Euclidean Algebras

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 $\mu_n$-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.