# Fragment Percolating and Many Body Localization Transition

Xiaolong Deng  Institut für Theoretische Physik, Leibniz Universität Hannover,
Appelstr. 2, 30167 Hannover, Germany

###### Abstract

Elements of eigenvectors obtained by exact diagonalization can be considered
as two dimensional lattice sites, in which dynamics of a given initial state
is seen as a percolating procedure on the lattice sites. Then one can use the
percolating procedure to generate a series of fragments of eigenvectors.
Combining the fraction ratio and entanglement entropy dynamics of the
generated fragment we examine the many-body localization transition in a
disordered hard-core bosonic model. It helps us to locate roughly the
transition point at the critical disorder strength $W_{c}\approx 12\sim 13$.
The scaling collapse of the fraction ratios gives us the localization length
exponent $\nu\approx 2.0$.

## Introduction.–

Interactions thermalize all of parts of a systemDeutsch1991 ; Srednicki1994 ,
while disorder localize non-interacting particlesAnderson1958 ; Mirlin2008 .
The interplay of disorder and interactions in quantum many-body systems can
lead to fascinating behaviors. One of prominent examples is many-body
localization (MBL)Basko2006 ; Gornyi2005 , in which quenched disorder prevents
thermalization of interacting systems. MBL provides a generic mechanism of
ergodicity breaking in quantum systems, and has attracted a huge attention in
recent years Nandshikore2015 ; Alet2018 ; Abanin2019 , including breakthrough
experiments Schreiber2015 ; Choi2016 ; Bordia2016 ; Smith2016 ; Lueschen2017 ;
Lukin2019 ; Rispoli2019 .

MBL systems are characterized by extensive number of local integral motions
(LIOMs)Serbyn2013b ; Huse2014 . The LIOMs forming a complete set can
completely determine the eigenstates. In the deep MBL phase two eigenstates
with the same total energy and particle number generally have different values
of LIOM. Thus, each eigenstate itself will form a fragmentation. Extensive
fragmentations break down the thermalization and lead to non-ergodicity. Very
recently, the idea of Hilbert space fragmentation has been explored
extensively not only in disorder-free systems Khemani2019 ; Sala2019 ;
Exp_fragmentation2020 , but also for the MBL problemsLoganChalker2018 ;
Pietracaprina2019 ; DeTomasi2019 ; Tarzia2020 ; Prelovsek2020 ; Krylov2020 .
In Ref. LoganChalker2018  the authors argue that the MBL transition can be
represented by a classical percolation model constructed in a graph of Fock
space. The percolation is decided by the competition between tunneling and
energy separation of Fock basis states. For the disordered Ising model they
get the scaling exponent $\nu=2$, which is consistent with the Harris
boundHarris1974 ; CCFS1986 . Based on a similar criterion as in
LoganChalker2018 , Ref. Pietracaprina2019  focus on a decimation scheme
through eliminating the irrelevant parts of Hilbert space. In Ref.
Prelovsek2020  the authors use rate equations to emphasizes the percolation
aspect of the MBL also. In their work they cluster the eigenstates with the
most largest overlaps into a fragment, and give the critical disorder
$W_{c}\approx 8$ for the random Heisenberg model.

Figure 1: The percolating procedure generates fragment structures of
eigenstates under different strengths of cutoffs $\mu$: (a) $\mu=0$, the
original structure with ascending energy indices; (b) $\mu=0.043$; (c)
$\mu=0.064$; and (d) $\mu=0.1664$. The $x$ and $y$ coordinates correspond to
the indices of eigen states and Fock states, respectively. Inside the fragment
the indices are sorted in ascending energy order. The calculation is done for
$L=14$, $N=7$, $V=1$ and $W=15$ with open boundary condition and a single
disorder.

Motivated by Refs LoganChalker2018 ; Pietracaprina2019 ; Prelovsek2020 , we
explore a bosonic MBL problem, which is interesting for a large variety of
physical problemsYan2013 ; Richerme2014 ; Jurcevic2014 ; Baier2016 ;
Leseleuc2019 . The fragment structure of the Hamiltonian eigenstates can be
self-consistently selected through a percolating procedureShklovskiiEfros .
Tracking the fragment generated by a given initial Fock state (i.e., track the
area that the state is percolating), we examine the MBL transition. As the MBL
transition is a kind of dynamical behaviorAbanin2019 , the static quantities,
such as level spacing and inverse participation ratio, might be difficult to
capture its dynamical feature. The fragments are dynamically generated in the
percolating procedure. The corresponding fraction ratios to the whole Hilbert
space and dynamics of entanglement entropy are naturally of dynamical
characteristics, which allow us to locate the MBL transition. We get
numerically the critical disorder strength $W_{c}\approx 12\sim 13$, which is
much larger than the ones obtained by the level spacingfootnote1 . More
interesting, the scaling of fraction ratios satisfies the Harris
criterionHarris1974 ; CCFS1986 . Our data gives $\nu\approx 2.0$, where $\nu$
is the localization length exponent.

## Model.–

We consider a 1D hard-core extended Bose-Hubbard model. The Hamiltonian is:

$\displaystyle\hat{H}=\sum_{i}-J(\hat{b}_{i}^{\dagger}\hat{b}_{i+1}+\mathrm{h.c.})+V\hat{n}_{i}\hat{n}_{i+1}+\epsilon_{i}\hat{n}_{i},$
(1)

where $\hat{b}_{i}$ are bosonic operators at site $i$
($(\hat{b}_{i}^{\dagger})^{2}=0$),
$\hat{n}_{i}=\hat{b}^{\dagger}_{i}\hat{b}_{i}$, $J=1$ and $V$ are,
respectively, the hopping amplitude and interaction strength, and
$\epsilon_{i}\in[-W,W]$ is a uniformly distributed random energy. Eq. (1) is
equivalent to a spin-$1/2$ random Heisenberg Hamiltonian, and has been studied
intensively as a standard MBL modelLuitz2015 . In the Fock basis
$\\{|j\rangle\\}$, Eq. (1) can be written as
$\hat{H}=\sum^{\mathcal{N}}_{j=1}\langle
j|\sum_{i}n_{i}n_{i+1}+\epsilon_{i}n_{i}|j\rangle|j\rangle\langle
j|+J\sum_{j,k}|j\rangle\langle k|,$ Then it is considered as a single-particle
Anderson problem in a high-dimensional lattice, where the first term is the
effective on-site disorder, and the second is the hopping between Fock basis.

Figure 2:  Relevant quantities (a) the fragment ratio of $N_{b}/\mathcal{N}$,
(b) the level spacing $\langle{r}\rangle_{b}$, and (c) the generalized fractal
dimensions $\langle{D_{2}}\rangle_{b}$ in the fragment generated from the Néel
state are calculated under different cutoffs. The parameters are the same as
in Fig. 1.

## Percolating procedure.–

By exact diagonalization of Eq. 1 one gets eigenenergies $E_{\alpha}$ and
eigenvectors $\psi_{\alpha}(j)$. The eigenvetors $\psi_{\alpha}(j)$ form a
matrix with the column eigenbasis $\\{|\alpha\rangle\\}$ and the row Fock
basis $\\{|j\rangle\\}$.

Now based on $\psi_{\alpha}(j)$ let us formulate a procedure in percolation
schemeShklovskiiEfros . Consider a two dimensional lattice formed by $x$-axis
(eigen basis) and $y$-axis (Fock basis). The lattice sites are the elements of
eigenvectors. We assume that the sites can be unblocked (active) or blocked
(inactive), which are decided by the amplitudes of elements. Bonds between
sites are hyperlinked either in $x$ or in $y$ direction, and unblocked. The
blocked sites do not permit flow of liquid in either $x$ or $y$ direction, but
the unblocked sites permit the flow of liquid in both directions. In other
words, blocked sites cannot be wet, nor can wet others, while the unblocked
sites are wet, and will instantly wet other unblocked sites. A given random
unblocked sites through this procedure may create two possibilities: either a
finite or an infinite number of unblocked sites will be wetfootnote2 . A
finite number of unblocked sites corresponds to a fragment of Hilbert space,
while the infinite number of unblocked sites corresponds to the whole Hilbert
space. Clearly the outcome will depend on the fraction of unblocked sites.
Here it will be decided by the disorder strength $W$. The random positions of
blocked and unblocked sites are also important. For a given disorder sample,
the distribution of blocked and unblocked sites are fixed and remains constant
during the percolation. In practice in our work, the fraction of unblocked and
blocked sites will be decided by a cutoff $\mu$, which will be discussed
later.

Figure 3: The fidelity $f$ (a) and the entropy dynamics (b) in the generated
fragments of the Néel state. The different colors represent different cutoffs
$\mu$. When $0\leq\mu\leq 0.2219$ the entropies are almost the same. The
parameters are the same as in Fig. 1.

We start from an initial Fock state. In the eigenbasis it means a cluster of
random-distributed unblocked sites in a 2D lattice. We choose a cutoff $\mu$
to numerically decide unblocked ($>\mu$) and blocked ($<\mu$) sites. From the
inverse participation ratio we know that the delocalized states have
$IPR=\sum_{j}|\psi_{\alpha}(j)|^{4}\approx\mathcal{N}|\psi_{\alpha}(j)|^{4}\propto
1/\mathcal{N}$, that is, $|\psi_{\alpha}(j)|\propto 1/\sqrt{\mathcal{N}}$. It
will be natural to start a cutoff from $\mu=1/\sqrt{\mathcal{N}}$, and
increase it gradually. In a long time limit the percolation will be in an
equilibrium. All of the connected unblocked sites may form a close subsystem.
If the Hilbert space fragmentation exists here, the norm of the generated
fragment should be equivalent to $1$. So, the norm
$f=\sum_{j,\alpha\in{\mathrm{block}}}|\psi_{\alpha}(j)|^{2}$ can be used to
measure the fidelity of the fragment in this block structure close to a
Hilbert space fragmentation. On the other hand, we are interested in the
entanglement entropy dynamics of the fragment. For an isolated finite system,
observing the unitary dynamics of an initial Fock state in the whole Hilbert
space should be equivalent to observe its dynamics in its Hilbert space
fragmentation. Therefore, we can define a kind of fidelity of the entropy
dynamics in the fragment comparing to the whole Hilbert space. Suppose
$S_{vN}(t)$ is the entropy dynamics in the whole Hilbert space and
$S^{b}_{vN}(t)$ is the entropy dynamics in the generated fragment for a given
cutoff. Then we introduce entropy ratios $\chi_{s,k}$ between $S^{b}_{vN}(t)$
and $S_{vN}(t)$. When the entropy is saturated in the long time limit, we
measure the entrop itself
$\chi_{s}=S^{b}_{vN}(t\to\infty)/S_{vN}(t\to\infty)$. However, before the
saturation, when the entropy is still increasing with time, we measure its
slope $\chi_{k}=k^{b}_{vN}/k_{vN}$. where the slope k is extracted from either
$\ln{S_{vN}(t)}\propto k\ln{t}$ for the delocalized state or $S_{vN}(t)\propto
k\ln{t}$ for the MBL stateZnidaric2008 ; Bardarson2012 ; Serbyn2013a . These
relative-error-like quantities will measure the fidelity of entropy dynamics
in the fragment.

