Intrinsic \(q\)-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras

TL;DR

This paper develops an intrinsic q-deformation of the vector derivative on radial algebras, establishing Fischer-type theorems and decompositions that extend classical results into the q-analogue setting.

Contribution

It introduces a new intrinsic q-deformation of the vector derivative on radial algebras, not derived from coordinate substitutions, and proves Fischer-type theorems with explicit resonance factors.

Findings

Constructed an intrinsic q-Cartan derivative using scalar variables.

Proved an exterior Fischer operator with explicit resonance factors.

Established a monogenic Fischer decomposition after localization.

Abstract

We construct an intrinsic q-deformation of the vector derivative on radial algebras. The construction is not obtained from a coordinate realization by replacing ordinary partial derivatives with one-variable Jackson derivatives; that coordinatewise procedure does not preserve radial subalgebras. Instead, for each distinguished vector variable $x$ and each finite set of auxiliary variables $Y\subset S\setminus\{x\}$, we define a q-Cartan derivative $\partial^Y_{x,q}$ on $R(\{x\}\cup Y)$ using the $x$-relative scalar variables $x^2$ and $\{x,y\}$, $y\in Y$. We prove two Fischer-type theorems. First, an exterior Fischer operator has a triangular anticommutator with explicit resonance factors; after inverting them one obtains a global Green operator and an exterior direct-sum decomposition. Second, using full left multiplication by $x$, we prove the monogenic Fischer decomposition after localization by finite-block determinants. We also describe the first denominator factors: the one-vector and two-vector factors are explicit, while the general determinant factors split by $x$-support. A degree-zero support-rank obstruction shows that a universal unlocalized theorem for all real $0<q<1$ cannot hold without excluding q-resonances.