Differential Equations and Mathematical Physics

Analysis on generalized Clifford algebras

H. Orelma

Tampere University, Tampere, 33100, Finland

Abstract: 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.

Keywords: Clifford–Kanzaki algebra, generalized Clifford algebra, Dirac operator, regular function.

UDC: 512.646.7:517.95

MSC: Primary 30G35; Secondary 35E05

Received: December 27, 2022
Revised: February 16, 2023
Accepted: February 27, 2023
First online: March 30, 2023

Language: English

DOI: 10.14498/vsgtu1973

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