In connection with the percolation, let us define the fraction ratio $\eta$ of
the size of generated fragment $N_{b}$ to the size of whole Hilbert space
$\mathcal{N}$: $\eta=\frac{N_{b}}{\mathcal{N}}$. $\eta$ is equivalent to the
probability of a random unblocked site wetting $N_{b}$ other unblocked sites.
In infinite lattice when $\eta=0$, the blocked sites prevent the liquid
spreading far from the initial site. On the contrary, as $\eta$ approaches
$1$, any unblocked site will result in all sites wet. In disorder ensemble
there are the typical fraction ratio
$\eta_{typ}=\exp(\langle\ln{N_{b}}\rangle)/{\mathcal{N}}$, and the averaged
fraction ratio
$\eta_{avg}=\langle{N_{b}}\rangle/{\mathcal{N}}$LoganChalker2018 . Generally
$\eta_{typ}$ and $\eta_{avg}$ give similar behaviors. However, in our
calculations $\eta_{typ}$ and $\eta_{avg}$ give peculiar feature in the
ergodic phase and the MBL phase, respectively.

Figure 4:  The averaged entropy ratio $\chi_{k}$ as a function of the typical
fraction ratio $\eta_{typ}$ for various systems sizes $L$ at different
disorder strengths: (a) $W=4$; (b) $W=8$; (c) $W=12$; and (d) $W=20$. (a) and
(b) indicate a large amount of fractions $\eta_{typ}\to\infty$ when
$L\to\infty$. Starting from $W\approx 12$ all of fractions $\eta_{typ}\to 0$
with $L\to\infty$. The disorder samples are $2000,1000,500$ for $L=12,14,16$,
respectively.  Figure 5:  The averaged entropy ratio $\chi_{k}$ as a function
of the averaged fraction ratio $\eta_{avg}$ for various systems sizes $L$ at
different disorder strengths: (a) $W=4$; (b) $W=8$; (c) $W=12$; and (d)
$W=20$. (a) and (b) indicate all of fractions $\eta_{avg}\to\infty$ when
$L\to\infty$. Starting from $W=12$ (c) all of fractions $\eta_{avg}$ do not
depend on the system sizes. At $W=20$ (d) we have not seen the dependence. The
disorder samples are $2000,1000,500$ for $L=12,14,16$, respectively.

## Fragment in a single disorder.–

Combining $f$ , $\chi_{k,s}$ and $\eta$ we will discuss the fragment generated
from an initial state. Without loss of generality, we choose the Néel state.
We firstly choose the phase at a strong disoder $W=15$ to demonstrate our
method. Fig. 1 shows that the percolating procedure under different cutoffs
generates different fragment structures for the eigenvectors in a single
disorder. The $x$-axis and $y-$axis correspond to the indices of eigen and
Fock basis, respectively. At $\mu=0$ (Fig. 1 (a)), there is only a big block
structure with eigen and Fock basis in the ascending-energy ascending order.
When the cutoff is increased to some point, two fragments start to appear, and
they keep the two-fragment structure until $\mu=0.043$, see Fig. 1 (b). Then
another different structure with more fragments starts to appear, and keep
itself for a range of $\mu$. With increasing the cutoff further, more and more
fragments appear, see Fig. 1 (c) and (d). We emphasize that the same fragment
structure can be kept for a range of $\mu$. In Fig. 1 the structures are shown
at the maximal value of $\mu$ which still generate the same fragment
structure.

We investigate the relevant quantities under different cutoffs. Fig. 2 (a)
shows that the fraction ratio changes with different cutoffs. As the eigen
levels in the fragment are sorted, we can calculate the level spacing
$\langle{r}\rangle_{b}=\langle\frac{\min(E_{\alpha+1}-E_{\alpha},E_{\alpha}-E_{\alpha-1})}{\max(E_{\alpha+1}-E_{\alpha},E_{\alpha}-E_{\alpha-1})}\rangle_{b}$Huse2007
and the generalized fractal dimensions
$\langle{D_{2}}\rangle_{b}=-\ln\langle{IPR}\rangle_{b}/\ln{N_{b}}$Mirlin2008 ,
see Fig. 2 (b) and (c).

Through checking the fidelity of the series of fragments in Fig. 3 (a) we know
that our percolating procedure works very well. Comparing Fig. 3 (a) and (b)
we find that even with $f=0.85$ the fragment entropy dynamics is still close
to the dynamics in the whole Hilbert space. When $\mu<0.23$ all of generated
fragments give almost the same dynamics. Therefore, one may expect an as-
small-as possible fragment but it can still capture the physics in the whole
Hilbert space.

## MBL transition.–

We have seen the cutoff triggers the percolation in our procedure. However,
various disorder samples may induce different cutoffs, different percolated
fractions, and different fidelities. This means we cannot simply fix either
the cutoff or the fidelity to make average over disorder ensemble. For
individual disorder we have to scan the cutoff from $0$ to the maximal value
(supporting the minimum of the fragment fraction). Before making average we
take into account the relative errors of saturated entropy $1-\chi_{s}$. For
each sample we account only the smallest fraction within a given relative
error $1-\chi_{s}$, and then make average over samples. In this way, we get
the typical (averaged) fraction ratio $\eta_{typ}$($\eta_{avg}$). The relation
between $\chi_{k,s}$ and $\eta_{typ,avg}$ will help us to roughly locate the
transition.

In Fig. 4 we plot the averaged entropy ratio as a function of the typical
fraction ratio for various system sizes $L$. Fig. 4 (a) and (b) show that both
at small disorder strength $W=4$ and at finite disorder strength $W=8$,
$\eta_{typ}$ increases with system sizes $L$. This indicates that
$\eta_{typ}\to 1$ when $L\to\infty$, although the speed at $W=8$ might be much
slower than the one at $W=4$. It is interesting to note that when
$\eta_{typ}<0.1$ (see the left sides of the dashed lines in Fig. 4 (a) and
(b)) the fragment tends to be localized (i.e. $\eta_{typ}\to 0$ at
$L\to\infty$), and with a high entropy ratio for $W=8$. This unconventional
fragment is only seen in the regime $W\in[4,8]$ of $\eta_{typ}$. The averaged
fraction ratio $\eta_{avg}$, however, gives us clearly the tendency to the
delocalization for all of fragments at $W=4,8$, see Fig. 5 (a) and (b).

With the disorder strength increasing, the localization effect becomes
stronger. From the curves of $\chi_{k}-\eta_{typ}$ and $\chi_{k}-\eta_{avg}$
we can roughly determine the MBL transition point. At the transition, the
unconventional regime starts to extend in the whole $\eta_{typ}$. All of
$\eta_{typ}\to 0$ when $L\to\infty$, see Fig. 4 (c). In this sense it
indicates that at the transition point may be a localized phase. Comparing to
$\eta_{typ}$, Fig. 5 (c) shows that $\eta_{avg}$ does not depend on the system
sizes at the transition point. This indicates that there might be a universal
distribution of $\eta_{avg}$ at the critical $W_{c}$ (see Fig. 5 (c)).
Increase further the disorder, however, $\eta_{avg}$ does not show the clear
dependence on the system sizes, at least at $W=20$ (Fig. 5 (d)). This is
consistent with the observation in Ref. Pietracaprina2019 .

We have obtained the critical disorder $W_{c}$ from the growth of the entropy
dynamics $\chi_{k}$. Now we would like from the other side to check the
transition using the saturated entropy dynamics $\chi_{s}$. For a given
$\chi_{s}$ we do the scaling collapse of $\eta_{typ}$ for various systems, by
the means of the universal function $(W-W_{c})L^{1/\nu}$, where $\nu$ is the
localization length exponent. It is known that a disorder-driven phase
transition in $d$ dimensional space satisfies the Harris bound $\nu\geq
2/d$Harris1974 ; CCFS1986 . In MBL transitions a similar bound is also
derivedChandran2015 ; Khemani2017 . In our scaling collapse we choose the
values of $\chi_{s}$ close to $1$, for instance $\chi_{s}=0.98$. It is very
interesting that our data yields $W_{c}\approx 13$ and $\nu\approx 2.0$, see
Fig. 6. Our result on the scaling exponent is consistent with Ref.
LoganChalker2018 .

Figure 6:  The typical fraction ratio at $\chi_{s}=0.98$ as a function of
disorder strength for various systems sizes $L$. The main panel shows the
scaling collapse with $W_{c}\approx 13.0$ and $\nu\approx 2.0$. The inset are
the raw data before collapsing. The disorder samples are $5000,2000,1000,500$
for $L=10,12,14,16$, respectively.

## Conclusions and outlook.–

Given an initial Fock state we can generate the fragments of the eigenvectors
of a Hamiltonian by the means of a percolating procedure. Using this method we
have investigated the hard-core extended Bose-Hubbard model with on-site
disorder. Combing the fraction and entropy ratios of the fragments we have
shown that the MBL transition may take place at about $W_{c}\approx 12\sim
13$, and the fraction ratios have a scaling collapse with $\nu\approx 2$.

This percolating procedure can also be applied to power-law
interacting/hopping lattice modelsBurin2006 ; Yan2013 ; Yao2013 ; Burin2015a ;
Burin2015b ; Deng2020 . In future it will be interesting to study whether the
generated fragment depends on the power-law exponent.

## Acknowledgements.–

We are grateful to Luis Santos and Wei-Han Li for collaboration on related
work and for very useful discussions. We acknowledge the support of the German
Science Foundation (DFG) (SA 1031/11, SFB 1227, and Excellence Cluster
QuantumFrontiers).

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# Cocycle perturbation on Banach algebra

YU Jing Wu  Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>and  Luo Yi Shi  Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>
# Cocycle perturbation on Banach algebra

YU Jing Wu  Tianjin Vocational Institute
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>and  Luo Yi Shi  Department of Mathematics
Tianjin Polytechnic University
Tianjin 300160
P.R. CHINA<EMAIL_ADDRESS>
###### Abstract.

Let $\alpha$ be a flow on a Banach algebra $\mathfrak{B}$, and $t\longmapsto
u_{t}$ a continuous function on $\mathbb{R}$ into the group of invertible
elements of $\mathfrak{B}$ such that
$u_{s}\alpha_{s}(u_{t})=u_{s+t},s,t\in\mathbb{R}$. Then
$\beta_{t}=$Ad$u_{t}\circ\alpha_{t},t\in\mathbb{R}$ is also a flow on
$\mathfrak{B}$. $\beta$ is said to be a cocycle perturbation of $\alpha$. We
show that if $\alpha,\beta$ are two flows on nest algebra (or quasi-triangular
algebra), then $\beta$ is a cocycle perturbation of $\alpha$. And the flows on
nest algebra (or quasi-triangular algebra) are all uniformly continuous.

###### Key words and phrases:

cocycle perturbation, inner perturbation, nest algebra, quasi-triangular
algebra.

###### 2000 Mathematics Subject Classification:

47D03(46H99, 46K50, 46L57)

Supported by NCET(040296), NNSF of China(10971079)

##  1\. Introduction

In the quantum mechanics of particle systems with an infinite number of
degrees of freedom, an important problem is to study the differential equation

$\frac{d\alpha_{t}(A)}{dt}=S\alpha_{t}(A)$

under variety of circumstances and assumptions. In each instance the $A$
corresponds to an observable, or state, of the system and is represented by an
element of some suitable Banach algebra $\mathfrak{B}$. And $S$ is an operator
on $\mathfrak{B}$, $\\{\alpha_{t}\\}_{t\in\mathbb{R}}$ is a group of bounded
automorphisms on $\mathfrak{B}$. The Function

$t\in\mathbb{R}\longmapsto\alpha_{t}(A)\in\mathfrak{B}$

describes the motion of $A$. The dynamics are given by solutions of the
differential equation which respect certain supplementary conditions of
continuity. Thus it is worth to study the group of bounded automorphisms on
$\mathfrak{B}$. More details see [1].

