Abstract
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.
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- 1.
This is not always assumed in this article.
- 2.
To generalize to nonassociative algebras we would need a different technique, see, for example, the paper on Cayley-Dickson calculus [8] which makes due with the weaker property of power associativity.
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Acknowledgments
We would like to express our gratitude to the organizers of The 8th Annual UNCG Regional Mathematics and Statistics Conference for providing a forum to express our research. We also thank Liberty University for a hospitable workspace during the Fall 2012 semester, the reviewer of this paper for several pedagogical improvements, and William J. Cook for useful insights into the algebra of our problem. We are of course responsible for any mistakes or oversights [1–18].
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Cook, J.S., Leslie, W.S., Nguyen, M.L., Zhang, B. (2013). Laplace Equations for Real Semisimple Associative Algebras of Dimension 2, 3 or 4.. In: Rychtář, J., Gupta, S., Shivaji, R., Chhetri, M. (eds) Topics from the 8th Annual UNCG Regional Mathematics and Statistics Conference. Springer Proceedings in Mathematics & Statistics, vol 64. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-9332-7_8
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