Quaternion division algebra - Imported Knowlpedia Content

Definition

The quaternion division algebra is the four-dimensional real vector space

\mathbb H = \{a+bi+cj+dk\mid a,b,c,d\in\mathbb R\}

with bilinear multiplication determined by

i^2=j^2=k^2=ijk=-1.

Equivalently, $ij=k=-ji$ , $jk=i=-kj$ , and $ki=j=-ik$ . With this multiplication, $\mathbb H$ is an associative, noncommutative division ring whose center is $\mathbb R$ . It is also a unital algebra over the real numbers. The standard embedding $\mathbb C$ $\hookrightarrow\mathbb H$ sends the complex imaginary unit to $i$ .

Conjugation, norm, and inverse

For $q=a+bi+cj+dk$ , define

\overline q=a-bi-cj-dk.

Quaternionic conjugation reverses products:

\overline{pq}=\overline q\,\overline p.

The norm is

|q| = \sqrt{q\overline q} = \sqrt{a^2+b^2+c^2+d^2}.

It is multiplicative, so $|pq|=|p||q|$ . Every nonzero quaternion therefore has the two-sided inverse

q^{-1}=\frac{\overline q}{|q|^2},

which proves directly that $\mathbb H$ is a division algebra.

Scalar and vector parts

Write $q=a+\mathbf v$ , where $a\in\mathbb R$ and $\mathbf v\in\mathbb R^3$ is identified with $bi+cj+dk$ . Then

(a+\mathbf v)(b+\mathbf w) = (ab-\mathbf v\mathbin{\cdot}\mathbf w) +(a\mathbf w+b\mathbf v+\mathbf v\times\mathbf w).

Thus the failure of commutativity records the cross product: the commutator of two purely imaginary quaternions is twice their vector cross product.

Unit quaternions

The elements of norm one form a group isomorphic to $\operatorname{Sp}(1)$ and $\operatorname{SU}(2)$ . Conjugation $v\mapsto qvq^{-1}$ by a unit quaternion preserves the three-dimensional space of imaginary quaternions and its Euclidean norm. This gives the double covering $\operatorname{Sp}(1)\to\operatorname{SO}(3)$ , with kernel $\{\pm1\}$ .

Remarks

The algebra $\mathbb H$ is not the quaternion group $Q_8$ . The latter is the finite multiplicative subgroup $\{\pm1,\pm i,\pm j,\pm k\}$ inside $\mathbb H^\times$ . Quaternionic vector spaces also require a choice of left or right scalar multiplication because $\mathbb H$ is noncommutative.

References

John H. Conway and Derek A. Smith, On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry, A K Peters, 2003. DOI record. Relevant: Chapters 1–3, quaternion multiplication, conjugation, and norm.