A flow $\alpha$ on $\mathfrak{B}$ is a group homomorphism of the real line
$\mathbb{R}$ into the group of bounded automorphisms on $\mathfrak{B}$ (i.e.
$t\longmapsto\alpha_{t}$) such that

$\lim_{t\rightarrow t_{0}}||\alpha_{t}(B)-\alpha_{t_{0}}(B)||=0$

for each $t_{0}\in\mathbb{R}$ and each $B\in\mathfrak{B}$. If there exists a
$h\in\mathfrak{B}$ such that $\alpha_{t}(B)=e^{th}Be^{th},\forall
B\in\mathfrak{B},t\in\mathbb{R}$, then we call $\alpha$ an inner flow. We say
that a flow $\alpha$ is uniformly continuous if

$\lim_{t\rightarrow t_{0}}||\alpha_{t}-\alpha_{t_{0}}||=0$

for each $t_{0}\in\mathbb{R}$.

If $\alpha$ is a flow on $\mathfrak{B}$ and if $u$ is a continuous map of
$\mathbb{R}$ into the group of invertible elements $G(\mathfrak{B})$ of
$\mathfrak{B}$ such that $u_{s}\alpha_{s}(u_{t})=u_{s+t},s,t\in\mathbb{R}$,
then we call $u=(u_{t})_{t\in\mathbb{R}}$ an $\alpha$-cocycle for
$(\mathfrak{B},\mathbb{R},\alpha)$. Let
$\beta_{t}=$Ad$u_{t}\circ\alpha_{t},t\in\mathbb{R}$ (i.e.
$\beta_{t}(B)=u_{t}\alpha_{t}(B)u_{t}^{-1},\forall B\in\mathfrak{B}$), then
$\beta$ is also a flow on $\mathfrak{B}$, and $\beta$ is said to be a cocycle
perturbation of $\alpha$.

If $\alpha$ is a flow on $\mathfrak{B}$, let $D(\delta_{\alpha})$ be composed
of those $B\in\mathfrak{B}$ for which there exists a $A\in\mathfrak{B}$ with
the property that

$A=\lim_{t\rightarrow 0}\frac{\beta_{t}(B)-B}{t}.$

Then $\delta_{\alpha}$ is a linear operator on $D(\delta_{\alpha})$ defined by
$\delta_{\alpha}(B)=A$. We call $\delta_{\alpha}$ the infinitesimal generator
of $\alpha$. By [1,Proposition 3.1.6], $\delta_{\alpha}$ is a closed
derivation. i.e. the domain $D(\delta_{\alpha})$ is a dense subalgebra of
$\mathfrak{B}$ and $\delta_{\alpha}$ is closed as a linear operator on
$D(\delta_{\alpha})$ and satisfies
$\delta_{\alpha}(AB)=\delta_{\alpha}(A)B+A\delta_{\alpha}(B),$ for $A,B\in
D(\delta_{\alpha})$. We call $\beta$ an inner perturbation of $\alpha$, if
$\alpha,\beta$ are two flows on $\mathfrak{B}$,
$D(\delta_{\alpha})=D(\delta_{\beta})$ and there exists $h\in\mathfrak{B}$
such that $\delta_{\beta}=\delta_{\alpha}+$ad$ih$ (where $i$ is the imaginary
unit, and ad$ih(B)\triangleq i(hB-Bh),\forall B\in\mathfrak{B}$). Moreover
$D(\delta_{\alpha})=\mathfrak{B}$ if and only if $\alpha$ is uniformly
continuous. More details see [1,2].

The problem we considered is classifying cocycle of flows on Banach algebras.
Such a problem has been considered in the $C^{*}$algebra cases, notably by
Kishimoto[3-10]. We refer the reader to [3] for a detailed study of the
general results concerning cocycles and invariants for cocycle perturbation of
flows on $C^{*}$-algebras. In particular, it is shown there that let $\alpha$
be a flow on a $C^{*}$-algebra $\mathfrak{A}$, if $u=(u_{t})_{t\in\mathbb{R}}$
a $\alpha$-cocycle for $(\mathfrak{A},\mathbb{R},\alpha)$, and $u$ is
differentiable (i.e. $\lim_{t\rightarrow
t_{0}}\frac{u_{t}-u_{t_{0}}}{t-t_{0}}$ exist for any $t_{0}\in\mathbb{R}$) and
$h=-i(du_{t}/dt)|_{t=0}\in\mathfrak{A}$, then the infinitesimal generator of
the flow $\beta_{t}=Adu_{t}\circ\alpha_{t}$ is given by
$\delta_{\beta}=\delta_{\alpha}+$ad$ih$. i.e. $\beta$ is an inner perturbation
of $\alpha$. Moreover for any $\alpha$-cocycle $u=(u_{t})_{t\in\mathbb{R}}$,
there is $w\in G(\mathfrak{A})$ and a differentiable $\alpha$-cocycle
$v=(v_{t})_{t\in\mathbb{R}}$ (i.e. $v_{t}$ is an $\alpha$-cocycle and
differentiable) such that $u_{t}=wv_{t}\alpha_{t}(w^{-1})$. In section 2, we
consider the cocycle of flows on Banach algebras and obtain some similar
results to [3].

It is well-known that a flow $\alpha$ on $\mathfrak{B}$ may not be uniformly
continuous even if $\mathfrak{B}$ is a $C^{*}$algebra [1,2]. In section 3, we
study the flows on nest algebra $\tau(\mathcal{N})$ and quasi-triangular
algebra $Q\tau(\mathcal{N})=\tau(\mathcal{N})+K$ [11]. We recall that a nest
$\mathcal{N}$ is a chain of closed subspaces of a Hilbert space $\mathfrak{H}$
containing $\\{0\\}$ and $\mathfrak{H}$ which is in addition closed under
taking arbitrary intersections and closed spans. The nest algebra
$\mathcal{T(N)}$ associated with $\mathcal{N}$ is the set of all $T\in
B(\mathfrak{H})$ which leave each element of the nest invariant. For instance,
if $\mathfrak{H}$ is separable with orthonormal basis
$\\{e_{n}\\}_{n=1}^{\infty}$ and
$\mathfrak{H}_{n}$=span$\\{e_{1},\cdots,e_{n}\\}$, then
$\mathcal{N}=\\{0,\mathfrak{H}\\}\cup\\{\mathfrak{H}_{n}\\}_{n=1}^{\infty}$ is
a nest. In this case, $\mathcal{T(N)}$ is simply the set of all operators
whose matrix representation with respect to this basis is upper triangular. It
is obvious, $\tau(\mathcal{N})$ and $Q\tau(\mathcal{N})$ are typical Banach
algebras. We obtain that all of the flows on $\tau(\mathcal{N})$ (or
$Q\tau(\mathcal{N})$) are uniformly continuous. Moreover all of the flows are
cocycle perturbation to each other.

##  2\. Cocycle perturbations

Let $\mathfrak{B}$ be a Banach algebra, $\alpha$ be a flow on $\mathfrak{B}$
and $u$ an $\alpha$-cocycle, then
$\beta_{t}=$Ad$u_{t}\circ\alpha_{t},t\in\mathbb{R}$ is a cocycle perturbation
of $\alpha$. In this section, we obtain that $\beta$ is an inner perturbation
of $\alpha$ if and only if $u$ is differentiable (Theorem 2.4.). Moreover for
any $\alpha$-cocycle $u$ , there is a differentiable $\alpha$-cocycle $v$ and
a inventible element $w$ in $\mathfrak{B}$ such that
$u_{t}=wv_{t}\alpha_{t}(w^{-1})$ (Theorem 2.8.).

We will use the following Lemmas [1, Proposition 3.1.3, Theorem 3.1.33].

Lemma 2.1[1]: Let $\\{\alpha_{t}\\}_{t\in\mathbb{R}}$ be a flow on the Banach
algebra $\mathfrak{B}$. Then there exists an $M\geq 1$ and
$\xi\geq$inft≠0($t^{-1}$log$||\alpha_{t}||$) such that $||\alpha_{t}||\leq
Me^{\xi|t|}.$

Lemma 2.2[1]: Let $\alpha$ be a flow on a Banach algebra $\mathfrak{B}$ with
infinitesimal generator $\delta_{\alpha}$. For each $P\in\mathfrak{B}$ define
the bounded derivation $\delta_{P}$ by $D(\delta_{P})=\mathfrak{B}$ and
$\delta_{P}(B)=i[P,B]\triangleq i(PB-BP),\forall B\in\mathfrak{B}$. Then
$\delta+\delta_{P}$ generates a flow on $\mathfrak{B}$ given by

$\alpha_{t}^{P}(B)=\alpha_{t}(B)+\sum_{n\geq
1}i^{n}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{n-1}}dt_{n}[\alpha_{t_{n}}(P),[\cdots[\alpha_{t_{1}}(P),\alpha_{t}(B)]]],\forall
B\in\mathfrak{B},t\in\mathbb{R}.$

Lemma 2.3: Let $\mathfrak{B}$ be a Banach algebra with unit $\mathbf{1}$,
$\alpha$ be a flow on $\mathfrak{B}$ and let $\delta$ denote the infinitesimal
generator of $\alpha$. Furthermore, for each $P\in\mathfrak{B}$ define
$\delta_{P}$ as in Lemma 2.2.

Then $\delta+\delta_{P}$ generates a flow on $\mathfrak{B}$ given by

$\alpha_{t}^{P}(B)=\alpha_{t}(B)+\sum_{n\geq
1}i^{n}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{n-1}}dt_{n}[\alpha_{t_{n}}(P),[\cdots[\alpha_{t_{1}}(P),\alpha_{t}(B)]]].$

Moreover, one has

$\alpha_{t}^{P}(B)=u^{P}_{t}\alpha_{t}(B){(u^{P}_{t}})^{-1},$

where $u_{t}^{p}$ is a one-parameter family of invertible elements, determined
by

$u^{P}_{t}=1+\sum_{n\geq
1}i^{n}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{n-1}}dt_{n}\alpha_{t_{n}}(P)\cdots\alpha_{t_{1}}(P)$

which satisfies the $\alpha$-cocycle relation

$u^{P}_{t+s}=u^{P}_{t}\alpha_{t}(u^{P}_{s}).$

All integrals converge in the strong topology. The integrals define norm-
convergent series of bounded operators and there exists an $M\geq 1$,
$\xi\geq$inft≠0($t^{-1}$log$||\alpha_{t}||$) such that

$||\alpha_{t}^{P}(B)-\alpha_{t}(B)||\leq
Me^{\xi|t|}(e^{M|t|||P||}-1),||u^{P}_{t}-\mathbf{1}||\leq
Me^{\xi|t|}(e^{M|t|||P||}-1)$ .

Proof : The first statement of the proposition can be obtained from Lemma 2.2.
We just give the proof of the last statement of the Lemma.

There exists an $M\geq 1$ and $\xi\geq$inft≠0($t^{-1}$log$||\alpha_{t}||$)
such that $||\alpha_{t}||\leq Me^{\xi|t|},$ by Lemma 2.1. Let

$M_{t}=\left\\{\begin{array}[]{ll}Me^{\xi|t|}&\mbox{$|t|>1$;}\\\
M&\mbox{$|t|\leq 1$.}\end{array}\right.$

Next we consider $u^{P}_{t}$ defined by the series. The $n$-th term in this
series is well defined and has norm less than
$\frac{|t|^{n}}{n!}M_{t}^{n}||P||^{n}$. Thus $u^{P}_{t}$ is a norm-continuous
one-parameter family of elements of $\mathfrak{A}$ with $u^{P}_{0}=1$ and
$||u_{t}||\leq e^{|t|M_{t}||P||}$. Consequently, $u^{P}_{t}$ is invertible for
all $t\in[-t_{0},t_{0}]$ for some $t_{0}>0$ and $(u^{P}_{t})^{-1}$ is a norm-
continuous one-parameter family of elements of $\mathfrak{B}$ for all
$t\in[-t_{0},t_{0}]$.

