split-complex numbers in nLab

Contents

Contents

Definition

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

A split-complex number zz may be represented as

z=x+jy z= x+ j y

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

z *=x−jy z^* = x- j y

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

zz *=x 2−j 2y 2=x 2−y 2 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(1 - j)(1 + j)=1 - j^2=0). Using the diagonal basis

e=1−j2 e = \frac{1 - j}{2}

e *=1+j2 e^* = \frac{1 + j}{2}

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

(ae+be *)(ce+de *)=(ac)e+(bd)e * (a e+b e^*)(c e+d e^*) =(a c) e+(b d) e^*

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

• tessarines

• split quaternions

• split octonions

References

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