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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 existence of $*$-representations of certain algebras related to extended Dynkin graphs} %\thanks{This work is partically supported by grant DFFD Ukraine, grant \No 01.07/071} \author{Kostyantyn Yusenko} \email{kay@imath.kiev.ua} \newtheorem{lemma}{Lemma} \newtheorem{theo}{Theorem} \newtheorem{prop}{Proposition} \newtheorem{rem}{Remark} \newtheorem{coll}{Corollary} \newtheorem{defi}{Definition} \subjclass[2005]{Primary 47A62, 17B10, 16G20} \keywords{Operator algebras, additive spectral problem, extended Dynkin graphs, $*$-representations, Coxeter functors. \\ This work was partially supported by the State Foundation of Fundamental Research of Ukraine, grant no. 01.07/071} \begin{abstract} For associated with the extended Dynkin graphs $*$-algebras we investigate parameters set, for which there exist representations. Given result in paper is the some topological properties of such sets and complete description of set related to graph $\tilde D_4$. \end{abstract} \maketitle \section*{Introduction} In \cite{FU,ZS04} (see also cited bibliography therein) the following problems were studied, let $M_i=\{0=\alpha_0^{(i)}<\alpha_1^{(i)}<\ldots<\alpha_{m_i}^{(i)}\}, \ i=1,\ldots,n$ be given finite subsets of $\Rp$ and $\gamma \in \Rp$. The problem is to determine whether there exist n-tuples of Hermitian operators $A_i=A_i^*, \, i=1,\ldots,n$ such that spectra $\sigma(A_i) \subset M_i$ and $$ A_1+A_2+\ldots+A_n=\gamma I, $$ 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\ldots 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+\ldots+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\ldots 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 $\Pa_{M_1,M_2\ldots M_n;\gamma}$ one can assign the connected non-oriented graph $\Gamma$ that has n branches connected in the common vertex (the root), such that i-th branch has $m_i$ vertices, $i=1,\ldots,n$. Starting with $\alpha_j^{(i)}$ we construct a function $\chi$ (we will call it character of algebra) on the set of vertices except for the root in the following way: $\chi_j^{(i)}$ (i-th branch, j-th vertex) equals to $\alpha_j^{(i)}$, the root of the tree is corresponded to $\gamma$. Character $\chi$ could be written as the vector $\chi=(\alpha_1^{(1)},\ldots,\alpha_{m_1}^{(1)};\ldots;$ $\alpha_1^{(n)},\ldots,\alpha_{m_n}^{(n)})$. An algebra $\Pa_{M_1,M_2\ldots M_n;\gamma}$ is uniquely given by graph $\Gamma$, character $\chi$, and $\gamma$, hence we will denote it by $\Pa_{\Gamma,\chi,\gamma}$ in the sequel. The additive spectral problem is equivalent to the following tasks to be solved: \begin{enumerate} \item a) to describe the set $\Sigma_\Gamma=\{(\chi;\gamma) |$ there there exists a representation of algebra $\Pb\}$\\ b) for each character $\chi$ to describe the set $\Sigma_{\Gamma,\chi}=\{\gamma \in \Rp |$ there exists a representation of algebra $\Pb\}$ \item for every pair $(\chi;\gamma) \in \Sigma_\Gamma$ to describe all irreducible $*$ representation of $\Pb$. \end{enumerate} Depending on the properties of the graph $\G$, the structure of representations of $\Pb$ is quite different. The 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 Dynkin graph 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 nor an extended graph then $\Pb$ contains a free algebra with two generators (in this case task (2) could be too complicated). Attention in this article is concentrated at studing sets $\Sigma_{\Gamma,\chi}$ in the case where $\Gamma$ 