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\documentclass{mfatshort} \usepackage[cp1251]{inputenc} \usepackage[english]{babel} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amssymb} \begin{document} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\Zp}{{\mathbb Z_+}} \newcommand{\R}{{\mathbb R}} \newcommand{\Rp}{{\R_+}} \newcommand{\Pa}{{\mathcal P}} \newcommand{\Pb}{{\Pa_{\Gamma,\chi,\gamma}}} \newcommand{\G}{{\Gamma}} \newcommand{\eps}{\varepsilon} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \renewcommand{\emptyset}{\varnothing} \newcommand{\RR}{{\mathcal R}} \newcommand{\RX}{{\RR\times X}} \newcommand{\ZZ}{\Z_{+,\,0}^\infty} \newcommand{\Sets}{{\Sigma_{\G,\chi}}} \newcommand{\SD}{{\Sigma_{{\tilde D_4},\chi}}} \title {On parameters sets of algebras $\mathcal P_{\Gamma,\chi,\gamma}$ when $\Gamma$ is an extended Dynkin diagram} \author{Kostyantyn Yusenko} \email{kay@imath.kiev.ua} \newtheorem{lemma}{Lemma} \newtheorem{theo}{Theorem} \maketitle \begin{abstract} For associated with an extended Dynkin diagrams $*$-algebras we investigate parameters set, for which there are representations. \end{abstract} \maketitle \section*{Introduction} We remind one classical spectral problem (see \cite{FU} and cited bibliography there). Let $M_i=\{0=\alpha_0^{(i)}<\alpha_1^{(i)}<\ldots<\alpha_{m_i}^{(i)}\}$ be $n$ given closed subset of $\Rp$ and $\gamma \in \Rp$. The problem is to determine whether there exist n-tuples of Hermitian operators $A_i=A_i^*$ such that spectra $\sigma(A_i) \subset M_i$ and $$ A_1+A_2+...+A_n=\gamma I, $$ and to describe all irreducible (up to the unitary equivalence) n-tuples of the operators. This problem could be reformated in terms of $*$-algebras and $*$-representations. Let us consider $*$-algebra \begin{align*} \mathcal A_{M_1,M_2...M_n;\gamma}=\mathbb C\langle a_1\ldots a_n| a_i=a_i^*, (a_i-\alpha_0^{(i)})\ldots(a_i-\alpha_{m_i}^{(i)})=0, \\ a_1+a_2+...+a_n=\gamma e \rangle. \end{align*} It is quite easy to show that such algebra is isomorphic to the algebra generated by projectors \begin{align*} \Pa_{M_1,M_2...M_n;\gamma}=\mathbb C\langle p_1^{(1)}, \ldots p_{m_1} ^{(1)}, \ldots p_1^{(n)}, \ldots p_{m_n} ^{(n)}| p_i^{(k)}=p_i^{(k)2}=p_i^{(k)*}, \\ \sum \limits_{i=1}^{n}\sum \limits_{k=1}^{m_i} \alpha_k^{(i)}p_k^{(i)}=\gamma e, p_j^{(i)}p_k^{(i)}=0 \rangle. \end{align*} %Consider positive function $\chi:\Gamma \rightarrow \Rp$ that defined everywhere on $\Gamma$ except the %root and returns $\alpha_j^{(i)}$ on i-th branch and j-th vertice. With each $\Pa_{M_1,M_2...M_n;\gamma}$ algebra we will associate connected non-oriented graph $\Gamma$ that has n branches reconverged in common vertice (the root). Every i-branch has $m_i$ vertices where we enter elemets of set $M_i$ in decreasing order from the root. In the root we enter $\gamma$. We will denote by $\chi$ a vector $\chi=(\alpha_1^{(1)},\ldots \alpha_{m_1}^{(1)};\ldots;$ $\alpha_n^{(1)},\ldots \alpha_{m_n}^{(n)})$ and will called it character of algebra. Since $\Pa_{M_1,M_2...M_n;\gamma}$ is uniquely given by graph $\Gamma$, character $\chi$, and $\gamma$ we will denote it $\Pa_{\Gamma,\chi,\gamma}$ in the sequel. To solve spectral problem means to solve following tasks: \begin{enumerate} \item a) to describe set $\Sigma_\Gamma=\{(\chi;\gamma) |$ there is representation of algebra $\Pb\}$\\ b) for each character $\chi$ to describe set $\Sigma_{\Gamma,\chi}=\{\gamma \in \Rp |$ there is representation of algebra $\Pb\}$ \item for each pair $(\chi;\gamma) \in \Sigma_\Gamma$ to describe all irreducible $*$ representation of $\Pb$. \end{enumerate} So-called Dynkin dyagrams are successfully used to classify $\Pb$ algebras. The main result of recent paper \cite{VMS05} shows that if $\Gamma$ is a Dynkin diagram of the type $A_n$, $D_n$, $E_6$, $E_7$, or $E_8$: \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(62.00,22.00)(-1,-6.00) \put(0.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(25.0,0.0){\circle*{1.00}} \put(30.0,0.0){\circle*{1.00}} \put(35.0,0.0){\circle*{1.00}} \put(40.0,0.0){\circle*{2.00}} \put(60.0,0.0){\circle*{2.00}} \put(0.0,0.0){\line(1,0){20}} \put(40.0,0.0){\line(1,0){20}} \put(5,5){\makebox(0,0)[b]{$\textstyle A_n$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(62.00,32.00)(-1,-6.00) \put(0.0,0.0){\circle*{2.00}} \put(0.0,20.0){\circle*{2.00}} \put(20.0,10.0){\circle*{2.00}} \put(25.0,10.0){\circle*{1.00}} \put(30.0,10.0){\circle*{1.00}} \put(35.0,10.0){\circle*{1.00}} \put(40.0,10.0){\circle*{2.00}} \put(60.0,10.0){\circle*{2.00}} \put(0.0,0.0){\line(2,1){20}} \put(0.0,20.0){\line(2,-1){20}} \put(40.0,10.0){\line(1,0){20}} \put(1,8){\makebox(0,0)[b]{$\textstyle D_n$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(62.00,22.00)(-1,-6.00) \put(0.0,0.0){\circle*{2.00}} \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(30.0,0.0){\circle*{2.00}} \put(40.0,0.0){\circle*{2.00}} \put(20.0,10.0){\circle*{2.00}} \put(0.0,0.0){\line(1,0){10}} \put(10.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(0,1){10}} \put(5,5){\makebox(0,0)[b]{$\textstyle E_6$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(62.00,22.00)(-1,-6.00) \put(0.0,0.0){\circle*{2.00}} \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(30.0,0.0){\circle*{2.00}} \put(40.0,0.0){\circle*{2.00}} \put(50.0,0.0){\circle*{2.00}} \put(20.0,10.0){\circle*{2.00}} \put(0.0,0.0){\line(1,0){10}} \put(10.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(1,0){10}} \put(40.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(0,1){10}} \put(5,5){\makebox(0,0)[b]{$\textstyle E_7$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(62.00,22.00)(-1,-6.00) \put(0.0,0.0){\circle*{2.00}} \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(30.0,0.0){\circle*{2.00}} \put(40.0,0.0){\circle*{2.00}} \put(50.0,0.0){\circle*{2.00}} \put(60.0,0.0){\circle*{2.00}} \put(20.0,10.0){\circle*{2.00}} \put(0.0,0.0){\line(1,0){10}} \put(10.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(1,0){10}} \put(40.0,0.0){\line(1,0){10}} \put(50.