\documentclass[a4paper,12pt]{article} \usepackage[cp1251]{inputenc} \usepackage[english]{babel} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{amssymb} % Title Page \title{} \author{} \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{\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}}} \newtheorem{theo}{Theorem} \newtheorem{coll}{Corollary} \newtheorem{lemma}{Lemma} \large{ {\bf \vskip 6cm \centerline{ On parameters sets of algebras $\mathcal P_{\Gamma,\chi,\gamma}$} \centerline{when $\Gamma$ is an extended Dynkin diagram} \vskip 2cm \centerline{ Kostyantyn Yusenko} \vskip 2cm \centerline{Kiev, Institute of Mathematics} \vskip 1.5cm } } \newpage \centerline{\textbf{1. $*$-Algebras related to graphs}} \vskip 1cm \large{ Consider $*$-algebras related to graphs \begin{align*} \Pb=\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*} } Such algebras \newpage \centerline{\textbf{2. Extended Dynkin graphs}} \vskip 1cm In this talk we consider $\Pb$ algebras related to extended Dynkin graphs and study whether there exist $*$-representations of such algebras. \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(100,40.00)(-1,-6.00) \put(25.0,10.0){\circle*{2.00}} \put(40.0,10.0){\circle*{2.00}} \put(55.0,10.0){\circle*{2.00}} \put(40.0,25.0){\circle*{2.00}} \put(40.0,-5.0){\circle*{2.00}} \put(25.0,10.0){\line(1,0){15}} \put(40.0,10.0){\line(1,0){15}} \put(40.0,10.0){\line(0,1){15}} \put(40.0,-5.0){\line(0,1){15}} \put(30,20){\makebox(0,0)[b]{$\textstyle \tilde D_4$}} \end{picture}} \] \vskip 1cm \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(100.00,40.00)(-1,-6.00) \put(0.0,5.0){\circle*{2.00}} \put(20.0,5.0){\circle*{2.00}} \put(40.0,5.0){\circle*{2.00}} \put(60.0,5.0){\circle*{2.00}} \put(80.0,5.0){\circle*{2.00}} \put(40.0,20.0){\circle*{2.00}} \put(40.0,35.0){\circle*{2.00}} \put(0.0,5.0){\line(1,0){20}} \put(20.0,5.0){\line(1,0){20}} \put(40.0,5.0){\line(1,0){20}} \put(60.0,5.0){\line(1,0){20}} \put(40.0,20.0){\line(0,1){15}} \put(40.0,5.0){\line(0,1){15}} \put(10,10){\makebox(0,0)[b]{$\textstyle \tilde E_6$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(100.00,40.00)(-1,-6.00) \put(-20.0,5.0){\circle*{2.00}} \put(0.0,5.0){\circle*{2.00}} \put(20.0,5.0){\circle*{2.00}} \put(40.0,5.0){\circle*{2.00}} \put(60.0,5.0){\circle*{2.00}} \put(80.0,5.0){\circle*{2.00}} \put(100.0,5.0){\circle*{2.00}} \put(40.0,20.0){\circle*{2.00}} \put(-20.0,5.0){\line(1,0){20}} \put(80.0,5.0){\line(1,0){20}} \put(0.0,5.0){\line(1,0){20}} \put(20.0,5.0){\line(1,0){20}} \put(40.0,5.0){\line(1,0){20}} \put(60.0,5.0){\line(1,0){20}} \put(40.0,5.0){\line(0,1){15}} \put(10,15){\makebox(0,0)[b]{$\textstyle \tilde E_7$}} \end{picture}} \] \[ \lower12pt\hbox{\unitlength2pt\begin{picture}(100.00,40.00)(-1,-6.00) \put(0.0,5.0){\circle*{2.00}} \put(20.0,5.0){\circle*{2.00}} \put(40.0,5.0){\circle*{2.00}} \put(60.0,5.0){\circle*{2.00}} \put(80.0,5.0){\circle*{2.00}} \put(100.0,5.0){\circle*{2.00}} \put(120.0,5.0){\circle*{2.00}} \put(140.0,5.0){\circle*{2.00}} \put(40.0,20.0){\circle*{2.00}} \put(100.0,5.0){\line(1,0){20}} \put(120.0,5.0){\line(1,0){20}} \put(80.0,5.0){\line(1,0){20}} \put(0.0,5.0){\line(1,0){20}} \put(20.0,5.0){\line(1,0){20}} \put(40.0,5.0){\line(1,0){20}} \put(60.0,5.0){\line(1,0){20}} \put(40.0,5.0){\line(0,1){15}} \put(10,15){\makebox(0,0)[b]{$\textstyle \tilde E_8$}} \end{picture}} \] In what follows $\Sets$ denotes the set of those $\gamma$ for which there exist $*$-representation of $\Pb$. \newpage \centerline{\textbf{3. Coxeter functors}} \vskip 1cm The powerful tool to investigate algebras $\Pb$ is Coxeter functors. There are two functors of linear $$S:\mathrm{Rep}\Pb\rightarrow \mathrm{Rep}\Pa_{\Gamma,\chi^*,\gamma},$$ $$ \chi^*=(\gamma-\alpha_{m_1}^{(1)},\ldots ,\gamma-\alpha_{1}^{(1)};\ldots;\gamma-\alpha_{m_n}^{(n)},\ldots,\gamma-\alpha_{1}^{(n)}),$$ and hyperbolic $$T:\mathrm{Rep}\Pb\rightarrow \mathrm{Rep}\Pa_{\Gamma,\chi^\prime,\gamma^\prime},$$ $$ \chi^\prime=(\alpha_{m_1}^{(1)}-\alpha_{m_1-1}^{(1)},\ldots,\alpha_{m_1}^{(1)};\ldots; \alpha_{m_n}^{(n)}-\alpha_{m_n-1}^{(n)},\ldots,\alpha_{m_n}^{(n)})$$ $$ \gamma^\prime=\alpha_{m_1}^{(1)}+\ldots+\alpha_{m_n}^{(n)}-\gamma;$$ \vskip 1cm These two functors allow us to construct all irreducible representations of $\Pb$ from the one dimensional representations. \vskip 1cm In what follows $\Phi^+$ denotes the composition $\Phi^+=S\circ T$. \newpage \centerline{\textbf{4. Invariant functionals}} \vskip 1cm 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*} It has been proved that if $\Gamma$ is an extended Dynkin diagram then there exists 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*} In case where $\gamma=\omega_\Gamma(\chi)$ the algebras $\Pa_{\G,\chi,\omega_\G(\chi)}$ are PI-algebra and their 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 $\tilde E_8$ diagrams. \ Usign these functionals an action of $\Phi^+$ on the pair $(\chi,\gamma)$ functor could be written in the following way: $$(\Phi^+)^n(\chi;\gamma)=(v(t,\chi);w_1(t,\chi))-\lambda(u(t,n);w_2(n)), \label{F}$$ 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 $v(t,n)$ and 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. \newpage \centerline{\textbf{5. Description of the set $\Sigma_{\tilde D_4,\chi}$}} \vskip 1cm In case where $\G=\tilde D_4$ we have the character $\chi_{\tilde D_4}=(\alpha_1;\alpha_2;\alpha_3;\alpha_4;)$ and $*$-algebra: $$\Pa_{\tilde D_4,\chi,\gamma}=\mathbb C \langle p_1,p_2,p_3,p_4 | p_i=p_i^2=p_i^*, \alpha_1 p_1+\alpha_2 p_2+\alpha_3 p_3+\alpha_4 p_4=\gamma e\rangle. $$ We assume that all components of the character satisfy the condition $\alpha_i<\frac{\alpha}{2}$, where $\alpha=\alpha_1+\alpha_2+\alpha_3+\alpha_4$. One can show that in case one components doesn't satisfy this condition we get algebra with the relation: $$\alpha_1^\prime p_1+\alpha_2^\prime p_2+\alpha_3^\prime p_3=\gamma^\prime e,$$ where the answer in known. Using considered technique we have described the set $\Sigma_{\tilde D_4,\chi}$ \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} \newpage \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 $ \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\}, $ \begin{align*} \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\}, \\ \end{align*} \begin{align*} \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} \newpage \centerline{\textbf{6. Infinity of the sets $\Sigma_{\Gamma,\chi}$}} \vskip 1cm When sets $\Sigma_{\Gamma,\chi}$ are infinite? \vskip 0.5 cm One can show 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 $I$, hence the sets $\Sigma_{\Gamma,\chi}$ are finite. \vskip 0.5 cm We prove that if all components of the character satisfy the following conditions: $$ \chi_i<\omega_\Gamma(\chi); $$ $$ \tilde{\chi_i}<\omega_\Gamma(\tilde{\chi}); $$ then the sets $\Sigma_{\Gamma,\chi}$ are infinite. To show this we build the infinite series of $\gamma$ for each there exist representation of $\Pb$. \vskip 0.5 cm Let's see how it works when $\Gamma=\tilde E_6$. \begin{enumerate} \item Consider the sets $A_{\chi_i}=\{v_i(j,\chi) | j=1,..,k_{\tilde E_6}=6\}$ and the set $A=\cup A_{\chi_i}$; \item Put $a=\min A$; \item Put $p$, $s$ such indexes that $v_p(s,\chi)=a$ \end{enumerate} \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} {\it"Proof"} Verify that it is possible to use $\Phi^+$ functor $(n-2)$ times. Show that we obtain the algebra with the relation: $$ \alpha_1^\prime p_1+\alpha_2^\prime p_2+\alpha_3^\prime p_3+\alpha_4^\prime p_4+\alpha_5^\prime p_5+\alpha_6^\prime p_6=\alpha_6^\prime e.$$ This procedure could be applied for all extended Dynkin graphs, hence we prove the following theorem: \begin{theo} Let $\Gamma$ be an extended Dynkin diagram. The set $\Sigma_{\Gamma,\chi}$ is infinite iff each component of character satisfies two conditions: $\chi_i<\omega_{\Gamma}(\chi)$ and $\tilde{\chi_i}<\omega_\Gamma(\tilde{\chi})$. \end{theo} \vskip 1cm \begin{coll}(Representation on the hyperplane $\gamma=\omega_{\G}(\chi)$) 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 exist a representation of algebra $\Pa_{\G,\chi,\omega_\G(\chi)}$ \end{coll} {\it"Proof"} According to Shulman's theorem 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{document}

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