nLab hypercomplex number

Idea

A hypercomplex number system is a finite-dimensional unital algebra over the field $\mathbb{R}$ of real numbers. A hypercomplex number is an element of one of these algebras.

Examples by dimension

• There is only one hypercomplex number system of dimension $0$, of course.

• There is also only one hypercomplex number system of dimension $1$, which is $\mathbb{R}$ itself.

• Up to isomorphism, there are three hypercomplex number systems of dimension $2$, each of which is a commutative algebra. Given the rule

  $$e^2 = a + b e$$

  (for $a, b$ fixed real numbers), the algebra $\mathbb{R}[e]$ may be identified as one of these three cases by the sign of $b^2 - 4 a$. They are:

  • the complex numbers (elliptic case, $b^2 - 4 a \gt 0$),

  • the dual numbers (parabolic case, $b^2 - 4 a = 0$),

  • the perplex numbers (hyperbolic case, $b^2 - 4 a \lt 0$).

  Notice that the complex numbers have 2 distinct continuous automorphisms, given, over $\mathbb{R}$, by $e \mapsto \pm e$ (see at automorphism of the complex numbers).

  On the other hand, the ring of dual numbers has a continuous automorphism $e \mapsto k e$ for each $k \in \mathbb{R} \setminus \{0\}$. But the latter space is homotopy equivalent to two points, corresponding to the connected components of, again, $e \mapsto \pm e$.

  Finally, $e \mapsto \pm e$ are the only two non-trivial continuous automorphisms of the perplex numbers. (For let $\phi$ be an automorphism of the perplex numbers. Observing that $\phi(e)^2 = \phi(e^2) = \phi(1) = 1$, it follows that $\phi(e) \in \{1,-1,e,-e\}$. But if $\phi(e) \in \{+1, -1\}$ then $\phi$ is not injective. So therefore $\phi(e) \in \{e, -e\}$.)

  Therefore, the Euler characteristic of the space of continuous automorphisms of each of the three 2d hypercomplex number systems is 2. Maybe with a bit of handwaving towards groupoid cardinality one might summarize this as saying that there are “3/2” 2d hypercomplex number systems.

• Famous hypercomplex number systems of dimension $4$ include the quaternions and the bicomplex numbers.

• In dimension $8$, try the octonions and the biquaternion?s.

• In dimension $16$, try the sedenions and the bioctonion?s.

Of course, these are not the only possibilities by any means. One can always form the direct product of two hypercomplex number systems to get a hypercomplex number systems with the sum of the dimensions. Another way to double the dimension is to form the tensor product with any of the hypercomplex number systems of dimension $2$; in particular, complexification (the tensor product with the complex numbers) is often denoted by the prefix ‘bi‑’. The Cayley–Dickson construction will double the dimension of any hypercomplex number system equipped with a (possibly trivial) involution. Another way to generate associative hypercomplex number systems is through Clifford algebras.

There is a thorough list of examples on the English Wikipedia.