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 $x\in A$,
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\ (\textbf{E1}) \ \ $(x,x,x)=0$, \ \ \ (\textbf{E2}) \ \ $(x^2,x^2,x^2)=0$, \ \ \ (\textbf{E3}) \ \ $x^2e=x^2$ \ \ and \ \ (\textbf{E4})\ \ $(xe)e=x$.
\\
We show that:

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

$\bullet$  $(\textbf{E1})\Longrightarrow (\textbf{E2})\Longrightarrow (\textbf{E3})\Longrightarrow (\textbf{E4})$.
\\In two-dimensional, we determine $A$ satisfying (\textbf{Ei}$)_{i\in\{1,2,3,4\}}$. We have
$$\begin{tabular}{|c|c|c|c|c|}
    \hline
    % after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
   A \  satisfies to & (E1)  & (E2)  & (E3)  & (E4)  \\\hline
   A \ isomorphic \ to & $\mathbb{R}$; $\mathbb{C}$ & $\mathbb{R}$; $\mathbb{C}$; $^{\star}\mathbb{C}$  & $\mathbb{R}$; $\mathbb{C}$; $^{\star}\mathbb{C}$  & $\mathbb{R}$; $\mathbb{C}$; $^{\star}\mathbb{C}$; $\mathcal{L}(1, -1, \gamma, 1)$\\
    \hline
  \end{tabular}
$$ We show
as well as
$(\textbf{E1})\Longrightarrow (\textbf{E2})\Longleftrightarrow (\textbf{E3})\Longrightarrow (\textbf{E4})$.
\\
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}).