in dictating factorization: since N(q_1 q_2) = N(q_1) N(q_2), any factorization of N(q) in \mathbb{[Z](/page/Z)} induces a corresponding factorization of q in \mathcal{H}, with the norms of the factors matching those in the integer decomposition.^[17] This property ensures that irreducibles are precisely those q where N(q) is prime, preventing non-trivial splits.^[18]

Division Algorithm

The Hurwitz quaternions \mathcal{H} form a Euclidean domain equipped with the Euclidean function given by the squared Euclidean norm N(q) = q \bar{q} = a^2 + b^2 + c^2 + d^2 for q = a + bi + cj + dk \in \mathcal{H}, where \bar{q} denotes the quaternion conjugate. This norm takes non-negative integer values and is multiplicative, satisfying N(q_1 q_2) = N(q_1) N(q_2) for all q_1, q_2 \in \mathcal{H}.^[17]^[14] For any q, d \in \mathcal{H} with d \neq 0, the division algorithm guarantees the existence of s, r \in \mathcal{H} such that q = s d + r and N(r) < N(d). To compute s and r, select s \in \mathcal{H} that minimizes N(q - s d); the corresponding remainder is then r = q - s d. This process enables the standard Euclidean algorithm for computing greatest common divisors in \mathcal{H}, where repeated application yields \gcd(q, d) = \gcd(d, r) until the remainder is zero.^[17]^[20]^[14] The Euclidean property arises from the geometric structure of \mathcal{[H](/page/H+)} as a lattice in the real vector space \mathbb{[H](/page/H+)} \cong \mathbb{R}^4. Specifically, the covering radius of this lattice—the maximum distance from any point in \mathbb{R}^4 to the nearest lattice point—is at most $1/\sqrt{2}. For x = q d^{-1} \in \mathbb{H} (possible since \mathbb{H} is a division ring), there thus exists s \in \mathcal{H} with \|x - s\| \leq 1/\sqrt{2}, where \|\cdot\| denotes the Euclidean norm on \mathbb{R}^4. Then,

r = q - s d = d (x - s),

N(r) = N(d (x - s)) = N(d) N(x - s) = N(d) \|x - s\|^2 \leq N(d) \cdot \frac{1}{2} < N(d),

establishing the strict inequality. Both left and right versions of the algorithm hold due to the non-commutativity of \mathbb{H}, but the ring structure ensures compatibility.^[17]^[20]^[21] In contrast, the Lipschitz ring \mathcal{L} = \{a + bi + cj + dk \mid a,b,c,d \in \mathbb{Z}\} lacks a general division algorithm, as its integer lattice in \mathbb{R}^4 has covering radius 1, allowing cases where \|x - s\| = 1 and N(r) = N(d). For instance, dividing $1 + i + j + k by 2 in \mathcal{L} yields quotients s (such as 0 or 1) for which the remainder r always satisfies N(r) = 4 = N(2), violating the strict inequality. This deficiency in \mathcal{L} motivated Hurwitz's 1896 construction of the denser lattice \mathcal{H}, which includes half-integer components to ensure the required geometric property.^[20]^[21]