A duality theorem for fractal sets and Cuntz algebras and their central extensions (Representations of Cuntz algebras and their applications in mathematical physics)

(1)

Aduality theorem for fractal

sets

and

Cuntz algebras

and their central

extensions

J.Lawrynowicz,K.Nouno

and

O.Suzuki

July

8,

2003

Abstract

In this paper we introduce concepts of“flower type” and ”branch type” for fractal

sets, atfirst and the concept ofthe central extensionofCuntz algebras secondlyand we

give the following results:

(1)$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}$ exists aduality

theorem between fractal sets of flower type and branch

type, which is called”flower-branch duality”(Theorem$\mathrm{I}$).

(2)$\mathrm{W}\mathrm{e}$introduce aconcept of central extensions

ofCuntz algebras(which will be called

Zunk algebra)andmakeaFockrepresentationonafractal set ofbranchtype(Proposition

$\mathrm{B})$

.

(3)$\mathrm{T}\mathrm{h}\mathrm{e}$ flower-branch duality

induces the duality theorem between the representations

ofCuntz algebras and Zunk algebras(Theorem$\mathrm{I}\mathrm{I}$).

It is suggested how the duality theorem can be applied to several topics both in

mathematics and physics. The details will be given in the forthcoming papers.([1],[6]

and [7]

1Introduction

In papers([8],[9]) we have treated fractal sets by use of representations of Cuntz algebras

and give the criterion whether they

are

equivalent or not. In this paper

we

introduce two

kindsoffractal sets, theone is calledof flower type which describes the condensating objects

and clusters. The other one is called of branch type which describes the growing objects,

developing cities, tree leaves, baccterias. Atfirst

we

show that there exists

aone

toone

corre-spondencebetween thesetwo classes, $\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\dot{\mathrm{h}}$

will be called ”flower-branch duality”(Theorem

$\mathrm{I})$

.

willbe concerned withtherepresentations. Wehave constructed representations

on fractal sets of flower type. Here

we

will make the Fock representations on fractal sets

of branch type. For this purpose

we

have to make acentral extension of

Cuntz

algebras,

which is called Zunk algebras. Here we have to make its central extension, when we want

to make the Fock representation of the Cuntz algebra. Because we have to prepare central

elements for the

vaccum

state. Hence

we are

led to

an

introduction ofcentral extensions of

Cuntz algebras at first and then we will make representations of Zunk algebras on fractal

数理解析研究所講究録 1333 巻 2003 年 99-108

(2)

sets of branch type(Proposition B). Then

we can

apply the duality theorem between these

representa,tions and

ca.n

discuss their equivalences(Theorem II).

2Aduality

theorem

for

ffactal

sets of flower

type

and

branch

type

In this section

we

prepare two kindsoffractal setswhich are called ”of flower type” and ”of

branch type” and prove Theorem I. At first

we

recall some basic facts

on

fractal sets([9]).

For the simplicity sake

we

restrict ourselves only to asystem ofpiecewise a$f$

fine

mappings

$\{\sigma j : j=1,2, .., N\}$ between acompact set $I\mathrm{f}_{0}$

.

Then

we

see

that there exist non-negative

numbers $\{\lambda;\}$ and positive numbers $\{\Lambda;\}$ $(0\leq \mathrm{A}_{\mathrm{j}} \leq\Lambda_{i})$ satisfying the condition:

$\lambda_{i}d(x, y)\leq d(\sigma_{j}(x), \sigma.\cdot(y))\leq\Lambda_{\mathrm{i}}d(x, y)(i=1,2, .., N)$

.

Here we

assume

that these conditions are extremal, namely they attain the equalities in

the both sides exactly. Here we will be concerned with mappings $\sigma_{\dot{l}}$ with $\Lambda_{1}$. _{$\leq 1$}, which

we

call contractible mapping simply. Acontractible mapping $\sigma_{i}$ with $\Lambda_{1}$. $<1(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}. \Lambda_{i}=1)$

is called essentially proper(resp. partially isometric). Moreover, an essentially proper

mapping $\sigma i$ is called proper, if $\lambda_{i}=\Lambda_{\mathrm{i}}$ holds. We call the mapping $\sigma j$ in the

case

where

$\lambda_{\mathrm{i}}=0$ degenerate and in the other

case

non-degenerate respectively. Next

we

define aself

similar fractal set. Here we make acomment on the construction of fractal sets. Although

the construction of the fractal sets defined by essentially proper mappings is unique,

we

still

have not definite construction principles of fractal sets for general contractible mappings.

