Quaternions form Algebra

Theorem

The set of quaternions $\Bbb H$ forms an algebra over the field of real numbers.

This algebra is:

$(1): \quad$ An associative algebra, but not a commutative algebra.

$(2): \quad$ A normed division algebra.

$(3): \quad$ A nicely normed $*$-algebra.

Proof

The quaternions $\Bbb H$ are formed by the Cayley-Dickson Construction from the complex numbers $\C$.

From Complex Numbers form Algebra, we have that $\C$ forms:

$(1): \quad$ An associative algebra

$(2): \quad$ A commutative algebra

$(3): \quad$ A normed division algebra

$(4): \quad$ A nicely normed $*$-algebra.

From Cayley-Dickson Construction forms Star-Algebra, $\Bbb H$ is a $*$-algebra.

From Cayley-Dickson Construction from Nicely Normed Algebra is Nicely Normed, $\Bbb H$ is a nicely normed $*$-algebra.

From Cayley-Dickson Construction from Commutative Associative Algebra is Associative, $\Bbb H$ is an associative algebra.

Now suppose $\Bbb H$ formed a commutative algebra.

Then from Cayley-Dickson Construction from Real Star-Algebra is Commutative, that would mean $\C$ is a real $*$-algebra.

But from Complex Numbers form Algebra it is explicitly demonstrated that $\C$ is not a real $*$-algebra.

So $\Bbb H$ can not be a commutative algebra.

Proof of Normed Division Algebra

Consider the element $\tuple {1, 0}$ of $\C^2$.

We have:

\(\ds \)

\(\)

\(\ds \tuple {x_1, x_2} \times \tuple {1, 0}\)

\(\ds \)

\(=\)

\(\ds \tuple {x_1 \times 1 - 0 \times x_2, x_1 \times 0 + x_2 \times 1}\)

\(\ds \)

\(=\)

\(\ds \tuple {x_1, x_2}\)

As $\times$ is commutative on $\C$, it follows that $\tuple {1, 0} \times \tuple {x_1, x_2} = \tuple {x_1, x_2}$.

So $\tuple {1, 0} \in \C^2$ functions as a unit.

That is, $\tuple {\C^2, \times}$ is a unitary algebra.

We define a norm on $\left({\C^2, \times}\right)$ by:

$\forall \mathbf a = \tuple {a_1, a_2} \in \C^2: \norm {\mathbf a} = \sqrt {a_1^2 + a_2^2}$

This is a norm because:

$(1): \quad \norm {\mathbf x} = 0 \iff \mathbf x = \mathbf 0$

$(2): \quad \norm {\lambda \mathbf x} = \size \lambda \norm x$

$(3): \quad \norm {x - y} \le \norm {x - z} + \norm {z - y}$

It also follows that:

$\norm {x \times y} = \size {x \times y} = \size x \times \size y = \norm x \times \norm y$

and so $\struct {\C^2, \times}$ is a normed division algebra.

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$\blacksquare$

Sources

• 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.26$: Extensions of the Complex Number System. Algebras, Quaternions, and Lagrange's Four Squares Theorem