Real Division Algebras with Left Unit Satisfying Some Identities.

Abstract

We study A, finite dimensional real division algebra with left unit $e$, satisfying: for all $x\in A$,
\
\ (\textbf{E1}) \ \ $(x,x,x)=0$, \ \ \ (\textbf{E2}) \ \ $(x^2,x^2,x^2)=0$, \ \ \ (\textbf{E3}) \ \ $x^{2}e=x^2$ \ \ and \ \ (\textbf{E4})\ \ $(xe)e=x$.
\
We show that:

$\bullet$ If $A$ satisfies to (\textbf{E1}), then e is the unit element of $A$.

$\bullet$  $(\mathbf{\text{E1}})⟹(\mathbf{\text{E2}})⟹(\mathbf{\text{E3}})⟹(\mathbf{\text{E4}})$.
\In two-dimensional, we determine $A$ satisfying (\textbf{Ei}${)}_{i\in \{1,2,3,4\}}$. We have
$$\text{Unknown environment 'tabular'}$$ We show
as well as
$(\mathbf{\text{E1}})⟹(\mathbf{\text{E2}})⟺(\mathbf{\text{E3}})⟹(\mathbf{\text{E4}})$.
\
We finally study the fused four-dimensional real division algebras satisfying (\textbf{Ei}${)}_{i\in \{1,2\}}$. We have shown that
those which verify (\textbf{E2}) are $\mathbb{H}$, ${}^{\star }\mathbb{H}$ and $\mathbb{C}\oplus \mathbb{B}$. and that $\mathbb{H}$ is the only fused algebra division with left unit satisfies to (\textbf{E1}).