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 special case where $\gamma=\omega_{\G}$ (see section \ref{om} for the difinition of $\omega_{\G}$). \section{Description of set $\Sigma_{\tilde D_4,\chi}$} As it was shown in \cite{YU05} the set $\Sigma_{\tilde D_4,\chi}$ can be reduced to the structure of set $\Sigma_{D_4,\chi}$. Complete description of set $\Sigma_{D_4,\chi}$ was given in \cite{KPS05}: $$ \Sigma_{D_4,(\alpha_1,\alpha_2,\alpha_3)}=\{0, (\alpha_1+\alpha_2+\alpha_3)/2\} \cup \{\sum_{i \in J} \alpha_i, J \subset \{1,2,3\}\}, \label{TD4} $$ Let $\alpha_i, i=1,\ldots,4$ denote i-th component of the character $\chi$ and $\alpha=\alpha_1+\alpha_2+\alpha_3+\alpha_4$. The set $\Sigma_{\tilde D_4,\chi}$ satisfy the following properties (see \cite{YU05}) \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} Third property means that $\SD$ is symmetric around $\frac{\alpha}{2}$, therefore we will study the set $\SD \cap[0,\frac{\alpha}{2})$. Notice that in the case where at least one components of the character $\chi_i\geqslant \alpha$ it isn't hard to show that corresponding projection equals to $0$ or $I$ in the representation, hence the structure of set $\SD$ is the same to the structure of $\Sigma_{D_4,\chi}$. Therefore it is interesting to study the case where all components of character $\chi$ are less than $\frac{\alpha}{2}$. In this case the set $\SD$ is quite different from $\Sigma_{D_4,\chi}$, furthermore it is infinite. The following propositions give complete description of the set $\SD$ (exact technique and proofs can be found in \cite{YU05}) \begin{lemma} \label{infD4} The set $\SD$ contains the infinite 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 \\ $\Sigma_0=\left\{\frac{\alpha}{2}-\frac{\alpha-2\alpha_4}{2(2n-1)} | n<\frac{\alpha_1}{\alpha_2+\alpha_3-\alpha_1-\alpha_4}, n \in \N\right\}$ \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 \\ $\Sigma_0=\left\{\frac{\alpha}{2}-\frac{\alpha_1}{2n} | n<\frac{\alpha_1}{\alpha_1+\alpha_4-\alpha_2-\alpha_3}, n \in \N\right\}$ \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} | \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(4\alpha_i-\alpha)<\alpha_i, 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} \begin{coll} By using the structure of $\SD$ given by the latter theorem one can show that the set $\SD$ contains the only limit point $\omega_{\tilde D_4}(\chi)$. \end{coll} \section{Coxeter functors and evolution of characters} \label{om} %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 for investigation algebras $\Pb$ is the reflection (Coxeter) functors. Namely there exist two functors \cite{KRS02} linear $S$ and hyperbolic $T$, which establish the equivalence of categories of $*$-representation of the algebras $\Pb$. These actions between categories give rise to the action on the pairs $(\chi;\gamma)$ as follows: $$ S:(\chi;\gamma)\longmapsto(\chi';\gamma'), $$ $$ {\chi'}=(\alpha_{m_1}^{(1)}-\alpha_{m_1-1}^{(1)},\ldots,\alpha_{m_1}^{(1)};\ldots; \alpha_{m_n}^{(n)}-\alpha_{m_n-1}^{(n)},\ldots,\alpha_{m_n}^{(n)}) \\ $$ $$ {\gamma'}=\alpha_{m_1}^{(1)}+\ldots+\alpha_{m_n}^{(n)}-\gamma; $$ $$ T:(\chi;\gamma)\longmapsto(\chi'';\gamma) $$ $$ \chi''=(\gamma-\alpha_{m_1}^{(1)},\ldots ,\gamma-\alpha_{1}^{(1)};\ldots;\gamma-\alpha_{m_n}^{(n)},\ldots,\gamma-\alpha_{1}^{(n)}), $$ %These two functors are being applied twice equal to identity. %therefore it is interesting to use their composition %$\Phi^+=S\circ T$. The main idea is to take the pair ($\chi;\gamma)$, such that the structure of the representations