For asystem of $\sigma j$ : $I\mathrm{f}_{0}\vdasharrow I\mathrm{f}_{0}(j=1,2,3,4)$, where

we

need not necesarry

assume

that they

are contractible, we put

$K= \bigcup_{i=0}^{\infty}(\bigcap_{n=*}^{\infty}.I\mathrm{f}_{n})$, where $I\{_{n}’=\cup\sigma_{\mathrm{j}}(I\{_{n-1}.)(n=1,2, \ldots)$

.

(2.1)

We notice the following invariant condition([4]):

$\bigcup_{j=1}^{N}\sigma_{j}(K)=I\{^{-}$

.

(2.2)

In the

case

of essentially proper mappings $\sigma \mathrm{j}$, the definition is equivalent to the given in

$(2,3)$([3]).

In this paper

we assume

that the following separation condition is satisfied:

$\sigma_{j}(I\mathrm{f}^{\mathrm{o}})\cap\sigma_{\mathrm{j}}(I\mathrm{t}^{\acute{\mathrm{o}}})=\phi$

,

(2.3)

where $B^{\mathrm{o}}$ implies the open kernel of$B$

.

proceed to the fractal sets of flower type and branch type.

(i) Afractal set offlower type

For asystem of contractible mappings $\sigma j$ : $I\dot{\mathrm{t}}_{0}\vdash*I\mathrm{f}_{0}(j=1,2, .., N)$ with the separation

condition$(2,3)$, we put

$I \mathrm{f}=\bigcap_{n=1}^{\infty}I\mathrm{f}_{n}$

,

where $I \dot{\mathrm{t}}_{\mathrm{i}}=\bigcup_{\mathrm{j}=1}^{N}\sigma_{j}(I\mathrm{f}_{j-1})$

,

(2.4)

(3)

(\"u)A fractal set ofbranch type

For asystem ofcontractible mappings $\sigma_{j}$ : $I\acute{\backslash }.0\vdasharrow K_{0}(j=1,2,$.., N), we choose areference

point $p_{0}$ in $I\mathrm{f}_{0}$ and define fundamental branches

by

$L_{0}= \bigcup_{j=1}^{N}L_{0|j}$, $L_{0|j}=\overline{p_{0\prime}\sigma_{\mathrm{j}}(p_{0})},$ (j _$=1,$2,..N) (2.5)

and

we

make alattice $L$

on

$I\mathrm{f}_{0}$ in the following

manner:

(1)$L$ is connected,

(2)$L= \bigcup_{n=0}^{\infty}L_{n}$, where _{$L_{n}=L_{n-1}\cup L_{n’},$}_{$L_{n’}= \bigcup_{j}\sigma_{j}(L_{n-1})$}, (2.6)

(3)$L$ satisfies the separation conditions:

$\mu(\sigma_{j}(L_{n-1})\cap\sigma_{k}(L_{n-1}))=0(k\neq j)$, $\mu(L_{n-1}\cap L_{n’})=0(n\neq n’)$,

where $\mu$ is the Lebesgue

measure on

$L$, which is called afractal set of branch type.

Then

we see

that they are connected through the following duality theorem:

Theorem $\mathrm{I}$($\mathrm{F}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}$

-Branch duality for self similar fractal sets)

The set offractal sets offlower type(resp. branch type) in $I\mathrm{f}_{0}$ is denoted by $\mathcal{K}(K_{0})(\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{p}$

.