Next one has

$\frac{du^{P}_{t}}{dt}=iu^{P}_{t}\alpha_{t}(P),$ $\lim_{t\rightarrow
0}(\frac{u^{P}_{t}-1}{t})=\frac{du^{P}_{t}}{dt}|_{t=0}=iP.$

Hence,

$\lim_{t\rightarrow 0}(\frac{(u^{P}_{t})^{-1}-1}{t})=\lim_{t\rightarrow
0}(u^{P}_{t})^{-1}\frac{1-u^{P}_{t}}{t}=-iP.$

To establish the $\alpha$-cocycle relation we first note that

$\frac{du^{P}_{t+s}}{ds}=iu^{P}_{t+s}\alpha_{t+s}(P)$

and $u^{P}_{t+s}|_{s=0}=u^{P}_{t}$.

But $\alpha_{t}(u^{P}_{s})=u^{\alpha_{t}(P)}_{s}$ and hence

$\frac{d}{ds}u^{P}_{t}\alpha_{t}(u^{P}_{s})=iu^{P}_{t}u^{\alpha_{t}(P)}_{s}\alpha_{s}(\alpha_{t}(P))=iu^{P}_{t}\alpha_{t}(u^{P}_{s})\alpha_{t+s}(P).$

Moreover, $u^{P}_{t}\alpha_{t}(u^{P}_{s})|_{s=0}=iu^{P}_{t}\alpha_{t}(p)$.
Thus $s\longmapsto u^{P}_{t+s}$ and $s\longmapsto
u^{P}_{t}\alpha_{t}(u^{P}_{s})$ satisfy the same first-order differential
equation and boundary condition for each $t\in\mathbb{R}$. Therefore, the two
functions are equal and can be obtained by iteration of the integral equation

$X_{t}(s)=u^{P}_{t}+i\int_{0}^{s}ds^{\prime}X_{t}(s^{\prime})\alpha_{t+s^{\prime}}(P).$

Hence, $u^{P}_{t+s}=u^{P}_{t}\alpha_{t}(u^{P}_{s}),t,s\in\mathbb{R}$. Since
$u^{P}_{t}$ is invertible for all $t\in[-t_{0},t_{0}]$ for some $t_{0}>0$, we
obtain that $u^{P}_{t}$ is a norm-continuous one-parameter family of
inventible elements. Thus $t\mapsto u^{P}_{t}\alpha_{t}(B)(u^{P}_{t})^{-1}$
defines a flow $\beta_{t}$ on $\mathfrak{B}$.

Let $\tilde{\delta}$ denote the infinitesimal generator of $\beta$. Next we
will prove $\tilde{\delta}=\delta+\delta_{P}$.

Choose $A\in D(\delta+\delta_{P})$, one has

$\delta(A)=\lim_{t\rightarrow 0}(\frac{\alpha_{t}(A)-A}{t}),$
$\displaystyle\tilde{\delta}(A)$ $\displaystyle=$
$\displaystyle\lim_{t\rightarrow 0}(\frac{\beta_{t}(A)-A}{t})$
$\displaystyle=$ $\displaystyle\lim_{t\rightarrow
0}(\frac{u^{P}_{t}\alpha_{t}(A)(u^{P}_{t})^{-1}-u^{P}_{t}A(u^{P}_{t})^{-1}}{t}$
$\displaystyle+$
$\displaystyle\frac{u^{P}_{t}A(u^{P}_{t})^{-1}-u^{P}_{t}A}{t}+\frac{u^{P}_{t}A-A}{t})$
$\displaystyle=$ $\displaystyle(\delta+\delta_{P})(A).$

Similarly, if $A\in D(\tilde{\delta})$, we
obtain$\tilde{\delta}(A)=(\delta+\delta_{P})(A)$. And then
$\tilde{\delta}=\delta+\delta_{P}$.

Thus one must have
$\alpha_{t}^{P}(B)=\beta_{t}(B)=u^{P}_{t}\alpha_{t}(B){(u^{P}_{t}})^{-1},$ for
any $B\in\mathfrak{B}$ by [1, Theorem 3.1.26]. Finally the estimates on
$\alpha_{t}^{P}(B)-\alpha_{t}(B)$ and $u^{P}_{t}-\mathbf{1}$ are
straightforward.

Theorem 2.4 : Let $\alpha$ be a flow on $\mathfrak{B}$,
$(u_{t})_{t\in\mathbb{R}}$ be an $\alpha$-cocycle ,
$\beta_{t}=$Ad$u_{t}\circ\alpha_{t}$. Then $\beta$ is an inner perturbation of
$\alpha$ if and only if $u_{t}$ is is differentiable.

Proof : $\Rightarrow$ It follows immediately from Lemma 2.3.

$\Leftarrow$ If $u_{t}$ is an $\alpha$-cocycle and differentiable with
$h=-i(du_{t}/dt)|_{t=0}\in\mathfrak{B}$, then $u_{t}$ is given by

$u_{t}=1+\sum_{n\geq
1}i^{n}\int_{0}^{t}dt_{1}\int_{0}^{t_{1}}dt_{2}\cdots\int_{0}^{t_{n-1}}dt_{n}\alpha_{t_{n}}(P)\cdots\alpha_{t_{1}}(P).$

Hence, $\beta$ is an inner perturbation of $\alpha$ by lemma 2.3.

Corollary 2.5 : Adopt the assumptions of the Lemma 2.3 and also assume that
$\alpha_{t}$ is an inner flow. i.e. there exists $h\in\mathfrak{B}$ such that
$\alpha_{t}(A)=e^{ith}Ae^{-ith},\forall A\in\mathfrak{B},t\in\mathbb{R}$.

Then

$\alpha^{P}_{t}(A)=\Gamma_{t}^{P}A(\Gamma_{t}^{P})^{-1},u_{t}^{P}=\Gamma_{t}^{P}e^{-ith},$

where $u_{t}^{P}$ is defined as in Lemma 2.3 and $\Gamma_{t}^{P}=e^{it(h+P)}$.
i.e. $\alpha^{P}_{t}$ is an inner flow.

Proof : If $\Gamma_{t}^{P}=e^{it(h+P)}$ and $X_{t}=\Gamma_{t}^{P}e^{-ith}$,
then

$\frac{dX_{t}}{dt}=i\Gamma_{t}^{P}Pe^{-ith}=iX_{t}\alpha_{t}(P)$

and $X_{0}=\mathbf{1}$. Thus, $X_{t}$ is the unique solution of the integral
equation

$X_{t}=\mathbf{1}+i\int_{0}^{t}dsX_{s}\alpha_{s}(P).$

This solution can be obtained by iteration and one finds $X_{t}=u_{t}^{P}$,
where $u_{t}^{P}$ is defined as in Lemma 2.3.
And$\alpha_{t}^{P}(A)=u^{P}_{t}\alpha_{t}(A){(u^{P}_{t}})^{-1}=\Gamma_{t}^{P}A(\Gamma_{t}^{P})^{-1}$.

Next we obtain that every $\alpha$-cocycle is similar to a differentiable
$\alpha$-cocycle.

Definition 2.6: Let $\alpha$ be a flow on $\mathfrak{B}$. $A\in\mathfrak{B}$
is called an analytic element for $\alpha$ if there exist a analytic function
$f:~{}~{}\mathbb{C}\rightarrow\mathfrak{B}$ such that $f(t)=\alpha_{t}(A)$ for
$t\in\mathbb{R}$.

Lemma 2.7: Let $\alpha$ be a flow on Banach algebra $\mathfrak{B}$ and $M,\xi$
are constants such that $||\alpha_{t}||\leq Me^{\xi|t|}$. For
$A\in\mathfrak{B}$, define

$A_{n}=\sqrt{\frac{n}{\pi}}\int\alpha_{t}(A)e^{-nt^{2}-\xi
t}dt,~{}~{}~{}n=1,2,\cdots.$

Then each $A_{n}$ is an entire analytic element for $\alpha_{t}$,
$||A_{n}||<M||A||$ for all n; and $A_{n}\rightarrow A$ in the weak topology as
$n\rightarrow\infty$. In particular, the $\alpha$ analytic elements form a
normal-dense subspace of $\mathfrak{B}$.

Proof :

$f_{n}(z)=\sqrt{\frac{n}{\pi}}\int\alpha_{t}(A)e^{-n(t-z)^{2}-\xi(t-z)}dt$

is well defined for all $z\in\mathbb{C}$, since $t\mapsto
e^{-n(t-z)^{2}}\in\mathbf{L}^{1}(\mathbb{R})$ for each $z\in\mathbb{C}$.

For $z=s\in\mathbb{R}$, we have

$\displaystyle f_{n}(s)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{n}{\pi}}\int\alpha_{t}(A)e^{-n(t-s)^{2}-\xi(t-s)}dt$
$\displaystyle=$
$\displaystyle\sqrt{\frac{n}{\pi}}\int\alpha_{t+s}(A)e^{-nt^{2}-\xi
t}dt=\alpha_{s}(A_{n}).$

But for $\eta\in\mathfrak{B}^{*}$ we have

$\eta(f_{n}(z))=\sqrt{\frac{n}{\pi}}\int\eta(\alpha_{t}(A))e^{-n(t-z)^{2}-\xi(t-z)}dt.$

Since $|\eta(\alpha_{t}(A))|\leq M||\eta||||A||$, it follows from the Lebesgue
dominated convergence theorem that $t\mapsto\eta(f_{n}(z))$ is analytic. Hence
each $A_{n}$ is analytic for $\alpha_{t}(A)$.

Next, one derives the estimate

$||A_{n}||\leq M||A||\sqrt{\frac{n}{\pi}}\int e^{-n(t)^{2}}dt\leq M||A||.$

Next note that

$\int e^{-nt^{2}-\xi t}dt=e^{\frac{\xi^{2}}{4n^{2}}}\sqrt{\frac{\pi}{n}}.$

Hence

$\eta(A_{n}-A)=\sqrt{\frac{n}{\pi}}\int e^{-nt^{2}-\xi
t}\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)dt$

for all $\eta\in\mathfrak{B}^{*}$. But for any $\varepsilon>0$ we may choose
$\delta>0$ such that $t<|\delta|$ implies
$|\eta(\alpha_{t}(A)-A)|<\varepsilon$. Further, we can choose $N$ large enough
that $\frac{\xi}{2N}<\frac{\delta}{2}$. It follows that if $n>N$ we have

$\displaystyle|\eta(A_{n}-A)|$ $\displaystyle\leq$
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|<\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)|dt$ $\displaystyle+$
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|\geq\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)|dt$

On the other hand,

$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|<\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)|dt$ $\displaystyle\leq$
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|<\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-A)|dt$ $\displaystyle+$
$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|<\delta}e^{-nt^{2}-\xi
t}|\eta(A)(1-e^{-\frac{\xi^{2}}{4n^{2}}})|dt$ $\displaystyle<$
$\displaystyle\varepsilon
e^{\frac{\xi^{2}}{4n^{2}}}+|1-e^{-\frac{\xi^{2}}{4n^{2}}}|\cdot||\eta||\cdot||A||Me^{\frac{\xi^{2}}{4n^{2}}}.$

And

$\displaystyle\sqrt{\frac{n}{\pi}}\int_{|t|\geq\delta}e^{-nt^{2}-\xi
t}|\eta(\alpha_{t}(A)-e^{-\frac{\xi^{2}}{4n^{2}}}A)|dt$ $\displaystyle\leq$
$\displaystyle\sqrt{\frac{n}{\pi}}||\eta||\cdot||A||M\int_{|t|\geq\delta}e^{-nt^{2}}dt$
$\displaystyle+$
$\displaystyle\sqrt{\frac{n}{\pi}}e^{-\frac{\xi^{2}}{4n^{2}}}||\eta||\cdot||A||\int_{|t|\geq\delta}e^{-nt^{2}-\xi
t}dt.$

So $|\eta(A_{n}-A)|\rightarrow 0$ when $n\rightarrow\infty$, for any
$\eta\in\mathfrak{B}^{*}$. Finally note that the norm closure and the weak
closure of convex set are the same, hence the $\alpha$ analytic elements form
a normal-dense subspace of $\mathfrak{B}$.

