%\documentclass{article}\documentclass[11pt]{amsart}%{article}%\usepackage[cp1251]{inputenc}%\usepackage[russianb]{babel}\usepackage{amssymb}\usepackage{latexsym}% If you are not using complex mathematical typesetting, comment out% the line below.\usepackage{amsmath}\usepackage{amsthm}\theoremstyle{plain}\newtheorem{lemma}{Lemma}\newtheorem{theorem}{Theorem}\theoremstyle{remark}\newtheorem*{remark}{Remark}\DeclareMathOperator{\Lie}{Lie}\DeclareMathOperator{\Span}{Span}\DeclareMathOperator{\Cl}{Cl}\DeclareMathOperator*{\limind}{lim\,ind}\renewcommand{\phi}{\varphi}\newcommand{\ot}{\otimes}\newcommand{\BC}{{\mathbb{C}}}\newcommand{\BN}{{\mathbb{N}}}\newcommand{\BR}{{\mathbb{R}}}\newcommand{\BT}{{\mathbb{T}}}\newcommand{\BZ}{{\mathbb{Z}}}\newcommand{\CA}{{\mathcal{A}}}\newcommand{\TCA}{{\widetilde{\mathcal{A}}}}\newcommand{\CE}{{\mathcal{E}}}\newcommand{\CH}{{\mathcal{H}}}\newcommand{\CL}{{\mathcal{L}}}\newcommand{\CX}{{\mathcal{X}}}\newcommand{\CZ}{{\mathcal{Z}}}\newcommand{\GA}{{\mathfrak{A}}}\newcommand{\TGA}{{\widetilde{\mathfrak{A}}}}\newcommand{\GZ}{{\mathfrak{Z}}}\begin{document}%\vspace{0.4in}%%\bigskip%\bigskip\title{GRADED LIE ALGEBRAS AND DYNAMICAL SYSTEMS}\author{A. M. Vershik}\thanks{\parindent=0ptSt. Petersburg Division,Steklov Mathematical Institute, RAS,\\27, Fontanka, St. Petersburg, 191011, Russia.\\Partially supported by grant RFBR 99-01-00098.\\This paper contains  the subject of my talk on theConference in Twente in December 2000.}%%}%\bigskip\maketitle\section*{Introduction}%\bigskipWe consider a class of infinite-dimensional Lie algebraswhich is associated to dynamical systems with invariantmeasures. There are two constructions of the algebras -- onebased on the associative cross product algebra which consideredas Lie algebra and then extended with nontrivial scalar two-cocycle;the second description is the specification of the constructionof the graded Lie algebras with continuum root system in spirit ofthe papers of Saveliev-Vershik \cite{VS1,VS2,V} which is a generalizationof the definition of classical Cartan finite-dimensional algebrasas well as Kac--Moody algebras.In the last paragraph we present the third construction for thespecial case of dynamical systems with discrete spectrum.The first example of such algebras was so called sine-algebraswhich was discovered independently in \cite{VS1} and \cite{FFZ} and had beenstudied later in \cite{GKL} from point of view Kac--Moody Lie algebras.In the last paragraph of this paper we also suggest a new examplesof such type algebras appeared from arithmetics: adding of $1$ inthe additive  group $Z_p$ as a transformation of the group  of$p$-adic integers. The set of positive simple roots in this caseis $Z_p$; Cartan subalgebra is the algebra of continuous functionson the group $Z_p$ and Weyl group of this Lie algebra contains theinfinite symmetric group. Remarkably this algebra is the inductivelimit of Kac--Moody affine algebras of type $A^1_{p^n}$.