nLab tessarines

Definition

The tessarines are an algebra over the real numbers. Every tessarine $t$ may be represented as

$$t = a_0 + a_1 i + a_2 j + a_3k$$

where the basis elements satisfy the following products:

$\times$	i	j	k
i	-1	k	-j
j	k	+1	i
k	-j	i	-1

and conjugation $t^* = a_0 - a_1 i - a_2 j - a_3k$.

These are closely related to the quaternions, as the generators satisfy similarly-looking relations.

The tessarines are also referred to as bicomplex numbers since they may be obtained using the generalization of the Cayley-Dickson construction applied on the complex numbers.

Related concepts

• quaternions

• split-quaternion

• split-complex numbers

References

• Elena Luna-Elizarrarás, Michael Shapiro, Daniele Struppa (2013) Bicomplex Holomorphic Functions: the algebra, geometry and analysis of bicomplex numbers. Birkhauser ISBN 978-3-319-24868-4

• Fidelis Zanetti de Castro, Marcos Eduardo Valle. A Broad Class of Discrete-Time Hypercomplex-Valued Hopfield Neural Networks. Neural Networks 122, February 2020, Pages 54-67. (doi)