$\mathcal{L}(I\mathrm{f}_{0})$. Then there exists

aone

to

one

mapping

$\phi$ : A$(I\mathrm{s}\mathrm{i}_{0})-*\mathcal{L}(I\dot{\iota}_{0})$ between fractal sets of

latticesets $L$ and fractal sets ofbranch type $L$

ProofofTheorem I

Suppose that afractal set of branch type is given. Then we can make the corresponding

dual fractal set of flower type in the following $\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}:\mathrm{P}\mathrm{u}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}$

$I\dot{\mathrm{C}}_{n}=\mathrm{t}\mathrm{h}\mathrm{e}$ closure of_{$\{L-\bigcup_{k=1}^{n}(L_{n})\}$}

.

(2.7)

By the construction of$L_{n}$,

we

see that

$R_{n+1}^{\cdot}=\cup\sigma_{j}I\mathrm{f}_{n}$ (2.8)

$I\mathrm{f}_{n+1}\subset I\mathrm{f}_{n}$

.

(2.9)

Then we have the desired fractal set

$K= \bigcap_{n=1}^{\infty}I\mathrm{f}_{n}$, (2.10) which satisfies the invariant condition$(2,2)$

.

The construction in the

converse

direction is

given in $(2,6)$

.

Hence

we

have proved the assertion.

In thecase where the fractal set of flower type is aproper fractal set, we can find more

direct correspondence. Taking the fact into account that each point $x$

can

be expressed as

follows

$x= \lim_{narrow\infty}\sigma_{i_{1}}\sigma_{\dot{\mathrm{t}}_{2}}\ldots\sigma_{1}.(nx_{0})$, (2.11)

(4)

where $x_{0}$is an arbitrary point of I\prime i,

we can make a

one

toonecorrespondence $\Phi$ :

$K\cross \mathrm{R}_{+}\vdasharrow$

L which is defined by

$\Phi(x)(t)=\bigcup_{n=1}^{\infty}\sigma_{i_{2}}\sigma_{i_{3}}\ldots\sigma_{i_{n}}(Lj_{1})(t)$ where

$t\in[n, n+1]$, _(2.12)

where

we

notice the $\dot{\mathrm{R}}_{+}$ is the union

of

intervals

$[nn+\}1]$ _which

_describe

_the _parameters_of

pieces oflines. We give

an

example of

califlower:

3 _{Central extensions}

_of

_Cuntz

_{algebras and}

_their

representa-tions

on

fractal

sets

of

_branch

_type

In this section we recall

some

basic facts

on

the

Cuntz

algebras and their central extensions

at first and discuss representations

_on

_self_{similar fractal} _sets.

The

Cuntz

algebra $O(N)$ is a $C^{*}$-algebra with

generators $\{S_{J}\}(j=1,2, .., N)$ _{with the}

followingcommutation relations([2]):

(1)$S_{j}^{*}S_{j}=1(j=1,2, .., N)$, ₍₂₎$\sum S_{j}S_{j}^{*}=1$

.

_(3.13)

These commutation relations give

an

algebraic description of the division of the total space

into N-parts. Next we proceed _{to central} _extensions _of_{the Cuntz} _{algebras. A} $C^{*}$-algebra

$Z(N)$ is _called _the _{Zunk algebra} _with _generators

$\{Sj\}(j=1,2, .., N)$ _with _the

_following

commutation

relations:

(1)$S_{j}^{*}S_{j}=1(j=1,2, .., N),$ ₍₂₎_{$\sum SjS^{*}j+Q=1$}_, _where

$Q^{*}=Q,$ $Q^{2}=Q$

.

_(3.14)

We

see

that the Zunk algebra is not simple and

we

see that it is

obtained

from the Cuntz

algebra by the central extension. We

can

make representations _of

_Cuntz

_algebras_{on fractal}

sets offlower type in awell known

_manner:

Proposition$\mathrm{A}$($\mathrm{H}\mathrm{a}\mathrm{u}\mathrm{s}\mathrm{d}\mathrm{o}\mathrm{r}\mathrm{f}\mathrm{f}$ representations

on

_{ffactal sets offlower} _type)$([8],[9])$

Let $I\dot{\{}$ be afractal

set offlowertype which

defined

by $\{\sigma_{j}\}(j=1,2, ..N)$

.