Theorem 2.8: If $u$ is an $\alpha$-cocycle for $\mathfrak{B}$. Then, given
$\varepsilon>0$, there is a differentiable $\alpha$-cocvcle $v$ and a $w\in
G(\mathfrak{B})$ such that

$||w-\mathbf{1}||<\varepsilon,u_{t}=wv_{t}\alpha_{t}(w^{-1}).$

Proof : The proof is similar to [3,Lemma 1.1] and is omitted.

Given two flows $\alpha$ and $\beta$ on a unitary Banach algebra
$\mathfrak{B}$ we say that $\beta$ is a conjugate to $\alpha$ if there is a
bounded automorphism $\sigma$ of $\mathfrak{B}$ such that
$\beta=\sigma\alpha\sigma^{-1}$. Conjugate, cocycle perturbation and inner
perturbation define three equivalence relations. We say that $\beta$ is
cocycle-conjugate to $\alpha$ if there is a bounded automorphism $\sigma$ of
$\mathfrak{B}$ such that $\beta$ is a cocycle perturbation of
$\sigma\alpha\sigma^{-1}$. This also defines an equivalence relation among the
flows.

We say that $\alpha$ is approximately inner if there is a sequence
$\\{h_{n}\\}$ in $\mathfrak{B}$ such that $\alpha_{t}=$ limAd $e^{th_{n}}$,
i.e., $\alpha_{t}(A)=$$\lim$Ad$e^{th_{n}}(A)$ for every $t\in\mathbb{R}$ and
$A\in\mathfrak{B}$, or equivalently, uniformly continuous in $t$ on every
compact subset of $\mathbb{R}$ and every $A\in\mathfrak{B}$. A flow on a
Banach algebra $\mathfrak{B}$ is said to be asymptotically inner if there is a
continuous function $h$ of $\mathbb{R}_{+}$ into $\mathfrak{B}$ such that
$\lim_{s\rightarrow\infty}\max_{|t\leq
1|}\parallel\alpha_{t}(A)-$Ad$e^{th(s)}(A)\parallel=0$ for any
$A\in\mathfrak{B}$.

Corollary 2.9: Let $\alpha$ and $\beta$ be two flows on Banach algebra
$\mathfrak{B}$. Then the following conditions are equivalent:

(i). $\beta$ is cocycle-conjugate to $\alpha$.

(ii). $\beta$ is an inner perturbation of $\sigma\alpha\sigma^{-1}$ for some
automorphism $\sigma$ of $\mathfrak{B}$, where $\sigma\alpha\sigma^{-1}$ is
the action $t\mapsto\sigma\alpha_{t}\sigma^{-1}$.

If one of the above conditions are satisfied and $\alpha$ is
inner(approximately or asymptotically inner), then so is $\beta$.

Proof: The first statement of the proposition can be obtained from
[3,Corollary 1.3]. We just give the proof of the last statement of the
proposition.

First we prove that if $\beta$ is an inner perturbation of $\alpha$, i.e.
$\delta_{\beta}=\delta\alpha+i$ad$P$, then it follows that if $\alpha$ is
inner(approximately or asymptotically inner), then so is $\beta$.

(a) If $\alpha$ is inner,then so is $\beta$ by Corollary 2.5.

(b) If $\alpha$ is approximately inner, then there is a sequence $\\{h_{n}\\}$
in $\mathfrak{B}$ such that

$\lim_{n\rightarrow\infty}\max_{|t|\leq
1}||\alpha_{t}(A)-Ade^{th_{n}}(A)||=0.$

$\alpha_{n,t}\triangleq Ade^{th_{n}}$, we construct $\alpha_{n}$-cocycle
$u_{n,t}$(resp. $u$) for $\alpha_{n,t}$(resp. $\alpha$) such that
$\frac{d}{dt}u_{n,t}|_{t=0}=iP$ by Lemma 2.3. Then since
$\beta_{t}=Adu_{t}\circ\alpha_{t}$ and $u_{n,t}\rightarrow u_{t}$, we obtain
that $Adu_{n,t}\circ\alpha_{n,t}\rightarrow\beta_{t}$. Besides,
$Adu_{n,t}\circ\alpha_{n,t}$ is inner by Corollary 2.5. Then $\beta$ is
approximately inner.

(c) If $\alpha$ is asymptotically inner, then there is a continuous function
$h$ of $\mathbb{R}_{+}$ into $\mathfrak{B}$ such that

$\lim_{s\rightarrow\infty}\max_{|t\leq
1|}\parallel\alpha_{t}(A)-Ade^{th(s)}(A)\parallel=0$

for any $A\in\mathfrak{B}$.

$\alpha_{s,t}\triangleq Ade^{th(s)}$, we construct $\alpha_{s}$-cocycle
$u_{s,t}$(resp. $u$) for $\alpha_{s,t}$(resp. $\alpha$) such that
$\frac{d}{dt}u_{s,t}|_{t=0}=iP$ by Lemma 2.3. Then since
$\beta_{t}=Adu_{t}\circ\alpha_{t}$ and $u_{s,t}\rightarrow u_{t}$, we obtain
that $Adu_{s,t}\circ\alpha_{s,t}\rightarrow\beta_{t}$. Besides,
$Adu_{s,t}\circ\alpha_{s,t}$ is inner and
$Adu_{s,t}\circ\alpha_{s,t}=Ade^{t(h(s)+P)}$ by Corollary 2.5. Then $\beta$ is
approximately inner.

Next we shall prove that if $\beta$ is conjugate to $\alpha$, i.e. there is a
bounded automorphism $\sigma$ of $\mathfrak{B}$ such that
$\beta=\sigma\alpha\sigma^{-1}$. Then if $\alpha$ is inner(approximately or
asymptotically inner), then so is $\beta$.

(a’) If $\alpha$ is inner, i.e. $\alpha_{t}(A)=e^{th}Ae^{-th}$. Then
$\beta_{t}(A)=\sigma(\alpha_{t}(\sigma(A)))=e^{t\sigma(h)}Ae^{-t\sigma(h)}$
i.e. $\beta$ is inner.

(b’) If $\alpha$ is approximately inner, then there exits a sequence
$\\{h_{n}\\}$ in $\mathfrak{A}$ such that

$\lim_{n\longrightarrow\infty}\max_{|t|\leq
1}||\alpha_{t}(A)-Ade^{th_{n}}(A)||=0.$

Because $\beta$ is conjugate to $\alpha$, i.e.
$\beta=\sigma\alpha\sigma^{-1}$, $\sigma$ is the bounded automorphism of
$\mathfrak{A}$, so there exists an $M>0$ such that $||\sigma||\leq
M,||\sigma^{-1}||\leq M$.

Therefore,

$||\beta_{t}(A)-Ade^{t\sigma(h_{n})}(A)||=||\sigma^{-1}(\beta_{t}(\sigma(A)))-\sigma^{-1}(e^{t\sigma(h_{n})}(\sigma(A)))||$

i.e.$\alpha$ is approximately inner.

(c’) If $\alpha$ is asymptotically inner, a similar argument shows that
$\beta$ is asymptotically inner.

Finally, if $\beta$ is cocycle-conjugate to $\alpha$, then there is a bounded
automorphism $\sigma$ of $\mathfrak{B}$ such that $\beta$ is a cocycle
perturbation of $\sigma\alpha\sigma^{-1}$. If $\alpha$ is inner(approximately
or asymptotically inner), then so is $\sigma\alpha\sigma^{-1}$, then so is
$\beta$.

##  3\. Cocycle perturbations on nest algebra
and quasi-triangular algebra

In this section $\mathfrak{H}$ denotes a Hilbert space, $\mathfrak{B}$ denotes
a nest algebra (or a quasi-triangular algebra ) on $\mathfrak{H}$. $id$
denotes the identity automorphism on $\mathfrak{B}$ .

Let us mention briefly some well-known results on this subject.

Proposition 3.1[11]: If $\sigma$ is an automorphism on $\mathfrak{B}$ and
$||\sigma-id||<1$,then there is an invertible element $T$ in $\mathfrak{B}$
such that $\sigma(A)=TAT^{-1}$ and $||T-I||<4||\alpha-id||$.

Proposition 3.2: Let $\alpha$ be a flow on $\mathfrak{B}$ and
$||\alpha_{t}-id||=O(t)$. Then there is a $P\in\mathfrak{B}$ such that
$\alpha_{t}(A)=e^{tP}Ae^{-tP}$, $\forall A\in\mathfrak{B}$.

Proof: When $||\alpha_{t}-id||=O(t)$, there is an $\varepsilon>0$ so that
$|t|<\varepsilon$ implies $||\alpha_{t}-id||<1$. By Proposition 3.1, there are
invertible elements $\\{T_{t}\\}$ in $\mathfrak{B}$ such that
$\alpha_{t}(A)=T_{t}AT_{t}^{-1}$ and $||T_{t}-I||<4||\alpha_{t}-id||$. By [1,
Proposition 3.1.1] $t^{-1}||\alpha_{t}-id||$ is bounded. So
$P_{n}=n(T_{\frac{1}{n}}-I)$ is bounded. So there is a weak∗ convergent sub-
net $\\{P_{n_{r}}\\}$ with limit $P$ in $\mathfrak{B}$. Let $\delta_{\alpha}$
denote the infinitesimal generator of $\alpha$, one obtains

$\delta_{\alpha}(A)=\lim_{r\rightarrow\infty}n_{r}(\alpha_{\frac{1}{n_{r}}}-id)AT_{\frac{1}{n_{r}}}=\lim_{r\rightarrow\infty}n_{r}(T_{\frac{1}{n_{r}}}A-AT_{\frac{1}{n_{r}}})=\lim_{r\rightarrow\infty}P_{n_{r}}A-AP_{n_{r}}=PA-
AP$

for $\forall A\in\mathfrak{B}$. Then by [1, Proposition 3.1.1]
$\alpha_{t}(A)=e^{tP}Ae^{-tP}$.

Proposition 3.3: Let $\alpha,\beta$ be two flows on $\mathfrak{B}$ and
$||\alpha_{t}-\beta_{t}||=O(t)$, then $\alpha$ is an inner perturbation of
$\beta$.

Proof: There exists an $M\geq 1$ and
$\xi\geq$inft≠0($t^{-1}$log$||\alpha_{t}||$) such that

$||\alpha_{t}||\leq Me^{\xi|t|},||\beta_{t}||\leq Me^{\xi|t|}$

Since $||\alpha_{t}\circ\beta_{-t}-id||\leq
Me^{\xi|t|}||\alpha_{t}-\beta_{t}||=O(t)$, we get that there is a
$P\in\mathfrak{B}$ such that $\alpha_{t}(A)=e^{tP}\beta_{t}(A)e^{-tP}$, for
$A\in\mathfrak{B}$ by Proposition 3.2. Finally
$\delta_{\alpha}=\delta_{\beta}+$ad$i(-iP)$ is straightforward. i.e. $\alpha$
is an inner perturbation of $\beta$.

Remark : By the definition of cocycle perturbation, it is obvious that if
$\beta$ is an cocycle perturbation of $\alpha$, then
$||\alpha_{t}-\beta_{t}||=O(t)$. Hence, cocycle perturbation and inner
perturbation are equivalent on $\mathfrak{B}$ by Lemma 2.3 and Proposition
3.3. From the next theorem we obtain that every flow on nest algebra or quasi-
triangular algebra are uniformly continuous.

