Associative Composition Algebra/Split-quaternions

The imaginary units v such that v ² = −1 lie on a two-sheeted hyperboloid in split quaternions

There are at least three portals leading to split quaternions: the dihedral group of a square, matrix products in M(2,R), and the modified Cayley-Dickson construction. The work of Max Zorn on split octonions showed the necessity of including split real AC algebras in the aufbau of the category.

The development through the dihedral group was started with a lemma in the Introduction, and is completed with exercises below.

Or one can start with a basis {1, i, j, k} taken from M(2,R), where the identity matrix is one, ${\begin{pmatrix}0&1\\1&0\end{pmatrix}}$ is j, ${\begin{pmatrix}0&1\\-1&0\end{pmatrix}}$ is i, and ${\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$ is k. Some practice with matrix multiplication shows they are anticommutative like division quaternions, but some products differ:

j² = +1 = k²,   j k = − i .

Then the real AC algebra of split-quaternions uses coefficients w, x, y, z ∈ R to express an element, its conjugate, and the quadratic form N:

$q=w+xi+yj+zk,\quad q^{*}=w-xi-yj-zk,\quad N(q)=w^{2}+x^{2}-y^{2}-z^{2}.$

Exercises

1. What are the involutions on a square ?

2. As reflections, what is the angle of incidence of the axes of reflection ?

3. The composition of these reflections has what angle of rotation ?

Computations

Insight into the structure and dynamics of split-quaternions is available through elementary computational exercises. These exercises use j² = +1 = k² and jk = −i, contrary to the quaternion group, which is expressed with the same letters i, j, k, but which here refer to the dihedral group of a square instead.

1. For r = j cos θ + k sin θ, show that r² = +1 = −r r*.
2. Compute ir .
3. Recall that <q, t> = (q t* + t q*)/2. Show <q, t> = real part of q t*
4. Definition: q and t are orthogonal when <q, t> = 0.
5. Show that for any theta, r and ir are orthogonal.
6. Let p = i sinh a + r cosh a. Show that p² = +1 for any a and r.
7. Let v = i cosh a + r sinh a. Show that v² = −1.
8. For a given a and r, show that p and v are orthogonal.
9. Let m = p exp(bp) = sinh b + p cosh b. Show that m m* = −1.
10. Let w = exp(bp) = cosh b + p sinh b. Show that m is orthogonal to w.
11. Show that m is orthogonal to v.
12. For any θ, a, and b defining r, p, w, v, and m, the set {m, w, v, ir} is an orthonormal basis.
13. If u is a unit, <qu, tu> = uu* <q, t>.