# Evolution of the magnetic field and flows of solar active regions with
persistent magnetic bipoles before emergence.

C.S. Alley  School of Information and Physical Sciences, The University of
Newcastle, Australia  H. Schunker  School of Information and Physical
Sciences, The University of Newcastle, Australia  [

###### Abstract

Magnetic active regions on the Sun are harbingers of space weather.
Understanding the physics of how they form and evolve will improve space
weather forecasting. Our aim is to characterise the surface magnetic field and
flows for a sample of active regions with persistent magnetic bipoles prior to
emergence. We identified 42 emerging active regions (EARs), in the Solar
Dynamics Observatory Helioseismic Emerging Active Region survey (Schunker et
al. 2016), associated with small magnetic bipoles at least one day before the
time of emergence. We then identified a contrasting sample of 42 EARs that
emerge more abruptly without bipoles before emergence. We computed the
supergranulation scale surface flows using helioseismic holography. We
averaged the flow maps and magnetic field maps over all active regions in each
sample at each time interval from 2 days before emergence to 1 day after. We
found that EARs associated with a persistent pre-emergence bipole evolve to
be, on average, lower flux active regions than EARs that emerge more abruptly.
Further, we found that the EARs that emerge more abruptly do so with a
diverging flow of $(3\pm 0.6)\times 10^{-6}$ s-1on the order of 50-100 ms-1.
Our results suggest that there is a statistical dependence of the surface flow
signature throughout the emergence process on the maximum magnetic flux of the
active region.

###### keywords:

Sun: magnetic fields; helioseismology; activity; sunspots

H<EMAIL_ADDRESS>
##  1 Introduction

Active regions are generally thought to be formed by coherent, arched magnetic
flux tubes rising through the interior to manifest as roughly east-west
aligned opposite polarity pairs at the surface of the Sun. In many dynamo
models they are important to convert the Sun’s global toroidal magnetic field
to poloidal CS2015 (8). Understanding the physics behind the emergence process
is important to constrain their origins and connection to the Sun’s large
scale global field, as well as for space weather forecasting.

It is not clear from what depth these flux tubes originate, nor what causes
them to rise. At the surface and below, the Sun’s magnetic field is embedded
in the convective flows, and as such can be modelled with the equations of
magnetohydrodynamics. Below the surface, the plasma pressure is generally
greater than the magnetic pressure. Understanding the dominant terms in these
equations, particularly the flows and the magnetic field, is important to
understand how active regions form.

Simulations of magnetic flux emergence show that thin flux tubes can rise a
priori from the base of the convection zone due to magnetic buoyancy in the
absence of convection (Fanetal1993, 14, 26, e.g.); or tubes of magnetic flux
can be formed within the convection zone itself and brought up by convective
upflows (Chenetal2017, 9, 18, e.g.). It may also be possible for active
regions to form without the presence of a flux tube (Brandenburg2005, 6, 7,
e.g.). For a full review see FanLRSP2021 (13).

The thin flux tube models in Fan2008 (12) predict a retrograde flow at the
peak of the flux tube just before emergence. In principle, local
helioseismology could be used to measure this flow. Local helioseismology
measures perturbations to the natural seismic oscillations driven by the
turbulent convection at the surface of the Sun and infers the three-
dimensional subsurface structure and dynamics in localised areas
(GizonBirch2005, 16, see ). Birchetal2013 (3) used helioseismic holography
LindseyBraun2000 (21) to measure the subsurface flows prior to the formation
of one hundred active regions observed by the Global Oscillation Network Group
(Harveyetal1998, 17, GONG;). They found that there were no statistically
significant flows below the surface, however near the surface they found a
statistically significant flow of about 15ms-1towards the emergence location
in the day preceding the active region formation.

Recently, it has become apparent that the near-surface convective flows
themselves are important in the emergence process (Weberetal2023, 27, see). By
comparing the observed surface flows at the time of active region emergence
with simulations, Birchetal2016 (5) showed that flux tubes cannot be rising
faster than about 100 ms-1through the upper convection zone, on the order of
the convective velocities themselves. Birchetal2019 (4) went on to show that,
on average, active regions preferentially emerge in east-west aligned
converging flow lanes, suggesting a connection to the Sun’s supergranulation
pattern. Schunkeretal2019 (24) showed that the growth of active regions
through the emergence process is consistent with the length and timescales of
supergranulation, supporting the idea that convection on these spatial scales
buffets the polarities as they emerge.

From a practical perspective, perhaps most importantly for space weather, it
would be useful to be able to predict the location and time an active region
will emerge, and whether or not it is likely to have low magnetic flux or high
magnetic flux. Some effort has been dedicated to this. Helioseismic studies
have not resulted in a significant subsurface detection (Kommetal2015, 20, see
e.g.), but Barnesetal2014 (1) showed that in the day before emergence the
strongest indication that an active region will emerge is the unsigned surface
magnetic field itself. This was also reflected in more recent efforts using
machine learning Dhurietal2020 (11).

In this paper, we examine the magnetic field prior to emergence for two
samples of active regions. In Section 2 we describe our database of emerging
active regions. We then outline the data reduction, computation of the surface
flows, how we measure the location of the polarities in each active region,
and the ensemble averaging of the flow and magnetic field maps (Section 3). In
Section 4 we describe the identification of active regions that show
persistent pre-emergence bipole magnetic fields. In Section 5 we show that
there is a distinct difference in the averaged surface flows and magnetic
field from $2$ days before the emergence to 0.6 days afterwards, and in
Section 6 we discuss the implications of our results and possible avenues to
make future progress.

##  2 Sample of observed emerging active regions

The Solar Dynamics Observatory Helioseismic Emerging Active Region (SDO/HEAR)
Survey currently consists of 182 emerging active regions (EARS)
(Schunkeretal2016, 25, 24, for a full list see) observed by the Helioseismic
and Magnetic Imager onboard the Solar Dynamics Observatory (Scherreretal2012,
22, SDO/HMI, ) between May 2010 (the start of science grade SDO/HMI
observations) and July 2014 (the declining phase of solar cycle 24). To avoid
complications in interpreting local helioseismology results (Schunker2010, 23,
e.g.), the active regions selected for the HEAR survey specifically emerge
into a relatively quiet area of the Sun’s surface at least $20^{\circ}$ from
any surrounding strong surface magnetic fields.

The emergence time for each active region, $t=0$ days (see A and B), is
defined as the time when the absolute flux, corrected for line-of-sight
projection, reaches 10% of its maximum value over a 36 hour interval following
the first appearance of the sunspot (or group) in the National Oceanic and
Atmospheric Administration (NOAA) record Schunkeretal2016 (25). We emphasise
that a small amount of flux from the EAR is present at the emergence time. A
negative time indicates pre-emergence and a positive time indicates post-
emergence for each EAR.

Each active region, labelled by its NOAA number, is paired with a
corresponding control region (CR). The CR is assigned a mock-emergence time
when the Stonyhurst coordinates of the CR was the same as the EAR at its
(real) emergence time (Schunkeretal2016, 25, see Table LABEL:tab:lbp and Table
LABEL:tab:nlbp in this paper, and ). The control region is necessary to ensure
that the signal is not due to systematics from the SDO/HMI instrument or due
to some line-of-sight observing effect.

##  3 Data reduction

Our data reduction closely follows what has already been published in
Schunkeretal2016 (25), Birchetal2016 (5) and Birchetal2019 (4). Here we
outline the relevant details for this study.

###  3.1 Mapping and tracking

The HMI observes the full-disk continuum intensity, line-of-sight magnetic
field and Doppler velocity at the photosphere with a cadence of 45 s. We track
the location of the EARs and CRs at the Carrington rotation rate over time
intervals (TI) of 6.825 h (547 frames with a cadence of 45 s).

Local helioseismology measures the travel time of a wave from one location on
the surface until it appears at another location on the surface. Waves travel
at the local sound speed, and generally waves that travel shorter distances do
so over a shorter time. The waves we are interested in for this study have
travel times less than about 6 hours, and so this is the length of time we
chose for a single time interval and the corresponding datacube.

These datacubes are labelled with their time interval (TI) relative to the
emergence time interval, TI+00. The emergence time is defined as $\tau=0.0$
days, and the midtime of time interval TI+00 corresponds to $\tau=0.1$ days
(Schunkeretal2019, 24, see Table B.1 in ). The beginning of each time interval
is spaced at 5.3375 h (320.25 min, 427 frames), and they have a 1.5 hour, or
120 frame, overlap (Schunkeretal2016, 25, see Figure 5 in ). Each active
region is tracked up to 7 days before and after the emergence time, depending
on their distance to central meridian at that time.

At each 45 s interval we projected the full-disk SDO/HMI observations onto
$60^{\circ}\times 60^{\circ}$ Postel projection maps. The projection is made
to a $512\times 512$ pixel grid with a pixel size of 1.39 Mm. The coordinates
of the map centre are the flux-weighted centre of the line-of-sight magnetic
field at the emergence time (Schunkeretal2016, 25, see ). In this article we
examine the magnetic field maps averaged over each 6.825 hour time interval to
correspond directly to the flows.

###  3.2 Computing the surface flows

Local helioseismology is a tool that uses the acoustic waves in the Sun to map
the three-dimensional subsurface structure and/or dynamics (for a review of
the different methods and key results see GizonBirch2005 (16)). For example,
perturbations to the travel-times of the waves can be interpreted as a linear
perturbation to the structure and dynamics of the interior of the Sun at the
depths where that particular wave has sensitivity. By selecting waves that are
sensitive to the near-surface of the Sun, we can infer the horizontal flows
which we can then be used to identify supergranulation structures
(Gizonetal2003, 15, e.g.).

We filtered the tracked and remapped Doppler velocities with a phase-speed
filter with a central phase speed of $17.49$ km s-1and a width of $2.63$ km
s-1(Couvidatetal2005, 10, filter 3 from Table 1 in ). This filter isolates
waves that are most sensitive to the 3 Mm just below the photosphere. We then
measured the north-south and east-west travel time differences using surface-
focusing helioseismic holography LindseyBraun2000 (21). We used an empirically
determined conversion constant of $-7.7$ ms-2 to convert from travel time
differences to surface flows Birchetal2016 (5).

We remove any remnant large scale velocities from the Sun’s bulk rotation or
orbital velocity of the SDO satellite (a constant offset) in the flows by
subtracting a plane fit to each map, excluding a central region of radius 70
Mm (50 pixels) which is our region of interest where the active regions
emerge. To remove high-wavenumber noise, we use a low-bandpass filter with a
value of 1 for $kR_{\odot}<140$, a raised cosine from 1 to 0 in the region
$140<kR_{\odot}<220$, and zero for $kR_{\odot}>220$. We then have surface flow
maps, $\mathbf{v_{x}}(x,y)$ and $\mathbf{v_{y}}(x,y)$ at a 5.3375 h time
interval, corresponding to the time- averaged magnetic field maps (described
at end of Section 3.1).