Theorem 3.4: Every flow $\mathfrak{B}$ are uniformly continuous. And any two
flows on $\mathfrak{B}$ are cocycle perturbation to each other.

Proof: In fact, if there is a flow $\alpha$ on $\mathfrak{B}$ which is not
uniformly continuous, let $\delta_{\alpha}$ be the infinitesimal generator of
$\alpha$. Then there is a $v\in G(\mathfrak{B})$ such that $v\notin
D(\delta)$. Let $v_{t}=v^{-1}\alpha_{t}(v)$,
$\beta_{t}(A)=v_{t}\alpha_{t}(A)(v_{t})^{-1},t\in\mathbb{R},A\in\mathfrak{B}$.
Then $\beta$ is an cocycle perturbation of $\alpha$. And $v_{t}$ is not
differentiable.

By Proposition 3.3 and Remark, $\beta$ is a inner perturbation of $\alpha$
i.e. there is differentiable $\alpha$-cocycle
$\\{u_{t}\\}\subseteq\mathfrak{B}$ such that
$\beta_{t}(A)=u_{t}\alpha_{t}(A){(u_{t}})^{-1}$.

But the commutant of $\mathfrak{B}$ is trivial [11]. Hence it follows that
$v_{t}=\lambda_{t}u_{t},t\in\mathbb{R}$, where $\lambda_{t}\in\mathbb{C}$.
$u_{t},v_{t}$ are $\alpha$-cocycle, it follow that
$\lambda_{t+s}=\lambda_{t}\lambda_{s}$ and $\lambda_{t}\rightarrow 1$, for
$t\rightarrow 0$. Therefore $\lambda_{t}$ is differentiable and $v_{t}$ is
differentiable. This is contrary to $v_{t}$ is not differentiable.

Finally if $\alpha,\beta$ are flows on $\mathfrak{B}$, then $\alpha,\beta$ are
uniformly continuous. So $\alpha,\beta$ are inner perturbation to
$\gamma_{t}\equiv id$. Because the inner perturbation defines an equivalence
relation on the flows, so $\alpha$ is an inner perturbation of $\beta$, and
$\alpha$ is a cocycle perturbation of $\beta$.

## References

  * [1] Bratteli, O. and Robinson, D. W., Operator Algebras and Quantum Statistical Mechanics, I, Springer, 1979.
  * [2] Sakai, S., Operator algebras in dynamical systems, Cambridge Univ. Press, Cambridge 1991.
  * [3] Kishimoto, A., Locally representable one-parameter automorphism groups of AF algebras and KMS states, Rep. Math. Phys. 45 (2000), 333–356.
  * [4] Kishimoto, A., UHF flows and the flip automorphism. Rev. Math. Phys. 13 (2001), no. 9, 1163–1181
  * [5] Kishimoto, A., Examples of one-parameter automorphism groups of UHF algebra. Commun. Math. Phys. 216 (2001), 395–428
  * [6] Kishimoto, A., Approximately inner flows on seperable $C^{*}$-algebras. Rev. Math. Phs. 14 (2002), 1065–1094.
  * [7] Kishimoto, A., Approximate AF flows. J. Evol. Equ. 5 (2005), 153–184.
  * [8] Kishimoto, A., The one-cocycle property for shifts. Ergod. Th. And. Dynam. Sys. 25 (2005), 823–859.
  * [9] Kishimoto, A., Multiplier cocycles of a flow on a $C^{*}$-algebra. J. Funct. Anal. 235 (2006), 271–296.
  * [10] Kishimoto, A., Lifting of an asymptotically inner flow for a separable $C^{*}$-algebra. Operator Algebras: The Abel Symposium 2004, 233–247, Abel Symp., 1, Springer, Berlin, 2006.
  * [11] Davidson K. R., Nest algebras, Longman group UK limited, Essex, 1988.


11institutetext: Stuttgart Media University, Stuttgart, Germany,
11email<EMAIL_ADDRESS>
http://wiss.iuk.hdm-stuttgart.de/

# Easy and complex: new perspectives for metadata modeling using RDF-star and
Named Graphs

Florian Rupp  Benjamin Schnabel  Kai Eckert

###### Abstract

The Resource Description Framework is well-established as a lingua franca for
data modeling and is designed to integrate heterogeneous data at instance and
schema level using statements. While RDF is conceptually simple, data models
nevertheless get complex, when complex data needs to be represented.
Additional levels of indirection with intermediate resources instead of simple
properties lead to higher barriers for prospective users of the data. Based on
three patterns, we argue that shifting information to a meta-level can not
only be used to (1) provide provenance information, but can also help to (2)
maintain backwards compatibility for existing models, and to (3) reduce the
complexity of a data model. There are, however, multiple ways in RDF to use a
meta-level, i.e., to provide additional statements about statements. With
Named Graphs, there exists a well-established mechanism to describe groups of
statements. Since its inception, however, it has been hard to make statements
about single statements. With the introduction of RDF-star, a new way to
provide data about single statements is now available. We show that the
combination of RDF-star and Named Graphs is a viable solution to express data
on a meta-level and propose that this meta-level should be used as first class
citizen in data modeling.

###### keywords:

data modeling, RDF-star, Named Graphs, meta-level

##  1 Introduction

Data can come in many different forms and there are nearly infinite ways to
model data describing the same information. Depending on the use case,
different forms might be preferable: a simple view on data is the easiest to
work with and facilitates data reuse. On the other hand, more complex
applications need advanced data models providing the full complexity of the
data with all available details. The Resource Description Framework (RDF) [1]
provides a common basis for the publication of data in form of statements on
the Web but makes no assumptions how these statements actually look like and
how entities and their relations are used to represent the desired
information. Thus, various modeling approaches can represent the same
information. The Semantic Web aims at providing interlinked data based on the
”follow your nose” principle [4], i.e.: users and applications can get data
about a resource by dereferencing its URI.

There are two possible solutions, when data is to be provided both in simple
and complex111While it depends on the perspective, we consider data complex
that uses several layers of properties and dependent classes to describe a
resource. forms for different applications:

  1. 1.

Different data representations can be provided; in this case, a mechanism is
required that allows the application to request one of these representations
(see Section 2).

  2. 2.

The data is provided in a simple data model and additional information is
provided on a meta-level in the form of statements that provide further
explanations.

In this article, we discuss the latter approach. We argue that there are
benefits of using such a meta-level approach versus the provision of different
data formats: there is only one representation required, it is straight-
forward to extend existing data representations while maintaining backwards
compatibility and it fits the ”follow your nose” principle of the Semantic Web
where applications simply can ask if there is further information about a
statement.

A meta-level can be created by providing a further statement describing a
single or multiple statements. To provide statements about statements in RDF,
a mechanism is needed to identify statements. The identification can happen in
two ways: either a single statement is identified (statement-level) or a group
of statements – usually called a graph – is identified and meta-statements are
used to further describe all statements in this graph (graph-level).

For statement-level data there always has been the optional RDF reification,
for graph-level statements, Named Graphs have been introduced. Particularly
RDF reification, however, has two major shortcomings: it is inefficient for
exchanging RDF data and writing queries to access statement-level metadata is
cumbersome [7].

Now there is a new way to provide statement-level metadata within RDF itself
with the recent specification of RDF-star222https://www.w3.org/2021/12/rdf-
star.html. With increasing support in RDF databases and tools and the
specification of various serializations including TriG-star333Throughout this
paper, we use TriG-star syntax in the examples. For reference this is a list
of the implied namespaces, ex is the default or empty namespace: rdf:
http://www.w3.org/1999/02/22-rdf-syntax-ns#; owl:
http://www.w3.org/2002/07/owl#; ex: http://example.org/; dcat:
http://www.w3.org/ns/dcat#; dct: http://purl.org/dc/terms/; foaf:
http://xmlns.com/foaf/0.1/; prov: http://www.w3.org/ns/prov#; dbp:
http://dbpedia.org/property/; dbo: http://dbpedia.org/ontology/; gn:
http://www.geonames.org/ontology#; gndo:
http://d-nb.info/standards/elementset/gnd#, it provides all that is needed to
use both statement-level metadata with RDF-star and graph-level metadata with
Named Graphs.

In this article, we explore the possibilities and limitations of RDF-star and
Named Graphs with respect to the creation of data models which are simple and
easy to use, but nevertheless contain the full complexity of the data on a
meta-level so that it can be used when needed. We start with an overview on
related work focusing on providing data to applications with different
requirements. In Section 3, we identify three abstract patterns that show how
common use cases can be modeled with metadata. In Section 4, further examples
about querying and working with the meta-level are given. The paper concludes
with a discussion of the advantages and disadvantages of data modeling with a
meta-level as first-class citizen.

##  2 Related Work

Different applications have different requirements towards data. Consider a
personal blog that wants to provide a thumbnail view of a book versus a
library that needs data about a book to include in its catalog. It is common
practice in many open data portals that data can be accessed in different
formats, e.g. Dublin Core for simple data vs. MARC for all the details. In the
Semantic Web, however, this poses a problem. If a URI for a book is resolved
automatically by an application, what data should be provided? In our paper
this is not directly solved, but arguably our meta-level approach allows
applications to decide if they want to access the additional information. For
each resource there can only be one data description. The provider of the data
needs to decide, how this data is modeled and which vocabularies are used. To
address this shortcoming, the Dublin Core community developed the concept of
_Application Profiles_ [8] to describe the data model for a specific
application. Based on the notion of Application Profiles, there is a recent
RFC draft to use HTTP for the negotiation of different data representations on
content level, similar to the well-known content negotiation for different RDF
serializations [18]. Using such mechanisms, however, requires proper support
in applications and it is not yet clear if this will actually be used in
practice.

Another aspect is the provision of metadata to describe the actual data,
mostly to provide information on the _provenance_ of the data [14, 5, 6, 16,
17, 13]. There are generally two aspects to consider: (1) how can the subject,
i.e., the data, be identified to allow statements about it, and (2) how can
the provenance of data be described. As we do not focus specifically on the
description of provenance, we are mainly concerned with the former.

Besides RDF-star [12] and Named Graphs [3], other approaches have been
proposed. The oldest way to do this is _RDF reification_ , which is available
since the very first versions of RDF, cf. [15]. However, making statements
about statements using reification is hard, mainly due to its verbose syntax
and the fact that every statement needs to be reified, even when many
statements share the same meta information. This might be a reason why
reification is rarely used in real world applications444 To get evidence of
reification usage in real world application we checked a given list of SPARQL
endpoints. In favor we use the Wikidata SPARQL endpoint list at
https://www.wikidata.org/wiki/Wikidata:Lists/SPARQL˙endpoints. This
examination was done automatically with a suitable SPARQL query. The list,
however, contains 139 entries and can be, according to its small size,
considered as a sample only. Merely 78 endpoints were reachable via GET or
POST requests (on July 20th 2022). 7 out of 78 remaining endpoints were not
suitable for this study, being no actual SPARQL endpoint or the domain has
been sold. Out of 72 endpoints 5 are using reification. At just 7,7% that’s a
small percentage. This supports the thesis that reification is rarely used.

_Singleton statements_ [11] use unique identifiers which are added to the
predicate statement. Following this method, the meta-level can be expressed by
using the predicate in combination with the assigned identifier. Singleton
properties lead to compact representations of the meta-level, but obviously
affect the querying of the data. There is support for singleton properties in
some triple stores, which makes the application more feasible. Even then it is
not recommended, for instance see the documentation of
GraphDB555https://graphdb.ontotext.com/documentation/free/devhub/rdf-sparql-
star.html, Singleton Properties.).

