  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.

Functional Analysis