To create ensemble averages, we treat all active regions as if they were in
the northern hemisphere, so that positive $\mathbf{y}$, is towards the pole
(north); negative $\mathbf{y}$ is towards the equator (south); positive
$\mathbf{x}$ is in the prograde (solar west) direction; and negative
$\mathbf{x}$ is in the retrograde (solar east) direction.

We reversed the magnetic field polarity of the regions in the southern
hemisphere, to account for Hale’s law when averaging EARs, so that the leading
polarity is always negative. Under the assumption that the magnetic field is
radial at the solar surface, we approximately corrected for the magnitude of
the magnetic field for the line-of-sight projection by dividing it by
$\cos\theta$, where $\theta$ is the angular distance to disk centre.

For active regions in the southern hemisphere, we flipped the averaged
magnetic field and flow maps in the latitudinal direction to account for the
pole-to-equator symmetry, and reversed the direction of the flows in the
north-south direction, so that the poleward flows are in the positive
$y$-direction. We then computed the divergence of the flows,
$\bf{\nabla}\cdot\bf{v}_{h}$ where $\bf{v}_{h}=(v_{x},v_{y})$, as the most
representative way to show the location of the supergranules, which are the
tops of convection cells.

###  3.3 Measuring the location of the bipoles

The magnetic field associated with an emergence can have significant proper
motion compared to the Carrington rotation rate. To analyse the evolution of
the flows associated with the EARs, we measured the location of the active
region magnetic field at each time interval.

We tracked the position of the centroid of the positive and negative polarity
in the active region as described in Schunkeretal2019 (24), and we outline the
process here. We first measured the location of the roughly circular
polarities with a threshold magnetic field strength of 20 G at time interval
TI+02 ($\tau=0.6$ days). We used a feature recognition algorithm (feature.pro
copyright 1997, John C. Crocker and David G. Grier) designed to determine the
centroid position of roughly circular features in an image to determine the
location of both polarities individually. Moving forward and backwards in
time, we repeated the process and selected the $x$ and $y$-centroid closest to
the polarity location in the previous time interval.

If the location of the bipole at some time interval is not defined (e.g. a
bipole cannot be detected), then we linearly interpolated for the $x$ and
$y$-centroid locations from the nearest time intervals. For times before a
clear bipole is detected, we extrapolate the first measured location of the
bipole. Similarly, for times after emergence when a clear bipole can no longer
be detected, e.g. after decay, we extrapolate the last measured location of
the bipole.

We shifted the averaged magnetic field and flow maps, using a bi-linear
interpolation (over the nearest four pixels), for each EAR to the point half-
way between the centroid locations at each time interval. The shifts are
typically on the order of up to $5$ pixels. The shifting of the maps at each
time interval removing the proper motion of the bipole is unique to the
analysis method in this paper, and is required to get a well-defined mean of
the absolute magnetic field prior to emergence.

Figure 1: Time averaged line-of-sight magnetic field maps for an example EAR
with a bipole associated with the emerging flux more than two days before the
emergence time (top row, AR 11702). And an example of an EAR without an
obviously associate pre-emergence bipole (bottom row, AR 11697). The greyscale
is saturated at $\pm 15$ G. Note that these maps have not been shifted to the
emergence location.

##  4 Identification of persistent magnetic bipoles before emergence.

Active regions are defined by their dark appearance in the intensity of light
from the surface of the Sun. This is due to the strong concentrations of
magnetic field, that appear in an east-west aligned pair of opposite
polarities on the solar surface, and increases in size and field strength as
more flux emerges (Schunkeretal2019, 24, see, for example, Figure 1 in). It
takes about 2 days on average from the time of emergence for an active region
to reach its maximum flux (Weberetal2023, 27, see Figure 2 in).

By inspection, we identified 42 EARs (listed in A) as having persistent
magnetic bipoles at least one day before emergence associated with the
eventual bipole structure of the active region. The persistent polarities are
characterised by their roughly east-west orientation and the proximity as a
pair as they are buffeted by the convection. The bipoles do not change
significantly in size (on the order of 10 Mm, see Figure 1, first panel) until
the main emergence process begins closer to $t=0$.

Some active regions, such as AR 11182 and AR 11969, show magnetic bipoles up
to 2.8 days (TI-12) before emergence. An example of an EAR with a clearly
associated pre-emergence bipole more than 2 days before emergence is AR 11702
shown in the top row of Figure 1.

For a contrasting sample, we selected an equal number (42) of EARs that, by
inspection, are not associated with any magnetic field bipoles prior to
$t=-0.3$ days (listed in B). One example is AR 11697 shown in the bottom row
of Figure 1. These two samples may constitute the extremes of a continuum
allowing us to clearly identify any fundamental differences in their
evolution.

Any EARs with dense, small-scale magnetic field within $\approx 50$ Mm radius
of the emergence location were excluded from either sample since they may
obscure the identification of any pre-emergence bipole. In Schunkeretal2016
(25), the authors defined a $P$-factor, where 0 represents an emergence into a
very quiet region; a $P$-factor of 1 or 2 indicates emergence into increasing
amounts of magnetic field nearby (but not directly at) the subsequent
emergence location; and a $P$-factor of 3 or higher indicates the region may
be compromised by pre-existing field at the emergence time and location. We
cross-checked the $P$-factor for the active regions identified in our two
samples, and found that neither of the samples we identified have a dominant
$P$-factor, and were mostly 0 or 1.

In summary, we have averaged line-of-sight magnetic field maps, $B(x,y)$, flow
maps, $v_{x}(x,y)$ and $v_{y}(x,y)$, and flow divergence maps
$\bf{\nabla}\cdot\bf{v}_{h}$ with a 5.3375 h cadence, centred on the bipole
location at each time interval for 42 EARs with persistent pre-emergence
bipoles, and 42 EARs without pre-emergence bipoles.

Figure 2: Averaged divergence flow maps over $N$ active regions with pre-
emergence bipoles at different time intervals. Blue represents converging
flows and red represents diverging flows. The arrows indicate the direction
and magnitude of the flows. Solid (dashed) black lines contour the $-20$ G
($+20$ G) of the averaged line-of-sight magnetic field maps. There is a
significant converging flow prior to emergence.  Figure 3: Averaged divergence
flow maps over $N$ active regions without pre-emergence bipoles at different
time intervals. Blue represents converging flows and red represents diverging
flows. The arrows indicate the direction and magnitude of the flows. Solid
(dashed) black lines contour $-20$ G ($+20$ G) of the averaged line-of-sight
magnetic field maps. There is no significant flow signal prior to emergence,
however, there is a significant diverging flow post-emergence.

##  5 Evolution of the average magnetic field and flows

We averaged the magnetic field maps and the flows for each sample of EARs
(with and without persistent pre-emergence bipoles). Figures 2 and 3 show the
average surface flows at each time interval for the two samples. The
equivalent maps for the control regions are in Figures 5 and 6 in C.

The sample of EARs with persistent pre-emergence bipoles (Figure 2) shows a
converging flow at the location of the bipoles (centre of the map) prior to
the emergence time. The growth of the bipole is shown in the contours of the
mean magnetic field. The sample without pre-emergence bipoles (Figure 3) does
not show a statistically significant converging flow, but does show a
statistically significant diverging flow from the time of emergence.

At each time interval we computed the spatially averaged magnetic flux within
a central disk of radius 35 Mm for each active region. We chose a fixed radius
that encompasses the contour of the average $|B_{\mathrm{los}}|=20$ G as the
active region grows in size to $\tau=0.6$ days (see Figure 2).

The top panel of Figure 4 shows the mean and standard error of the magnetic
flux over the sample of active regions with (blue), and without (orange), a
pre-emergence bipole. The expected difference in magnetic flux prior to the
emergence of the active regions is clear. Then, after about -0.3 days, the
averaged magnetic flux of the regions that emerge abruptly becomes higher than
for the regions with pre-emergence bipoles. This shows that the active regions
with pre-emergence bipoles evolve to have significantly lower flux than those
without pre-emergence bipoles.

We then averaged the surface flows in the central 11 Mm (8 pixels) radius for
each active region and time interval. We chose this radius by inspection of
the maps in Figure 3 to include only the flows associated with the emergence,
and note that it is on the order of supergranulation scales, but is
considerably smaller than the radius of the area over which the magnetic flux
was averaged (35 Mm). The lower three panels of Figure 4 show the evolution of
the averaged flow divergence, east-west velocity, and north-south velocity.
The averaged flow divergence follows a similar evolution for both populations
apart from an offset of about $3\times 10^{-6}$s-1. The mean flow in both
samples is converging up until 0.5 days before emergence,
$\langle\mathbf{\nabla}\cdot\mathbf{v_{h}}\rangle=(-2.9\pm 0.5)\times 10^{-6}$
s-1for regions with a pre-emerging bipole, and
$\langle\mathbf{\nabla}\cdot\mathbf{v_{h}}\rangle=(0.03\pm 0.4)\times 10^{-6}$
s-1for those without. From about 0.5 days before emergence until about 0.1
days after emergence, the divergence increases in both samples.

The flows for both samples are strongest in the east-west direction,
$\langle\mathbf{v_{x}}\rangle$, and peak close to the time of emergence at
about $70$ ms-1. The flows in the north-south direction,
$\langle\mathbf{v_{y}}\rangle$, are consistent with zero and do not vary
significantly, due to averaging over the antisymmetric flow in the north-south
direction (see Figure 2).

Most of the surface of the Sun consists of the diverging flows of granules and
supergranules. Statistically, it is more likely that a randomly selected
location on the surface will be a diverging flow, rather than the narrower
inflow lanes. This is reflected in Figures 5 and 6 (which shows the averaged
surface flow maps of the control regions), and the orange dashed curve in the
second panel of Figure 4. The averaged divergence signal close to the
emergence time in Figure 4 is equivalent in magnitude to other diverging flows
in the map (see C), showing that it is not significant compared to the
background signal. Because we averaged the flows over a relatively small
sample of active regions and the averaging area (diameter of about 20 Mm) is
the size of a supergranule, we have captured the evolution of
supergranulation-scale flows.

Figure 4:  Averaged magnetic flux and flows as a function of time for active
regions with pre-emergence bipoles (blue) and those without (orange). The top
panel shows the average magnetic flux within a central 35 Mm radius of the
emergence location as a function of time. The error bars show the standard
error in the average magnetic flux over the active regions at each time
interval. The dashed lines show the corresponding control regions with shaded
standard errors. Active regions with pre-emergence bipoles tend to evolve to
be lower magnetic flux active regions post-emergence. The second panel shows
the averaged flow divergence, the third panel shows the average East-West
flow, and the bottom panel shows the average North-South flow. The surface
flows are averaged in the central 11 Mm radius of the emergence location.

##  6 Discussion

From our classification of two samples of EARs based on their pre-emergence
bipole signatures we found that these samples also form distinct post-
emergence populations of active regions. The sample of active regions with
persistent pre-emergence bipoles evolve to be, on average, lower flux active
regions111We do note, however, that although the largest active region in our
sample, AR 11158, falls in this category, it is a double emergence where two
bipoles emerge close to one another and then recombine to form a large,
complex active region. In this study, we have effectively only followed the
central bipole., and those that emerge more abruptly are higher flux active
regions. It is not clear why this is the case, but suggests that there is some
flux-dependence in the growth and evolution of active regions.

