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e7-3-1.tex
\documentclass{article} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amsthm} \newtheorem{theorem}{Theorem} \title{On representations of the $*$-algebra related to $\tilde E_7$} \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 a linear relation and their representations were studied. To such algebra one can put into correspondence certain simply laced graph. Of special interest are the algebras to which there correspond extended Dynkin diagrams. Representations of four-tuples of projections whose sum is equal to~$2I$ (to this algebra there corresponds~$\tilde D_4$) were studied in~\cite{os_book}, representations of the $*$-algebra generated by three partial reflections whose sum is zero (the corresponding graph is $\tilde E_6$) were discussed in~\cite{anton}. We consider representations of the $*$-algebra 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 $(A+B)^2 = I$. According to \cite{sam_etal1} to this algebra there corresponds the $\tilde E_7$ extended Dynkin diagram. \begin{theorem} 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. \end{theorem} \begin{thebibliography}{9} \bibitem{sam_etal1} M.A.Vlasenko, A.S.Mellit, Yu.Samoilenko. \emph{On algebras generated by linearly connected generators with a given spectrum}, Funct. Anal. Appl. \textbf{38} (2004), no.~2. \bibitem{os_book} V.Ostroskyi, Yu.Samoilenko. \emph{Introduction to the theory of representations of finitely presented *-algebras. Representations by buonded operators}, Rev. Math.\& Math. Phys., 1999, vol. 11, pp. 1--261, Gordon \& Breach, London 1999. \bibitem{anton} A.S.Mellit. \emph{On the case where a sum of three partial maps is equal to zero}, Ukr. Math. J., \textbf{55} (2003), no.~9, pp.~1277--1283. \end{thebibliography} \end{document} %%% Local Variables: %%% mode: latex %%% TeX-master: t %%% End:
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