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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 the existence of $*$-representations of certain algebras related to extended Dynkin graphs} \author{Kostyantyn Yusenko} \email{kay@imath.kiev.ua} \newtheorem{lemma}{Lemma} \newtheorem{theo}{Theorem} \newtheorem{coll}{Corollary} \maketitle \begin{abstract} For associated with an extended Dynkin graphs $*$-algebras we investigate parameters set, for which there exist representations. Given result in paper is complete description the set related to graph $\tilde D_4$ and the conditions under which parameters sets are infinite. \end{abstract} \maketitle \section*{Introduction} In \cite{FU,ZS04} (see also cited bibliography therein) the following problems were studied. Let $M_i=\{0=\alpha_0^{(i)}<\alpha_1^{(i)}<\ldots<\alpha_{m_i}^{(i)}\}, \ i=1,..n$ be given finite subsets of $\Rp$ and $\gamma \in \Rp$. The problem is to determine whether there exist n-tuples of Hermitian operators $A_i=A_i^*, \, i=1,..n$ 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 reformulated in terms of $*$-algebras and their $*$-representations. Consider the following $*$-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 projections \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. To each algebra $\Pa_{M_1,M_2...M_n;\gamma}$ one can assign the connected non-oriented graph $\Gamma$ that has n branches reconverged in the common vertice (the root). Every i-branch has $m_i$ vertices where we put elemets of set $M_i$ in decreasing order from the root. In the root we put $\gamma$. Denote by $\chi$ the vector $\chi=(\alpha_1^{(1)},\ldots \alpha_{m_1}^{(1)};\ldots;$ $\alpha_1^{(n)},\ldots \alpha_{m_n}^{(n)})$ and call it a character of an algebra. In this paper we use two different indocations for components of the character: 1) usual indication $\chi_i$ which denotes $i$-th component of the character; 2) indication $\chi_i^{(j)}$ which denotes component that corresponds to $\alpha_i^{(j)}$. 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. Spectral problem is equivalent to the following tasks to be solved: \begin{enumerate} \item a) to describe the set $\Sigma_\Gamma=\{(\chi;\gamma) |$ there is the representation of algebra $\Pb\}$\\ b) for each character $\chi$ to describe the set $\Sigma_{\Gamma,\chi}=\{\gamma \in \Rp |$ there is the representation of algebra $\Pb\}$ \item for every pair $(\chi;\gamma) \in \Sigma_\Gamma$ to describe all irreducible $*$ representation of $\Pb$. \end{enumerate} Well known Dynkin dyagrams are successfully used to classify $\Pb$ algebras by their growth. The main result of recent paper \cite{VMS05} shows that if $\Gamma$ is a Dynkin graph of the type $A_n$, $D_n$, $E_6$, $E_7$, or $E_8$: \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(62.00,22.00)(-1,-6.00) \put(-30.0,10.0){\circle*{2.00}} \put(-10.0,10.0){\circle*{2.00}} \put(-5.0,10.0){\circle*{1.00}} \put(0.0,10.0){\circle*{1.00}} \put(5.0,10.0){\circle*{1.00}} \put(10.0,10.0){\circle*{2.00}} \put(30.0,10.0){\circle*{2.00}} \put(-30.0,10.0){\line(1,0){20}} \put(10.0,10.0){\line(1,0){20}} \put(-25,15){\makebox(0,0)[b]{$\textstyle A_n$}} \put(50.0,0.0){\circle*{2.00}} \put(50.0,20.0){\circle*{2.00}} \put(70.0,10.0){\circle*{2.00}} \put(75.0,10.0){\circle*{1.00}} \put(80.0,10.0){\circle*{1.00}} \put(85.0,10.0){\circle*{1.00}} \put(90.0,10.0){\circle*{2.00}} \put(110.0,10.0){\circle*{2.00}} \put(50.0,0.0){\line(2,1){20}} \put(50.0,20.0){\line(2,-1){20}} \put(90.0,10.0){\line(1,0){20}} \put(51,8){\makebox(0,0)[b]{$\textstyle