Our results are consistent with Birchetal2019 (4) who show that active regions
tend to emerge into regions of converging flow, but we add that the amplitude
and sign of the flow divergence is flux dependent. We find that the sample
with persistent pre-emergence bipoles emerges into strong converging flows and
these flows begin more thsn two days prior to emergence. Pre-emergence bipoles
confined to the converging flow lanes between supergranules is not surprising
since small, lower flux magnetic features are buffeted by the flows at the
surface of the Sun.

Secondly, Birchetal2016 (5) showed that the upward rise speed of flux tubes
that form active regions must be on the order of the convective flow
velocities ($\lesssim 100$ ms-1), based on the lack of any diverging flow
signature in a single observed active region. In our statistical analysis, we
have shown that some samples of active regions _are_ associated with a
diverging flow, although the magnitude is on the order of supergranulation
velocities ($\approx 100$ ms-1) and not as strong as the diverging flows
produced by simulations of a traditional thin flux tube (Birchetal2016, 5, up
to $500$ ms-1) .

Furthermore, this sample of EARs shows that higher flux active regions emerge
with stronger diverging flows. This may suggest that it is easier to form a
large active region where the converging (down) flow is weak, and/or that the
magnetic flux is brought up in the upflows of newly forming supergranules. On
the other hand, it could also suggest that tubes with higher magnetic flux
rise faster, driving a moderate surface diverging flow at the time of
emergence, however this can only be properly explored once the emergence
process of the full sample of active regions is (better) understood.

We suspect that these active regions with pre-emergence bipoles are
responsible for the conclusions of Barnesetal2014 (1) and Dhurietal2020 (11)
that the best predictor of an imminent active region emergence is the surface
magnetic field itself. It may also be that these active regions are at the
sites of active region ‘nests’ Isik+20 (19) or active longitudes (BerdUso2003,
2, e.g.) supplementing the surface small-scale magnetic field from below. A
larger sample of active regions will help to explore this idea.

We also note that we are limited by our resolution of 1.39 Mm per pixel, which
is four times coarser than the nominal HMI resolution. From inspection of some
of the full resolution line-of-sight magnetograms for these active regions,
the onset of the bipoles before emergence and their motions can be tracked
more precisely.

HottaIijima2020 (18) have successfully formed a small active region by placing
a flux tube at some depth in their three dimensional numerical
magnetohydrodynamic simulations where the near-surface convective flows have
brought the flux to the surface. The key to the formation of their active
region is placement in an upflow region sandwiched between two downflow
regions. The emergence in an upflow region is consistent with our sample of
EARs with higher flux, but without pre-emergence bipoles. These simulations
are computationally expensive, and while an equivalent statistical sample of
emerging active regions to compare with is prohibitive, such simulations are
crucial to probe the subsurface mechanisms of active region formation.

##  7 Conclusions

We have identified two distinct samples of emerging active regions: one sample
with a persistent magnetic bipole more than one day prior to emergence, and
one sample with no distinct magnetic signature until the emergence time. There
are 42 active regions in each sample, which may constitute the extremes of a
continuum.

We found that both samples of active regions are associated with converging
flows prior to emergence, but that the magnitude of the convergence was
statistically significantly different, with the sample of active regions with
a persistent pre-emergence bipole emerging into strongly converging flows. We
also found a statistical dependence on the post-emergence flux of the active
regions, where the sample of active regions with a persistent pre-emergence
bipole evolved into lower flux active regions, with an average flux of $(19\pm
1.3)~{}\times 10^{20}$ Mx, and those without evolved into stronger active
regions, with an average flux of $(24\pm 1.8)\times 10^{20}$ Mx.

Furthermore, we found that the higher flux active regions are associated with
a diverging flow at the time of emergence, whereas the lower flux active
regions did not show any significant flow signature. The ensemble averages of
the flows for both samples show the same profile of the diverging flows in
time, offset by about $3\times 10^{-6}$s-1(and about 30 ms-1in the averaged
east-west flows).

We have presented a new observational constraint for flux emergence models,
and a potential quantity to predict the maximum magnetic flux of an emerging
active region. A full interpretation of these intriguing results requires both
a more in-depth observational analysis of a broader sample of active regions
coupled with numerical simulations of rising flux tubes with a range of
magnetic fluxes. This will help to understand whether the flux dependence of
the flows in these samples we have identified are distinct or are the extremes
of a continuum of EARs governed by a single emergence mechanism.

We acknowledge the Awabakal people, the traditional custodians of the unceded
land on which this research was undertaken. Observations courtesy of NASA/SDO
and the HMI science teams.

#### Funding Statement

The HEARs data were originally processed at the German Data Center for SDO,
funded by the German Aerospace Center under grant DLR 50OL1701. HS is the
recipient of an Australian Research Council Future Fellowship Award (project
number FT220100330) and this research is partially funded by this grant from
the Australian Government.

#### Competing Interests

None.

#### Data Availability Statement

The HEARs data can be reproduced following the description in this paper and
more fully from Schunkeretal2016 (25). The results presented in this paper can
be fully reproduced following Section 3 and Section 4. Digital data is
available through private communication with the authors.

## References

  * (1) G. Barnes, A.. Birch, K.. Leka and D.. Braun  “Helioseismology of Pre-emerging Active Regions. III. Statistical Analysis”  In _ApJ_ 786, 2014, pp. 19  DOI: 10.1088/0004-637X/786/1/19
  * (2) S.. Berdyugina and I.. Usoskin  “Active longitudes in sunspot activity: Century scale persistence”  In _A &A_ 405, 2003, pp. 1121–1128  DOI: 10.1051/0004-6361:20030748
  * (3) A.. Birch et al.  “Helioseismology of Pre-emerging Active Regions. II. Average Emergence Properties”  In _ApJ_ 762, 2013, pp. 131  DOI: 10.1088/0004-637X/762/2/131
  * (4) A.. Birch, H. Schunker, D.. Braun and L. Gizon  “Average surface flows before the formation of solar active regions and their relationship to the supergranulation pattern”  In _A &A_ 628, 2019, pp. A37  DOI: 10.1051/0004-6361/201935591
  * (5) A.. Birch et al.  “An upper limit on the subsurface rise speed of solar active regions”  In _Science Advances_ , 2016  DOI: 10.1126/sciadv.1600557
  * (6) A. Brandenburg  “The Case for a Distributed Solar Dynamo Shaped by Near-Surface Shear”  In _ApJ_ 625, 2005, pp. 539–547  DOI: 10.1086/429584
  * (7) A. Brandenburg et al.  “Mean-field and direct numerical simulations of magnetic flux concentrations from vertical field”  In _A &A_ 562, 2014, pp. A53  DOI: 10.1051/0004-6361/201322681
  * (8) R. Cameron and M. Schüssler  “The crucial role of surface magnetic fields for the solar dynamo”  In _Science_ 347, 2015, pp. 1333–1335  DOI: 10.1126/science.1261470
  * (9) Feng Chen, Matthias Rempel and Yuhong Fan  “Emergence of Magnetic Flux Generated in a Solar Convective Dynamo. I. The Formation of Sunspots and Active Regions, and The Origin of Their Asymmetries”  In _ApJ_ 846.2, 2017, pp. 149  DOI: 10.3847/1538-4357/aa85a0
  * (10) S. Couvidat et al.  “Time-Distance Helioseismology: Inversion of Noisy Correlated Data”  In _ApJS_ 158.2, 2005, pp. 217–229  DOI: 10.1086/430423
  * (11) Dattaraj B. Dhuri, Shravan M. Hanasoge, Aaron C. Birch and Hannah Schunker  “Application and Interpretation of Deep Learning for Identifying Pre-emergence Magnetic Field Patterns”  In _ApJ_ 903.1, 2020, pp. 27  DOI: 10.3847/1538-4357/abb771
  * (12) Y. Fan  “The Three-dimensional Evolution of Buoyant Magnetic Flux Tubes in a Model Solar Convective Envelope”  In _ApJ_ 676.1, 2008, pp. 680–697  DOI: 10.1086/527317
  * (13) Y. Fan  “Magnetic fields in the solar convection zone”  In _Living Reviews in Solar Physics_ 18.1, 2021, pp. 5  DOI: 10.1007/s41116-021-00031-2
  * (14) Y. Fan, G.. Fisher and E.. Deluca  “The Origin of Morphological Asymmetries in Bipolar Active Regions”  In _ApJ_ 405, 1993, pp. 390  DOI: 10.1086/172370
  * (15) L. Gizon, T.. Duvall and J. Schou  “Wave-like properties of solar supergranulation”  In _Nature_ 421.6918, 2003, pp. 43–44  DOI: 10.1038/nature01287
  * (16) Laurent Gizon and Aaron C. Birch  “Local Helioseismology”  In _Living Reviews in Solar Physics_ 2.1, 2005, pp. 6  DOI: 10.12942/lrsp-2005-6
  * (17) J. Harvey, R. Tucker and L. Britanik  “High Resolution Upgrade of the GONG Instruments”  In _Structure and Dynamics of the Interior of the Sun and Sun-like Stars_ 418, ESA Special Publication, 1998, pp. 209
  * (18) H. Hotta and H. Iijima  “On rising magnetic flux tube and formation of sunspots in a deep domain”  In _MNRAS_ 494.2, 2020, pp. 2523–2537  DOI: 10.1093/mnras/staa844
  * (19) Emre Işık, Alexander I. Shapiro, Sami K. Solanki and Natalie A. Krivova  “Amplification of Brightness Variability by Active-region Nesting in Solar-like Stars”  In _ApJ_ 901.1, 2020, pp. L12  DOI: 10.3847/2041-8213/abb409
  * (20) Rudolf Komm et al.  “Sub-photosphere to Solar Atmosphere Connection”  In _Space Sci. Rev._ 196.1-4, 2015, pp. 167–199  DOI: 10.1007/s11214-013-0023-5
  * (21) C. Lindsey and D.. Braun  “Basic Principles of Solar Acoustic Holography - (Invited Review)”  In _Sol. Phys._ 192, 2000, pp. 261–284  DOI: 10.1023/A:1005227200911
  * (22) P.. Scherrer et al.  “The Helioseismic and Magnetic Imager (HMI) Investigation for the Solar Dynamics Observatory (SDO)”  In _Sol. Phys._ 275.1-2, 2012, pp. 207–227  DOI: 10.1007/s11207-011-9834-2
  * (23) H. Schunker  “Local helioseismology and the active Sun”  In _Astronomische Nachrichten_ 331, 2010, pp. 901  DOI: 10.1002/asna.201011423
  * (24) H. Schunker et al.  “Average motion of emerging solar active region polarities. I. Two phases of emergence”  In _A &A_ 625, 2019, pp. A53  DOI: 10.1051/0004-6361/201834627
  * (25) H. Schunker et al.  “SDO/HMI survey of emerging active regions for helioseismology”  In _A &A_, 2016  arXiv:1608.08005 [astro-ph.SR]
  * (26) Maria A. Weber, Yuhong Fan and Mark S. Miesch  “The Rise of Active Region Flux Tubes in the Turbulent Solar Convective Envelope”  In _ApJ_ 741.1, 2011, pp. 11  DOI: 10.1088/0004-637X/741/1/11
  * (27) Maria A. Weber, Hannah Schunker, Laurène Jouve and Emre Işık  “Understanding Active Region Origins and Emergence on the Sun and Other Cool Stars”  In _Space Science Reviews_ 219.8, 2023, pp. 63  DOI: 10.1007/s11214-023-01006-5

##  Appendix A NOAA active region numbers with associated pre-emergence
bipoles

As described in Schunkeretal2016 (25), we assigned a number, $P$, to indicate
the amount of pre-emergence flux (see Table LABEL:tab:lbp and LABEL:tab:nlbp
in A and B) based on a visual inspection of the mapped magnetograms. A
$P$-factor of 0 represents an emergence into a very quiet region; a $P$ factor
of 1 or 2 indicates emergence into increasing amounts of magnetic field nearby
(but not directly at) the subsequent emergence location; and a $P$-factor of 3
or higher indicates the region may be compromised by pre-existing field at the
emergence time and location. These were all evaluated by inspection of the
line-of-sight magnetic field maps, and does not appear to correlate with the
existence of pre-emergence bipoles.

