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

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 $\partial<sup>Y_{x,q}$ on $R({x}\cup Y)$ using the $x$-relative scalar variables $x<sup>2$ 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.

• The paper develops intrinsic q-deformation of the vector derivative for radial algebras, overcoming limitations of coordinatewise methods.
• It establishes two localized Fischer-type decompositions (exterior and monogenic) with explicit determinant formulas aiding spectral analysis.
• The study reveals resonance phenomena and detailed denominator structures, setting a foundation for advanced q-deformed harmonic analysis.

Intrinsic $q$-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras

Introduction and Motivation

This work develops an intrinsic $q$-deformation of the vector derivative for radial algebras, a noncommutative algebraic abstraction designed to generalize core Clifford analysis concepts without committing to coordinate representations, specific quadratic forms, or fixed dimensions. Traditional approaches for $q$-deformed Clifford analysis typically rely on coordinatewise substitutions of partial with Jackson derivatives, but these fail to preserve the structural constraints of radial subalgebras. The paper overcomes this barrier, constructing novel $q$-Cartan derivatives acting intrinsically on radial algebras and establishing new Fischer-type decomposition theorems within this noncommutative and coordinate-free framework.

Construction of the Intrinsic $q$-Radial Derivative

The authors construct, for each vector variable $x$ and finite set of auxiliary variables $Y$, a $q$-Cartan derivative $\partial^Y_{x,q}$ acting on the radial algebra $R(\{x\} \cup Y)$. Unlike coordinatewise $q$0-differential operators, this construction deforms the vector variable $q$1 as a whole via dilations on central scalar generators—$q$2, and the anticommutators $q$3 and $q$4—rather than on coordinate components.

A functorial property ensures that these finite $q$5-Cartan derivatives cohere under inclusion and yield a direct-limit operator $q$6 acting on the full radial algebra $q$7 for arbitrary variable sets $q$8. This operator is shown to properly encode the correct deformation behavior, reducing to the classical radial vector derivative as $q$9, and intrinsically respects the structural relations of the radial algebra.

Fischer-Type Decompositions: Exterior and Monogenic

The main analytic consequences are two Fischer-type decomposition theorems, paralleling and generalizing classical results from harmonic and Clifford analysis. The first (exterior) Fischer decomposition employs an exterior creation operator and analyzes a triangular anticommutator, whose resonance (i.e., denominator) factors are made explicit. This localizes the module to nonresonant sectors, yielding a fully explicit Green operator and exterior direct sum decomposition. The triangularity and explicit computation of anticommutator eigenvalues are crucial for algorithmic applications and spectral analysis.

The second result is a monogenic Fischer decomposition, formulated in terms of full left multiplication by $q$0 (the monogenic creation operator) and its interaction with the $q$1-Cartan derivative. Again, explicit determinant expressions characterize the required localizations as the support/rank of the involved polynomials increases. For each finite degree and auxiliary set, a decomposition holds after localizing by explicit block determinants, ultimately yielding a direct-limit global Fischer decomposition for the intrinsic $q$2-radial Dirac kernel and its image under $q$3-multiplication.

Resonance Phenomena and Denominator Structure

Critical analysis is devoted to understanding the denominator structure of these decompositions, generalizing classical resonance phenomena associated with Fischer decomposition in the presence of dimension and harmonic-degree constraints. Explicit formulas are provided for determinant factors in the one- and two-vector cases, as well as their higher-rank generalizations via support factorization. The authors identify obstructions intrinsic to the $q$4-deformed setting: for generic $q$5, universal unlocalized decompositions are impossible when exact-support denominators vanish ("$q$6-resonance"), even for arbitrarily large formal dimension. As proven, genuine resonance roots appear for support sets of even cardinality, establishing the necessity of localization or restriction to nonresonant parameter regions.

Implications and Future Directions

This research provides a canonical intrinsic formalism for $q$7-deformed radial derivatives, establishing analytic and algebraic decompositions fundamental to symbolic Clifford analysis and quantum harmonic analysis. The explicit control over determinant denominators and identification of resonance sets informs both computation and the deeper understanding of noncommutative, $q$8-deformed invariant theory.

Practical implications include algorithmic tools for $q$9-Dirac and $q$0-Laplace equations, particularly in contexts where symmetry and noncommutativity are essential (e.g., quantum Euclidean spaces, orthogonal and Hermitian Clifford frameworks). Theoretically, the results illuminate the precise limitations imposed by $q$1-resonance, guiding further investigation into the structure of support-localized modules, their noncommutative syzygies, and potential extensions to quantum group settings.

Future research directions involve:

• Complete factorization and combinatorics of universal exact-support determinants for arbitrary auxiliary sets,
• Identification and geometric interpretation of nonresonance regions for unlocalized Fischer decompositions,
• Extension of the present framework to other $q$2-deformed noncommutative algebras, including quantum groups and related enveloping algebras,
• Investigating applications to systems of $q$3-difference equations with noncommutative symmetries and the associated harmonic analysis.

Conclusion

This work systematically develops intrinsic $q$4-Cartan derivatives for radial algebras and demonstrates localized Fischer decomposition theorems at both exterior and monogenic levels. The explicit denominator and resonance analysis marks a significant step toward a general $q$5-deformed, coordinate-independent Clifford analysis. The paper settles the general existence question for decompositions up to localization and clarifies essential structural obstructions, serving as a foundation for ongoing study into $q$6-deformations in noncommutative harmonic analysis and their algebraic underpinnings.