D_n$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(62.00,22.00)(-1,-6.00) \put(-20.0,0.0){\circle*{2.00}} \put(-10.0,0.0){\circle*{2.00}} \put(0.0,0.0){\circle*{2.00}} \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(0.0,10.0){\circle*{2.00}} \put(-20.0,0.0){\line(1,0){10}} \put(-10.0,0.0){\line(1,0){10}} \put(0.0,0.0){\line(1,0){10}} \put(10.0,0.0){\line(1,0){10}} \put(0.0,0.0){\line(0,1){10}} \put(-15,5){\makebox(0,0)[b]{$\textstyle E_6$}} \put(50.0,0.0){\circle*{2.00}} \put(60.0,0.0){\circle*{2.00}} \put(70.0,0.0){\circle*{2.00}} \put(80.0,0.0){\circle*{2.00}} \put(90.0,0.0){\circle*{2.00}} \put(100.0,0.0){\circle*{2.00}} \put(70.0,10.0){\circle*{2.00}} \put(50.0,0.0){\line(1,0){10}} \put(60.0,0.0){\line(1,0){10}} \put(70.0,0.0){\line(1,0){10}} \put(80.0,0.0){\line(1,0){10}} \put(90.0,0.0){\line(1,0){10}} \put(70.0,0.0){\line(0,1){10}} \put(55,5){\makebox(0,0)[b]{$\textstyle E_7$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(62.00,22.00)(-1,-6.00) \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(30.0,0.0){\circle*{2.00}} \put(40.0,0.0){\circle*{2.00}} \put(50.0,0.0){\circle*{2.00}} \put(60.0,0.0){\circle*{2.00}} \put(70.0,0.0){\circle*{2.00}} \put(30.0,10.0){\circle*{2.00}} \put(10.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(1,0){10}} \put(40.0,0.0){\line(1,0){10}} \put(50.0,0.0){\line(1,0){10}} \put(60.0,0.0){\line(1,0){10}} \put(30.0,0.0){\line(0,1){10}} \put(15,5){\makebox(0,0)[b]{$\textstyle E_8$}} \end{picture}} \] then $\Pb$ is finite dimensional, if $\Gamma$ is an extended Dinkyn graphs of the type $\tilde D_4, \tilde E_6, \tilde E_7, \tilde E_8$ \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(72.00,32.00)(-1,-6.00) \put(-10.0,10.0){\circle*{2.00}} \put(5.0,10.0){\circle*{2.00}} \put(20.0,10.0){\circle*{2.00}} \put(5.0,25.0){\circle*{2.00}} \put(5.0,-5.0){\circle*{2.00}} \put(-10.0,10.0){\line(1,0){15}} \put(5.0,10.0){\line(1,0){15}} \put(5.0,10.0){\line(0,1){15}} \put(5.0,-5.0){\line(0,1){15}} \put(-5,20){\makebox(0,0)[b]{$\textstyle \tilde D_4$}} \put(40.0,5.0){\circle*{2.00}} \put(50.0,5.0){\circle*{2.00}} \put(60.0,5.0){\circle*{2.00}} \put(70.0,5.0){\circle*{2.00}} \put(80.0,5.0){\circle*{2.00}} \put(60.0,15.0){\circle*{2.00}} \put(60.0,25.0){\circle*{2.00}} \put(40.0,5.0){\line(1,0){10}} \put(50.0,5.0){\line(1,0){10}} \put(60.0,5.0){\line(1,0){10}} \put(70.0,5.0){\line(1,0){10}} \put(60.0,5.0){\line(0,1){10}} \put(60.0,15.0){\line(0,1){10}} \put(45,10){\makebox(0,0)[b]{$\textstyle \tilde E_6$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(72.00,22.00)(-1,-6.00) \put(-30.0,0.0){\circle*{2.00}} \put(-20.0,0.0){\circle*{2.00}} \put(-10.0,0.0){\circle*{2.00}} \put(0.0,0.0){\circle*{2.00}} \put(10.0,0.0){\circle*{2.00}} \put(20.0,0.0){\circle*{2.00}} \put(30.0,0.0){\circle*{2.00}} \put(0.0,10.0){\circle*{2.00}} \put(-30.0,0.0){\line(1,0){10}} \put(-20.0,0.0){\line(1,0){10}} \put(0.0,0.0){\line(1,0){10}} \put(-10.0,0.0){\line(1,0){10}} \put(10.0,0.0){\line(1,0){10}} \put(20.0,0.0){\line(1,0){10}} \put(0.0,0.0){\line(0,1){10}} \put(-25,5){\makebox(0,0)[b]{$\textstyle \tilde E_7$}} \put(40.0,0.0){\circle*{2.00}} \put(50.0,0.0){\circle*{2.00}} \put(60.0,0.0){\circle*{2.00}} \put(70.0,0.0){\circle*{2.00}} \put(80.0,0.0){\circle*{2.00}} \put(90.0,0.0){\circle*{2.00}} \put(100.0,0.0){\circle*{2.00}} \put(110.0,0.0){\circle*{2.00}} \put(60.0,10.0){\circle*{2.00}} \put(40.0,0.0){\line(1,0){10}} \put(50.0,0.0){\line(1,0){10}} \put(60.0,0.0){\line(1,0){10}} \put(70.0,0.0){\line(1,0){10}} \put(80.0,0.0){\line(1,0){10}} \put(90.0,0.0){\line(1,0){10}} \put(100.0,0.0){\line(1,0){10}} \put(60.0,0.0){\line(0,1){10}} \put(45,5){\makebox(0,0)[b]{$\textstyle \tilde E_8$}} \end{picture}} \] then the algebra $\Pb$ is infinite dimensional and of polynomial growth, and finally if $\Gamma$ neither a Dynkin graph or an extended graph 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 graph. 