Then

we

have _the

following

representation $\pi$ : _{$O(N)arrow B(L_{D}^{2}(I\dot{\mathrm{t}}))$}

:

$\pi(S_{j})f(x)=\{$ $\Phi_{j^{1/2}}(\sigma_{j}^{-1}(x))f(\sigma_{j}^{-1}(x))$ $x\in\tau_{j}(I\dot{\mathrm{t}})$

0 $x\not\in\tau_{j}(K)(j=1,2, .., N)$

(5)

$\pi(S_{j}^{*})f(x)=\Phi_{j}^{-1/2}(x)f(\sigma_{j}(x))(j=1,2, ..N)$

The representation _defined in above is called aregular representation. We can prove the

following

proposition:

Proposition(The

_Kakutani’

$\mathrm{s}$ dichotomy $\mathrm{t}$heorem ([5]))

Let $I\dot{\mathrm{t}}$

and $I\mathrm{i}’$ be two

self similar

fractal

sets of flower type with the

same

number of

generators $N$ which

are

defined

on

compact sets $K_{0}$ and $K_{0}’$ respectively. Then

we

see

that the Hausdorff representations _are _{equivalent, if and only if they satisfy the}

_following

conditions:

$\lambda_{i,j}^{D}=\lambda_{\iota,j}^{\prime D’}(i,\dot{g}=1,2)$

.

(3.15) Then, for two Hausdorff $\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}:\pi_{i}$ : $O(N)-B(L_{D}^{2}(I\mathrm{f}_{i}))(i=1,2)$,

we

can

find

aunitary operator $U$ : $L_{D}^{2}(I\mathrm{f}_{1})\mapsto L_{D}^{2}(I\mathrm{t}_{2})$ such that

$\pi_{1}(S)U=U\pi_{2}(S)$ holds for any

$S\in O(N)$

Remark The equivalence _{does not imply that} _$D=D’$ _holds.

Next we proceed to the construction of the representations of Zunk algebras

on

fractai

sets ofbranch type. For this we prepare theso called Haar basis $e_{i_{1},i_{2},..,i_{n}:}$

$e_{i_{1},i_{2},..,i_{n}}=\{$ 1

$x\in\sigma_{i_{2}}\sigma_{i_{3}}\ldots\sigma_{i_{n}}(L_{i_{1}})$

0 $x\not\in\sigma_{2}.\cdot\sigma_{j_{3}}\ldots\sigma_{*_{n}}.(L_{i_{1}})$ (3.16)

After the normalization with respect _{to the Borel}

_measure

$L^{2}(L, d\mu)$, we have asystem of

orthnormal basis and have the Hilbert space H. Then

we can

prove the following theorem:

Proposition $\mathrm{B}$(The Fock representation

on afractal set ofbranch type) Let $L$ be afractal set of branch type.

Then

we

have the following Fock representation

$\rho_{b}$ : $Z(4)\vdasharrow B(\mathrm{H}):\mathrm{C}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}$the

vaccum

$|0>\mathrm{w}\mathrm{e}$ define $Q|0>$ $=$ $0>$ (3.17) $T_{j}|0>$ $=$ $e_{\mathrm{j}}$ (3.18) $T_{j}(e_{i_{1,}:_{2,\prime}..;_{\hslash}})$ $=$ $e_{j,:_{1^{1}2\prime\cdot\cdot\prime}\mathrm{i}_{n}\prime}.$, (3.19) $Q^{*}|0>$ $=$ $0>$ (3.20) $T_{j}^{*}(e_{i_{1\prime}i_{2\prime\cdot\cdot\prime}:_{n}})$ _$=$ $\{$

$e_{2,..\prime}.\cdot i_{n}$ $j=i_{1}$

0 $j\neq i_{1}$

The proof is easy and maly be omitted. Next

we

proceed to the duality theorem to

repre-sentations which arise from the

Theorem

I. Then

we

can

prove the

following theorem

Theorem $\mathrm{I}\mathrm{I}(\mathrm{F}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}$

-Branch

duality

theorem

for representations ofHausdorff

type)

Let $L$ be afractal set of branch type

and let $K$ be the corresponding dual

fractal

_set.

