Dunkl regularity over alternative $*$-algebras
Giulio Binosi, Alessandro Perotti
TL;DR
This work develops a unifying framework that characterizes slice-regularity and related hypercomplex function theories through Dunkl operators on real alternative *-algebras. By introducing Dunkl-Cauchy-Riemann kernels $D_\mathcal{P}$ on hypercomplex subspaces $M$ and organizing spaces via partitions $\\mathcal P$ of $[n]$, it embeds monogenic, slice-regular, Moisil-Teodorescu, and generalized partial-slice theories as Dunkl-regular subcases, often under the key multiplicity constraint $\sum k_i= (1-n)/2$. The paper provides detailed operator relations, including the fundamental link $ar ext{partial}_M-ar\theta_M = -ar x^{-1} ilde\Gamma_M$, and establishes a rich taxonomy of function spaces $\,\mathcal{F}_\mathcal P(\Omega)$ with classifications tied to partition structures, yielding insights into equivalences and non-equivalences across algebras such as $\b C$, $\bb H$, $\bb O$. These results unify Dunkl theory with hypercomplex analysis, enabling embedding of diverse function theories within Dunkl monogenic function theory and offering new tools for hypercomplex analysis on Clifford algebras, quaternions, octonions, and beyond.
Abstract
We characterise slice-regularity of functions over a real alternative *-algebra using operators that arise in Dunkl operator theory. We present a unifying perspective on hypercomplex analysis by defining a family of function spaces in the kernel of Dunkl-Cauchy-Riemann operators. Each of these function spaces, whose elements are called Dunkl-regular functions, refines Dunkl monogenic function theory and Dunkl harmonic analysis on Euclidean spaces. This approach allows a wide variety of hypercomplex function theories to be embedded as subcases of Dunkl monogenic function theory. This paves the way for further interactions between Dunkl theory and hypercomplex analysis.