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 exist $*$-representations of $\Pb$ on the hyperplane. \section{Some statements and Coxeter functors} %In our work we take advantage of structure of the sets $\Sets$ where $\G$ is a Dynkin graph. %The desc %Article \cite{KPS05} gives full description of $\Sets$ for each Dynkin %graph and shows that such sets are finite. %this requires the knowledge of structure of the set $\Sigma_{D_4,\chi}$: % $$ % \Sigma_{D_4,(\alpha_1,\alpha_2,\alpha_3)}=\{0\} \cup \{\sum_{i \in J} \alpha_i, J \subset \{1,2,3\}\} \cup \{(\alpha_1+\alpha_2+\alpha_3)/2\} %\label{TD4} % $$ %\section{Coxeter functors} The powerful tool to investigate algebras $\Pb$ is Coxeter functors. there exist two functors (\cite{KRS02}) linear $S$ and hyperbolic $T$ which act between categories of $*$-representation of the algebras $\Pb$. Their actions on the pair $(\chi;\gamma)$ could be written as follows: $$ S:(\chi;\gamma)\longmapsto(\chi^*;\gamma), $$ $$ \chi^*=(\gamma-\alpha_{m_1}^{(1)},\ldots ,\gamma-\alpha_{1}^{(1)};\ldots;\gamma-\alpha_{m_n}^{(n)},\ldots,\gamma-\alpha_{1}^{(n)}),$$ $$ T:(\chi;\gamma)\longmapsto(\overline{\chi};\overline{\gamma}) $$ $$\overline{\chi}=(\alpha_{m_1}^{(1)}-\alpha_{m_1-1}^{(1)},\ldots,\alpha_{m_1}^{(1)};\ldots; \alpha_{m_n}^{(n)}-\alpha_{m_n-1}^{(n)},\ldots,\alpha_{m_n}^{(n)}) \\ $$ $$ \overline{\gamma}=\alpha_{m_1}^{(1)}+\ldots+\alpha_{m_n}^{(n)}-\gamma; $$ These two functors are being applied twice equal to identity, therefore it is interesting to use their composition $\Phi^+=S\circ T$. %we will use the composition %In studing sets $\Sigma_{\Gamma,\chi}$ we use the fact that when $\Gamma$ is an extended Dynkin graph then there exists invariant functionals %$\omega_{\Gamma}$ (see \cite{OS05}). Let $\omega(\chi)$ be a positive functional on $\Gamma$. We say that $\omega(\chi)$ is an invariant functional if the following conditions holds: \begin{align*} S:(\chi,\omega(\chi))\longmapsto(\chi',\omega(\chi')), \\ T:(\chi,\omega(\chi))\longmapsto(\chi'',\omega(\chi'')). \end{align*} The results of the recent paper (see \cite{OS05}) prove that if $\Gamma$ is an extended Dynkin graph then there is the only invariant functional: \begin{gather*} \omega_{\tilde D_4}(\chi)=\frac12(\alpha_1^{(1)}+\alpha_1^{(2)}+\alpha_1^{(3)}+\alpha_1^{(4)}),\\ \omega_{\tilde E_6}(\chi)=\frac13(\alpha_1^{(1)}+\alpha_2^{(1)}+\alpha_1^{(2)}+\alpha_2^{(2)}+\alpha_1^{(3)}+\alpha_2^{(3)}),\\ \omega_{\tilde E_7}(\chi)=\frac14(\alpha_1^{(1)}+\alpha_2^{(1)}+\alpha_3^{(1)}+\alpha_1^{(2)}+\alpha_2^{(2)}+\alpha_3^{(2)}+2\alpha_1^{(3)}),\\ \omega_{\tilde E_8}(\chi)=\frac16(\alpha_1^{(1)}+\alpha_2^{(1)}+\alpha_3^{(1)}+\alpha_4^{(1)}+\alpha_5^{(1)}+2\alpha_1^{(2)}+2\alpha_2^{(2)}+3\alpha_1^{(3)}).