(6)

Then the $\mathrm{f}\mathrm{o}11\mathrm{o}\mathrm{w}\mathrm{i}_{1\mathrm{l}}\mathrm{g}$ duality is

induced between

the representations:For arepresentation _of

Hausdorff type, $\pi_{fl\mathrm{o}we\mathrm{r}}$ : $O(N)\succ\not\simeq L^{2}(K, d\mu_{I\backslash }^{D}.),$ there exists an

central extellSion $Z(N)$ _of

the Cuntz algebra $O(N)$ so that _there _{is a} _{representation}

$\pi_{b\mathrm{r}anch}$ : _{$Z(N)\succ’ L^{2}(L, d\mu_{L})$}

.

The

converse

is also true. Proofof

Theorem

II

We give

a

proofof

Theorem

$\mathrm{I}\mathrm{I}.$ At first

we

assume

that the Fock representation _{is given}

_on

a

fractal

_{set of branch type. We make the} _fractal _{set of}_flower _type. _Then

_we

can

extend

therepresentation _to _the _{representation} _of

_Hausdorff

_{type in the}

_following

$\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}:\mathrm{C}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}$

the Hilbert space $\mathrm{H}(L)$ spanned by the basis

$\{e_{\alpha} : \alpha=(\alpha_{1}, \alpha_{2}, \ldots)\in\prod_{n=1}^{\infty}\{1,2, \ldots, N\}\}$

of infinite paths. Then

we

see that the representation _{above defined}

_can

_be _written

_as

follows:

$\pi(S_{\mathrm{j}})e_{\alpha}=e_{(j,\alpha_{1},\alpha_{2},\ldots)}$, _{$\pi(S_{j}^{*})e_{\alpha}=\{$} $e_{(\alpha_{2\prime}\alpha_{3\prime}\ldots)}$ $j=\alpha_{1}$ 0 $j\neq\alpha_{1}$

The

converse

direction can be given in the similar

manner

and may be omitted.

4 Applications

In this section we

demonstrate

how

_{we can}

_apply _{the duality} _theorem _{to several topics in}

mathematicsand_{physics. We treat the}_{following three}_{topics:(l)Infinite}

_dimensional

Clifford

algebras, (2)$\mathrm{L}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{i}\mathrm{c}\mathrm{e}$

models on

fractal

sets and (3)$\mathrm{C}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{l}\mathrm{e}\mathrm{x}$ anaysis. The applications

can

be performed as $\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{s}:\mathrm{W}\mathrm{e}$ realize

the complex systems _{as infinite}

_dimensional

_objects _and

represent _them by use of those of

Cuntz

algebras

on

fractal sets. We approach the systems

from

_{finite dimensional}

_{systems by} _{approximation.} _This

_can

_{be done by the}

representations

of Zunk algebras. By use of the _{duality theorems,}

_we

_can

_discuss _{the original complex}

systems.

$(\alpha)\mathrm{I}\mathrm{n}\mathrm{f}\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}\mathrm{e}$ dimensional Clifford

algebras ([6])

We define the infinite

_{dimensional Clifford}

_{algebras by}

_use

_of _the _inductive _{limit of} _finite

dimensional

Clifford algebras. For example,

we can

choose the exahusions in the following

$\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}:\mathrm{A}\mathrm{t}$first wenotice that the

Clifford algebra$Cl_{2N+1}(\mathrm{C})$ can be realized

on

the matrice

space $M$($2^{N}$ : C) in the

inductive

manner.

The$Cl_{3}(\mathrm{C})$

can

be given by the Pauli matrices: $\sigma_{1}=(\begin{array}{ll}0 \mathrm{l}1 0\end{array})\sigma_{2}=(\begin{array}{ll}0 i-i 0\end{array})\sigma_{3}=(\begin{array}{ll}1 00 -1\end{array})$

(4.21)

For

the generators $A\mathrm{j}(j=1,2, .., 2p-1)$ of$\mathrm{C}_{2p-1}(\mathrm{C})$, putting

$(\begin{array}{ll}A_{j} 00 -A_{j}\end{array})(\begin{array}{ll}0 I_{2}I_{2} 0\end{array})(\begin{array}{ll}0 iI_{2}-iI_{2} 0\end{array})(j=1,2, .., p)$ ,

(4.22)

(7)

we have the generators of$C_{2p+1}(\mathrm{C})$

.