Table 1: Emerging active regions with associated pre-emergence bipoles and their associated control region tracking locations and emergence time (Schunkeretal2016, 25, 24, adapted from Table A.1. in each of). The left panel of the table lists the NOAA active region number, emergence time, Carrington latitude, Carrington longitude, central meridian distance (CMD) at the time of emergence and the $P$-factor. The middle panel lists the emergence time and Carrington longitude of the control region. The right panel lists the difference in $B$-angle, $\Delta B=B0\mathrm{(CR)}-B0\mathrm{(EAR)}$, and the rounded difference in days $\Delta T=t_{0}(\mathrm{CR})-t_{0}(\mathrm{EAR})$. Active regions with a maximum flux larger than the median of the entire HEARS are marked with an asterisk. AR | emergence time | lat. | lon. | CMD | $P$ | CR emergence time | CR lon. | $\Delta B0$ | $\Delta T$
---|---|---|---|---|---|---|---|---|---
# | [TAI] | [∘] | [∘] | [∘] |  | [TAI] | [∘] | [∘] | [days]
11066 | 2010.05.02_23:48:00 | -26.6 | 208.2 | -16.8 | 0 | 2010.05.10_23:48:00 | 102.4 | 0.8 | 7
11072* | 2010.05.20_17:12:00 | -15.1 | 314.4 | -36.1 | 0 | 2010.05.22_17:12:45 | 288.0 | 0.2 | 2
11074 | 2010.05.29_01:36:00 | 18.6 | 285.4 | 45.3 | 1 | 2010.05.31_01:36:00 | 258.9 | 0.2 | 1
11075 | 2010.05.28_13:48:00 | -20.2 | 229.4 | -17.2 | 1 | 2010.06.11_13:48:00 | 123.5 | 1.7 | 14
11098 | 2010.08.10_23:12:00 | 13.9 | 300.9 | -41.2 | 3 | 2010.08.08_23:12:00 | 327.4 | -0.1 | -1
11103 | 2010.09.01_10:12:00 | 26.2 | 85.4 | 26.7 | 4 | 2010.09.10_00:00:00 | 111.8 | 0.1 | 8
11136 | 2010.12.24_08:24:00 | -21.4 | 30.4 | 34.3 | 0 | 2011.01.01_08:23:15 | 56.7 | -0.9 | 7
11141* | 2010.12.30_22:36:00 | 34.5 | 267.9 | -1.4 | 2 | 2010.12.28_22:36:00 | 294.3 | 0.3 | -2
11143 | 2011.01.06_01:12:00 | -22.1 | 145.6 | -43.3 | 3 | 2011.01.08_01:12:45 | 119.2 | -0.2 | 2
11148 | 2011.01.17_02:24:00 | -27.7 | 65.2 | 21.8 | 0 | 2011.01.22_02:24:45 | 359.3 | -0.4 | 5
11158* | 2011.02.11_01:24:00 | -19.3 | 35.9 | -38.8 | 1 | 2011.02.13_01:23:15 | 9.6 | -0.1 | 1
11182 | 2011.03.27_04:12:00 | 13.2 | 201.5 | -12.0 | 4 | 2011.03.25_04:12:00 | 227.8 | -0.1 | -1
11198 | 2011.04.21_14:00:00 | -25.9 | 272.1 | 33.9 | 1 | 2011.04.23_14:00:45 | 245.7 | 0.2 | 2
11318* | 2011.10.11_20:12:00 | 20.9 | 94.9 | -12.6 | 2 | 2011.10.19_00:00:00 | 68.5 | -0.5 | 7
11385 | 2011.12.22_04:12:00 | -30.5 | 225.3 | -21.9 | 2 | 2011.11.29_04:12:00 | 80.4 | 3.0 | -22
11414 | 2012.02.04_09:24:00 | -5.4 | 35.7 | 10.8 | 0 | 2012.02.06_09:23:15 | 9.4 | -0.1 | 1
11446 | 2012.03.22_17:24:00 | 24.5 | 103.3 | -18.1 | 0 | 2012.03.14_17:23:15 | 208.8 | -0.2 | -8
11510 | 2012.06.18_20:36:00 | -16.2 | 17.8 | -18.8 | 2 | 2012.06.20_06:21:00 | 271.9 | 0.1 | 1
11531* | 2012.07.25_11:12:00 | 14.4 | 308.4 | 36.3 | 2 | 2012.07.30_11:12:00 | 242.3 | 0.4 | 4
11547 | 2012.08.16_09:36:00 | 5.4 | 297.4 | -44.7 | 3 | 2012.08.18_09:36:00 | 270.9 | 0.1 | 1
11549 | 2012.08.18_14:12:00 | -17.8 | 324.1 | 11.0 | 1 | 2012.08.12_06:21:00 | 350.5 | -0.3 | -6
11626 | 2012.12.03_01:36:00 | 12.5 | 299.0 | -49.2 | 3 | 2012.12.05_01:36:00 | 272.6 | -0.3 | 2.0
11640* | 2012.12.29_15:24:00 | 27.8 | 319.3 | -38.9 | 0 | 2012.12.31_15:24:00 | 292.9 | -0.2 | 2.0
11675* | 2013.02.16_06:36:00 | 12.5 | 34.2 | -43.5 | 0 | 2013.02.18_06:36:45 | 7.9 | -0.1 | 2.0
11702* | 2013.03.21_02:12:00 | 8.3 | 14.9 | 9.5 | 0 | 2013.03.23_02:12:00 | 348.5 | 0.1 | 2.0
11750* | 2013.05.15_01:48:00 | -10.3 | 359.8 | 0.5 | 3 | 2013.05.24_01:48:45 | 240.7 | 1.0 | 9.0
11776* | 2013.06.18_12:24:00 | 11.7 | 252.1 | -11.5 | 1 | 2013.06.16_12:24:45 | 278.5 | -0.2 | -2.0
11784* | 2013.07.01_11:24:00 | -14.8 | 52.7 | -39.3 | 3 | 2013.07.03_11:24:45 | 26.2 | 0.2 | 2.0
11813* | 2013.08.06_20:00:00 | -13.1 | 320.7 | -10.2 | 0 | 2013.08.11_01:20:15 | 264.9 | 0.3 | 4.2
11821 | 2013.08.14_06:24:00 | 1.3 | 245.4 | 12.7 | 1 | 2013.08.10_17:20:15 | 292.2 | -0.2 | -3.5
11829* | 2013.08.20_17:00:00 | 4.2 | 190.0 | 42.4 | 3 | 2013.08.23_17:00:00 | 150.3 | 0.1 | 3.0
11831* | 2013.08.21_06:48:00 | 13.5 | 165.2 | 25.2 | 2 | 2013.08.24_06:47:15 | 125.5 | 0.1 | 3.0
11833 | 2013.08.22_08:48:00 | 19.8 | 96.9 | -28.7 | 4 | 2013.08.26_12:00:00 | 42.3 | 0.1 | 4.1
11867* | 2013.10.09_05:00:00 | 23.2 | 180.3 | -33.7 | 0 | 2013.10.25_05:00:45 | 329.2 | -1.2 | 16.0
11878 | 2013.10.19_15:24:00 | -9.9 | 110.1 | 33.7 | 3 | 2013.10.24_12:00:00 | 46.0 | -0.4 | 4.9
11915* | 2013.12.03_05:48:00 | -29.6 | 206.9 | -1.5 | 2 | 2013.11.26_00:00:00 | 302.3 | 0.9 | -7.2
11946* | 2014.01.04_10:36:00 | 9.8 | 99.9 | -44.3 | 3 | 2013.12.26_17:20:15 | 214.7 | 1.0 | -8.7
11962 | 2014.01.19_07:48:00 | -37.2 | 279.6 | -28.6 | 0 | 2014.01.21_07:47:15 | 253.3 | -0.2 | 2.0
11969* | 2014.01.30_19:24:00 | -10.5 | 159.8 | 2.8 | 1 | 2014.01.17_12:00:00 | 335.1 | 1.1 | -13.3
11992 | 2014.02.25_20:36:00 | -20.2 | 137.1 | -36.8 | 3 | 2014.02.23_20:35:15 | 163.5 | 0.0 | -2.0
12039 | 2014.04.15_15:12:00 | 23.9 | 234.8 | -16.0 | 1 | 2014.04.18_15:12:45 | 195.2 | 0.2 | 3.0
12105 | 2014.06.28_23:24:00 | -7.1 | 307.8 | -39.7 | 2 | 2014.06.26_23:24:00 | 334.3 | -0.2 | -2.0