\section{Lie algebra generated by automorphism}\subsection{Associative algebra $\CA(\CX,T)$}Let $(\CX,\mu)$ be a separable compactum with Borel probabilitymeasure $\mu$ which is positive for any open set $A\subset\CX$and $T$ is a measure preserving homeomorphism of $\CX$.It is old and well-known construction of $W^*$-algebra (von Neumann)and $C^*$-algebra (Gel'fand) generated by $(\CX,\mu,T)$, see for example\cite{D,ZM}. Algebraically this is an associative algebra $\CA(\CX,T)$which is semidirect product of $C(\CX)$ and $C(\BZ)$ with the actionof $\BZ$ on $\CX$ and consequently on $C(\CX)$. As a linear space thisis direct sum\[\CA(\CX,T)=\bigoplus_{n\in\BZ}C(\CX)\otimes U^n\]where $C(\CX)$ is Banach space of all continuous functions on $\CX$,and $U=U_T$ is linear operator $(U_Tf)(x)=f(Tx)$, $f\in C(\CX)$.  Themultiplication of the monomials is defined by formula\[(\phi\otimes U^n)\cdot(\psi\otimes U^m)=(\phi\cdot U^n\psi)\otimes U^{n+m},\qquad n,m\in\BZ,\quad\phi,\psi\in C(\CX).\]Involution on $\CA(\CX,T)$ is the following:\[(\phi\ot U^n)^*=(U^{-n}\bar\phi)\ot U^{-n}.\]Completion of $\CA(\CX,T)$ with respect to the appropriate$C^*$-norm gives a corresponding $C^*$-algebra.It is possible to include to this construction a 2-cocycle of the action of$\BZ$ with values in $C(\CX)$ to obtain another $C^*$-algebra which areunsplittable extensions (see \cite{ZM, VSh}), but we restrict ourselfto the case of the trivial cocycle.If we use a measure $\mu$ as a state on $\CA(\CX,T)$ and construct$*$-representation corresponding to this state then $W^*$-closure ofimage of $\CA(\CX,T)$ gives us $W^*$-algebra generated by triple$(\CX,\mu,T)$.There are two classical representations of the algebra$\CA(\CX,T)$ --- Koopmans representation (in $L^2_\mu(\CX)$) andvon~Neumann one in $L^2_{\mu\times m}(\CX\times\BZ)$ ($m$ is Haarmeasure on $\BZ$). This area called ``algebraic theory of dynamicalsystems'' and there are many papers on this. Extremally popular is so calledrotation algebras (also called as "quantum torus") which is associative$C^*$-algebra generated by irrational rotation of the unit circle.{\it We want to point out that there is another remarkable algebraicobject which is associated with dynamical systems---some Lie algebraswhich are similar to the classical Cartan Lie algebras and to affineKac-Moody algebras.} Some nontrivial central extension included in thedefinition plays very imporatant role in the whole theory. Upto nowonly  the Lie algebras corresponing to rotation algebras wereconsidered; it was  discovered independently in \cite{VS1} and \cite{FFZ}(see also \cite{GKL} were  the shift on d-dimensional torus was considered)and called by physisits "sine algebras" - all those dynamical systems have a{\it discrete spectrum},In whole generality  the Lie algebras generated by an arbitrarydynamical system with invariant measure was briefly defined in \cite{VS2}and more systematically in \cite{V}. It is interesting that we startednot form the generaltheory of dynamical systems as in the first definition below  but from the notions presented in our series of papers withM.~Saveliev \cite{VS1,VS2} were we had defined so called``$\BZ$-graded  Lie algebras  with continuous root systems''.Those algebras which we will discuss here were one ofthe type of the examples and a special case of $\BZ$-gradedLie algebras with general root systems.  