Intrinsic {Right} q-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras
Abstract
We define intrinsic right q-radial vector derivatives on radial algebras of abstract vector variables. For a finite parameter set Y, a scalar Jackson calculus on x2 and the mixed anticommutators {x,yi} is combined with the exterior decomposition relative to x. Relabelling covariance and compatibility under inclusions of parameter sets give a direct-limit operator on arbitrary radial algebras. {Under the dimension specialization Q=qM, its classical limit agrees in every M-dimensional Clifford realization with the standard right Clifford derivative; for an exterior blade one obtains [x∧ yI]∂x=(−1)|I|(M−|I|)yI.} Two Fischer-type constructions are considered. The anticommutator with exterior creation is triangular and admits an explicit Green inverse after localization at its diagonal factors. {For right q-monogenic elements, the derivative is paired with right multiplication RxF=Fx. The homogeneous operators ∂Y,Rx,qRx have generically nonzero determinants, which yields determinant-localized Fischer decompositions and explicit projectors.} The one- and two-vector determinants are computed explicitly. {After Q=qM, the two-vector determinant is nonzero for every 0<q<1 and every positive integer M. For arbitrary finite support, a support filtration factors the determinant into exact-support terms; in degree zero, support rank p contributes the factor [m−p]q+p.}
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BibTeX
@misc{barseghyan2026intrinsic,
title = {{Intrinsic Right $q$-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras}},
author = {Diana Barseghyan and Juan Bory-Reyes and Baruch Schneider and Yifan Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2605.00775},
eprint = {2605.00775},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.00775},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Intrinsic {Right} q-Radial Vector Derivatives and Localized Fischer Decompositions on Radial Algebras
AU - Diana Barseghyan
AU - Juan Bory-Reyes
AU - Baruch Schneider
AU - Yifan Zhang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2605.00775
UR - https://arxiv.org/abs/2605.00775
AB - We define intrinsic right q-radial vector derivatives on radial algebras of abstract vector variables. For a finite parameter set Y, a scalar Jackson calculus on x^2 and the mixed anticommutators {x,y_i} is combined with the exterior decomposition relative to x. Relabelling covariance and compatibility under inclusions of parameter sets give a direct-limit operator on arbitrary radial algebras. {Under the dimension specialization Q=q^M, its classical limit agrees in every M-dimensional Clifford realization with the standard right Clifford derivative; for an exterior blade one obtains [x∧ y_I]∂_x=(−1)^|I|(M−|I|)y_I.} Two Fischer-type constructions are considered. The anticommutator with exterior creation is triangular and admits an explicit Green inverse after localization at its diagonal factors. {For right q-monogenic elements, the derivative is paired with right multiplication R_xF=Fx. The homogeneous operators ∂^Y,R_x,qR_x have generically nonzero determinants, which yields determinant-localized Fischer decompositions and explicit projectors.} The one- and two-vector determinants are computed explicitly. {After Q=q^M, the two-vector determinant is nonzero for every 0<q<1 and every positive integer M. For arbitrary finite support, a support filtration factors the determinant into exact-support terms; in degree zero, support rank p contributes the factor [m−p]_q+p.}
ER -