##  Appendix B NOAA active region numbers without associated pre-emergence
bipoles

Table 2: Emerging active region and control region tracking locations and emergence time (Schunkeretal2016, 25, 24, adapted from Table A.1. in each of). The left panel of the table lists the NOAA active region number, emergence time, Carrington latitude, Carrington longitude, central meridian distance (CMD) at the time of emergence and the $P$-factor. The middle panel lists the emergence time and Carrington longitude of the control region. The right panel lists the difference in $B$-angle, $\Delta B=B0\mathrm{(CR)}-B0\mathrm{(EAR)}$, and the rounded difference in days $\Delta T=t_{0}(\mathrm{CR})-t_{0}(\mathrm{EAR})$. Active regions with a maximum flux larger than the median of the entire HEARS are marked with an asterisk. AR | emergence time | lat. | lon. | CMD | $P$ | CR emergence time | CR lon. | $\Delta B0$ | $\Delta T$
---|---|---|---|---|---|---|---|---|---
# | [TAI] | [∘] | [∘] | [∘] |  | [TAI] | [∘] | [∘] | [days]
11070 | 2010.05.05_03:24:00 | 20.7 | 195.0 | -1.5 | 1 | 2010.05.09_00:00:00 | 89.3 | 0.4 | 3
11079* | 2010.06.08_08:24:00 | -26.0 | 118.5 | 14.5 | 1 | 2010.06.10_08:23:15 | 92.1 | 0.2 | 1
11080* | 2010.06.10_02:12:00 | -23.1 | 109.2 | 28.3 | 2 | 2010.06.12_02:12:00 | 82.8 | 0.2 | 2
11081* | 2010.06.11_07:12:00 | 24.0 | 100.5 | 35.6 | 1 | 2010.06.13_07:12:00 | 74.0 | 0.2 | 1
11086 | 2010.07.04_08:36:00 | 17.8 | 152.0 | 32.2 | 1 | 2010.07.06_08:36:00 | 125.5 | 0.2 | 1
11122 | 2010.11.06_01:12:00 | 13.8 | 261.3 | -11.3 | 0 | 2010.11.08_01:12:45 | 235.0 | -0.2 | 2
11174* | 2011.03.16_20:12:00 | 21.3 | 10.7 | 21.0 | 2 | 2011.03.14_20:12:00 | 37.0 | -0.0 | -2
11194 | 2011.04.13_05:12:00 | -31.8 | 8.9 | 20.3 | 3 | 2011.04.15_05:11:15 | 342.5 | 0.1 | 1
11199* | 2011.04.25_18:36:00 | 21.2 | 187.3 | 4.5 | 2 | 2011.05.09_18:36:00 | 2.3 | 1.4 | 14
11209 | 2011.05.08_04:48:00 | 34.8 | 358.9 | -19.6 | 1 | 2011.05.10_04:48:00 | 332.5 | 0.2 | 1
11211 | 2011.05.08_15:24:00 | -13.6 | 16.2 | 3.4 | 1 | 2011.05.03_15:24:00 | 82.3 | -0.5 | -4
11273 | 2011.08.16_13:24:00 | -17.1 | 111.0 | -19.8 | 2 | 2011.09.08_13:24:00 | 44.9 | 0.6 | 22
11297* | 2011.09.13_17:48:00 | -17.6 | 152.3 | 33.9 | 1 | 2011.09.08_17:48:45 | 218.3 | 0.0 | -4
11300* | 2011.09.17_03:48:00 | 24.2 | 92.3 | 19.0 | 0 | 2011.09.24_00:00:00 | 65.9 | -0.2 | 6
11311* | 2011.10.03_16:36:00 | -12.8 | 177.2 | -37.9 | 0 | 2011.10.23_16:36:00 | 273.3 | -1.4 | 20
11322* | 2011.10.15_14:24:00 | -27.0 | 103.5 | 45.5 | 1 | 2011.10.01_14:24:00 | 37.5 | 0.9 | -13
11331* | 2011.10.22_18:36:00 | 10.1 | 5.6 | 42.3 | 1 | 2011.10.20_18:36:00 | 32.0 | 0.2 | -2
11334* | 2011.10.30_00:36:00 | 11.3 | 187.9 | -39.8 | 2 | 2011.10.28_00:36:00 | 214.3 | 0.2 | -2
11397 | 2012.01.12_22:36:00 | -20.5 | 277.1 | -43.3 | 1 | 2012.01.30_22:36:45 | 92.8 | -1.6 | 18
11416* | 2012.02.08_18:24:00 | -18.5 | 287.6 | -39.8 | 1 | 2012.02.16_18:23:15 | 182.2 | -0.4 | 7
11431* | 2012.03.04_13:12:00 | -28.7 | 16.3 | 15.4 | 1 | 2012.03.09_13:11:15 | 310.5 | 0.0 | 4
11437 | 2012.03.16_16:12:00 | -34.3 | 167.7 | -33.4 | 1 | 2012.03.14_16:12:45 | 194.1 | -0.0 | -1
11560* | 2012.08.29_11:36:00 | 2.9 | 125.4 | -43.8 | 1 | 2012.08.21_11:35:15 | 231.1 | -0.2 | -8
11561 | 2012.08.30_01:48:00 | -12.4 | 132.5 | -28.9 | 1 | 2012.09.10_01:48:45 | 347.2 | 0.1 | 11
11570 | 2012.09.11_19:00:00 | -12.8 | 10.4 | 16.9 | 0 | 2012.09.13_18:59:15 | 344.0 | -0.0 | 1
11624 | 2012.11.27_12:12:00 | 20.7 | 32.5 | -29.0 | 1 | 2012.11.23_00:00:00 | 247.5 | 0.6 | -4
11645* | 2013.01.02_20:12:00 | -13.3 | 290.4 | -12.4 | 0 | 2012.12.29_20:12:00 | 343.1 | 0.5 | -4.0
11696* | 2013.03.11_10:24:00 | 4.4 | 90.5 | 317.8 | 1 | 2013.03.20_12:00:00 | 331.0 | 0.2 | 9.1
11697 | 2013.03.13_13:00:00 | 14.7 | 107.7 | 2.8 | 1 | 2013.03.22_12:00:00 | 349.6 | 0.2 | 9.0
11699* | 2013.03.17_00:24:00 | -15.8 | 91.4 | 32.3 | 0 | 2013.03.05_12:00:00 | 243.2 | -0.1 | -11.5
11706 | 2013.03.27_01:24:00 | -6.5 | 268.7 | -18.0 | 1 | 2013.04.03_01:23:15 | 176.4 | 0.4 | 7.0
11707 | 2013.03.28_11:48:00 | -10.7 | 229.0 | -38.8 | 0 | 2013.03.26_11:48:00 | 255.4 | -0.1 | -2.0
11718* | 2013.04.05_15:24:00 | 22.0 | 109.6 | -50.6 | 0 | 2013.04.03_15:24:00 | 136.0 | -0.1 | -2.0
11786 | 2013.07.02_00:00:00 | -32.1 | 53.7 | -31.4 | 0 | 2013.07.04_00:00:00 | 27.2 | 0.2 | 2.0
11824* | 2013.08.17_07:36:00 | -14.8 | 194.8 | 2.4 | 1 | 2013.08.26_12:00:00 | 73.4 | 0.3 | 9.2
11849* | 2013.09.19_13:00:00 | 20.9 | 75.3 | -38.2 | 1 | 2013.09.16_12:00:00 | 115.5 | 0.1 | -3.0
11910* | 2013.11.27_13:12:00 | 1.5 | 276.3 | -7.1 | 1 | 2013.11.25_13:11:15 | 302.7 | 0.3 | -2.0
11978* | 2014.02.10_07:24:00 | 5.6 | 34.0 | 15.3 | 1 | 2014.01.31_07:24:00 | 165.7 | 0.6 | -10.0
12041 | 2014.04.15_15:36:00 | -20.7 | 262.3 | 11.7 | 0 | 2014.04.13_12:00:00 | 290.7 | -0.2 | -2.2
12078 | 2014.05.31_00:48:00 | -18.4 | 327.4 | -43.2 | 1 | 2014.05.28_12:00:00 | 1.0 | -0.3 | -2.5
12118 | 2014.07.17_17:24:00 | 7.0 | 113.3 | 13.9 | 0 | 2014.07.16_12:00:00 | 129.5 | -0.1 | -1.2
12119* | 2014.07.18_11:12:00 | -22.1 | 66.8 | -22.8 | 1 | 2014.07.22_11:12:00 | 13.9 | 0.4 | 4.0

##  Appendix C Averaged flow maps of control regions for both samples

Figure 5: Averaged divergence flow maps of the control regions associated with
pre-emergence bipole regions. Blue represents converging flows and red
represents diverging flows. The arrows indicate the direction and magnitude of
the flows. Solid (dashed) black lines contour the $-20$ G ($+20$ G) of the
averaged line-of-sight magnetic field maps. There are no significant flows.
Figure 6: Averaged divergence flow maps of the control regions associated with
active regions without pre-emergence bipoles. Blue represents converging flows
and red represents diverging flows. The arrows indicate the direction and
magnitude of the flows. There is a diverging flow at the centre of the map
near the artificial emergence time, however, neither the size nor magnitude is
significantly different than other surrounding regions. We note that most of
the Sun’s surface consists of supergranulation cells of diverging flows, and
so this is statistically not unexpected.


# Some examples of nonassociative coalgebras and supercoalgebras

###### Abstract.

Locally finiteness of some varieties of nonassociative coalgebras is studied
and the Gelfand-Dorfman construction for Novikov coalgebras and the Kantor
construction for Jordan super-coalgebras are given. We give examples of a non-
locally finite differential coalgebra, Novikov coalgebra, Lie coalgebra,
Jordan super-coalgebra, and right-alternative coalgebra. The dual algebra of
each of these examples satisfies very strong additional identities. We also
constructed examples of an infinite dimensional simple differential coalgebra,
Novikov coalgebra, Lie coalgebra, and Jordan super-coalgebra over a field of
characteristic zero.

Daniyar Kozybaev111 Department of Mathematics, Eurasian National University,
Nur-Sultan, Kazakhstan<EMAIL_ADDRESS>Ualbai
Umirbaev222Department of Mathematics, Wayne State University, Detroit, MI
48202, USA; Department of Mathematics, Al-Farabi Kazakh National University,
Almaty, 050040, Kazakhstan; and Institute of Mathematics and Mathematical
Modeling, Almaty, 050010, Kazakhstan, e-mail<EMAIL_ADDRESS>and Viktor
Zhelyabin333 Institute of Mathematics of the SB of RAS, Novosibirsk, 630090,
Russia, e-mail<EMAIL_ADDRESS>
Mathematics Subject Classification (2020): 17B62, 17C70, 17D25, 17D15.

Key words: coalgebra, coderivation, differential algebra, Novikov algebra, Lie
algebra, Jordan superalgebra, right-alternative algebra.

##  1\. Introduction

The notion of coalgebra is dual to the notion of algebra. The theory of
(co)associative coalgebras has been developed for a long time within the
framework of the theory of Hopf algebras [18]. The study of coalgebras,
bialgebras, and Hopf algebras received a new impetus when the term ”quantum
group”, along with revolutionary new examples, was launched by V. Drinfeld in
1986 [3]. Lie bialgebras, which are simultaneously Lie algebras and Lie
coalgebras, were introduced as one of the most important notions of quantum
group theory [4]. The study of Lie coalgebras, investigated earlier by W.
Michaelis [11], was intensified. It is well known that the dual of an
associative coalgebra is an associative algebra and the dual of a Lie
coalgebra is a Lie algebra. In 1994 J. Anquella, T. Cortes, and F. Montaner
[1] called a coalgebra $C$ an $\mathfrak{M}$-coalgebra if the dual algebra
$C^{*}$ belongs to the variety of algebras $\mathfrak{M}$. This allows to
define alternative, Jordan, Malcev, left-symmetric, Novikov coalgebras, and so
on.

The Fundamental Theorem on Coalgebras asserts that every finitely generated
associative coalgebra over a field is finite-dimensional. An analogue of this
result is true for alternative and Jordan coalgebras [1], for structurable
coalgebras [20], for Jordan copairs [24], for right alternative Malcev
admissible coalgebras and binary (-1,1)-coalgebras [15].