of $\Pb$ is known, and apply $S$ and $T$ functors to build the whole series of algebras which has the same structure of the representations as the algebra we started with. To make use of this technique we first study the evolution of the pair $(\chi;\gamma)$ under the action of Coxeter functors. In the general case for an arbitrary graph $\G$ the formulas of the evolution could be complicated, but for the case where $\G$ is an extended Dynkin graph the results of \cite{OS05} give an explicit formulae for the powers of $(ST)^{k_{\G}}$ functor. %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 the set of characters. 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 \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 the case where $\gamma=\omega_\Gamma(\chi)$ the algebras $\Pa_{\G,\chi,\omega_\G(\chi)}$ are PI-algebra (see \cite{Mel05}) and their irreducible representations are of dimension not greater that 2,3,4 and 6 respectively for the $\tilde D_4$, $\tilde E_6$, $\tilde E_7$, and $\tilde E_8$ graphs. Put $p_{\G}=2,3,4,6$ for $\G=\tilde D_4$, $\tilde E_6$, $\tilde E_7$, and $\tilde E_8$ respectively. An evolution of the pair $(\chi;\gamma)$ under the action of the powers of $(ST)^{p_{\G}(p_{\G}-1)}$ functor could be written as follows: \begin{theo}(see \cite{OS05}) Let $\Gamma$ be an extended Dynkin graph then following formula holds \begin{equation} (ST)^{p_{\G}(p_{\G}-1)k}(\chi;\gamma)=(\chi-kp_{\G}(\omega_{\G}(\chi)-\gamma)\chi_{\G};\gamma-kp^2_{\G}(\omega_{\G}(\chi)-\gamma)), \label{F}%(\Phi^+)^n(\chi;\gamma)=(v(t,\chi)-u(t,n)\lambda;w_1(t,\chi)-w_2(n)\lambda), \label{F} \end{equation} where $\chi_{\G}$ is special character on $\G$ (see \cite{OS05-2}). \end{theo} Applying $(ST)$ functor to (\ref{F}) we can obtain the evolution of the pair $(\chi;\gamma)$ under an action of an arbitrary power $k \in \N$ of $(ST)^k$ (in what follows we denote by $\left(\chi(k);\gamma(k)\right)$ an image of the pair $(\chi;\gamma)$ under an action of $(ST)^k$ functor) \begin{prop} The action of $(ST)^k$ functor on the pair $(\chi;\gamma)$could be written in the following way: \begin{equation} (ST)^{k}(\chi;\gamma)=(f_{1,k}(\chi)-(\omega_{\G}(\chi)-\gamma)f_{2,k}(\chi_{\G});\psi_{1,k}-(\omega_{\G}(\chi)-\gamma)\psi_{2,k}), \label{F2} \end{equation} where the characters $f_{1,k}(\chi)$ and $f_{2,k}(\chi_{\G})$, and the numbers $\psi_{1,k}$ and $\psi_{2,k}$ satisfy the following properties: \begin{itemize} \item[\it{(i)}] if $k_1\equiv k_2 \pmod{p_ {\G}(p_{\G}-1)}$ then $f_{1,k_1}(\chi)=f_{1,k_2}(\chi)$ and $\psi_{1,k_1}=\psi_{1,k_2}$; \label{orbit} \item[\it{(ii)}] the components of $f_{2,k}(\chi_{\G})$ and the numbers $\psi_{2,k}$ are defined in the following way: $$ f_{2,k}(\chi_{\G})_i^{(j)}=\left[\frac{(\chi_{\G})_{i}^{(j)}}{p_{\G}-1} k\right], \, \, \psi_{2,k}=\left[\frac{p_{\G}}{p_{\G}-1} k\right], $$ \item[\it{(iii)}] $f_{1,p_{\G}(p_{\G}-1)k}=\chi$, $f_{2,p_{\G}(p_{\G}-1)k}=kp_{\G}\chi_{\G}$, $\psi_{1,k}=\gamma$, and $\psi_{2,k}=kp_{\G}^2$. \end{itemize} % components of the character $f_{2,k}(\chi_{\G})$ are %defined in the following way: %$$ % f_{2,k}(\chi_{\G})_i^{(j)}=\left[\frac{(\chi_{\G})_{i}^{(j)}}{(\chi_{\G})_{m_j}^{(j)}} % k\right], %$$ %(for example the character $f_{2,k}(\chi_{\G})$ equals %$\left(\left[\frac{k}{2}\right],k;\left[\frac{k}{2}\right],k;\left[\frac{k}{2}\right],k\right)$ %where $\G=\tilde E_6$), and characters $f_{2,k}(\chi_{\G})$ %satisfy the condition: %\begin{equation} % k_1\equiv k_2 (\mod p_ {\G}(p_{\G}-1)) \Rightarrow % f_{1,k_1}(\chi)=f_{1,k_2}(\chi). \label{orbit} %\end{equation} \end{prop} \begin{proof} Direct