Below we will describeexplicitly the modern and detailed version of the constructionof Lie algerbas generated by an arbitrary discretedynamical systems with invariant measure and then will givethe link between various definitions. One can hope thatthis type of algebras can give a new type of invariants of the dynamical systems as well as new examples of calssical andquantum integrable systems. \subsection{Lie algebras $\widetilde\GA(\CX,T)$}Most interesting case is the case when $T$ is minimal (= each orbit of$T$ is dense in $\CX$) and ergodic with respect to measure $\mu$ (=there are no nonconstant $T$-invariant measurable functions).  Weassume this in further considerations.It is known that if $T$ is minimal (=each $T$-orbit is densein $\CX$) then $C^*$-algebra is simple (= hasno proper two-sided ideals), see \cite{ZM}. Algebra $\CA(\CX,T)$with brackets\[[a,b]=ab-ba\]will be denoted as $\Lie\CA(\CX,T)$; it is still $\BZ$-graded Liealgebra and the brackets of monomials are\[[\phi\ot U^n,\psi\ot U^m]=(\phi\cdot U^n\psi-\psi\cdot U^m\phi)\ot U^{n+m}.\]This algebra has a center.\begin{lemma}The center of $\Lie\CA(\CX,T)$ is the set of constants functionsin zero component subspace:$\GZ=c\ot U^0$, $c\in\BC$.  The complement linear subspace\[\CA_0(\CX,T)=\bigoplus_{n<0}C(\CX)\ot U^n\oplusC_0(\CX)\ot U^0 \oplus\bigoplus_{n>0}C(\CX)\ot U^n,\]where $C_0(\CX)=\bigl\{\,\phi\in C_(\CX):\int_\CX\phi(x)d\mu=0\,\bigr\}$,is Lie subalgebra which is isomorphic to quotient $\CA(\CX,T)\big/\GZ$over center $\GZ$.\end{lemma}\begin{remark}The center is not ideal of associative algebra consequently thereis no ``associative'' analogue of this lemma and $\CA_0(\CX,T)$is not a subalgebra of $\CA(\CX,T)$.\end{remark}Now we define a 2-cocycle on $\CA_0(\CX,T)$ with the scalar valuesand one-dimensional central expansion of it.\begin{lemma}The following formula defines 2-cocycle on $\CA_0(\CX,T)$:\[\alpha(\phi\ot U^n,\psi\ot U^m)=n\int_{\CX}\phi\cdot U^n\psi\,d\mu\cdot\delta_{n+m},\]so\[\alpha(\phi\ot U^n,\psi\ot U^m)=\begin{cases}n\int_{\CX}\phi\cdot U^n\psi\,d\mu & \text{if }m=-n,\\0 & \text{if }m\ne-n. \end{cases}\]\end{lemma}\begin{proof}We need to check that $\alpha([x,y],z)+\alpha([y,z],x)+\alpha([z,x],y)=0$.Let $k+l+n=0$. Then\begin{multline*}\alpha\bigl([\phi\ot U^k,\psi\ot U^l],\gamma\ot U^n\bigr)+\dots=\alpha\bigl((\phi\cdot U^k\psi-\psi\cdot U^l\phi)\ot U^{k+l},   \gamma\ot U^{-k-l}\bigr)+\dots \\=(k+l)\int_\CX\bigl(\phi\cdot U^k\psi U^{-n}\cdot U^{-n}\gamma    -\psi\cdot U^l\phi\cdot U^{-n}\gamma\bigr)d\mu+\dots=0\end{multline*}(Dots mean cyclic permutation of indices, we used here the invarianceof measure$\mu$ under $T$.)\end{proof}\begin{remark}Cocycle $\alpha$ is not cohomologous to zero because it is easyto check that $\alpha(x,y)$ can not be represented as $f([x,y])$,with any linear functional $f$.of $C(\CX)$.\end{remark}Let us identify scalars $c$  which are extensions of $\CA_0(\CX,T)$with scalars $c\in C(\CX)\ot U^0\subset\CA(\CX,T)$. So we can consideragain linear space $\CA(\CX,T)$ as one dimensional nontrivial extensionof Lie algebra $\CA_0(\CX,T)$.  