Lie coalgebras are not locally finite and the first example of an infinite
dimensional finitely generated Lie coalgebra was given in [11]. In 1995 A.
Slinko [17] found some necessary and sufficient conditions for Lie coalgebras
to be locally finite. A connection between Jordan and Lie (super)coalgebras,
which is an analog of the well known Kantor-Koecher-Tits construction for
usual (super)algebras, was found in [21, 22]. M.E. Goncharov and V.N.
Zhelyabin [6, 7] showed that every Malcev coalgebra embeds into a Lie
coalgebra with triality. Unlike Jordan coalgebras, Jordan super-coalgebras are
not locally finite [22].

In 2000 D. Kozybaev [9] constructed an example of a non-locally finite right-
symmetric coalgebra and an example of a non-locally finite right-alternative
coalgebra. I. Shestakov reported [16] that the example of a right-alternative
coalgebra given in [9] is incorrect. This report attracted the attention of
the authors to these old examples. First of all we noticed that the left-
symmetric analogue of the non-locally finite right-symmetric coalgebra from
[9] is a Novikov coalgebra. Moreover, we noticed that the commutator coalgebra
of this coalgebra is exactly the non-locally finite Lie coalgebra given by W.
Michaelis [11]. In order to understand the nature of these examples we started
to study codifferential coalgebras.

To any associative and commutative differential algebra $A$ one relates the
following three algebras:

(1) a Novikov algebra obtained from $A$ by the Gelfand-Dorfman construction;

(2) a Lie algebra obtained as the commutator algebra of the Novikov algebra
mentioned in (1); and

(3) a Jordan superalgebra obtained from $A$ by the Kantor construction.

The notion of coderivation allows us to define the notion of a
(co)differential coalgebra. We constructed a very easy example $C$ of a non-
locally finite associative and commutative differential coalgebra. Using this
example we constructed three examples of non-locally finite coalgebras. We
define an analogue of the Gelfand-Dorfman construction for coalgebras and
using this we give an example of a non-locally finite Novikov coalgebra
obtained from $C$, and the dual algebra of this coalgebra satisfies the
identity $(xy)z=0$. This example coincides with Kozybaev’s example mentioned
above. Moreover, the commutator coalgebra of this Novikov coalgebra is exactly
the non-locally finite Lie coalgebra given by Michaelis [11]. This Lie
coalgebra is metabelian. Using $C$ and an analogue of the Kantor construction
for super coalgebras, we give an example of a non-locally finite Jordan super-
coalgebra. The dual of this super-coalgebra satisfies the super identities
$xy=yx,xz=zx$, and $(zz_{1})(z_{2}z_{3})=0$ for even variables $x,y$ and for
odd variables $z,z_{1},z_{2},z_{3}$.

We repeated the same route starting from the simple differential algebra
$(F[x],\partial=\frac{d}{dx})$ over a field $F$ of characteristic zero. The
graded dual of this algebra is a simple infinite dimensional differential
coalgebra. Recall that a coalgebra is called simple if it does not have any
proper subcoalgebras. Applying the Gelfand-Dorfman construction to this
coalgebra, we get an example of a simple infinite dimensional Novikov
coalgebra. This coalgebra is the graded dual of the Novikov-Witt algebra
$\mathcal{L}_{1}$ [10]. The commutator Lie coalgebra of this coalgebra is also
simple and coincides with the graded dual of the Witt algebra $W_{1}$. And
finally, using the Kantor construction, we constructed an example of a simple
infinite dimensional Jordan super-coalgebra.

We also noticed that the example of a non-locally finite right-alternative
coalgebra from [9] can be fixed only by interchanging the indexes $3n-2$ and
$3n-1$ on lines 3 and 4 of the formula (12). The fixed example is given in
Section 6. A much more complicated example of a non-locally finite right-
alternative coalgebra was recently given in [16].

The paper is organized as follows. In Section 2 we give some necessary
terminology of coalgebras, notations, and general statements. In Section 3 we
give examples of non-locally finite differential, Novikov, and Lie coalgebras.
Simple infinite dimensional differential, Novikov, and Lie coalgebras are
given in Section 4. Section 5 is devoted to Jordan super-coalgebras and
Section 6 is devoted to right-alternative coalgebras.

##  2\. Coalgebras and coderivations

Let $F$ be an arbitrary field. For any vector space $V$ over $F$ denote by

$V^{*}={\rm Hom}_{F}(V,F)$

its dual vector space, i.e., the vector space of all linear forms on $V$.

Denote by

$V^{\otimes n}=\underbrace{V\otimes\ldots\otimes V}_{n}$

the $n$-th tensor power of the vector space $V$ over $F$.

The map

$\rho:(V^{*})^{\otimes n}\to(V^{\otimes n})^{*}$

defined by

$\rho(f_{1}\otimes\ldots\otimes f_{n})(\sum_{i_{1}\ldots
i_{n}}e_{i_{1}}\otimes\ldots\otimes e_{i_{n}})=\sum_{i_{1}\ldots
i_{n}}f_{1}(e_{i_{1}})\ldots f_{n}(e_{i_{n}})$

is injective. For this reason we can assume that

$(V^{*})^{\otimes n}\subseteq(V^{\otimes n})^{*}.$

If $\phi:V\to U$ is a linear map of vector spaces then the transpose
$\phi^{*}:U^{*}\to V^{*}$ of $\phi$ is defined by

$\phi^{*}(u^{*})(v)=u^{*}(\phi(v)),\ \ \ v\in V,u^{*}\in U^{*}.$

A vector space $C$ over $F$ with a linear map

$\Delta:C\to C\otimes_{F}C$

is called a coalgebra. The map $\Delta$ is called its comultiplication. We
often call the pair $(C,\Delta)$ a coalgebra in order to emphasize the
comultiplication in question. For any $a\in C$, using the Sweedler notation
(see [18]), we write

$\Delta(a)=\sum_{a}a_{(1)}\otimes a_{(2)}.$

If $C$ is a coalgebra, then

$(fg)(a)=\rho(f\otimes g)(\Delta(a))=\sum_{a}f(a_{(1)})g(a_{(2)}),\ \ \ \
f,g\in C^{*},a\in C,$

defines a product on $C^{*}$ and this product turns $C^{*}$ into an algebra.
Denote this product by

$\displaystyle m_{\Delta}:C^{*}\otimes C^{*}\to C^{*}.$

The algebra $C^{*}$ or $(C^{*},m_{\Delta})$ is called the dual algebra of the
coalgebra $(C,\Delta)$.

A coalgebra $(C,\Delta)$ is called coassociative if

$(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta=0,$

i.e., for any $a\in C$ we have

$\sum_{a}(\Delta(a_{(1)})\otimes a_{(2)}-a_{(1)}\otimes\Delta(a_{(2)})=0.$

It is well known that a coalgebra $(C,\Delta)$ is coassociative if and only if
its dual algebra $C^{*}$ is associative. Moreover, a coalgebra $(C,\Delta)$ is
a Lie coalgebra if and only if its dual $C^{*}$ is a Lie algebra [11].
Following these results, the definition of coalgebras from any variety of
algebras was given in [1]:

Let $\mathfrak{M}$ be an arbitrary variety of algebras. A coalgebra
$(C,\Delta)$ is called an $\mathfrak{M}$-coalgebra if its dual algebra $C^{*}$
belongs to $\mathfrak{M}$.

Let $V$ be a vector space and let $\tau:V\otimes V\mapsto V\otimes V$ be the
ordinary flip, i.e., a linear map with $\tau(x\otimes y)=y\otimes x$ for all
$x,y\in V$.

A coalgebra $(C,\Delta)$ is called cocommutative if

$\Delta=\tau\Delta,$

i.e.,

$\sum_{a}a_{(1)}\otimes a_{(2)}=\sum_{a}a_{(2)}\otimes a_{(1)}$

for any $a\in C$.

Let $(C,\Delta)$ be an arbitrary coalgebra. A subspace $B$ of $C$ is called
subcoalgebra of the coalgebra $(C,\Delta)$ if $\Delta(B)\subseteq B\otimes B$.

A subcoalgebra $B$ of a coalgebra $(C,\Delta)$ is called proper if
$B\neq\\{0\\},C$. A coalgebra $(C,\Delta)$ without proper subcoalgebras is
called simple [14].

It is well known [1] that $C$ admits a $C^{*}$-bimodule structure. The left
and right actions of $C^{*}$ on $C$ are defined by

$\alpha\cdot a=\sum_{(a)}a_{(1)}\alpha(a_{(2)}),\ \
a\cdot\alpha=\sum_{(a)}\alpha(a_{(1)})a_{(2)},\ \ \alpha\in C^{*},a\in C.$

Moreover, a vector subspace $B$ of a coalgebra $C$ is a subcoalgebra if and
only if $B$ is a $C^{*}$-subbimodule of $C$. Therefore the intersection of any
set of subcoalgebras of $C$ is again a subcoalgebra.

Let $S$ be a subset of a coalgebra $C$. The smallest subcoalgebra $Coalg(S)$
of $C$ that contains $S$ is called the subcoalgebra generated by $S$. In other
words, $Coalg(S)$ is the $C^{*}$-subbimodule of $C$ generated by $S$. If $S$
is a finite set then $Coalg(S)$ is called finitely generated.

A coalgebra $(C,\Delta)$ is called locally finite if every finitely generated
subcoalgebra of $C$ is finite dimensional.

A linear map $d:C\mapsto C$ is called a coderivation of the coalgebra
$(C,\Delta)$ if

$\Delta d=(d\otimes id+id\otimes d)\Delta,$

i.e.,

$\Delta(d(a))=\sum_{a}d(a_{(1)})\otimes a_{(2)}+a_{(1)}\otimes d(a_{(2)}),\ \
a\in C.$

A triple $(C,\Delta,d)$ is called a (co)differential coalgebra if $(C,\Delta)$
is a coalgebra and $d$ is its coderivation. A subspace $B$ of a differential
coalgebra $C$ is called a subcoalgebra if $B$ is a subcoalgebra of
$(C,\Delta)$ and $d(B)\subseteq B$, i.e., $B$ is a codifferentially closed
subcoalgebra.

######  Lemma 1.

Let $d$ be a coderivation of the coalgebra $(C,\Delta)$. Then its transpose
$d^{*}$ is a derivation of the dual algebra $C^{*}$, i.e.,

$d^{*}(fg)=d^{*}(f)g+fd^{*}(g),\ \ f,g\in C^{*}.$

Proof. Let $f,g\in C^{*}$, $a\in C$. Then

$(d^{*}(fg))(a)=(fg)(d(a))=\rho(f\otimes
g)\Delta(d(a))=\sum_{a}f(d(a_{(1)}))g(a_{(2)})+f(a_{(1)})g(d(a_{(2)}))$
$=\sum_{a}(d^{*}(f))(a_{(1)})g(a_{(2)})+f(a_{(1)})(d^{*}(g))(a_{(2)})=(d^{*}(f)g+fd^{*}(g))(a),$

which proves the statement of the lemma. $\Box$

######  Corollary 1.

If $(C,\Delta,d)$ is an associative and commutative differential coalgebra
then $(C^{*},m_{\Delta},d^{*})$ is an associative and commutative differential
algebra.