calculations using formula (\ref{F}). \end{proof} \begin{rem} Property (i) means that the orbits of the $f_{1,0}(\chi)$ and $\psi_{1,0}$ under the action of $(ST)$ functor is finite and its length equal to $p_{\G}(p_{\G}-1)$. The formulas for the charactes $f_{1,k}$ and numbers $\psi_{1,k}$ are complicated (unlike characters $f_{2,k}(\chi_{\G})$ and numbers $\psi_{2,k}$) and we do not give the list of their evolutions (this list contains 30 items in the case where $\G=\tilde E_8$). Nevertheless using (\ref{F}) one can compute their values. \end{rem} \section{Technique} Notice that knowing the set $\Sets\cap[0,\omega_{\G}(\chi)]$ one can restore the whole set $\Sets$, indeed to get the set $\Sets \backslash [0,\omega_{\G}(\chi)]$ we will make use of $S$ functor to the pair $(\chi';\gamma)$, where $\gamma$ is taken from $\Sigma_{\G,\chi'}\cap[0,\omega_{\G}(\chi'))$, and we will obtain the pair $(\chi;\tilde \gamma)$ but with $\tilde \gamma \notin [0,\omega_{\G}(\chi)]$. \begin{defi} Let $\pi: \Pa_{\Gamma,\chi,\gamma} \rightarrow L(\mathcal{H})$ be a finite dimensional $*$-representation in Gilbert space $\mathcal{H}$. The generalized dimension of $\pi$ we call the vector $d$ defined in the following way: $$ d_0=dim(\mathcal{H}), \,\,\ d_i^{(j)}=dim(Im(\pi(p_i^{(j)}))), \, j=1,\ldots,n, \,\, i=1,\ldots,m_j. $$ \end{defi} One can extend the action of the functors $S$ and $T$ on the set of general dimensions (see for example \cite{KRS02}). Following formulas (\ref{F2}) Let us fix $\gamma<\omega_{\G}(\chi)$ and apply functor $(ST)$ to the pair $(\chi;\gamma)$ until we get (in finite numbers of steps) one of the three following situations: %$(\chi(k);\gamma(k))$ denotes $(ST)^k(\chi;\gamma)$, following formulas %(\ref{F2}) on some step $k$ one of three cases is possible: \begin{itemize} \item[(a)] $\gamma(k)=0$. In this case there exist $*$-representation of the algebra $\Pa_{\G,\chi(k),\gamma(k)}$ with generalized dimension $d(k)$ defined by $$d(k)_0=0, \ \ d(k)_i^{(j)}=0.$$ \item[(b)] $\chi(k)_1^{(j)}=\gamma(k)$, for $j \in J \subset \{1,\ldots,n\}.$ In this case there exist $*$-representation of the algebra $\Pa_{\G,\chi(k),\gamma(k)}$ with generalized dimension $d(k)$ defined by \begin{align*} d(k)_0=1, \ \ & d(k)_1^{(j)}=1 \ \ if \ \ j \in J, \\ & d(k)_i^{(j)}=0 \ \ if \ \ i=1, \ j \notin J \ \ or \ \ i\neq 1. \end{align*} Notice that this case corresponds to the case where $\chi(k_1)_1^{(j)}=0$, for $j \in J.$ \item[(c)] $\chi(k)_i^{(j)}<0$, for $j \in J' \subset \{1,\ldots,n\}$ and $i \in J'' \subset \{1,\ldots,m_j\}.$ In this case we go back to the first step $\tilde k<k$, where all components of the character $\chi(\tilde k)$ are positive, and construct new graph $\tilde \G$ by deleting vertices from $G$ that correspond to the sets $J'$ and $J''$ (branch from $J'$ and vertex form $J''$). Since we know the exact structure of the set $\Sigma_{\tilde \G,\tilde \chi}$ (see \cite{KPS05}), where $\tilde \chi$ is the restriction of $\chi(\tilde k)$ on $\tilde \G$, we can determine for $\gamma(\tilde k)$ to lie in $\Sigma_{\tilde \G,\tilde \chi}$, and if the latter is true to build corresponding vector of generalized dimension $d(\tilde k)$. \end{itemize} So to each pair $(\chi;\gamma) \in \Sigma_{\G}$ we can associated the vector of generalized dimension by above rule, hence we can split the set $\Sets\cap[0,\omega_{\G}(\chi)]$ into the set $\Sigma_{d_i}$ indexed by the set of generalized dimensions. \begin{lemma} Index $d_i$ takes on a finite number of posible