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(0,1){10}} \put(5,5){\makebox(0,0)[b]{$\textstyle E_8$}} \end{picture}} \] then $\Pb$ is finite dimensional, if $\Gamma$ is an extended Dinkyn diagrams of the type $\tilde D_4, \tilde E_6, \tilde E_7, \tilde E_8$ \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(72.00,32.00)(-1,-6.00) \put(0.0,15.0){\circle*{2.00}} \put(15.0,15.0){\circle*{2.00}} \put(30.0,15.0){\circle*{2.00}} \put(15.0,30.0){\circle*{2.00}} \put(15.0,0.0){\circle*{2.00}} \put(0.0,15.0){\line(1,0){15}} \put(15.0,15.0){\line(1,0){15}} \put(15.0,15.0){\line(0,1){15}} \put(15.0,0.0){\line(0,1){15}} \put(5,25){\makebox(0,0)[b]{$\textstyle \tilde D_4$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(72.00,22.00)(-1,-6.00) \put(0.0,0.0){\circle*{2.00}} \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(30.0,0.0){\circle*{2.00}} \put(40.0,0.0){\circle*{2.00}} \put(20.0,10.0){\circle*{2.00}} \put(20.0,20.0){\circle*{2.00}} \put(0.0,0.0){\line(1,0){10}} \put(10.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(0,1){10}} \put(20.0,10.0){\line(0,1){10}} \put(5,10){\makebox(0,0)[b]{$\textstyle \tilde E_6$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(72.00,22.00)(-1,-6.00) \put(0.0,0.0){\circle*{2.00}} \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(30.0,0.0){\circle*{2.00}} \put(40.0,0.0){\circle*{2.00}} \put(50.0,0.0){\circle*{2.00}} \put(60.0,0.0){\circle*{2.00}} \put(30.0,10.0){\circle*{2.00}} \put(0.0,0.0){\line(1,0){10}} \put(10.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(1,0){10}} \put(40.0,0.0){\line(1,0){10}} \put(50.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(0,1){10}} \put(5,5){\makebox(0,0)[b]{$\textstyle \tilde E_7$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(72.00,22.00)(-1,-6.00) \put(0.0,0.0){\circle*{2.00}} \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(30.0,0.0){\circle*{2.00}} \put(40.0,0.0){\circle*{2.00}} \put(50.0,0.0){\circle*{2.00}} \put(60.0,0.0){\circle*{2.00}} \put(70.0,0.0){\circle*{2.00}} \put(20.0,10.0){\circle*{2.00}} \put(0.0,0.0){\line(1,0){10}} \put(10.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(1,0){10}} \put(40.0,0.0){\line(1,0){10}} \put(50.0,0.0){\line(1,0){10}} \put(60.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(0,1){10}} \put(5,5){\makebox(0,0)[b]{$\textstyle \tilde E_8$}} \end{picture}} \] then algebra $\Pb$ is infinite dimensional and of polynomial growth, and finally if $\Gamma$ neither a Dynkin diagram or an extended diagram then $\Pb$ contains a free algebra with two generators (in this case task (2) appears to be too complicated). Attention in this article is concentrated at studing sets $\Sigma_{\Gamma,\chi}$ in case when $\Gamma$ is an extended Dynkin diagram. Given result here is full description of the set $\Sigma_{\tilde D_4,\chi}$, the conditions under which sets $\Sigma_{\Gamma,\chi}$ are infinite, and the conditions under which there are $*$-representations of $\Pb$ on hyperplane. \section{Some statements} In our work we take advantage of structure of the sets $\Sets$ where $\G$ is a Dynkin diagram. Article \cite{KPS05} gives full description of $\Sets$ for each Dynkin diagram and shows that such sets are finite. To get full description of the set $\Sigma_{D_4,(\alpha_1,\alpha_2,\alpha_3)}$ we will use following formula from \cite{KPS05} (as well as another formulas