The $q$-Dirac Operator on Quantum Euclidean Space
Swanhild Bernstein, Martha Lina Zimmermann, Baruch Schneider
TL;DR
This work extends Clifford analysis to quantum Euclidean space by introducing a $q$-Dirac operator that factors the $U_q(\mathfrak{o})$-invariant Laplacian $\Delta_q^R$ using a dedicated $q$-Clifford algebra $\mathcal{C}\ell_{0,n}^q$. Because noncommutative multiplication prevents a single algebra from simultaneously representing the Laplacian and the squared radius, the authors develop two distinct $q$-Clifford structures, enabling definitions of $q$-monogenic functions and a Fischer-type inner product that leads to a twisted Fischer decomposition of homogeneous Clifford-valued polynomials. Key contributions include the explicit construction of right $q$-monogenic spaces, the $q$-Fischer inner product, and a decomposition $\mathcal{P}_k = \sum_{s=0}^k (^L\underline{x})^s \mathcal{M}_{k-s}^R$, along with the identification of an $\mathfrak{osp}(n|2n)$-type symmetry in the $q$-setting. The results lay foundational groundwork for quantum Clifford analysis and noncommutative harmonic analysis on quantum spaces, with connections to nonstandard quantum groups such as $U'_q(\mathfrak{so}_n)$ and Howe duality frameworks.
Abstract
This paper provides the foundations of quantum Clifford analysis in $q$-commutative variables with symmetric difference operators. We consider a $q$-Dirac operator on the quantum Euclidean space that factorizes the $U_q(\frak{o})$-invariant Laplacian $Δ_q.$ Due to the non-commutativity of the multiplication, we need a special Clifford algebra $C\ell_{0,n}^q.$ We define $q$-monogenic functions as null solutions of the $q$-Dirac operator and $q$-spherical monogenic functions. We define an inner Fischer product and decompose the space of homogeneous polynomials of degree $k.$