One Hat Cyber Team

View File Name : e7-3.tex

\documentclass{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amsthm}

\title{On representations of $*$-algebra related to the $\tilde E_7$
  extended Dynkin diagram}

\author{Vasyl Ostrovskyi}


\begin{document}
\maketitle
In several recent papers (see e.g. \cite{sam_etal1} and references
therein) algebras generated by families on idempotents satisfying
linear relation were related to a
star-shaped graph $\Gamma$. In the case where $\Gamma$ is a Dynkin
diagram the algebra is finite-dimensional, and the algebra has
polynomial growth if and only if it corresponda to an extended Dynkin
diagram (see \cite{sam_etal2}) for more details. Representations of
the $*$-algebra corresponding to the $\tilde D_4$ were studied in
\cite{os_book}, representations of the $*$-algebra corresponding to
$\tilde E_6$ were discussed in \cite{anton}.

We consider representations of the $*$-algebra corresponding to the
$\tilde E_7$ diagram. It is generated by the projections $P_1$, $P_2$,
$P_3$, $Q_1$, $Q_2$, $Q_3$, $R$, such that $P_jP_k =Q_jQ_k
=\delta_{jk} =0$ and
\[
P_1+2P_2+3P_3+Q_1+2Q_2+3Q_3+R= 4I,
\]
or equivalently, the pair of self-adjoint operators $A$, $B$ having their
spectrum in the set $\{\pm 1/2, \pm3/2\}$ such that $(A+B)^2 = I$.

Below is our main result.

Representations of the $\tilde E_7$ algebra are the following
\begin{align*}
A= \frac{1}{8\phi }\begin{pmatrix} -16-\beta &2\gamma &\sqrt{4\phi
        ^4-\beta ^2} &0 \\
        2\gamma &16+\beta &0 &-\sqrt{4\phi^4-\beta^2} \\
        \sqrt{4\phi^4-\beta^2}&0&-16+\beta &2\gamma  \\
        0&-\sqrt{4\phi^4-\beta^2}&2\gamma &16-\beta\end{pmatrix}
\\
B=\frac1{8\phi} \begin{pmatrix} 16-\beta &2\gamma \omega &-
        \sqrt{4\phi^4 -\beta^2}&0 \\
        2\gamma\bar\omega&-16+\beta &0&\sqrt{4\phi^4-\beta^2} \\
        -\sqrt{4\phi^4-\beta^2}&0&16+\beta &2\gamma \omega  \\
        0&\sqrt{4\phi^4-\beta^2}&2\gamma \bar\omega&-16-\beta
        \end{pmatrix}
\end{align*}
where $\gamma^2=-\phi^4+20\phi^2-64$,
  $\beta^2=16\phi^2-\gamma^2(1+\omega)(1+\bar\omega)$, $|\omega|=1$,
  $\phi\in [2,4]$.

For $\phi=2$ the representation decomposes into direct sum of
four one-dimensional represenations, $A=\pm 1/2$, $B=\mp 3/2$ and
$A=\pm 3/2$, $B= \mp 1/2$.

For $\phi = 2\sqrt{2}$, $\omega=1$ the representations decomposes into
direct sum of two one-dimensional representations $A=B=\pm 1/2$ and
one irreducible two-dimensional representaion
\[
A=\begin{pmatrix} -\sqrt{2} & 1/2 \\ 1/2 &\sqrt{2}\end{pmatrix}, \quad
B = \begin{pmatrix} \sqrt{2} & 1/2 \\ 1/2 & -\sqrt{2} \end{pmatrix}
\]
with the spectrum of $A$ and $B$ equal to $\{-3/2, 3/2\}$

For $\phi=4$ the representation decomposes into direct sum of two
irreducible two-dimensional representations,
\begin{gather*}
A=\begin{pmatrix} 1/2&0\\0&-3/2\end{pmatrix} , \quad B =
\begin{pmatrix} 0 & \sqrt{3}/2 \\\sqrt{3}/2 & 1\end{pmatrix}
\\
\sigma(A)=\{1/2, -3/2\}, \quad \sigma(B)= \{-1/2, 3/2\}
\end{gather*}
and
\begin{gather*}
A=\begin{pmatrix} -1/2&0\\0&3/2\end{pmatrix} , \quad B =
\begin{pmatrix} 0 & -\sqrt{3}/2 \\-\sqrt{3}/2 & -1\end{pmatrix}
\\
\sigma(A)=\{-1/2, 3/2\}, \quad \sigma(B)= \{1/2, -3/2\}
\end{gather*}

For all other pairs $(\phi, \omega)$, $\phi \in (2,4)$, $|\omega| =1$
the representation is irreducible with $\sigma(A) = \sigma(B) =
\{\pm1/2,\pm3/2\}$.

Listed above are all irreducible representations up to a unitary
equivalence.

\begin{thebibliography}{9}

\bibitem{sam_etal1}
Yu.Samoilenko et al

\bibitem{sam_etal2}
Yu.Samoilenko et al

\bibitem{anton}
A.Mellit

\bibitem{os_book}
V.Ostroskyi, Yu.Samoilenko

\end{thebibliography}
\end{document}

%%% Local Variables:
%%% mode: latex
%%% TeX-master: t
%%% End: