Laplace Equations for Real Semisimple Associative Algebras of Dimension 2, 3 or 4.

We study calculus on an associative algebra \(\mathcal{A}\) of dimension n = 1, 2, 3, and 4 over the real numbers. Following the approach of numerous authors from Scheffers (Mathematisch-physikalische Klasse 46:120-134, 1894) to Vladimirov and Volovich (USSR Acad Sci 59(1):3-27, 1984) we take \(\mathcal{A}\)-linearity of the differential as the defining condition of \(\mathcal{A}\)-differentiability. The necessary condition for \(\mathcal{A}\)-differentiability is the generalized Cauchy Riemann equations. The Cauchy Riemann equations are seen as conditions to place the Jacobian matrix in the left regular representation of the algebra. We introduce the \(\mathcal{A}\)-Laplace equation as an n-th order partial differential equation solved by \(\mathcal{A}\)-differentiable functions. In contrast, Wagner constructed \(\frac{n(n-1)} {2}\) Laplace equations for the case of Frobenius algebras. We discuss the distinction between our approach and that of Wagner. Explicit Cauchy Riemann equations and the \(\mathcal{A}\)-Laplacian are given for each of the nine associative semisimple real algebras of dimensions 1, 2, 3, or 4.