generalized dimension. \end{lemma} \begin{proof} Each $d_i$ corresponds to case (a), (b) or (c). Clear that in the cases (a), (b) the number of different generalized dimension is finite. In case when $\tilde \G$ is the proper subgraph of the extended Dynkin graph (see \cite{KPS05}) the number of different generalized dimension is also finite, hence it proofs the case where $d_i$ corresponds to (c) and lemma holds. \end{proof} \begin{lemma} The sets $\Sigma_{d_i}$ are eather finite or has limit point $\omega_{\G}(\chi)$. \end{lemma} \begin{proof} Direct calculation using (\ref{F2}). \end{proof} %Using brute force To describe the set $\Sets\cap[0,\omega_{\G}(\chi)]$ Although we have exact algorithm to get the explicit formulas turns out to be combinatory hard for extended Dynkin graph $\G\neq\tilde D_4$ (case $\G=\tilde D_4$ is simpler, and complete description was done in previous section), so using (\ref{F2}) we will study structure properties of $\Sigma_{\G,\chi}$. %In what follows we also use the following simple lemma: %\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 seen 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{Structure properties of set $\Sigma_{\Gamma,\chi}$} According to Lemma \ref{infD4} the set $\Sigma_{\tilde D_4,\chi}$ is infinite if and only if all components of character $\chi$ satisfy the condition $\chi_i<\frac\alpha2$ (in other words it means that $\chi_i<\omega_{\tilde D_4}(\chi)$). Let's study a similar question for all extended Dynkin graphs. Let $\chi_i$ be the i-th component of the character $\chi$ and ${\chi'_i}$ be the corresponding component of the character $\chi'$ obtained as an action of functor $S$ on the pair $(\chi,\gamma)$. \begin{theo} \label{inf} Let $\Gamma$ be an extended Dynkin graph. The set $\Sigma_{\Gamma,\chi}$ is infinite if and only if all components of character satisfies two conditions: $\chi_i<\omega_{\Gamma}(\chi)$ and ${\chi'_i}<\omega_\Gamma(\chi')$. \end{theo} \begin{proof} We prove this theorem in few steps. At first we show that the conditions $\chi_i<\omega_\Gamma(\chi)$ and $\chi'_i<\omega_\Gamma(\chi')$ are both necessary for the set $\Sigma_{\Gamma,\chi}$ to be infinite. \begin{lemma} If at least one component of character $\chi$ or character $\chi'$ satisfy the condition $\chi_i\geq \omega_\Gamma(\chi)$ or ${\chi'_i} \geq \omega_\Gamma(\chi')$ then the set $\Sigma_{\Gamma,\chi}$ is finite. \end{lemma} \begin{proof} It isn't hard to check that the projection corresponding to such component of $\chi$ that satisfy the condition is equal to $0$ or to $I$ in the representation. Hence the set $\Sigma_{\Gamma,\chi}$ has the same structure as the set $\Sigma_{\tilde \Gamma,\chi}$ where $\tilde \G$ is the proper subgraph of $\G$. Since the set $\Sigma_{\Gamma,\chi}$ is always finite when $\Gamma$ is a Dynkin graph (see \cite{KPS05}) the set $\Sigma_{\Gamma,\chi}$ is also finite. \end{proof} \begin{rem} Conditions $\chi_i<\omega_\Gamma(\chi)$ and $\tilde{\chi_i}<\omega_\Gamma(\tilde{\chi})$ are equivalent if $\G=\tilde D_4$, but are not generally speaking if $\G\neq\tilde D_4$. For example let's consider the character $\chi=(5,6;7,8;8,9)$ on graph $\G=\tilde E_6$, all components of $\chi$ satisfy $\chi_i<\omega_{\tilde E_6}(\chi)$ but for coresponding character $\chi'=(1,6;1,8;1,9)$ they do not. \end{rem} To prove that the conditions $\chi_i<\omega_\Gamma(\chi)$ and $\chi'_i<\omega_\Gamma(\chi')$ are sufficient we will consider special procedure (on the example where $\G=\tilde E_6$) which allows us to build infinite series in $\Sigma_{\Gamma,\chi}$. Let $\Gamma=\tilde E_6$ and the latter inequalities hold, consider sets $$A_{\chi_1^{(j)}}=\{(f_{1,k}(\chi))_{1}^{(j)} | k=1,\ldots,p_{\tilde E_6}(p_{\tilde E_6}-1)=6\}$$ - the sets of orbits of the components $f_{1,0}(\chi)_1^{(j)}$ under an action of $(ST)$ functor, and consider the set $$A=\bigcup\limits_{j=1}^{3} A_{\chi_1^{(j)}}.