\\ \end{gather*} Recall that in case when $\gamma=\omega_\Gamma(\chi)$ the algebras $\Pa_{\G,\chi,\omega_\G(\chi)}$ are PI-algebra (see \cite{VMS05}) and irreducible representations are of dimension not greater that 2,3,4 and 6 respectively for $\tilde D_4$, $\tilde E_6$, $\tilde E_7$, and $\tilde E_8$ graphs. Usign these functionals it has been shown that an action of $\Phi^+$ functor could be written in the following way: \begin{theo}(see \cite{OS05}) Let $\Gamma$ be an extended Dynkin graph then following formulae holds \begin{equation} (\Phi^+)^n(\chi;\gamma)=(v(t,\chi)-u(t,n)\lambda;w_1(t,\chi)-w_2(n)\lambda), \label{F} \end{equation} where $\lambda=\omega_\Gamma(\chi)-\gamma$, $t=[\frac{n}{k_\Gamma}]$, $k_{\tilde D_4}=2, k_{\tilde E_6}=6, k_{\tilde E_7}=12, k_{\tilde E_8}=30$. Notice that the vector $v(t,n)$ (as well as numbers $w_1(t,n)$) depends on $\chi$ and independent on $\lambda$ and the components of the vector $u(t,n)$ are strictly increasing with n. \end{theo} \begin{lemma} Let $u_{p1}^{(s1)}(t,n)$ and $u_{p2}^{(s1)}(t,n)$ be the components of the vector $u(t,n)$ (here by component $u_i^{(j)}(t,n)$ we mean a component that corresponds to $\alpha_i^{(j)}$ in the character $\chi$) then $u_{p1}^{(s1)}(t,n)=u_{p2}^{(s2)}(t,n)$ iff $p1=p2$ and $u_{p1}^{(s1)}(t,n)<u_{p2}^{(s2)}(t,n)$ iff $p1<p2$. \end{lemma} \begin{proof} Direct calculations. \end{proof} Using the result of the Theorem 1 we can describe the sets $\Sigma_{\Gamma,\chi}$. Indeed, one can fix $\gamma$ and apply functor $\Phi^+$ until we can easily check are there representations of obtained algebra. This requires the knowledge of structure of the set $\Sigma_{\tilde \Gamma,\chi}$. Set $\Sigma_{\tilde D_4,\chi}$ was described using this technique (see next chapter), and the following complete description of the set $\Sigma_{D_4,\chi}$ (see \cite{KPS05}) \begin{equation} \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} \end{equation} but for other graphs this task appears to be combinatory hard and just one infinite series was found. \begin{lemma} \label{com} Let $\mathcal A_n$ be an algebra generated by n-tuples of projections $p_i^2=p_i^*=p_i$ and with the relation $$\alpha_1 p_1+\alpha_2 p_2+\cdots+\alpha_{n}p_{n}=\alpha_{n}e,$$ ($\alpha_i$ are positive real numbers and $e$ is unit element in $\mathcal A_n$), then such algebra has at least one representation. \end{lemma} \begin{proof} It is easily proved that the collection of the projections $P_i=0, i\neq n$ and $P_n=I$ form the representation of $\mathcal A_n$. \end{proof} \section{Description of the set $\Sigma_{\tilde D_4,\chi}$} Denote the character $\chi$ on $\tilde D_4$ by $\chi=(\alpha_1;\alpha_2;\alpha_3;\alpha_4)$ and $\alpha=\alpha_1+\alpha_2+\alpha_3+\alpha_4$. 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 study the set $\SD \cap[0,\frac{\alpha}{2})$. Notice that all components of the character $\chi$ are required to be less than $\frac{\alpha}{2}$. Indeed, if some coefficient is equal or greater than $\frac{\alpha}{2}$ then corresponding projection is equal to $0$ or $I$ in the representation and problem simplifies since instead of $\tilde D_4$ graph we obtain $D_4$ graph where solution is known. Having written the action of $(\Phi^+)^n$ functor on pair $(\chi;\gamma)$ (and applied (\ref{TD4}) and Lem.\ref{com}) we build the following series in $\SD$. \begin{lemma} \label{infD4} The set $\SD$ contains the infinity series $\Sigma_\infty$ with limit point $\frac\alpha2$ and the finite series $\Sigma_0$. This two series are described by the 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} Proofs for these statements can be found in \cite{YU05}. \section{Infinity of the set $\Sigma_{\Gamma,\chi}$} According to Lemma \ref{infD4} the set $\Sigma_{\tilde D_4,\chi}$ is infinite if and only if $\chi_i<\frac\alpha2$ (in other words it means that $\chi_i<\omega_{\tilde D_4}$). Let's study a similar question for all extended Dynkin graphs. Let $\chi_i$ be the component of the character $\chi$ and $\tilde{\chi_i}$ be the corresponding component of the character $\tilde{\chi}$ obtained as an action of the functor $S$ on the pair $(\chi,\gamma)$. 