Let $A$ be an algebra over $F$ with the multiplication

$m:A\otimes A\to A,$

i.e., $m(a\otimes b)=ab$ for all $a,b\in A$. Let

$m^{*}:A^{*}\mapsto(A\otimes A)^{*}$

be the transpose of $m$. Unfortunately, the image $m^{*}(A^{*})$ of $m^{*}$
does not always belong to $A^{*}\otimes A^{*}\equiv\rho(A^{*}\otimes
A^{*})\subseteq(A\otimes A)^{*}$. The structure of the dual coalgebra
$(A^{\circ},\Delta^{\circ})$ is a little complicated (see [1, 12]).

Let

$\displaystyle A=\bigoplus_{i\in\mathbb{Z}}A_{i},\ \ A_{i}A_{j}\subseteq
A_{i+j},$

be a $\mathbb{Z}$-graded algebra such that there exists an integer $m$ with
$A_{i}=0$ for all $i<m$ and $A_{i}$ is finite dimensional for all $i\geq m$.
In this case

$\displaystyle(A\otimes A)_{k}=\bigoplus_{k=i+j}A_{i}\otimes A_{j}$

is finite dimentional for all $k$ and

$\displaystyle(A\otimes A)_{k}^{*}=\bigoplus_{k=i+j}\rho(A_{i}^{*}\otimes
A_{j}^{*}).$

The coproduct $\Delta=\rho^{-1}m^{*}$ turns the graded space

$\displaystyle\bigoplus_{i\in\mathbb{Z}}A_{i}^{*}$

into a coalgebra. This coalgebra is called the graded dual of the graded
algebra $A$.

The following lemma is useful for studying subcoalgebras.

######  Lemma 2.

Let $V$ be a vector space and let $W$ be a subspace of $V$. Let
$a=\sum_{i=1}^{n}a_{i}\otimes b_{i}$ and let the vectors $a_{1},\ldots,a_{n}$
be linearly independent. Assume that $a\in W\otimes W$. Then
$b_{1},b_{2},\ldots,b_{n}\in W$.

Proof. Let $\alpha_{1},\ldots,\alpha_{n}$ be a system of linear forms dual to
$a_{1},\ldots,a_{n}$, i.e., $\alpha_{i}(a_{j})=\delta_{ij}$ for all $i,j$,
where $\delta$ is the Kronecker delta function.

Set $\phi=\alpha_{1}\otimes\mathrm{id}:V\otimes V\mapsto V$. Then

$\phi(\sum e_{i}\otimes f_{i})=\sum_{i}\alpha_{1}(e_{i})f_{i}.$

Obviously, $\phi(W\otimes W)\subseteq W$ and
$\phi(a)=\sum_{i}\alpha_{1}(a_{i})b_{i}=b_{1}$. Since $a\in W\otimes W$ it
follows that $\phi(a)\in W$. Consequently, $b_{1}\in W$. Similarly, we get
$b_{2},\ldots,b_{n}\in W$. $\Box$

##  3\. Examples of non locally finite differential, Novikov, and Lie
coalgebras

An algebra $A$ is called a Novikov algebra if it satisfaies the following
identities:

(1) $\displaystyle x(yz)-(xy)z=y(xz)-(yx)z,$ (2) $\displaystyle(xy)z=(xz)y.$

Recall that an algebra satisfying the identity (1) is called left-symmetric.
Left-symmetric algebras are Lie-admissible, i.e., if $A$ is a left-symmetric
algebra then $A$ with respect to the commutator $[x,y]:=xy-yx$ is a Lie
algebra. This algebra is called the commutator algebra of $A$ and is denoted
by $A^{(-)}$.

The identity (1) can be written as

(3) $\displaystyle(x,y,z)=(y,x,z),$

where $(x,y,z):=(xy)z-x(yz)$ is the associator of elements $x,y,z$.

The Gelfand-Dorfman construction [5]. Let $A$ be an associative and
commutative algebra with a derivation $d$. Define a new multiplication
$(\circ)$ on $A$ by

$x\circ y=xd(y),$

where $x,y\in A$. Then $(A,\circ)$ is a Novikov algebra.

Moreover, the vector space $A$ with respect to the bracket

$\displaystyle[x,y]=xd(y)-yd(x)$

is a Lie algebra. Obviously, $(A,[\cdot,\cdot])$ is the commutator algebra of
the Novikov algebra $(A,\circ)$.

Thus, any associative and commutative differential algebra $A$ generates the
Novikov algebra $(A,\circ)$ and the Lie algebra $(A,[\cdot,\cdot])$. Over a
field of characteristic zero every Novikov algebra can be embedded into a
Novikov algebra $(A,\circ)$ for a suitable associative commutative
differential algebra $A$ [2]. The class of Lie algebras embeddable into Lie
algebras of the type $(A,[\cdot,\cdot])$ is not described yet.

######  Proposition 1.

A pair $(C,\Delta)$ is a Novikov coalgebra if and only if the following
(co)identities hold:

(4)
$\displaystyle(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta=(\tau\otimes\mathrm{id})(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta,$
(5)
$\displaystyle(\Delta\otimes\mathrm{id})\Delta=(\mathrm{id}\otimes\tau)(\Delta\otimes\mathrm{id})\Delta.$

Proof. Let $(C,\Delta)$ be a coalgebra and let $\alpha,\beta,\gamma\in C^{*}$
and $c\in C$. Then

$(\alpha,\beta,\gamma)(c)=(\alpha\otimes\beta\otimes\gamma)((\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta(c))$

and

$(\beta,\alpha,\gamma)(c)=(\alpha\otimes\beta\otimes\gamma)((\tau\otimes\mathrm{id})(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta(c)).$

Consequently,

$\displaystyle[(\alpha,\beta,\gamma)-(\beta,\alpha,\gamma)](c)$
$\displaystyle=(\alpha\otimes\beta\otimes\gamma)[(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta-(\tau\otimes\mathrm{id})(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta](c)).$

This implies that the identity (3) in $C^{*}$ is equivalent to the identity
(4) in $C$.

Similarly, the identity (2) in $C^{*}$ is equivalent to the identity (5) in
$C$. $\Box$

The Gelfand-Dorfman construction for coalgebras. Let $(C,\Delta,d)$ be an
associative and commutative differential coalgebra. Define on the space $C$ a
new comultiplication $\Delta_{N}$ by

$\Delta_{N}=(\mathrm{id}\otimes d)\Delta.$

This means

$\Delta_{N}(a)=\sum_{a}a_{(1)}\otimes d(a_{(2)})$

for any $a\in C$. Set also

$\displaystyle\Delta_{L}=\Delta_{N}^{(-)}=(1-\tau)\Delta_{N},$

i.e.,

$\Delta_{N}(a)=\sum_{a}(a_{(1)}\otimes d(a_{(2)})-d(a_{(2)})\otimes a_{(1)})$

for any $a\in C$.

######  Proposition 2.

$(1)$ The coalgebra $(C,\Delta_{N})$ is a Novikov coalgebra and the product in
its dual algebra is defined by

$\alpha\circ\beta=\alpha d^{*}(\beta),\ \ \ \ \alpha,\beta\in C^{*}.$

$(2)$ The coalgebra $(C,\Delta_{L})$ is a Lie coalgebra and the bracket in its
dual algebra is defined by

$[\alpha,\beta]=\alpha d^{*}(\beta)-d^{*}(\alpha)\beta,\ \ \ \alpha,\beta\in
C^{*}.$

Proof. By Corollary 1, the dual $(C^{*},m_{\Delta},d^{*})$ of the diferential
coalgebra $(C,\Delta,d)$ is a differential algebra. We have

$(fg)(a)=\rho(f\otimes g)(\Delta(a)),$

where $f,g\in C^{*},\ a\in C$. By the Gelfand-Dorfman construction, the
algebra $(C^{*},\circ)$ is a Novikov algebra, where $f\circ g=fd^{*}(g)$ for
all $f,g\in C^{*}$.

On the other hand,

$(f\circ
g)(a)=(fd^{*}(g))(a)=\sum_{a}f(a_{(1)})d^{*}(g)(a_{(2)})=\sum_{a}f(a_{(1)})g(d(a_{(2)}))=\rho(f\otimes
g)(\Delta_{N}(a)),$

i.e.,

$f\circ g=fd^{*}(g)=\rho(f\otimes g)\Delta_{N}.$

Hence $(C^{*},\circ)$ is the dual algebra of the coalgebra $(C,\Delta_{N})$.
Since $(C^{*},\circ)$ is a Novikov algebra it follows that $(C,\Delta_{N})$ is
a Novikov coalgebra.

The second statement of the lemma can be checked similarly. $\Box$

Example 1. Let $C$ be a vector space with a linear basis

$\displaystyle e,f_{1},f_{2},\ldots,f_{n},\ldots.$

Define a comultiplication $\Delta:C\to C\otimes C$ on $C$ by

$\displaystyle\Delta(e)=e\otimes e,\ \Delta(f_{i})=f_{i}\otimes e+e\otimes
f_{i},\ i\geq 1.$

Define also a linear map $d:C\to C$ by

$\displaystyle d(e)=0,\,d(f_{i})=f_{i+1},\,i\geq 1.$

######  Lemma 3.

The triple $(C,\Delta,d)$ is an associative and commutative differential
coalgebra.

Proof. Obviously $(C,\Delta)$ is cocommutative. Direct calculations give that

$\displaystyle(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta(e)=e\otimes
e\otimes e-e\otimes e\otimes e=0,$
$\displaystyle(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta(f_{i})=f_{i}\otimes
e\otimes e+e\otimes f_{i}\otimes e$ $\displaystyle+e\otimes e\otimes
f_{i}-f_{i}\otimes e\otimes e-e\otimes f_{i}\otimes e-e\otimes e\otimes
f_{i}=0,$

i.e., $(C,\Delta)$ is coassociative.

We also have

$\Delta(d(e))=0=d(e)\otimes e+e\otimes d(e),$
$\Delta(d(f_{i}))=\Delta(f_{i+1})=f_{i+1}\otimes e+e\otimes f_{i+1}=$
$(d(f_{i})\otimes e+d(e)\otimes f_{i+1})+(f_{i+1}\otimes d(e)+e\otimes
d(f_{i})),\,i\geq 1.$

This means that $d$ is a coderivation of the coalgebra $(C,\Delta)$. $\Box$

By Lemma 3, $(C,\Delta,d)$ is an associative and commutative differential
coalgebra. Consequently, $(C^{*},m_{\Delta},d^{*})$ is an associative and
commutative differential algebra. Following the tradition in the theory of
ordinary differential algebras, we denote the derivative of $x\in C^{*}$ by
$x^{\prime}$, i.e., $x^{\prime}=d^{*}(x)$.