$$ Put $a=\min A$ and $l$, $m$ corresponding to $a$ indexes, i.e. such indexes that $(f_{1,m}(\chi))_{l}=a$. Consider sequence $\gamma_n=\omega_{\tilde E_6}(\chi)-\frac{a}{\varphi_n}$, where $\varphi_n=(f_{2,p_{\tilde E_6}(p_{\tilde E_6}-1)n+m}(\chi_{\tilde E_6}))_l)=\left[\frac{6n+m}{2}\right]$, and $n \in \N$. The following proposition holds \begin{lemma} The algebra $\Pa_{\tilde E_6,\chi,\gamma_n}$ has a representation for every natural $n$. \end{lemma} \begin{proof} Fix $n \in \N$ and apply functor $(ST)^{6n+m-2}$ to the pair $(\chi;\omega_{\tilde E_6}(\chi)-\frac{a}{\varphi_n})$. To check that this action is correct we have to show that on each step $k\leqslant 6n+m-2$ 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 the component $\chi(k)_i^{(j)}$ on step $k\leqslant 6n+m-2$. Using folmula (\ref{F2}) we get: \begin{equation*} \begin{array}{rcl} \chi(k)_i^{(j)} & = & (f_{1,k\bmod 6}(\chi))_i^{(j)}-(f_{2,k}(\chi_{\tilde E_6}))_i^{(j)}\frac{a}{\varphi_n}\geqslant \\ &\geqslant& a\left(1-\left[\frac{k}{2}\right] \left[\frac{6n+m}{2}\right]^{-1} \right) \geqslant 0 \end{array} \end{equation*} To complete the proof it remains to note that $\chi(6n+m)_l=0$, hence corresponding component $\chi(6n+m-2)_l=\gamma(6n+m-2)$. According to Lemma 1 the algebra $\Pa_{\tilde E_6,\chi(6n+m-2),\gamma(6n+m-2)}$ has a representation. \end{proof} Notice that the same procedure was used to build infinite series in the case $\G=\tilde D_4$ (see Lemma 2). This procedure could be slightly modified for the cases $\Gamma=\tilde E_7$ and $\Gamma=\tilde E_8$, hence the theorem holds \end{proof} An interesting question is when there exist a $*$-representation of algerbas $\Pa_{\G,\chi,\gamma}$ in the case $\gamma=\omega_\G(\chi)$. When $\G=\tilde D_4$ this question was studied in \cite{K04} and the answer is: a representation exists if all components of the character $\chi$ satisfy the condition $\chi_i<\omega_{\tilde D_4}(\chi)$. The similar answer appers to be correct for all extended Dynkin graphs. \begin{coll}(Representation in the case $\gamma=\omega_{\G}(\chi)$) 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} Since the conditions of Theorem \ref{inf} are satisfied there exist the series in $\Sets$ with the limit point $\omega_\G(\chi)$. By Shulman's theorem (see \cite{SHU01}) the sets $\Sets$ are closed, therefore the set $\Sets$ contains the point $\omega_\G(\chi)$. \end{proof} \begin{theo} Let $\Gamma$ be an extended Dynkin graph. If the set $\Sets$ is infinite then it contains the only limit point. \end{theo} \begin{proof} This theorem is obvious collorally of Lemma 2 and Lemma 3. % According to our technique (see Section 1) the set $\Sets$ could be given in the following form: %$$ % \Sets\cap[0,\omega_{\G}(\chi))=\bigcup\limits_{i}\Sigma_i, %$$ %where $\Sigma_i$ \ldots. %The length of orbit of $f_{1,0}(\chi)$ is finite and if $\tilde \G$ is proper subgraph of extended Dynkin graph then the set $\Sigma_{\tilde\G}$ is %finite (see \cite{KPS05}), hence index $i$ takes on a limited number of values. Notice also that according to building $\Sigma_i$ this set is either %finite or tends to the point $\omega_{\G}(\chi)$. 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