nLab hypercomplex number

Hypercomplex numbers

Hypercomplex numbers


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 00, of course.

  • There is also only one hypercomplex number system of dimension 11, which is \mathbb{R} itself.

  • Up to isomorphism, there are three hypercomplex number systems of dimension 22, each of which is a commutative algebra. Given the rule

    e 2=a+be e^2 = a + b e

    (for a,ba, b fixed real numbers), the algebra [e]\mathbb{R}[e] may be identified as one of these three cases by the sign of b 24ab^2 - 4 a. They are:

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

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

    Finally, e±ee \mapsto \pm e are the only two non-trivial continuous automorphisms of the perplex numbers. (For let ϕ\phi be an automorphism of the perplex numbers. Observing that ϕ(e) 2=ϕ(e 2)=ϕ(1)=1\phi(e)^2 = \phi(e^2) = \phi(1) = 1, it follows that ϕ(e){1,1,e,e}\phi(e) \in \{1,-1,e,-e\}. But if ϕ(e){+1,1}\phi(e) \in \{+1, -1\} then ϕ\phi is not injective. So therefore ϕ(e){e,e}\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 44 include the quaternions and the bicomplex number?s.

  • In dimension 88, try the octonions and the biquaternion?s.

  • In dimension 1616, 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 22; 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.

Last revised on November 16, 2020 at 21:39:22. See the history of this page for a list of all contributions to it.