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Concept
Hurwitz quaternion
Summary
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd integers; a mixture of integers and half-integers is excluded). The set of all Hurwitz quaternions is
That is, either a, b, c, d are all integers, or they are all half-integers.
H is closed under quaternion multiplication and addition, which makes it a subring of the ring of all quaternions H. Hurwitz quaternions were introduced by .
A Lipschitz quaternion (or Lipschitz integer) is a quaternion whose components are all integers. The set of all Lipschitz quaternions
forms a subring of the Hurwitz quaternions H. Hurwitz integers have the advantage over Lipschitz integers that it is possible to perform Euclidean division on them, obtaining a small remainder.
Both the Hurwitz and Lipschitz quaternions are examples of noncommutative domains which are not division rings.
As an additive 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.
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