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arXiv · 2107.04896

Euclidean Algebras

Xingde Dai·Wei Huang
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Abstract

We introduce a new example of unital commutative $n$-dimensional group algebra $\mathbb{R}_n$ for $n \geq 2$. The algebra $\mathbb{R}_n$ and the complex numbers $\mathbb{C}$ are astonishingly alike. The zero divisor set of the algebra has Lebesgue $μ_n$-measure zero. The formula for the Haar measure is established. Also, the analytic function theory in $\mathbb{R}_n,$ for $n=2k$ that similar to the classical theory in $\mathbb{C}$ is introduced. This includes the Cauchy-Riemann equations, mean-value theorem and Louisville theorem.

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Xingde Dai, Wei Huang. 2021-09-06. Euclidean Algebras. https://arxiv.org/abs/2107.04896

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