(Discrete) Almansi Type Decompositions: An umbral calculus framework based on $\mathfrak{osp}(1|2)$ symmetries

TL;DR

This paper develops an umbral calculus framework based on $rak{osp}(1|2)$ symmetries to extend Almansi and Fischer decompositions in Clifford analysis, unifying discrete and continuous settings.

Contribution

It introduces an umbral calculus formalism for hypercomplex variables linked to $rak{osp}(1|2)$ symmetries, enabling a new formulation of Almansi decomposition in Clifford analysis.

Findings

Provides an algebraic framework connecting Clifford analysis with Lie superalgebra symmetries.

Establishes a generalized Almansi decomposition for Clifford-valued polynomials.

Shows the relevance of $rak{sl}_2(r)$ symmetries in quantum mechanics separation of variables.

Abstract

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials $\BR[\underline{x}]$ shall be described in terms of the generators of the Weyl-Heisenberg algebra. The extension of $\BR[\underline{x}]$ to the algebra of Clifford-valued polynomials $\mathcal{P}$ gives rise to an algebra of Clifford-valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra $\mathfrak{osp}(1|2)$. This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of Almansi decomposition in Clifford analysis (c.f. \cite{Ryan90,MR02,MAGU}) that corresponds to a meaningful generalization of Fischer decomposition for the subspaces $\ker (D')^k$. We will discuss afterwards how the symmetries of $\mathfrak{sl}_2(\BR)$ (even part of $\mathfrak{osp}(1|2)$) are ubiquitous on the recent approach of \textsc{Render} (c.f. \cite{Render08}), showing that they can be interpreted in terms of the method of separation of variables for the Hamiltonian operator in quantum mechanics.