In the classical Clifford analysis the Laplace operator is factorized by the Cauchy–Riemann operator \(\Delta =\overline{D}D\). The consequence is all components of a monogenic function are harmonic functions. In more general situation, suppose that we are given a linear partial differential equation. We wish to find a generalized Clifford algebra such that all components of a generalized monogenic function taking values in that algebra satisfy the given partial differential equation. For instance, in order to represent biharmonic functions in the theory of plane elasticity, in 1934 L. Sobrero introduced a hypercomplex algebra which is generated by the imaginary e with the rule \((e^2+1)^2=0\). In this paper we introduce an extension of the idea of L. Sobrero to construct some generalized Clifford algebras in order to cover more partial differential equations in higher dimensions.

中文翻译：

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与某些偏微分方程相关的广义Clifford代数

在经典的Clifford分析中，Laplace运算符由Cauchy–Riemann运算符\（\ Delta = \ overline {D} D \）分解。结果是单基因函数的所有成分都是谐波函数。在更一般的情况下，假设给定一个线性偏微分方程。我们希望找到一个广义的Clifford代数，以使采用该代数的广义单基因函数的所有分量都满足给定的偏微分方程。例如，为了表示平面弹性理论中的双谐波函数，1934年L. Sobrero引入了一个超复杂的代数，它由虚数e生成，规则为（（（e ^ 2 + 1）^ 2 = 0 \）。在本文中，我们介绍了L. Sobrero概念的扩展，以构造一些广义的Clifford代数，以覆盖更高维的更多偏微分方程。