for diagrams $E_6$, $E_7$, $E_8$ could be found there): $$ \Sigma_{D_4,(\alpha_1,\alpha_2,\alpha_3)}=\{0\} \cup \{\sum_{i \in J} \alpha_i, J \subset \{1,2,3\}\} \cup \{(\alpha_1+\alpha_2+\alpha_3)/2\} \label{TD4} $$ \begin{lemma} \label{com} Let $\mathcal A_n$ is an algebra generated by n-tuples of projectors $p_i^2=p_i^*=p_i$ and with relation $$\alpha_1 p_1+\alpha_2 p_2+\cdots+\alpha_{n}p_{n}=\alpha_{n}e,$$ ($\alpha_i$ here are positive real numbers and $e$ is unit element in $\mathcal A_n$), Then such algebra has at least one representation. \end{lemma} \section{Coxeter functors} A powerful tool to investigate algebras $\Pb$ is Coxeter functors. Such functors act between categories of $*$-representation of the algebras $\Pb$. In \cite{KRS02} two functors were investigated: linear $S$ and hyperbolic $T$. Their actions on pair $(\chi;\gamma)$ could be written as follows: $$ S:(\chi;\gamma)\longmapsto(\chi^*;\gamma), $$ $$ \chi^*=(\gamma-\alpha_{m_1}^{(1)},\ldots ,\gamma-\alpha_{1}^{(1)};\ldots;\gamma-\alpha_{m_n}^{(n)},\ldots,\gamma-\alpha_{1}^{(n)}),$$ $$ T:(\chi;\gamma)\longmapsto(\overline{\chi};\overline{\gamma}) $$ $$\overline{\chi}=(\alpha_{m_1}^{(1)}-\alpha_{m_1-1}^{(1)},\ldots,\alpha_{m_1}^{(1)};\ldots; \alpha_{m_n}^{(n)}-\alpha_{m_n-1}^{(n)},\ldots,\alpha_{m_n}^{(n)}) \\ $$ $$ \overline{\gamma}=\alpha_{m_1}^{(1)}+\ldots+\alpha_{m_n}^{(n)}-\gamma; $$ In studing sets $\Sigma_{\Gamma,\chi}$ we will use the fact that when $\Gamma$ is an extended Dynkin diagram then there exist so called invariant functionals $\omega_{\Gamma}$ (see \cite{OS05}). For positive functional $\omega$ on $\Gamma$ to be invariant means that following conditions holds \begin{align*} S:(\chi,\omega(\chi))\longmapsto(\chi',\omega(\chi')), \\ T:(\chi,\omega(\chi))\longmapsto(\chi'',\omega(\chi'')). \end{align*} Given in \cite{OS05} invariant functionals have form: \begin{gather*} \omega_{\tilde D_4}(\chi)=\frac12(\alpha_1+\alpha_2+\alpha_3+\alpha_4),\\ \omega_{\tilde E_6}(\chi)=\frac13(\alpha_1+\alpha_2+\beta_1+\beta_2+\delta_1+\delta_2),\\ \omega_{\tilde E_7}(\chi)=\frac14(\alpha_1+\alpha_2+\alpha_3+\beta_1+\beta_2+\beta_2+2\delta),\\ \omega_{\tilde E_8}(\chi)=\frac16(\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5+2\beta_1+2\beta_2+3\delta).\\ \end{gather*} Taking advantage of this functionals and using $(ST)^n$ functor on pair $(\chi;\gamma)$ we are able to check $\gamma$ to lie in the set $\Sigma_{\Gamma,\chi}$, and so to build series in $\Sigma_{\Gamma,\chi}$. \section{Description of the set $\Sigma_{\tilde D_4,\chi}$} Denote character $\chi$ on $\tilde D_4$ by $\chi=(\alpha_1;\alpha_2;\alpha_3;\alpha_4)$ and by $\alpha$ we denote $\alpha_1+\alpha_2+\alpha_3+\alpha_4$. At first we state some characteristics of the set $\Sigma_{\tilde D_4,\chi}$. \begin{lemma}(\cite{KRS02}) \begin{enumerate} \item $\SD \subset [0, \alpha];$ \item $\SD \ni \sum_{i \in J} \alpha_i, J \subset\{0,1,2,3,4\};$ \item $\tau \in \SD \Longleftrightarrow \alpha-\tau \in \SD;$ \end{enumerate} \end{lemma} Since $\SD$ is symmetric around $\frac{\alpha}{2}$ we will study set $\SD \cap[0,\frac{\alpha}{2})$. Notice that all components of character $\chi$ are required to be less than $\frac{\alpha}{2}$. Indeed, if some coefficient is equal or more than $\frac{\alpha}{2}$ then corresponding projector is equal to $0$ or $I$ in the representation and instead of $\tilde D_4$ graph we obtain $D_4$ graph where solution is known. Having written the action of $(ST)^n$ functor on pair $(\chi;\gamma)$ (and applied (\ref{TD4}) and Lem.