We notice that

$Cl_{2N+1}(\mathrm{C})\underline{\simeq}M(2^{N}, \mathrm{C})$ (4.23)

We

can

introduce the infinite dimensional Clifford algebra by use of the inductive limit:

$Cl(\infty : \mathrm{C})=li\ovalbox{\tt\small REJECT} Cl_{2N+1}(\mathrm{C})$

.

(4.24)

By this construction

we are

temptated to introduce afractal method to the infinite

dimen-sional Clifford algebras. In fact

we can

realize the algebras in terms of fractal sets ofPeano

floer type which are defined by the four contra tible mappings $\{\sigma_{i,j}|i,j=1,2\}$ between the

unit rectangle $I\mathrm{f}_{0}(=\{(x, y)|0\leq x\leq 1,0\leq y\leq 1\}$ with the separate condition. Considering

thedual fractal set ofbranch type,

we can

realize the representations ofasequence offinite

dimensional

Clifford algebras by

use

of the Zunk representations.

.

–

$M_{1}(\mathrm{C})$

$NI_{2}(\mathrm{C})$ $M_{4}(\mathrm{C})$

In fact we can construct the representation in the following

manner.

At first we notice

the following fact:

$Cl(\infty : \mathrm{C})\subset Z(4)$

.

(4.25)

Then restricting the representation given in Proposition, we have the following

Theorem III(Duality theorem for infinite dimensional Clifford algebras) (1)$\mathrm{E}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}):\mathrm{W}\mathrm{e}$have arepresentation

$\pi_{b}$ : $Cl$($\infty$ : C) $|arrow B(L^{2}(L))$

.

(2)$(\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}):\mathrm{T}\mathrm{w}\mathrm{o}$representations of

(1) are unitary equivalent, ifand only if the dual

representations $\pi_{f}$ : $O(4)\mathrm{I}arrow B(L_{D}^{2}(I\dot{\mathrm{t}}))$ satisfy the Kakutani’s condition on $K$ in the case

where $I\dot{\mathrm{t}}$ is aproper fractal

set.

By this theorem

we

may discuss the

Clifford

analysis for $Cl(\infty :\mathrm{C})$

.

The detail will be

given in ([6]).

$(\beta)$ Lattice models

on

fractal sets([l])

We

can

treat theinteracting lattice models offermionic $N$-spin particles in terms of fractal

geometry and discuss their phase transitions by use of $\mathrm{t}1_{1}\mathrm{e}$ duality theorem. We consider

the following standard(i.e. free) lattice rnodel on the lattice of positive integers $\mathrm{N}$:

$\mathcal{H}_{0}=\beta\sum a_{n}^{(i)}a_{n}^{(i)\dagger}$, (4.26)

105

(8)

(j)

where $a_{n}$

are

the

annihilation

operators

of

frmionic

type at the site $n(n=1,2, ..)$ _for _the

spin _{$j(j=1,2, .., N)$} _and $a_{m}^{(i)\dagger}$

are

the corresponingcreation operator. The _{algebra is called}

the

fermionic

algebrawith spin $N$and is

denoted

_by

$AF(N).$ _We_will_treat

_{interacting lattice}

models in

terms

of

fractal

geometry. This

can

be performed by use of the representation _of

the Zunk algebra.

Theorem

$\mathrm{I}\mathrm{V}$(

$\mathrm{D}\mathrm{u}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{t}\mathrm{y}$

theorem

for

lattice

models

on

ffactal sets)

Let $L$ be

afractal

set of branch type and let $\pi$ : _{$Z(N)-\rangle B(L^{2}(L))$}

be arepresentation.