These approaches all address statement-level metadata. There are many cases
where an identification of single statements is not needed and even not
helpful. Probably the best example is provenance information. Usually, many
statements share the same provenance when they are created together by a
single process.

For the separation of different descriptions of the same resource, as an
alternative, _Proxy entities_ [9] or resource maps in OAI-ORE [10] have been
proposed, that use proxies as distinct placeholder entities that all represent
the same actual entity but with different URIs to be able to distinguish
statements about the entity from different sources. This raises the complexity
of the data and requires applications to correctly interpret the proxy
entities.

A problem with Named Graphs is that organizing structures, for example nested
graphs or subgraphs can only be represented via additional data describing
_graph relationships_. This can be used to identify provenance information in
hierarchical graphs [5]. A proposal for the structured use of several named
graphs are _nano publications_ [2] that also explicitly support provenance.

##  3 Meta Modeling Patterns

Based on our experience with data applications, we can identify three abstract
modeling patterns that illustrate the benefits of a meta-level. For each
pattern, we formulate a problem statement, the solving approach, an example
and a conclusion. The three patterns are:

  * •

Pattern 1: RDF provenance modeling in the meta-level.

  * •

Pattern 2: Extending an existing data model.

  * •

Pattern 3: Shifting proxy entities to the meta-level.

We will not distinguish statement-based and graph-based metadata. With RDF-
star and Named Graphs, there are now viable solutions for both levels and it
is best decided by the data modeler or the data provider which one is more
suitable. This of course requires that consuming applications understand both
approaches, i.e, TriG-star needs to be supported as serialization.

###  3.1 Pattern 1: RDF provenance modeling in the meta-level

#### Problem statement:

The provision of provenance data _in RDF_ is straight forward, for example
using PROV or Dublin Core [6]. Nevertheless provenance data is not often
provided on a meta-level _for RDF_ data. Instead, entities are introduced that
either represent the data (e.g. a dcat:CatalogRecord) or that act as a
placeholder for the actual entity for the description from one specific source
(e.g. a ore:Proxy). Both approaches are problematic when data from different
sources is to be merged due to both approaches must now fit into the same data
model.

Even unfavorable is the mixing of data about a resource and about its
description. Let’s say, we want to model the provenance of a statement that
provides a title for an entity <E>. In the following example, the information
is added directly to the instance of <E>.

    <E> rdf:type Entity_E ;
        dct:title "title" ;
        prov:wasDerivedFrom <source> .

But what is derived here? Is it the title of the resource or the resource
itself? As all statements describe their subject, it is clearly the resource
<E> and not the title or other descriptive data about the resource.

#### Solving approach:

Provenance is a prime example where Named Graphs should be used as usually
many statements share the same provenance:

    <entity> ex:ID <ID> .

    :data {
       <entity>
         ex:data "data" ;
    }

    :data
       prov:wasDerivedFrom <entity> ;
       ...

#### Example DBpedia:

The DBpedia project666https://www.dbpedia.org/. is one of the largest Linked
Open Datasets on the Web. It converts Wikipedia articles into RDF. The data is
expressed using the DBpedia ontology. The information from which Wikipedia
article data was derived is expressed using the prov:wasDerivedFrom predicate.
Additional provenance information is given by further statements such as
dbo:wikiPageID giving the ID of the Wikipedia page, dbo:wikiPageRevisionID
stating the revision ID or the number of characters of the original article in
dbo:wikiPageLength. Indeed, the provenance information is related to the
article, but is mixed with the description of the entity the article is about:

    <entity>
       dbp:size "63" ;
       dbp:built "1889" ;
       prov:wasDerivedFrom <article> ;
       dbo:wikiPageID "123" ;
       dct:date "2022-05-21" .

Here the statement level is not sufficient thus multiple statements share the
same provenance. A cleaner solution is a model with a meta-level. It can also
be argued that the provenance information of the data is only important for a
small subset of applications, for example when ensuring the data quality such
as the relevancy and timeliness of an article. Here is the example data using
Named Graphs:

    :data {
       <entity>
         dbp:size "63" ;
         dbp:built "1889" ;
    }

    <article> dbo:wikiPageID "123" .

    :data
       prov:wasDerivedFrom <article> ;
       dct:date "2022-05-21" ;

#### Conclusion:

The meta-level is the best and most suitable way how to model provenance data
in RDF: This avoids levels of indirections, i.e. the usage of proxy resources
or specific constructs such as the notion of records. The provenance
information can be kept separate from the data and may only be queried if
needed, which makes the data model lightweight. While Named Graphs are usually
the best fit for provenance data, it has to be noted that RDF-star can also be
used – even additionally – to provide the provenance for a single statement.
This first pattern addresses the most common use case for metadata. With the
following two patterns, we would like to extend its use to new applications.

###  3.2 Pattern 2: Extending an existing data model

#### Problem statement:

Data models usually evolve over time and changes to the model are inevitable.
This poses a problem when these changes are not backwards compatible and break
existing applications. We therefore propose to use the meta-level to extend
and improve an existing data model.

For example, an existing knowledge graph could be enriched to provide
additional confidence values. Sometimes an existing data model is wrong,
unfitting or simply corrupted resulting in loss of context.

For illustration, we assume we have a simple data model where a relation of
two entities <A> and <B> can be expressed:

    <A> :conformsTo <B> .

Now this model is to be enriched with additional information such as a
confidence score. To achieve this, however, various approaches are possible
without using the meta-level, for example n-ary entities. This would require a
remodeling which is a high risk of breaking existing applications, for
instance:

    <A> :hasConformanceStatement <C> .
    <C> a :ConformanceStatement;
        :conformingTo <B>;
        :confidence 0.8 .

#### Solving approach:

We propose a modeling of this new information in the meta-level without
touching the existing model to ensure backwards compatibility. The additional
information is integrated by using RDF-star.

    <A> :conformsTo <B> .
    << <A> :conformsTo <B> >> :confidence 0.8 .

#### Example 1 DBpedia:

All data within DBpedia is derived from Wikipedia articles. Among other
information, many articles have a thumbnail image related to its article. In
DBpedia a thumbnail is attached to an article with the predicate
dbo:thumbnail. However, Wikipedia is a Web page where the thumbnail is an HTML
image tag. In DBpedia the src-attribute of the tag is converted to the URI
describing the thumbnail. Furthermore, an HTML image tag is described by its
caption attribute and an alternative text (alt attribute). Both are included
in DBpedia via dbp:caption and dbp:alt. However, the relation to which
thumbnail a caption or alternative text conforms is gone.777According to the
DBpedia ontology, this should actually be modeled with an n-ary entity, but at
least in the current version of DBpedia, the described problem exists. Due to
this simplification the relation is lost. This becomes really disadvantageous
if articles have multiple thumbnails where it is impossible to resolve these
relationships. While this obviously should be fixed in the data, it might be
that there are applications using the data that rely on the current data
representation. With RDF-star, however, we can provide additional information
to deliver the relationship information to applications that need it, without
changing the asserted statements in the RDF data. One way would be to add the
caption to the relationship:

    <entity> dbo:thumbnail <thumbnail> ;
      dbp:caption "Portrait of X" .

    << <entity> dbo:thumbnail <thumbnail> >>
      dbp:caption "Portrait of X" .

This is arguably again not ideal as a caption should probably refer to a
thumbnail, not a thumbnail assignment. On the other hand, if the same
thumbnail is used in different contexts with different captions, the above
solution is good. For the sake of the argument however, let us assume that the
caption should actually be assigned to the thumbnail. So the data should look
like this:

    <entity> dbo:thumbnail <thumbnail> .
    <thumbnail> dbp:caption "Portrait of X" .

To actually fix the data, in this case the subject of the original statement
needs to be changed. We could provide a vocabulary for such cases that would
be universally understandable by applications supporting it, i.e. the
replaceSubjectBy-statement:

    <entity> dbo:thumbnail <thumbnail> ;
      dbp:caption "Portrait of X" .

    << <entity> dbp:caption "Portrait of X" >>
      ex:replaceSubjectBy <thumbnail> .

This means that an application should replace the subject of the original
statement (<entity>) with the new subject <thumbnail>. As can be seen here,
RDF-star opens many interesting ways to provide a history or change requests
for statements. Consider the following example where statements are related to
each other:

    <entity> dbo:thumbnail <thumbnail> .
    <thumbnail> dbp:caption "Portrait of X" .

    << <thumbnail> dbp:caption "Portrait of X" >>
        ex:replaced << <entity> dbp:caption "Portrait of X" >> .

In this case, the actually wrong statement <entity> dbp:caption "Portrait of
X". would only be available as part of the RDF-star triple stating that it has
been replaced. Nevertheless, it is still part of the graph and old
applications could still use it.

#### Example 2 GeoNames:

GeoNames888http://www.geonames.org/ is a graph for geographical data. It
includes alternative names, but not information on historical names. This
example shows how historical names of a city, or a place can be added to the
current data and extended including more specific information. Currently in
GeoNames there is only a distinction of gn:name and gn:alternateName. The
alternate name can contain a language tag, but it does not indicate when a
name was used in case of historical names. For example: The German city of
Chemnitz used to be called ”Karl-Marx-Stadt” between May 10th, 1953 and July
1st, 1990.

The listing below shows an excerpt of the RDF entry in
GeoNames999https://sws.geonames.org/2940132.:

    <https://sws.geonames.org/2940132/>
      gn:name "Chemnitz" ;
      gn:alternateName "Chemnitz"@de ;
      gn:alternateName "Chemnitz"@en ;
      gn:alternateName "Karl-Marx-Stadt"@de .

We could add the missing information about the time span, for example by using
the Common Authority File (Gemeinsame Normdatei, GND) of the German National
Library, which provides the following data:

    <https://d-nb.info/gnd/2015221-8>
      gndo:preferredNameForThePlaceOrGeographicName
        "Karl-Marx-Stadt";
      gndo:dateOfEstablishment "10.05.1953" .
      gndo:dateOfTermination "31.05.1990" ;

We can take this data from the GND and apply it to GeoNames to add the
additional data to the entities, such as date of establishment and date of
termination:

    <https://sws.geonames.org/2940132/>
      gn:alternateName "Karl-Marx-Stadt"@de .

    << <https://sws.geonames.org/2940132/>
       gn:alternateName "Karl-Marx-Stadt"@de >>
      ex:valid_from "09.05.1953"^^xsd:date ;
      ex:valid_to "01.06.1990"^^xsd:date .

This way, the data is still provided in a very simple manner that nevertheless
is useful for many applications that do not need the additional information,
for example for named entity resolution. Nevertheless, the additional
information can be provided in a modular way if it is needed.

#### Conclusion:

RDF-star is a great fit when additional information about a statement needs to
be provided. In particular when a data model is already used in applications
it is an adaption to provide more complex data avoiding incompatible changes.
By doing so, a simple model retains still simple by providing an additional
context in the meta-level. The backwards compatibility is obtained as well
(see also Section 4).

###  3.3 Pattern 3: Shifting complex relations to the meta-level

#### Problem statement:

Whenever additional data about a relationship between two entities is needed,
an additional _n-ary_ entity can be created for representation. As entities
(subjects and objects in statements) are substantives, this results in graphs
with many nominalized relationships, for example:

    <E> :hasSubject <SubjectAssignment1> .
    <SubjectAssignment1>
        :hasHeading "Data Modeling" ;
        :fromVocabulary <TopicsVocabulary> .

Here, a complex entity of the class SubjectAssignment is used to represent the
subject of entity <E>. The problem is that this structure feels cumbersome for
many applications that might only be interested in the subject heading and
that do not care about the vocabulary the subject heading is coming from. Even
if uncontrolled subject headings (free tags) are used, the intermediate
subject assignment is still required as the data model is created this way.

