The $q$-Dirac Operator on Quantum Euclidean Space

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 $U_q(\mathfrak{o})$-invariant Laplacian ${\mathrm{\Delta }}_q\mathrm{.}$ Due to the non-commutativity of the multiplication, we need a special Clifford algebra $C{\mathrm{\ell }}_{0,n}^q\mathrm{.}$ 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 $k\mathrm{.}$