Denote a new Lie algebra by$\TGA(\CX,T)$.So, Lie algebra $\TGA(\CX,T)$ as linear space is the same as $\CA(\CX,T)$but the brackets in $\TGA(\CX,T)$ differ from the brackets in$\CA(\CX,T)$:\begin{equation}\label{eq1}[\phi\ot U^n,\psi\ot U^m]=(\phi\cdot U^n\psi-\psi\cdot U^m\phi)\ot U^{n+m}  +\int_{\CX}\phi\cdot U^n \psi\,d\mu \cdot \delta_{n+m}\end{equation}It means that the center of $\TGA(\CX,T)$ is again scalars$\BC\cdot1\subset C(\CX)\ot U^0\subset \Lie \CA(\CX,T)$, but nowsubspace $\CA_0(\CX,T)$ is not Lie subalgebra and the central extension isnot trivial.Lie algebra $\TGA(\CX,T)$ is $\BZ$-graded Lie algebra.  We willgive a new definition of it in a framework of Lie algebras withcontinuous root systems.  We will call the subspace of $\TGA(\CX,T)$which consists of $\Span\{\gamma\ot U^{-1}\}\oplus\Span\{\phi\ot U^0\}\oplus\Span\{\psi\ot U^1\}$, $\phi,\psi,\gamma\in C(\CX)$,a ``local subalgebra''.  Here are the brackets for local part of$\TGA(\CX,T)$ are\begin{equation}\label{eq2}\begin{split}[\phi_1\ot U^0,\phi_2\ot U^0] &{}=0,\\[\phi\ot U^0,\psi\ot U^{\pm1}]&{}=\pm\bigl((\phi-U\phi)\cdot\psi\bigr)\ot U^{\pm1}=\bigl((I-U)\phi\cdot\psi\bigr)\ot U^{\pm1}, \\[\phi\ot U^{+1},\psi\ot U^{-1}]&{}=(\phi\cdot U\psi-\psi\cdot U\phi)\ot U^0  +\int_\CX(\phi\cdot U\psi)d\mu\cdot c.\end{split}\end{equation}The middle term of local algebra ($\{\phi\ot U^0; \phi \in C(\CX) \}$)is by definition Cartan subalgebra.This gives the first---``dynamical''---description of the Lie algebra$\TGA(\CX,T)$.\subsection{Lie algebras with root system $(\CX,T)$}Definition of Lie algebra will be followed to Kac--Moody pattern butwith important changes.  First of all we define a \emph{localalgebra}.  Let $\phi\in C(\CX)$; we consider three types of uncountablymany generators: $X_{-1}(\phi)$, $X_0(\phi)$, $X_{+1}(\phi)$ where$\phi$ runs over $C(\CX)$.  The list of relations is as follows:\begin{equation}\label{eq3}\begin{split}[X_0(\phi),X_0(\psi)]&\mbox{}=0,\\[X_0(\phi),X_{\pm1}(\psi)]&\mbox{}=\pm X_{\pm1}(K\phi\cdot\psi),\\[X_{+1}(\phi),X_{-1}(\psi)]&{}=X_0(\phi\cdot\psi),\end{split}\end{equation}where product ($\cdot$) is the product in associative algebra $C(\CX)$and $K$ is a linear operator in $C(\CX)$ which is called Cartanoperator:\begin{equation}\label{eq4}(K\phi)(x)=2\phi(x)-\phi(Tx)-\phi(T^{-1}x).\end{equation}It is evident that Jacobi identity is true (if it makes sense) in thelocal algebra\[\GA_{-1}\oplus\GA_0\oplus\GA_{+1},\]where $\GA_i=\Span\{X_i(\phi),\,\phi\in C(\CX)\}$, $i=-1,0,+1$.($\GA_0\simeq C(\CX)$ is Cartan subalgebra.)  The further steps are thesame as in Kac--Moody theory \cite{K}.We define free Lie algebra which is generated by the local algebra andfactorize it over the maximal ideal which has zero intersection with$\GA_0$.  The resulting Lie algebra is denoted by $\GA(\CX,T)$.  Thefact is that this algebra is the same as $\TGA(\CX,T)$ of subsection 2.We omit the verification that $\GA(\CX,T)$ is the graded Lie algebrawith the graded structure (as a linear space) as follow\[\bigoplus_{n\in\BZ}\GA_n\]and each $\GA_n\simeq C(\CX)$ (see \cite{V}).So we can denote theelements of $\GA_n$ as $X_n(\phi)$, $\phi\in C(\CX)$.\begin{theorem}The following formulas give the canonical isomorphism $\tau$ between$\TGA(\CX,T)$ and dense part of $\GA(\CX,T)$:\[\begin{split}\tau(\phi\ot U^n)&{}=\begin{cases}X_{-n}(U^{-n}\phi), & (n>0) \\X_0(\phi-U^{-1}\phi), & n=0,\quad \int_\CX\phi\,d\mu=0\\X_n(\phi) & (n>0) \end{cases}\\\tau(\mathbf{1}\ot U^0)&{}=X_0(\mathbf{1}).