Hurwitz quaternion

In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of an odd integer; a mixture of integers and half-integers is not allowed). The set of all Hurwitz quaternions is

H = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z} \;\mbox{ or }\, a,b,c,d \in \mathbb{Z} + \tfrac{1}{2}\right\}.

H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H.

A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers. The set of all Lipschitz quaternions

L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}

forms a subring of the Hurwitz quaternions H.

Structure of the ring of Hurwitz quaternions

As a group, H is free abelian with generators {(1 + i + j + k)/2, i, j, k}. It therefore forms a lattice in R⁴. This lattice is known as the F₄ lattice since it is the root lattice of the semisimple Lie algebra F₄. The Lipschitz quaternions L form an index 2 sublattice of H.

The group of units in L is the order 8 quaternion group Q = {±1, ±i, ±j, ±k}. The group of units in H is a nonabelian group of order 24 known as the binary tetrahedral group. The elements of this group include the 8 elements of Q along with the 16 quaternions {(±1 ± i ± j ± k)/2} where signs may be taken in any combination. The quaternion group is a normal subgroup of the binary tetrahedral group U(H). The elements of U(H), which all have norm 1, form the vertices of the 24-cell inscribed in the 3-sphere.

The Hurwitz quaternions form an order (in the sense of ring theory) in the division ring of quaternions with rational components. It is in fact a maximal order; this accounts for its importance. The Lipschitz quaternions, which are the more obvious candidate for the idea of an integral quaternion, also form an order. However, this latter order is not a maximal one, and therefore (as it turns out) less suitable for developing a theory of left ideals comparable to that of algebraic number theory. What Adolf Hurwitz realised, therefore, was that this definition of Hurwitz integral quaternion is the better one to operate with. This was one major step in the theory of maximal orders, the other being the remark that they will not, for a non-commutative ring such as H, be unique. One therefore needs to fix a maximal order to work with, in carrying over the concept of an algebraic integer.

The lattice of Hurwitz quaternions

The (arithmetic, or field) norm of a Hurwitz quaternion, given by $a^2+b^2+c^2+d^2$ , is always an integer. By a theorem of Lagrange every nonnegative integer can be written as a sum of at most four squares. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion. More precisely, the number c(n) of Hurwitz quaternions of given positive norm n is 24 times the sum of the odd divisors of n. The generating function of the numbers c(n) is given by the level 2 weight 2 modular form

2E_2(2\tau)-E_2(\tau) = \sum_nc(n)q^n = 1+24q+24q^2+96q^3+24q^4+ 144q^5+\cdots

A004011

where

q=e^{2\pi i \tau}

and

E_2(\tau) = 1-24\sum_n\sigma_1(n)q^n

is the weight 2 level 1 Eisenstein series (which is a quasimodular form) and σ₁(n) is the sum of the divisors of n.

Factorization into irreducible elements

A Hurwitz integer is called irreducible if it is not 0 or a unit and is not a product of non-units. A Hurwitz integer is irreducible if and only if its norm is a prime number. The irreducible quaternions are sometimes called prime quaternions, but this can be misleading as they are not primes in the usual sense of commutative algebra: it is possible for an irreducible quaternion to divide a product ab without dividing either a or b. Every Hurwitz quaternion can be factored as a product of irreducible quaternions. This factorization is not in general unique, even up to units and order, because a positive odd prime p can be written in 24(p+1) ways as a product of two irreducible Hurwitz quaternions of norm p, and for large p these cannot all be equivalent under left and right multiplication by units as there are only 24 units. However if one excludes this case then there is a version of unique factorization. More precisely, every Hurwitz quaternion can be written uniquely as the product of a positive integer and a primitive quaternion (a Hurwitz quaternion not divisible by any integer greater than 1). The factorization of a primitive quaternion into irreducibles is unique up to order and units in the following sense: if

p₀p₁...p_n

and

q₀q₁...q_n

are two factorizations of some Hurwitz quaternion into irreducible quaternions where p_k has the same norm as q_k for all k, then

q₀ = p₀u₁

q₁ = u–1
1p₁u₂

...

q_n = u–1
np_n

for some units u_k.

References

John Horton Conway, Derek Alan Smith (2003), On quaternions and octonions: their geometry, arithmetic, and symmetry, A K Peters Ltd., ISBN 978-1-56881-134-5

Hurwitz quaternion

Structure of the ring of Hurwitz quaternions

The lattice of Hurwitz quaternions

Factorization into irreducible elements

See also

References