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In [[mathematics]], a '''Hurwitz quaternion''' (or '''Hurwitz integer''') is a [[quaternion]] whose components are ''either'' all [[integer]]s ''or'' all [[half-integer]]s (halves of an odd integer; a mixture of integers and half-integers is not allowed). The set of all Hurwitz quaternions is
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:<math>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\}.</math>
 
It can be confirmed that ''H'' is closed under quaternion multiplication and addition, which makes it a [[subring]] of the [[ring (mathematics)|ring]] of all quaternions '''H'''.
 
A '''Lipschitz quaternion''' (or '''Lipschitz integer''') is a quaternion whose components are all [[integer]]s. The set of all Lipschitz quaternions
 
:<math>L = \left\{a+bi+cj+dk \in \mathbb{H} \mid a,b,c,d \in \mathbb{Z}\right\}</math>
 
forms a subring of the Hurwitz quaternions ''H''.
 
As a [[group (mathematics)|group]], ''H'' is [[free abelian group|free abelian]] with generators {(1 + ''i'' + ''j'' + ''k'')/2, ''i'', ''j'', ''k''}. It therefore forms a [[lattice (group)|lattice]] in '''R'''<sup>4</sup>. This lattice is known as the [[F4 lattice|''F''<sub>4</sub> lattice]] since it is the [[root lattice]] of the [[semisimple Lie algebra]] [[F4 (mathematics)|''F''<sub>4</sub>]]. 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 (ring theory)|order]] (in the sense of [[ring theory]]) in the [[division ring]] of quaternions with [[rational number|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 ideal]]s 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 [[field norm|(arithmetic, or field) norm]] of a Hurwitz quaternion, given by <math>a^2+b^2+c^2+d^2</math>, is always an integer. By a [[Lagrange's four-square theorem|theorem of Lagrange]] every nonnegative integer can be written as a sum of at most four [[square (algebra)|squares]]. Thus, every nonnegative integer is the norm of some Lipschitz (or Hurwitz) quaternion. A Hurwitz integer is a [[prime element]] if and only if its norm is a [[prime number]].
 
==See also==
* [[Gaussian integer]]
* [[Eisenstein integer]]
* The Lie group [[F4 (mathematics)|F<sub>4</sub>]]
* The [[E8 lattice|E<sub>8</sub> lattice]]
 
==References==
*John Horton Conway, Derek Alan Smith (2003), [http://books.google.co.uk/books?id=E_HCwwxMbfMC On quaternions and octonions: their geometry, arithmetic, and symmetry], A K Peters Ltd., ISBN 978-1-56881-134-5
 
[[Category:Quaternions]]

Revision as of 14:35, 27 February 2014

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