\end{split}\]\end{theorem}\begin{proof} The kernel of $\tau$ is $\mathbf{0}$.Let us check that $\tau([a,b])=[\tau a,\tau b]$.  It is enough to testmonomials only.\begin{multline*}\bigl[\tau(\phi\ot U^0),\tau(\psi\ot U)\bigr]=\bigl[\phi-U^{-1}\phi)\ot U^0,\psi\ot U\bigr]\\=\psi\bigl(\phi-U^{-1}\phi-U(\phi-U^{-1}\phi)\bigr)\ot U=(\psi\cdot K\phi)\ot U\\=X_{+1}(K\phi\cdot\psi)=\tau\bigl([\phi\ot U^0,\psi\ot U]\bigr);\end{multline*}\begin{multline*}\bigl[X_0(\phi),X_{-1}(\psi)\bigr]=\bigl[(\phi-U^{-1}\phi)\ot U^0, U^{-1}\psi\ot U^{-1}\bigr]\\=\bigl(U^{-1}\psi(\phi-U^{-1}\phi-U^{-1}\phi+U^{-2}\phi)\bigr)\ot U^{-1}=\bigl(U^{-1}(\psi(U\phi-2\phi+U^{-1}\phi))\bigr)\ot U^{-1}\\=-U^{-1}(\psi\cdot K\phi)\ot U^{-1}=-X_{-1}(K\phi\cdot\psi);\end{multline*}\begin{multline*}\bigl[X_{+1}(\psi),X_{-1}(\gamma)\bigr]=\bigl[\psi\ot U,U^{-1}\gamma\ot U^{-1}\bigr]\\=(\psi\gamma-U^{-1}\gamma\cdot U^{-1}\psi)\ot U^0=\bigl(\psi\gamma-U^{-1}(\psi\gamma)\bigr)\ot U^0=X_0(\psi\gamma);\end{multline*}\[\bigl[X_{+1}(\mathbf{1}),X_{-1}(\mathbf{1})\bigr]=c X_0(\mathbf{1})=c\mathbf{1}.\]\end{proof}\begin{remark} 1. We calculated the bracket only for elements oflocal algebra, but this is enough because it generates all algebra.\noindent 2. The $\tau$-image of $\TGA(\CX,T)$ is not all $\GA(\CX,T)$but dense part of $\GA(\CX,T)$, for example, the set of functions$\phi-U\phi$ is dense in $C_0(\CX)$ only, but $\GA(\CX,T)$ is theextension of the image of $\TGA(\CX,T)$ and we can consider $\GA(\CX,T)$as some kind of completion of $\TGA(\CX,T)$.\noindent 3. Using isomorphisms $\tau$ we can rewrite the definition of cocycleof $\GA(\CX,T)$ ($\tau(\phi\ot U^{-n})=X_{-n}(U^{-n}\phi)$, $n>0$) so\[\alpha\bigl(X_n(\phi),X_m(\psi)\bigr)=\begin{cases} 0, & \text{if $n+m\ne0$,}\\n\int_{\CX}\phi\psi\,d\mu, & n+m=0. \end{cases}\]\end{remark}Lie algebra $\GA(\CX,T)$ defined above is our main object.\begin{proof}If $n>0$ then $\alpha (X_n(\phi),X_{-n}(\psi))=n\int_\CX\phi\cdot U^n(U^{-n}\psi)d\mu=n\int_\CX\phi\psi\,d\mu$.If $n<0$ the $\alpha (X_{-n}(\phi),X_{n}(\psi))=-n\int_\CX U^{-n}\phi\cdot U^{-n}\psi\,d\mu=-n\int_\CX\phi\psi\,d\mu$($U$ is an unitary  operator).\end{proof}This consists with the initial formula\[\bigl[X_{+1}\bigl(\phi\bigr),X_{-1}\bigl(({\phi})^{-1}\bigr)\bigr]=X_0(\mathbf{1})=\mathbf{1}\cdot c\]for $n=\pm1$.Now we can rewrite the brackets for all monomials (not only for local part).Assume $n,m>0$.{\renewcommand{\theequation}{+,+}\begin{equation}\bigl[X_n(\phi),X_m(\psi)\bigr]=\bigl[\phi\ot U^n, \psi\ot U^m\bigr]=X_{n+m}(\phi\cdot U^n\psi-\psi\cdot U^m\phi);\end{equation}}{\renewcommand{\theequation}{+,$-$}\begin{equation}\begin{split}\bigl[X_n(\phi)&,X_{-m}\psi\bigr]=\bigl[\phi\ot U^n, U^{-m}\psi\ot U^{-m}\bigr]\\&{}=\bigl(\phi\cdot U^{n-m}\psi-U^{-m}(\phi\psi)\bigr)\ot U^{n-m}\\&{}=\begin{cases}X_{n-m}\bigl(\phi\cdot U^{n-m}\psi-U^{-m}(\phi\psi)\bigr), & n>m>0,\\X_{0}\bigl((1-U^{-m})(1-U^{-1})^{-1}\phi\psi\bigr), & n=m,\\X_{n-m}\bigl(U^{-n}\phi(\psi\cdot U^m\phi-U^{-n}\psi)\bigr), & 