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

Abstract

We construct intrinsic right q-radial vector derivatives on radial algebras. Here intrinsic means that the construction uses only the abstract radial variables and their central scalar invariants, without choosing coordinates, a Clifford representation, a quadratic form, or an ambient dimension, and that it is compatible with enlargement of the finite parameter set. Throughout we use the right-handed convention, whose classical limit is the standard right action \([F]\partial_x\). A coordinatewise replacement of ordinary partial derivatives by Jackson derivatives does not preserve even the one-vector radial subalgebra, so the deformation must act on the mixed scalar invariants associated with a distinguished vector variable. After extending scalars to \(\B=\R[q,q^{-1},Q,(1-q)^{-1}]\), where \(Q\) records the formal dimension through the specialization \(Q=q^M\), we define for every finite set \(Y\subset S\setminus\{x\}\) a relative right \(q\)-vector derivative \(\partial^{Y,\mathrm R}_{x,q}\) on \(R_{\B}(\{x\}\cup Y)\). These operators are equivariant under relabelling and compatible with inclusions of finite sets; hence they induce a direct-limit operator on the scalar extension of every radial algebra.