Analysis on generalized Clifford algebras

In this article, we study the analysis related to generalized Clifford algebras $\mathcal{C}_n(\underline{a})$, where $\underline{a}$ is a non-zero vector. If $\{e_1,\dots,e_n\}$ is an orthonormal basis, the multiplication is defined by relations
\begin{align*} e_j^2=a_je_j-1,\\ e_ie_j+e_je_i=a_ie_j+a_je_i, \end{align*}
for $a_j=e_j\cdot\m{a}$. The case $\underline{a}=\underline{0}$ corresponds to the classical Clifford algebra. We define the Dirac operator as usual by $D=\sum_je_j\partial_{x_j}$ and define regular functions as its null solution. We first study the algebraic properties of the algebra. Then we prove the basic formulas for the Dirac operator and study the properties of regular functions.