Then

we

$\mathrm{h}$

ave

(1)$(\mathrm{E}\mathrm{x}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}):\mathrm{W}\mathrm{h}\mathrm{e}\mathrm{n}M=2^{N},$

we

have asubalgebra $AF(N)$ _of $\mathcal{Z}(M)$ with generators

{

$a_{n}^{(i)},$

$a_{n}^{(i)\mathrm{t}_{|n,m=1,2,3,..,i,j=1,2,..N\}}}.$

Hence

we

have the

_Hamiltonian

$\mathcal{H}_{L}=\beta\sum\pi(a_{n}^{(i)})\pi(a_{n}^{(j)\uparrow})$,

(4.27) which is called the standatd

_Hamiltonian

_on

$L$

.

(2)

_{(Equivalence):Let}

$\pi’$:

$Z(N)\vdasharrow B(L^{2}(L’))$ be another_{representation.} _Then

the

dynam-ical systems _define _by _the

_Hamiltonian

_are _unitary _equivalent

if and only if the Kakutani,s

conditions

are

satisfied

on

_{the corresponding}

_fractal

_{sets of}_flower _type.

Here the dynamical system is defined by

$\dot{\iota}\frac{dx}{dt}=[x, H_{L}]$

.

(4.28)

On

the

base

of this theorem,

we can

treat the phase _transitions _by_considering

_deformations

of the corresponding

fractal

sets of

flower

type.

$(\gamma)$Complex anaysis ([7])

Finally

we

_{shall show apossibility} _of_treating _{complex analysis by}

_use

_of

the fractal

geom-etry. In this paragraph,

we

_{will be concerned with} _the

_following

_{two topics.}

(1)$\mathrm{T}\mathrm{h}\mathrm{e}$

boundary behavior

of

_{aholomorphic function}

The

one

of the $\mathrm{i}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}\dot{\mathrm{a}}\mathrm{n}\mathrm{t}$ subjects

in complex anaysis _{is to consider the}

_behavior

_of _holo

morphic

_functions

_on _{the natural}

_boundaries.

_This _can _be _done

in the following

manner.

At first

we

take aholomorphic function and consider its ,,$\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{c}\mathrm{k}$

Komplex”. This is defined

in the following $\mathrm{m}\mathrm{a}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{r}:\mathrm{L}\mathrm{e}\mathrm{t}f$ be aholomorphic(or

meromorphic)

_{function on}

$D$ with the

$\partial D$

as

anatural

boundary of$f.$ We choose

areference

_point

$z_{0}$ in $D$ and

we

put $f(z_{0})=c$

.

We consider the pointsset

$\{z_{n}\}$, where _{$f(z_{n})=c$}

.

(4.29)

Following

the analytic

continuation

of $f$ from $z_{0},$

we

$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$the point

$z_{1}$ and

continuating

it further,

we

have the _{sequence. Here we} _give

_an

_example _{for the} _holomorphic

function

$exp$ $z^{2}|$:

(9)

$arrow$

Then _{following the construction} _{rule of the} _fractal _{set of}_branch _type,

_we

_can _{define the}

fractal set $L(f : c)$

.

We consider the dual _fractal _set $K(f : \mathrm{c})$

.

Then

we

may identify _this

set

as

the cluster set of the value $c$

.

Herrce, considering the dual fractal set,

we

may treat

the

behavior

of

_{holomorphic functions on}

_{the boundary. We may} _{expect to} _give _aproof_of

the

_{Picard theorem or}

_Nevannlinna

_{theory in} _terms _of_the _{fractal geometry. For example}

we

_{can formulate}

_the

_following

_problem:

Problem

$,,\mathrm{I}\mathrm{f}$the

Hausdorff dimension

of$I\dot{\mathrm{t}}(f :c)$ is positive for

some

$c$, does the Picard theorem not

hold ?”

(2)$\mathrm{T}\mathrm{h}\mathrm{e}$ moduli of

Riemann surfaces

The_{second application is to the}_{moduli structure}_of_{Riemann surfaces.} _Here _we

_assume

_that

the universal covering of the

Riemann

surface$R_{g}(g>1)$ is _the _unit _disck $D$

.

