Generalizations and Applications

Abstract

In this section we will discuss several generalizations of the notion of (slice) regularity to the case of algebras other than $\mathbb{H}.$ We will begin with the algebra of octonions $\mathbb{O}$, where the foundational results mimic very closely those described in this book. We will then consider the case of functions defined on the Clifford algebra $\mathbb{R}_{3}$ (sometimes referred to as Cl(0, 3) in the literature) with values in that same algebra. We will see that in this case, it is not possible to fully reconstruct an analogous theory of regularity due to some peculiarities in the algebraic structure of $\mathbb{R}_{3}$. The basic theory of regularity over $\mathbb{H}, \mathbb{O},$ and $\mathbb{R}_{3}$ is surveyed in [60]. It was partly because of the difficulties encountered in the study of functions defined on a Clifford algebra that the next significant generalization that we will discuss is the notion of (slice) monogeneity, which is defined for functions on the Euclidean space ${\mathbb{R}}^{m+1}$ and with values in $\mathbb{R}_{m} = Cl(0,m).$ The theory of such functions is fully discussed in the recent [36] and will be presented here only for completeness. Finally, we will mention the new and exciting general approach due to Ghiloni and Perotti, [72], which allows a powerful generalization of some of our ideas to the setting of real alternative algebras.