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%&latex
\documentclass{mfatshort}
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\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{\Pgr}{{\Pa_{\Gamma,\chi,\gamma}}}$
\newcommand{\eps}{\varepsilon}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\renewcommand{\emptyset}{\varnothing}
\newcommand{\RR}{{\mathcal R}}
\newcommand{\RX}{{\RR\times X}}
\newcommand{\ZZ}{\Z_{+,\,0}^\infty}
\newcommand{\SD}{{\Sigma_{{\tilde D_4},\chi}}}


\title[]
    {On infinity of parameters set, for which $*$-representations of algebra $\mathcal P_{\Gamma,\chi,\gamma}$ in case when $\Gamma$ is extended Dynkin diagram exist}

%    Information for first author
\author{Kostyantyn Yusenko}
\address{}
\curraddr{}
\email{kay@imath.kiev.ua}


\keywords{}

\begin{abstract}
For associated with extended Dynkin diagrams $*$-algebras we study conditions under which the set of parameters, for which there are representation, is infinite.

\end{abstract}

\maketitle


\section*{Introduction}

We remind one classical problem. Let $M_i=\{0=\alpha_0^{(i)}<\alpha_1^{(i)}<\ldots<\alpha_{m_i}^{(i)}\}, i=1,...n$ to be finite sets in $\Rp$ and $\gamma \in \Rp$ to be given. We are to describe all irreducible, with accuracy up to the unitary equvivalence, n-tuples of self-adjoint operators  $A_i, i=1,..n$ such that spectra $\sigma(A_i) \subset M_i$ and following equations between operators holds
$$
   A_1+A_2+...+A_n=\gamma I.
$$  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^*, R_i(a_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
 \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*}

With each $\Pa_{M_1,M_2...M_n;\gamma}$ algebra we will associate connected graph $\Gamma$ that has n branches that reconverged in  common vertice. Each branche has $m_i$ vertices. Consider positive function $\chi:\Gamma \rightarrow \Rp$ that defined everywhere on graph except common vertice.




\end{document}