Hence we can

represent it by the

_{Decktransformationen}

_group $\Gamma(R_{\mathit{9}})$ as _{$R_{g}=D/\Gamma(R_{g})$}. We denote the

generators of $\Gamma(R_{g})$ by _{$\{g_{j}|j=1,2, .., 2g)\}$}

.

Taking areference

point $p_{0}$ of

afundamental

region and making the

fundamental

branch $L_{j}=\overline{p_{0)}g_{j}(p_{0})}(j=1,2, .., 2g)$, we can _make _a

fractal set $L(R_{g})$ of

branch

type by the _construction

method. $(\infty)$ (0) $(0\mathrm{Y})\mathrm{t}1)\gamma_{(\infty)}$ $\mathrm{t}11\gamma_{\infty}^{\mathrm{X}\mathrm{X}}()\mathrm{Y}_{(0\mathrm{I}}^{\mathrm{Y}}11]$ $(*_{0)}^{\mathrm{X}}(1)\mathrm{b}_{(\infty)}^{-}$

Then We

can

introduce

arepresentation:

$\pi_{b}$ : $Z(2g)|arrow B(L^{2}(L(R_{g}))$

.

(4.30)

Byuse ofthe dualitytheorem

we

may discuss the moduli spaces through the anaysis

on

the

fractal

set of flower type. For example, we can formulate the following problem: Problem

(1)$\mathrm{L}\mathrm{e}\mathrm{t}R_{g}$ and $R_{g}’$ be two Riemann surfaces. Then we can we show the

biholomorphic

equivalence _thorough _{the unitary equivalence of}_{the dual} _{representation}

$\pi_{f}$ : $O(2g)\succ+B(L^{2}(K(R_{g}))?$ (4.31)

(10)

References

[1] F.C. Alvarado, J.Lawrynowicz _and

_O.Suzuki:

_{Anoncommutative}

_differential

geometric

method to fractal _{geometry(II)(Duality for} _fractal _{sets and lattice models}

_on

fractal

sets).

[2] J.

Cuntz:

Simple $C^{*}$-algebrasgenerated

by _{isometries, Comm.} _{Math. Phys. 57 (1977),}

173-185.

[3]

_K.Falcorner:

_Fratal

_Geometry,

_{Mathematical foundations}

_and _{Applications,John} Wiley and Sons, U.S.A.(1990)

[4]

S

and

S. Ishimura:Fractal

Mathematics,

$\mathrm{T}\mathrm{o}\mathrm{k}\mathrm{y}\triangleright \mathrm{T}\mathrm{o}\mathrm{s}\mathrm{h}\mathrm{o}$, Japan(in japanese) (1990)

[5]

_S.Kakutani:

_On _{equivalence of} _infinite _product

_measures

_Ann.

Math., 47 (1948),

214-224.

[6]

J.Lawrynowicz,

and

O.Suzuki:

A fractal method to infinite

dimensional

Clifford

alge-bras, in Preparation.

[7]

J.Lawrynowicz,

_K.Nono _and

_O.Suzuki:

_Afractal _{method to complex} _anaysis, _in _prepa

ration.

[8] _{M.Mori, O.Suzuki} _and

_Y.Watatani:

_{Representations of} _Cuntz _algebras

_on

fractal sets,

in Preparation.

[9] M.Mori,_to

O.Suzuki

and

Y.Watatani:

Anoncommutative

differentiaJ geometric method fractal _{geometry(I)(Representations of}_Cuntz _{algebras of}_Hausdorff_type

_on

_fractal

sets), submitted to the Proc. ofInt.

ISSAC

Con.(2001).

[10]

R.Nevannlina:

Uniformizierung,

Springer Verlag(1953).

Julian

Lawrynowicz

Kiyoharu

_Nouno

Institute

of Physics Department of

Mathematics

$\mathrm{u}1$

.

Pomorska

149/153

_{Fukuoka University}

University of Lodz

of

$\mathrm{E}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{a}_{1}^{\mathrm{I}}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$