0<n<m;\end{cases}\end{split}%\eqno(+,-)\end{equation}}{\renewcommand{\theequation}{$-$,$-$}\begin{equation}\begin{split}\bigl[X_{-n}(\phi)&,X_{-m}\psi\bigr]=\bigl[U^{-n}\phi\ot U^{-n},U^{-m}\psi\ot U^{-m}\bigr]\\&=(U^{-n}\phi\cdot U^{-n-m}\psi-U^{-m}\psi\cdot U^{-n-m}\phi)\ot U^{-n-m}\\&=X_{-n-m}(\psi\cdot U^m\phi-\phi\cdot U^n\psi)\\&=-X_{-n-m}(\phi\cdot U^n\psi-\psi\cdot U^m\phi)\end{split}\end{equation}}{\renewcommand{\theequation}{0,+}\begin{equation}\begin{split}\bigl[X_0(\phi),&X_n(\psi)\bigr]=\bigl[(\phi-U^{-1}\phi)\ot U^0, \psi\ot U^n\bigr]\\&=\bigl(\psi\cdot(\phi-U^{-1}\phi+U^{n-1}\phi-U^n\phi)\bigr)\ot U^n=X_n(K_n\phi\cdot\psi),\end{split}\end{equation}}where $K_n=I-U^{-1}+U^{n-1}-U^n=(I-U^{-1})(I-U^n)$;{\renewcommand{\theequation}{0,$-$}\begin{equation}\begin{split}\bigl[X_0(\phi),&X_{-n}\psi\bigr]=\bigl[(\phi-U^{-1}\phi)\ot U^0, U^{-n}\psi\ot U^{-n}\\&=\bigl(U^{-n}\psi\cdot(\phi-U^{-1}\phi-U^{-n}\phi+U^{-n-1}\phi)\bigr)\ot U^{-n}\\&=X_{-n}((U^{-n}\psi)\cdot (U^n\phi-U^{n-1}\phi-\phi+U^{-1}\phi)=-X_n(K_n\phi\cdot\psi)\end{split}\end{equation}}We can now observe that the formulas (+,+) and ($-$,$-$) are the same,as well as (0,+) and (0,$-$).\begin{theorem}The formulas for the brackets of monomials in the subalgebra $\GA(\CX,T)$are the following:\noindent\emph{1)} $\bigl[X_n(\phi),X_m(\psi)\bigr]=\pm X_{n+m}(\phi\cdot U^{\pm n}\psi-\psi U^{\pm m}\phi)$,\\where the sign is ``$+$'' if $n,m>0$ and ``$-$'' if $n,m<0$.\noindent\emph{2)} $\bigl[X_0(\phi),X_{\pm n}(\psi)\bigr]=\pm X_n(K_n\phi\cdot\psi)$.\noindent\emph{3)} $\displaystyle\bigl[X_n(\phi), X_m(\psi)\bigr]=\begin{cases}X_{n+m}\bigl(U^m\psi(\phi\cdot U^n\psi-U^m\phi)\bigr), &{\scriptstyle  \begin{cases}\scriptstyle m<0,\\ \scriptstyle n+m>0\end{cases}}\\X_0\bigl(\frac{1-U^{-m}}{1-U^{-1}}(\phi\psi)\bigr)+n\int_\CX\phi\psi\,d\mu &{\scriptstyle \begin{cases}\scriptstyle n+m=0,\\ \scriptstyle n\ne0\end{cases}}\\X_{n+m}\bigl(U^{-n}\phi(\psi\cdot U^{-m}\phi-U^{-n}\psi)\bigr), &{\scriptstyle \begin{cases}\scriptstyle m<0,\\ \scriptstyle n+m<0\end{cases}}\end{cases}$\noindent\emph{4)} $\bigl[X_0(\phi),X_0(\psi)\bigr]=0$.\qed\end{theorem}Lie algebra $\GA(\CX,T)$ does not associate with associativealgebra; cocycle $\alpha$ has nothing to do with associatedcrossproduct of subsection 1.1.  The role of central extensionis very important.We defined Lie algebra $\GA(\CX,T)$ ($\simeq\TGA(\CX,T)$) in a newterms, compare with (3).  This manner gives us the formulas forlocal part (1--2)which are similar to classical ones (Cartan simple algebras andKac--Moody algebras).  But the formulas for general monomialsare more complicated than dynamical (see subsection 1.1) description.\section{General Lie algebras with continuous root systemsand new examples of $\GA(\CX,T)$}\subsection{General definition}We recall \cite{VS1,VS2,V} the  definition of graded Lie algebraswith continuous root system.Suppose $\CH$ is a commutative associative Lie $\BC$-algebra withunity (Cartan subalgebra) and $K:\CH\hookleftarrow$ is a linear operator(Cartan operator).  