\ref{com}) we build the following series in $\SD$. \begin{lemma} \label{infD4} Set $\SD$ contains infinity series $\Sigma_\infty$ with limit point $\frac\alpha2$ and finite series $\Sigma_0$. This two series are described by following rule: \begin{enumerate} \item when $\alpha_2+\alpha_3>\alpha_1+\alpha_4$ then $\Sigma_\infty=\left\{\frac{\alpha}{2}-\frac{\alpha_1}{2n} | n \in \N\right\}$ and \\ $\Sigma_0=\left\{\frac{\alpha}{2}-\frac{\alpha-2\alpha_4}{2(2n-1)} | n<\frac{1}{\alpha_2+\alpha_3-\alpha_1-\alpha_4}, n \in \N\right\}$ \item when $\alpha_2+\alpha_3<\alpha_1+\alpha_4$ then $\Sigma_\infty=\left\{\frac{\alpha}{2}-\frac{\alpha-2\alpha_4}{2(2n-1)} | n \in \N\right\}$ and \\ $\Sigma_0=\left\{\frac{\alpha}{2}-\frac{\alpha_1}{2n} | n<\frac{1}{\alpha_1+\alpha_4-\alpha_2-\alpha_3}, n \in \N\right\}$ \item when $\alpha_2+\alpha_3=\alpha_1+\alpha_4$ then $\Sigma_\infty=\left\{\frac{\alpha}{2}-\frac{\alpha_1}{n} | n \in \N\right\}$ and $\Sigma_0=\emptyset$. \end{enumerate} \end{lemma} \begin{theo} \label{tSD} $$ \SD \cap [0;\alpha/2) =\Sigma_\infty \cup \Sigma_0 \cup \Sigma_1 \cup \Sigma_2^i \cup \Sigma_3 \cup \Sigma_4 \cup \Sigma_5^j, \quad i=2,3,4, \quad j=1,2,3, $$ where \begin{align*} \Sigma_1=\left\{\frac{\alpha}{2}-\frac{\alpha}{2(4n-1)} |\right. & \left.n<\frac{\alpha_4}{4\alpha_4-\alpha}, n<\frac{\alpha-\alpha_1}{\alpha-4\alpha_1},n \in \N\right\}, \\ \Sigma_2^i=\left\lbrace \frac{\alpha}{2}-\frac{\alpha_i}{2n} | i=2,3,4;\right.& n<\frac{\alpha_i}{2\alpha_i+2\alpha_4-\alpha}, n<\frac{\alpha_i}{\alpha_i-\alpha_1} \\ &\left.n<\frac{\alpha_i}{4\alpha_i-\alpha},n \in \N \right\},\quad i=2,3,4,\\ \Sigma_3=\left\{\frac{\alpha}{2}-\frac{\alpha-2\alpha_1}{2(2n+1)} |\right. & \left. n<\frac{\alpha-\alpha_1}{\alpha-4\alpha_1}, n<\frac{\alpha_2+\alpha_3}{2(\alpha_4-\alpha_1)},n \in \N\right\},\\ \Sigma_4=\left\{\frac{\alpha}{2}-\frac{\alpha}{2(4n+1)} |\right. & \left. n<\frac{\alpha-\alpha_4}{4\alpha_4-\alpha}, n<\frac{\alpha_1}{\alpha-4\alpha_1},n \in \N \right\}, \\ \Sigma_5^i=\left\{\frac{\alpha}{2}-\frac{\alpha-2\alpha_i}{2(2n+1)} |\right.& n<\frac{\alpha_1}{\alpha-2\alpha_i-2\alpha_1}, n<\frac{\alpha_i}{\alpha-4\alpha_i},\\ & \left. n<\frac{\alpha-\alpha_4-\alpha_i}{2(\alpha_4-\alpha_i)}, n \in \N \cup \{0\}\right\}, \quad i=1,2,3. \end{align*} \end{theo} In work \cite{YU05} this statements are given with full proofs. \section{Infinity of the set $\Sigma_{\Gamma,\chi}$} Lemma \ref{infD4} says that the set $\Sigma_{\tilde D_4,\chi}$ is infinite if and only if when $\chi_i<\frac\alpha2$ (in other words it means that $\chi_i<\omega_{\tilde D_4}$). Let's study the similar question for another extended Dynkin diagrams. Let $\chi_i$ be component of the character $\chi$ and $\tilde{\chi_i}$ be corresponding component of the character $S(\chi,\gamma)$. It isn't hard to check that if at least one component of character satisfies one of the inequalities $\chi_i\geq \omega_\Gamma(\chi)$ or $\tilde{chi_i} \geq \omega_\Gamma(S(\chi))$ then corresponding projector is equal to $0$ or to $I$ and so considered set is finite (since the set $\Sigma_{\Gamma,\chi}$ is always finite when $\Gamma$ is an Dynkin diagram). So the conditions $\chi_i<\omega_\Gamma(\chi)$ and $\tilde{\chi_i}<\omega_\Gamma(S(\chi))$ are both necessary for the set $\Sigma_{\Gamma,\chi}$ to be infinite. It turns out that they are also sufficient. \begin{theo} \label{inf} Let $\Gamma$ be an extended Dynkin diagram. The set $\Sigma_{\Gamma,\chi}$ is infinite if and only if each component of character satisfies two conditions: $\chi_i<\omega_{\Gamma}(\chi)$ and $\tilde{\chi_i}<\omega_\Gamma(S(\chi))$. \end{theo} \begin{proof} Similarly to $D_4$ diagram we will prove that for each extended Diagram $\G$ there is infinity series of values $\gamma$, for which there are representations of $\Pb$, that tends to point $\omega_\G(\chi)$. . . . \end{proof} \begin{theo}(Representations on hyperplane) Let $\G$ be an extended Dynkin diagram, and $\chi$ be the character on $\G$ such that the conditions of Th.\ref{inf} are satisfied. Then there is a representation of algebra $\Pa_{\G,\chi,\omega_\G(\chi)}$ \end{theo} \begin{proof} Note that due to Shulman's theorem \cite{SHU01} sets $\Sets$ are closed. Since the conditions of Th.\ref{inf} are satisfied there is a series in $\Sets$ with limit points $\omega_\G(\chi)$ and so set $\Sets$ contain point $\omega_\G(\chi)$. \end{proof} Author thanks to his supervisor Ostrovskii V.L. for his helpful advices. \begin{thebibliography}{99} \bibitem{FU} Fulton William, {\it Eigenvalues, invariant factors, hightes weights, and Shubert calculus}, Bull. Amer. Math. Soc. (N.S.). (2000).37, no. 3.--P.209-249. \bibitem{ZS04} Samoilenko Yu. S., Zavodovsky M. V., {\it Theory of operators and involutive respresentation}, Ukr. Math. Bull. (2004).1, no. 4.--P.537-552. \bibitem{KRS02} S.A. Kruglyak, V.I. Rabanovich, Yu.S. Samoilenko {\it On sums of projectors}, Func. analis and its application. (2002).36 , 1. P.20-35. \bibitem{VMS05} M. A. Vlasenko, A. S. Mellit, and Yu. S. Samoilenko, {\it On algebras generated with linearly dependent generators that have given spectra}, Funct. Anal. Appl. {\b 39} (2005), no. 3, to appear. \bibitem{KPS05} S. Kruglyak, S. Popovich, Yu. Samoilenko,{\it The spectral problem and $*$-representations of algebras associated with Dynkin graphs}, Journal of algebra and its application ...... \bibitem{SHU01} V.S. Shulman, {\it On representations of limit relations}, Methods of Functional Analysis and Topology. (2001), no.4, pp.85-86. \bibitem{OS05} Ostrovskyi V. L., {\it Some remarks on families of operators related to extended Dynkin diagrams}, MFAT, to appear. \bibitem{YU05} Yusenko K. A., {\it On quadruples of projectors connected with linear equation}, Ukr. Math. J., to appear. \end{thebibliography} \end{document}
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