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Lipschitz quaternion

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Lipschitz 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\}.

It can be confirmed that 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.

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 (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. A Hurwitz integer is a prime element if and only if its norm is a prime number.

See also

References

  • John Horton Conway, Derek Alan Smith (2003), ISBN 978-1-56881-134-5
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