The local algebra $[K]$ is, as a linear space, a direct sum\[\CH_{-1}\oplus\CH_0\oplus\CH_{+1},\qquad \CH_{i}\simeq\CH,\quad i=0,\pm1\]with brackets:\[\begin{split}X_i(\phi)\in\CH_i,\quad i=0,&\pm1,\quad\phi,\psi\in\CH\\\bigl[X_0(\phi),X_0(\psi)\bigr]&{}=0,\\\bigl[X_0(\phi),X_{\pm1}(\psi)\bigr]&{}=\pm X_{\pm1}(K\phi\cdot\psi)\\\bigl[X_{+1}(\phi),X_{-1}(\psi)\bigr]&{}=X_0(\phi\cdot\psi)\end{split}\]The local algebra $\CH_{-1}\oplus\CH_0\oplus\CH_{+1}$ generatesgraded Lie algebra $\GA(\CH,K)$.in the same spirit as in Subsection~1.2 (and as in the theory ofLKM-algebras).Then we obtain $\GA(\CX,T)$ from Section~1.The spectrum of commutative algebra $\CH$ (if it exists) is root systemof $\GA(\CH,K)$by definition (see \cite{V}), more exactly the set ofsimple positive roots. But it could be no spectra (say, $\CH$ is thealgebra of rational functions) so we have Lie algebras without simpleroots but with Cartan operator.The condition of constant or polynomial growth of the dimension (inan appropriate sense) puts essential restriction on the operator $K$.\begin{remark}Let $E\subset\CH$ is an invariant under $K$ subalgebra of $\CH$.Then $\GA(E,K)$ is Lie subalgebra of $\GA(\CH,K)$.  In particular, if$E_1\subset E_2\subset\dots$, $\cup_i E_i=\CH$, is a sequence of$K$-invariant subalgebras of $\CH$ then$\GA(\CH,K)=\cup_{i=1}^\infty\GA(E_i,K)$.\end{remark}\subsection{New examples of algebras of type $\GA(\CX,T)$}The first nontrivial example of algebras of type $\GA(\CX,T)$ wasso called sine-algebra.  We will not consider it because it was donebefore from different point of view (see \cite{VS2,FFZ,GKL,V}).It was defined independently in \cite{VS2} and \cite{FFZ}. We givenow general example of similar type.Let $(\CX,T,\mu)$ be an ergodic system with discrete spectrum. Itmeans that operator $U=U_T$ has spectral decomposition\[Uf=\sum_{\lambda}\lambda f_\lambda\chi_\lambda,\text{ where }f=\sum_\lambda f_\lambda\chi_\lambda,\]sum is over eigenvalues of $U$ ($\lambda\in\BT^1$), and $\chi_\lambda$is the eigenfunction corresponding to $\lambda$.  It is well-known(von Neumann theorem) that such system can be realized on thecompact abelian group $G=\CX$ with Haar measure $m=\mu$ and $T$ istranslation on some element $g_0\in G$, then $\chi_\lambda$ is acharacter of $G$ ($\chi_\lambda\in G^\wedge$) and\[\lambda=\chi_\lambda(g_0)\in\BT^1.\]Sine-algebra corresponds to the case $\CX=\BT^1$ and $T$ is translation onirrational number $\theta\in S^1=\BR/BZ$.  In \cite{GKL} was consideredalso the case $\CX=\BT^d\ni\theta$.The case\[\CX=Z_p,\qquad Tx=x+1,\]where $Z_p$ is additive group of $p$-adic integers, $p$ is a prime, and$T$ is adding of unity, is more interesting from our point of view.The measure $\mu=m$ is Haar (additive) measure on $Z_p$.The group of characters $Z_p^\wedge=Q_{p\infty}$ is a group of all rootsof unity of the degree $p^n$, $n\in\BN$, and the characters$\mu\in Q_{p\infty}$ (as function on $Z_p$ with values in $\BT^1$)are eigenfunctions of operator $U=U_T$.We give the description of the general case when operator $U=U_T$has discrete spectrum. The specific property of algebra $\GA(\CX,T)$ inthis case is existence of the natural \emph{linear basis} in $\GA(\CX,T)$.Suppose $G$ is abelian (additive) compact group and $G^\wedge$ is acountable group of the (multiplicative) characters on $G$.   