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View File Name : extDyn.tex

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\newcommand{\Pb}{{\Pa_{\Gamma,\chi,\gamma}}}
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\title
    {On parameters sets of algebras $\mathcal P_{\Gamma,\chi,\gamma}$ when $\Gamma$ is an extended Dynkin diagram}

%    Information for first author
\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

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\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 nonoriented 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 $E_8$:

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\[
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\[
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\]
then $\Pb$ is finite dimensional, if $\Gamma$ is an extended Dinkyn diagrams of the type $\tilde D_4, \tilde E_6, \tilde E_7, \tilde E_8$

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\[
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\[
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then algebra $\Pb$ is infinite dimensional and of polinomial 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 poster is concentrated at studing sets $\Sigma_{\Gamma,\chi}$ in case when $\Gamma$ is an extended Dynkin diagram.
Given result here is full description of set $\Sigma_{\tilde D_4,\chi}$ and  the conditions under which $\Sigma_{\Gamma,\chi}$ are infinite.

\section{Some beforehanded statements}

  Given in this section statements will be taken advantage of in the sequel.

\begin{lemma}
  The set $\Sigma_{\Gamma,\chi}$ is finite when $\Gamma$ is a Dynkin diagram.
\end{lemma}


\begin{theo}(see \cite{KPS05}) \label{D4}
  Following formula gives full description of the set $\Sigma_{D_4,(\alpha_1,\alpha_2,\alpha_3)}$

  $$
   \Sigma_{D_4,\chi}=\{0\} \cup \{\sum_{i \in J} \alpha_i,  J \subset \{1,2,3\}\} \cup \{(\alpha_1+\alpha_2+\alpha_3)/2\} \label{TD4}
  $$

\end{theo}


\begin{lemma} \label{com}
   Let $P_i, i=1,\ldots,n$ such projector that
     $$\alpha_1 P_1+\alpha_2 P_2+\cdots+\alpha_{n} P_{n}+P_n=\alpha_{n}I,$$
  $\alpha_i$ here is positive real numbers. Then projector $P_n$ commutes with projectors $P_i, i=1,\ldots,n-1$.

\end{lemma}




\section{Coxeter functors}

A powerfull tool to investigate algebras $\Pb$ is Coxeter functors. Such functors act between categories of $*$-representation of the algebras $\Pb$. In \cite{KR03} two functors were investigated: linear $S$ and hiperbolic $T$, and 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 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 folowing 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}

\begin{enumerate}
  \item $\SD \subset [0, \alpha];$
  \item $\SD \ni \sum_{i \in J} \alpha_i,  J \subset\{0,1,2,3,4\};$
  \item $\tau \in \SD \Longleftrightarrow \alpha-\tau \in \SD;$
  \item $\frac\alpha2 \in \SD$.
 \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 coeficient is equal or more than $\frac{\alpha}{2}$ then corresponding projector is equal to $0$ or $I$ in the representation and from $\tilde D_4$ graph we obtain $D_4$ graph where solution is known.

 Having written the action of $(ST)^n$ functor on pair $(\chi;\gamma)$ (and applied Th.\ref{D4} 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 finited series $\Sigma_0$. This two series are desrcibed 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}


\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. It is easy to check that if at least one component of character $\chi_i\geq \omega_\Gamma(\chi)$ 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 condition $\chi_i<\omega_\Gamma(\chi)$ is necessary for the set $\Sigma_{\Gamma,\chi}$ to be infinite. It turns out that it is also sufficient and for each extended diagram we are able to build infinite series.


\begin{theo}
  Let $\Gamma$ be an extended Dynkin diagram. The set $\Sigma_{\Gamma,\chi}$ is infinite if and only if each component of character $\chi_i<\omega_{\Gamma}(\chi)$
\end{theo}





\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{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}
{\it S. Kruglyak, S. Popovich, Yu. Samoilenko }, The spectral
problem and $*$-representations of algebras associated  with
Dynkin graphs //arXiv:math.RT/0407418~v2.

\bibitem{KR03}
S. A. Kruglyak and A. V. Roiter, {\it Locally scalar representations of graphs in the category of Hilbert spaces}, Preprint 2003.4, Inst. Math. Nat. Acad. Sci. Ukraine, 2003, (Russian).


\bibitem{OS05}
Ostrovskyi V. L., {\it Some remarks on families of operators related to extended Dynkin diagrams}, MFAT, to appear.

\bibitem{YU05}
Kostyantyn Yusenko, {\it On fourtuples of projectors connected with linear equation}, Ukr. Math. J., to appear.


\end{thebibliography}


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