Euclidean Algebras

TL;DR

This paper introduces a new class of commutative Euclidean group algebras $\

Contribution

It constructs and analyzes a new family of Euclidean algebras $\

Findings

The algebra $\

The zero divisor set has Lebesgue measure zero.

Analytic function theory analogous to complex analysis is developed.

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.