nLab split-complex numbers

Definition

The algebra obtained from the generalization of the Cayley-Dickson construction applied on the complex numbers applied to the real numbers with parameter $\gamma=-1$.

A split-complex number $z$ may be represented as

$$z= x+ j y$$

where $j^2=1$ (in contrast with the imaginary unit $i^2=-1$ in the complex numbers). Conjugation is similarly given by

$$z^* = x- j y$$

A consequence is that the product $z z^*$ is not non-negative anymore, since

$$z z^* = x^2 - j^2 y^2 = x^2 - y^2$$

meaning in particular that zero-divisors exist (for example $(1 - j)(1 + j)=1 - j^2=0$). Using the diagonal basis

$$e = \frac{1 - j}{2}$$

$$e^* = \frac{1 + j}{2}$$

of idempotent elements, hence for which $e^2=e$ and $(e^*)^2=e^*$, and fulfilling $e e^*=e^* e=0$ results in multiplication given by

$$(a e+b e^*)(c e+d e^*) =(a c) e+(b d) e^*$$

which yields that the algebra $\mathbb{R}[j]=\mathbb{R}[e]$ of split-complex numbers is isomorphic to the algebra $\mathbb{R}\oplus\mathbb{R}$ with pointwise multiplication.

Related concepts

• tessarines

• split quaternions

• split octonions

References

• Robert Brown. On generalized Cayley-Dickson algebras. Pacific Journal of Mathematics 20, no. 3 (1967): 415-422. (doi)