We fixthe element $\lambda\in G$ with dense set of powers: $\Cl\{\lambda^n,n\in\BZ\}=G$.  Then $T_\lambda g=Tg=g+\lambda$, $U_T=U$,$(Uf)(g)=f(g+\lambda)$, $f\in C(G)$.  Each character $\chi\in G^\wedge$is an eigenfunction of $U$ with eigenvalue $\chi(\lambda)\in\BT^1$.\begin{theorem}Linear basis  in the Lie algebra $\GA(G,T)$ is the set$\{Y_{\chi,n}:\chi\in G^\wedge, n\in\BZ\}$ with the following brackets\setcounter{equation}{4}\begin{equation}\label{eq5}\bigl[Y_{\chi,n},Y_{\chi_1,n_1}\bigr]=\bigl(\chi_1(\lambda)^n-\chi(\lambda)^{n_1}\bigr) Y_{\chi\chi_1,n+n_1}  +\delta_{n+n_1}\tilde{\delta}_{\chi\chi_1}\cdot n\cdot c,\end{equation}where $\delta_n=\begin{cases} 1, & n=0\\ 0, &n\ne0\end{cases}$,$\tilde{\delta}_\chi=\begin{cases} 1, & \chi=\mathbf{1}\\   0, &\chi\ne\mathbf{1}\end{cases}$.Algebra $\GA(G,T)$ is $\BZ\times G^\wedge$-graded algebra; thesubalgebra $\{c1:c\in\BC\}$in Cartan subalgebra $\GA_0=C(G)$ is thecenter of $\GA(G,T)$.\end{theorem}\begin{proof}Assume $Y_{\chi,n}=\chi\ot U^n$ as an element of $\GA_n$, where $\chi$is the character of $G$ as a function $G\to\BT^1$. It is easy to checkthat the brackets (see formula \eqref{eq1} in Section~1) give usformula \eqref{eq5}.  Note that the center is not direct summand, sowe have\[\bigl[Y_{\chi,n},Y_{\chi^{-1},-n}\bigr]=n\cdot c.\qed\]\renewcommand{\qed}{}\end{proof}This is the third description of our algebra $\GA(\CX,T)$ with linear basis;this description is valid for discrete spectrum only.The group $G$ is the set of simple roots for $\GA(G,T)$ and ``Dynkin''diagram is the set of arrows $G\ni g\to g+\lambda\in G$.  In oppositeto Kac--Moody case our algebras $\GA(\CX,T)$ have no imaginary roots.\subsection{The case of $p$-adic integers}Return back to the case $G=Z_p$, $Tx=x+1$.  In this case $\GA(Z_p,T)$ is$\BZ\times Q_{p\infty}$-graded algebra. Cartan subalgebra is space $C(Z_p)$.Consider finite dimensional subspaces $L_n\subset C(Z_p)$ of functionsdepending on the points of the quotient $Z_p\to\BZ/p^n$; it is clearthat subspace $L_n$ is $U_T$-invariant, so$\CL_n=\bigoplus_{m\in\BZ}L_n\ot U^m$is a subalgebra of $\GA$.\begin{theorem}The Lie algebra $\CL_n$ is canonically isomorphic to the algebra$\CA^{(1)}_{p^n}$.  Consequently, $\GA(Z_p,T)$ is (completion) of theinductive limit of Kac--Moody Lie algebras\footnote{More exactly,inductive limit contains only linear combinations of monomials of type$\phi\ot U^n$, where $\phi$ are cylindric functions; so it is enough toextend the set of $\phi$ onto arbitrary continuous functions whichmeans to make a completion.} \[ \GA(Z_p,T)\supset \limind_n\CL_n.\qed\]\renewcommand{\qed}{}\end{theorem}\begin{remark} It is possible to define Weyl group $W$ for this algebra.Group $W$ contains the group of permutations of the coordinates in $Z_p$.It is very instructive to study the link between theory of Kac--Moodyaffine algebras and our theory looking on this example.\end{remark}\begin{thebibliography}{XXX}\bibitem[SV1]{VS1} M. Saveliev, A. Vershik. \it Continuum analogues ofcontragradient Lie algebras. \rm CMP \bf126 \rm(1989), 367--378.\bibitem[SV2]{VS2} M. Saveliev, A. Vershik. \it New examples ofcontinuum graded Lie algebras. \rm Phys. Lett. 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