#### Solving approach:

Similar to pattern 2, we propose to use a meta-level, but this time from the
beginning, so that the additional and perhaps even optional information can be
pushed to the meta-level:

    <E> dc:subject "Data Modeling" .
    << <E> dc:subject "Data Modeling" >>
        :fromVocabulary <TopicsVocabulary> .

#### Example DCAT:

In this example we demonstrate how such n-ary entities can be avoided directly
at the time of designing ontologies by shifting entities to the meta-level.
DCAT is an ontology enabling publishers to describe datasets and its
properties. The datasets can be listed in a dcat:Catalog entity. The
dcat:Dataset entity holds meta information of the actual data, which is linked
to as a dcat:Distribution entity. When listing a dataset in the catalog, the
optional entity dcat:CatalogRecord may be used to express metadata about the
listing such as the issue date.

As of the current DCAT 2 ontology
specification101010https://www.w3.org/TR/vocab-dcat-2/., the
dcat:CatalogRecord is provided for the purpose of adding additional metadata
for the description of the listing of a resource, e.g., datasets in the
catalog. The following example lists a dataset in a catalog. To add metadata
such as the issued date or the title, the CatalogRecord entity is applied
providing these information.

    ex:catalog dcat:record ex:catalogRecord .

    ex:dataset a dcat:Dataset .

    ex:catalogRecord a dcat:CatalogRecord ;
      dct:issued "05.04.2022" ;
      dct:title "record title" ;
      dct:description "record description" ;
      foaf:primaryTopic ex:dataset .

However, the dcat:CatalogRecord entity adds metadata only to the actual
listing of the catalog and the dataset. This can be shortened by rewriting it
in the following RDF-star syntax to shift the dcat:CatalogRecord entity to the
meta-level. The attributes of this entity can be used to describe the
provenance of the relation directly:

    ex:catalog dcat:dataset ex:dataset .

    << ex:catalog dcat:dataset ex:dataset >>
      a dcat:CatalogRecord ;
      dct:issued "05.04.2022" ;
      dct:title "record title" ;
      dct:description "record description" .

This information can be queried easily from the meta-level using SPARQL-star.
Here we give an example on how to query the issued date of the CatalogRecord
denoted before:

    SELECT ?date WHERE {
      << ex:catalog dcat:dataset ex:dataset >> dct:issued ?date .
    }

As RDF data can easily be split in several junks of statements, the metadata
could also be separated from the core data, containing only the RDF-star
statements.

#### Conclusion:

On the one hand, complexity can be shifted to the meta-level and the data
model is improved towards simpleness. In the course of this, the data may only
be queried when needed reducing the size of query results. On the other hand,
the simple data model might seem to be oversimplified to a data modeler. If
the vocabulary scheme belongs to the subject heading and is not optional, it
might seem arbitrary to put this information to the meta-level. And if the
majority of applications need the complete data, the querying and using of the
meta-level can feel more cumbersome than a more complex data model. To
conclude, there are situations where the data modeled in an entity really
should be on the same level as the rest of the actual data. In other
situations, using the meta-level might just be the ideal solution where a
simple model can be provided for simple applications and additional
information is available if needed.

##  4 Querying and constructing the meta-level with SPARQL-star

In the last section we have shown how data using the meta-level in its data
model could look like. For a broad adoption of the proposed patterns, it is
important that working with the data is effortless and data on the meta-level
can easily be found and processed when needed. For this, two aspects should be
considered:

  1. 1.

How can an application find out if there is data on the meta-level?

  2. 2.

How can data using the meta-level be transformed to other data models that
potentially do not use the meta-level?

#### How can an application find out if there is data on the meta-level?

For this question, it is important to adjust some expectations that an
application might have when dealing with RDF and Linked Data. A modern RDF
application that supports data on the meta-level is required

  1. 1.

to support Named Graphs and RDF-star as well as at least TriG-
star111111https://w3c.github.io/rdf-star/cg-spec/2021-04-13.html“#trig-star as
standard format for any responses from a Linked Data server as well as for
data dumps,

  2. 2.

to expect Named Graphs to be used for the actual data, so the internal
organisation of data, e.g. for provenance tracking, must support graph
hierarchies, such as a provenance context [5],

  3. 3.

to check for the existence of a meta-level in form of Named Graphs and/or RDF-
star triples.121212https://w3c.github.io/rdf-star/cg-
spec/2021-04-13.html“#dfn-triple

The following SPARQL-star query returns all RDF-star triples:

    SELECT DISTINCT ?s WHERE {
      ?s ?p ?o .
      FILTER( isTRIPLE(?s) )
    }

The function isTRIPLE is new in SPARQL-star and returns TRUE if the parameter
is an RDF-star triple. Together with a query for the existence of Named
Graphs, this enables an application to check if a meta-level is available.

#### How can data using the meta-level be transformed to other data models
that potentially do not use the meta-level?

The ability to transform data from one data model to another by means of
CONSTRUCT queries is one of the most powerful features of RDF. This is not
restricted by the use of meta-level data, as the following queries
demonstrate:

In Pattern 3 (Section 3.3) we have shown how to shift the meta information
encapsulated in an n-ary entity to the meta-level. Using SPARQL-star and
CONSTRUCT queries, it is possible to rebuild the former data model, e.g. for
the dcat:CatalogRecord. To extract the subject or object of a triple
(annotated with RDF-star), the SPARQL-star functions SUBJECT and OBJECT are
used:

    CONSTRUCT {
      ?catalog :record ?record .
      ?record :primaryTopic ?dataset .
      ?record ?pred ?obj .
    } WHERE {
      ?star ?pred ?obj
      FILTER(isTRIPLE(?star)) .
      BIND(SUBJECT(?star) as ?catalog) .
      BIND(OBJECT(?star) as ?dataset) .
      BIND(IRI(CONCAT(STR(?dataset), "/record")) as ?record) .
    }

Here, string functions are used to coin an IRI for the new n-ary entity
?record, by appending /record to the IRI of ?dataset. This is to avoid a blank
node, but it depends on the application if and how this should done.

The other way round is also possible, i.e., to create a simple data model with
meta-level data from a complex data model:

    CONSTRUCT {
      ?star ?pred ?obj .
    } WHERE {
      ?catalog :record ?record .
      ?record :primaryTopic ?dataset .
      ?record ?pred ?obj .
      BIND(TRIPLE(?catalog, :dataset, ?dataset) as ?star).
    }

This time, the SPARQL-star function TRIPLE is used to create an RDF-star
triple from a subject, a predicate, and an object.

##  5 Discussion

With this paper, we ask the following question: can and should meta-level
concepts be used for data modeling in RDF? Besides Named Graphs and RDF-star,
several concepts have been introduced to construct a meta-level as well.
However, they all have weaknesses regarding e.g., feasibility or performance.
Named Graphs have been around for a long time now and are well-supported in
SPARQL. With RDF-star, we now see a serious contender for an RDF extension
that allows statements about statements in a concise and straight-forward way,
also with a SPARQL extension. So yes, we can use it for data modeling. But
should we?

We identified three patterns that are suitable for a more detailed discussion
of the potential benefits of meta-level modeling: (1) using the meta-level for
actual metadata, i.e. data about RDF data like provenance data; (2) using the
meta-level for the extension of existing data models; and (3) using the meta-
level from the get-go to get simpler and more modular data models.

The first pattern is arguably the most trivial one, but addresses the reasons
why Named Graphs and RDF-star have been developed in the first place. While at
least Named Graphs exist for quite some while now, we still see only rarely
data models that make use of them. Instead, mostly workarounds within vanilla
RDF are used to artificially create a meta-level when it is needed. This
should change and data about RDF data should be where it belongs: in the meta-
level.

With the second pattern, we showed that existing simple data models can be
extended by additional meta statements, with two main scenarios in mind:
First, additional data can be added to provide more complex information to
applications that need it. Second, this can be used to provide corrections and
missing information to a data model without compromising existing
applications. Adding the additional information in the meta-level does not
only preserve backwards compatibility of the data model. It also keeps the
core model simple and allows a modular distribution of additional data for the
applications that need it. This observation leads to the third pattern where
we proposed to use the meta-level to create simple and modular data models
from the beginning.

We have further shown in several examples from real world data (including
DCAT, DBpedia, and GeoNames), how the application of these patterns could look
like and how they would affect the provided data. And finally, we have shown
that working with the meta-level is straight-forward, as long as applications
expect a meta-level and support Named Graphs and RDF-star, with TriG-star as
preferred serialization. Particularly also the transformation of data into and
out of the meta-level using CONSTRUCT queries is possible.

For the question, if meta-level modeling should be used, we conclude with the
following thoughts:

#### Is the provision and management of meta-level data possible?

This depends on the technology stack that is used. Named Graphs are usually
not a problem for triple stores but might lead to problems when the meta-level
is to be preserved in other systems. For data serializations, a compatible
file format such as TRIG must be used. File-based representations of graphs
(i.e., one RDF file per graph) are also possible, but require additional
infrastructure and/or tooling to make sure all meta information is properly
preserved. RDF-star is very new, but at least a transparent serialization via
RDF reification is possible.

#### Is there a clear decision for Named Graphs and/or RDF-star?

As stated in the paper, it depends on the use case and sometimes even on the
data. It might be worthwhile to just use one of the mechanisms to reduce
complexity within a project. Alternatively, both can be used, but for
different parts of the data model, for instance Named Graphs to provide
provenance information and RDF-star to provide additional information on
entity links. And finally, this could be left to the implementation and
applications would have to expect such information being provided with either
of the mechanisms.

#### Can we still ”follow the nose”?

One of the main advantages of RDF is that it is easy to get data about a
resource and use subsequent queries to find out more or even just to check if
there are any more information. This is in our opinion the main reason why
meta-level information is not used as often as it should. Applications often
expect Turtle or even still RDF/XML as RDF serializations. They might not be
able to deal with TriG. With SPARQL, Named Graphs can be queried. However, the
application needs to know that there is useful information attached to a Named
Graph. This is similar for RDF-star. The application needs to ask (or
understand from the serialization) if there are additional statements about a
statement. On the other hand, checking for the existence of Named Graphs or
RDF-star triples is possible in SPARQL and SPARQL-star. So this is only a
matter of getting used to it.

##  6 Conclusion and Future Work

Using the meta-level actively for data modeling opens interesting new
perspectives on the task. In this paper we formulated three abstract patterns
on how and when to use the meta-level in the modeling process actively.
Therefore we gave several examples on real world applications.

Usually, the data modeler has to decide if the model should be optimized for
data consumption, i.e. as easy to digest as possible and only as complex as
necessary. However, more often than not, it is rather created with the most
complex possible data in mind. Often data that does not even exist (yet) in
reality - but certainly would be great to have at some point so the data model
should better be ready. Meta-level modeling actually provides a means to
define a simple core data model - particularly with simple, direct relations.
Additional data can be provided in a modular way, with additional RDF data
that can contain additional statements further describing resources.

For future work, we aim at using the meta-level in our own projects on
production level to gain further experience with it and to support a wider
adoption of Named Graphs and RDF-star in public data models. We are interested
in the potential of meta-level modeling for the representation of changes in
data to support data transparency and better provenance tracking. Currently we
are exploring the creation of an ontology to capture the abstract entities
appearing in meta-modeling, based on the patterns we identified so far and
potentially more patterns that will arise with further applications.

## Acknowledgements

This research was partially supported by the Volkswagen Foundation (Project:
Consequences of Artificial Intelligence on Urban Societies, Grant 98555) and
the German Research Foundation (Project: Specialized Subject Service for
Jewish Studies, Grant 286004564). We thank Magnus Pfeffer for his valuable
feedback.

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