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 projection 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 a Dynkin graph). Thus the conditions $\chi_i<\omega_\Gamma(\chi)$ and $\tilde{\chi_i}<\omega_\Gamma(\tilde{\chi})$ are both necessary for the set $\Sigma_{\Gamma,\chi}$ to be infinite. Let $\Gamma=\tilde E_6$ and the latter inequalities hold, consider sets $A_{\chi_i}=\{v_i(j,\chi) | j=1,..,k_{\tilde E_6}=6\}$ and set $A=\cup A_{\chi_i}$. Put $a=\min A$ and $p$, $s$ such indexes that $v_p(s,\chi)=a$. The following proposition holds \begin{theo} \label{inf} Algebra $\Pa_{\tilde E_6,\chi,\gamma_n}$, where $\gamma_n=\omega_{\tilde E_6}(\chi)-\frac{a}{u_p(s,n)}$ has the representation for every natural $n>2$. \end{theo} \begin{proof} Fix $n \in \N$ and apply functor $\Phi^+$ $(n-2)$ times on the pair $(\chi;\omega_{\tilde E_6}(\chi)-\frac{a}{u_p(s,n)})$. For this action to be correct we have to show that on each step $k\leqslant n-2$ we will get the pair $(\chi(k);\gamma(k))$ where $\gamma(k)$ and all components of the $\chi(k)$ are positive. Indeed, consider for example $j$-th component of $\chi(k)$ on step $k\leqslant n-2$. Having made use of folmula (\ref{F}) we get: $$ \chi_j(k)=v_j(k\,\,mod\,\,6,\chi)-u_j(k\,\,mod\,\,6,n)\frac{a}{u_p(s,n)}\geqslant a\left(1-u_p(k\,\,mod\,\,6,n)\frac{1}{u_p(s,n)}\right) \geqslant 0$$ To conclude the proof, it remains to note that $\chi_p^{(s)}(n)=0$, hence corresponding component $\chi_p^{(s)}(n-2)=\gamma(n-2)$. According to lemma 2 algebra $\Pa_{\tilde E_6,\chi(n-2),\gamma(n-2)}$ has a representation. \end{proof} Using considered procedure for other extended Dynkin graphs proofs the following theorem: \begin{theo} Let $\Gamma$ be an extended Dynkin graph. The set $\Sigma_{\Gamma,\chi}$ is infinite if and only if each component of character satisfies two conditions: $\chi_i<\omega_{\Gamma}(\chi)$ and $\tilde{\chi_i}<\omega_\Gamma(\tilde{\chi})$. \end{theo} %\begin{proof} %Similarly to $\tilde D_4$ graph we will prove that for each extended graph $\G$ there is infinity %series of values $\gamma$, for which there exist representations of $\Pb$, that tends to point $\omega_\G(\chi)$. %Here we will show that %\end{proof} An interesting question is when there exist representations of algerbas $\Pa_{\G,\chi,\omega_\G(\chi)}$. This question was studied by Kyrychenko A.A. for $\tilde D_4$ graph and the answer is: representaion exist if the components of the character $\chi$ satisfy the conditions of the previous theorem. The following corollory shows that for other extended graphs the result is the same. \begin{coll}(Representation on the hyperplane) Let $\G$ be an extended Dynkin graph, 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{coll} \begin{proof} Note that due to Shulman's theorem (see \cite{SHU01}) the sets $\Sets$ are closed. Since the conditions of Th. 4 are satisfied there is the series in $\Sets$ with the limit point $\omega_\G(\chi)$. Hence the set $\Sets$ contains point $\omega_\G(\chi)$. \end{proof} The author is grateful to his supervisor Ostrovskii V.L. for constant attention to this work. \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. 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