Involutions of Bicomplex Numbers

TL;DR

This paper identifies six involutions in the algebra of bicomplex numbers, correcting previous literature, and characterizes n-involutions, providing new insights into the structure and invertibility within this algebra.

Contribution

It corrects the count of involutions in bicomplex numbers and characterizes n-involutions, offering a new perspective on their algebraic structure.

Findings

Six involutions of bicomplex numbers identified, contrary to four previously stated.

Eight n-involutions exist only for n=2 and n=4.

New characterization of invertible elements in bicomplex algebra.

Abstract

An involution of a real commutative algebra $A$ is a real-linear homomorphism $f : A \rightarrow A$ such that $f^2 = \mathrm{Id}$. We show that there are six involutions of the algebra of bicomplex numbers, contrary to the actual number of four stated in the literature. We also characterize $n$-involutions satisfying the additional property $f^n = \mathrm{Id}$ for some integer $n \geq 2$. We show there are eight $n$-involutions and they occur only for $n = 2$ and $n= 4$. We use our result to give a new characterization of the invertible elements of the algebra of bicomplex numbers.