######  Proposition 3.

The differential coalgebra $(C,\Delta,d)$ is not locally finite and the
differential algebra $(C^{*},m_{\Delta},d^{*})$ satisfies the differential
identity

$x^{\prime}y^{\prime}=0.$

Proof. Obviously the codifferential subcoalgebra $B$ of $C$ generated by
$f_{1}$ contains $f_{i}$ for all $i$. Since

$\displaystyle\Delta(f_{1})=f_{1}\otimes e+e\otimes f_{1}\in B\otimes B$

it follows that $e\in B$ by Lemma 2. Therefore $B=C$ is infinite dimensional.

Let $\alpha,\beta\in C^{*}$ and $c\in C$. Then

$\displaystyle(d^{*}(\alpha)d^{*}(\beta))(c)=\sum_{c}\alpha(d(c_{(1)}))\beta(d(c_{(2)})).$

Consequently

$\displaystyle(d^{*}(\alpha)d^{*}(\beta))(e)=\alpha(d(e))\beta(d(e))=0$

and

$\displaystyle(d^{*}(\alpha)d^{*}(\beta))(f_{i})=\alpha(d(f_{i}))\beta(d(e))+\alpha(d(e))\beta(d(f_{i}))=0$

since $d(e)=0$.

This means that $C^{*}$ satisfies the differential identity
$x^{\prime}y^{\prime}=0$. $\Box$

Example 2. Consider the comultiplication $\Delta_{N}=(\mathrm{id}\otimes
d)\Delta$ on $C$. By Proposition 2, $(C,\Delta_{N})$ is a Novikov coalgebra.
We have

$\Delta_{N}(e)=(\mathrm{id}\otimes d)\Delta(e)=e\otimes d(e)=0$

and

$\Delta_{N}(f_{i})=(\mathrm{id}\otimes d)\Delta(f_{i})=e\otimes
d(f_{i})=e\otimes f_{i+1}$

for all $i\geq 1$.

######  Theorem 1.

The Novikov coalgebra $(C,\Delta_{N})$ is not locally finite and the Novikov
algebra $C^{*}$ satisfies the identity

$(xy)z=0.$

Proof. Let $B$ be the subcoalgebra of $(C,\Delta_{N})$ generated by $f_{1}$.
If $f_{i}\in B$ then

$\Delta_{N}(f_{i})=(\mathrm{id}\otimes d)\Delta(f_{i})=e\otimes f_{i+1}\in
B\otimes B$

implies that $e,f_{i+1}\in B$ by Lemma 2. Consequently, $B=C$ is infinite
dimensional.

Let $\alpha,\beta,\gamma\in C^{*}$. Then

$((\alpha\beta)\gamma)(e)=(\alpha\otimes\beta\otimes\gamma)(\Delta\otimes\mathrm{id})\Delta(e)=0,$
$((\alpha\beta)\gamma)(f_{i})=(\alpha\otimes\beta\otimes\gamma)(\Delta\otimes\mathrm{id})\Delta(f_{i})=(\alpha\otimes\beta\otimes\gamma)(\Delta(e)\otimes
f_{i+1})=0,$

since $\Delta(e)=0$. Hence the algebra $C^{*}$ satisfies the identity
$(xy)z=0.$ $\Box$

Example 2 is the left-symmetric analogue of the non-locally finite right-
symmetric coalgebra from [9].

Example 3. Now consider the Lie coalgebra $(C,\Delta_{L})$. Recall that
$\Delta_{L}=\Delta_{N}^{(-)}$ and, consequently,

$\Delta_{L}(e)=(d\otimes id+id\otimes d)\Delta(e)=0,$
$\Delta_{L}(f_{i})=(d\otimes id+id\otimes d)\Delta(f_{i})=e\otimes
f_{i+1}-f_{i+1}\otimes e$

for all $i\geq 1$.

######  Theorem 2.

The Lie coalgebra $(C,\Delta_{L})$ is not locally finite and the Lie algebra
$C^{*}$ satisfies the identity

$[[x,y],[z,t]]=0.$

Proof. Let $B$ be the subcoalgebra of $(C,\Delta_{L})$ generated by $f_{1}$.
If $f_{i}\in B$ then

$\Delta_{L}(f_{i})=e\otimes f_{i+1}-f_{i+1}\otimes e\in B\otimes B$

implies that $e,f_{i+1}\in B$ by Lemma 2. Consequently, $B=C$ is infinite
dimensional.

Let $\alpha,\beta,\gamma,\delta\in C^{*}$. Then

$([\alpha,\beta])(e)=(\alpha\otimes\beta)\Delta_{L}(e)=0$

and

$\displaystyle([[\alpha,\beta],[\gamma,\delta]])(f_{i})=(\alpha\otimes\beta\otimes\gamma\otimes\delta)(\Delta_{L}\otimes\Delta_{L})\Delta_{L}(f_{i})$
$\displaystyle=(\alpha\otimes\beta\otimes\gamma\otimes\delta)(\Delta_{L}(e)\otimes\Delta_{L}(f_{i+1})-\Delta_{L}(f_{i+1})\otimes\Delta_{L}(e))=0$

since $\Delta_{L}(e)=0$. Therefore $[[\alpha,\beta],[\gamma,\delta]]=0$.
$\Box$

Example 3 is Michaelis’s example of a non-locally finite Lie coalgebra from
[11].

##  4\. Infinite dimensional coalgebras without proper subcoalgebras

Let $F$ be a field of characteristic zero and let $F[x]$ be the algebra of
polynomials over $F$ in one variable $x$. Then $(F[x],m,\partial)$, where $m$
is the polynomial multiplication and $\partial=\frac{d}{dx}$, is a simple
differential algebra. Consider the natural grading

$\displaystyle F[x]=F1\oplus Fx\oplus\ldots\oplus Fx^{n}\oplus\ldots.$

The following example is the graded dual of the differential algebra
$(F[x],m,\partial)$.

Example 4. Let

$\displaystyle C=Fx_{0}\oplus Fx_{1}\oplus\ldots\oplus Fx_{n}\oplus\ldots,$

where $x_{i}\in F[x]^{*}$ is defined by $x_{i}(x^{j})=\delta_{ij}$ for all
$i,j$. It is easy to check that the dual comultiplication

$\displaystyle\Delta=\rho^{-1}m^{*}:C\to C\otimes C$

is defined by

$\displaystyle\Delta(x_{n})=\sum_{i=0}^{n}x_{i}\otimes x_{n-i}$

and the coderivation

$\displaystyle d=\partial^{*}:C\to C$

is defined by

$\displaystyle d(x_{n})=(n+1)x_{n+1}$

for all $n\geq 0$.

Consequently, $(C,\Delta,d)$ is an associative and commutative differential
coalgebra.

######  Proposition 4.

The differential coalgebra $(C,\Delta,d)$ is simple.

Proof. Let $B$ be a nonzero subcoalgebra of $C$ and let

$\displaystyle f=x_{n}+\sum_{i<n}\alpha_{i}x_{i}\in B.$

Then

$\displaystyle\Delta(f)=x_{n}\otimes 1+\sum_{i<n}x_{i}\otimes f_{i}\in B,\ \
f_{i}\in C.$

By Lemma 2 this implies that $x_{n}\in B$. Since $B$ is differentially closed
it follows that $x_{i}\in B$ for all $i\geq n$. Morever,

$\displaystyle\Delta(x_{n})=\sum_{i=0}^{n}x_{i}\otimes x_{n-i}\in B$

imples that $x_{0},\ldots,x_{n-1}\in B$ by Lemma 2. Consequently, $B=C$.
$\Box$

Applying the Gelfand-Dorfman construction to the differential algebra
$(F[x],m,\partial)$ we get a Novikov algebra $(F[x],\circ)$, where $f\circ
g=f\partial(g)$. This Novikov algebra was denoted by $\mathcal{L}_{1}$ in [10]
and is the first algebra in the list of left-symmetric Witt algebras
$\mathcal{L}_{n}$. Notice that the commutator algebra of $\mathcal{L}_{n}$ is
the Witt algebra $W_{n}$ [10]. Since $\mathcal{L}_{1}$ is a Novikov algebra,
we call $\mathcal{L}_{1}$ the Novikov-Witt algebra. Recall that

$\displaystyle\mathcal{L}_{1}=F\partial\oplus Fx\partial\oplus\ldots\oplus
Fx^{n}\partial\oplus\ldots$

is the algebra of all derivations of $F[x]$ with respect to the product

$\displaystyle x^{i}\partial\circ x^{j}\partial=jx^{i+j-1}\partial,\ \ i,j\geq
0.$

The following example is the graded dual of the Novikov-Witt algebra
$\mathcal{L}_{1}$.

Example 5. Let $(C,\Delta,d)$ be the differential coalgebra from Example 4.
Applying the Gelfand-Dorfman construction, we get a Novikov coalgebra
$(C,\Delta_{N})$, where

$\displaystyle\Delta_{N}=(\mathrm{id}\otimes d)\Delta,$

i.e.,

$\displaystyle\Delta_{N}(x_{n})=\sum_{i=0}^{n}(n-i+1)x_{i}\otimes x_{n-i+1}$

######  Theorem 3.

The Novikov coalgebra $(C,\Delta_{N})$ is simple.

Proof. Let $B$ be a nonzero subcoalgebra of $C$ and let

$\displaystyle f=x_{n}+\sum_{i<n}\alpha_{i}x_{i}\in B.$

Then

$\displaystyle\Delta_{N}(f)=(n+1)1\otimes x_{n+1}+\sum_{i<n+1}f_{i}\otimes
x_{i}\in B,\ \ f_{i}\in C.$

By Lemma 2 this implies that $x_{n+1}\in B$. Consequently, we can assume that
$x_{n}\in B$ for some $n\geq 0$. Then

$\displaystyle\Delta_{N}(x_{n})=\sum_{i=0}^{n}(n-i+1)x_{i}\otimes x_{n-i+1}\in
B$

implies that $x_{0},\ldots,x_{n+1}\in B$ by Lemma 2. Consequently, $B=C$.
$\Box$

The following example is the graded dual of the Witt algebra $W_{1}$.

Example 6. Let $(C,\Delta,d)$ be the Novikov coalgebra from Example 5. Then
the commutator coalgebra $(C,\Delta_{L})$ is a Lie coalgebra, where

$\displaystyle\Delta_{L}=\Delta_{N}^{(-)}=(1-\tau)\Delta_{N},$

i.e.,

$\displaystyle\Delta_{L}(x_{n})=\sum_{i=0}^{n}(n-i+1)x_{i}\otimes
x_{n-i+1}-(i+1)x_{i+1}\otimes x_{n-i}$
$\displaystyle=\sum_{i=0}^{n+1}(n+1-2i)x_{i}\otimes x_{n+1-i}.$

######  Theorem 4.

The Lie coalgebra $(C,\Delta_{L})$ is simple.

Proof. Let $B$ be a nonzero subcoalgebra of $C$ and let

$\displaystyle f=x_{n}+\sum_{i<n}\alpha_{i}x_{i}\in B.$

Then, as in the proof of Theorem 3, we get $x_{n+1}\in B$. Applying Lemma 2 to
the inclusion $\Delta_{L}(x_{n})\in B\otimes B$, we get $x_{0}\in B$.
Consequently, $B=C$. $\Box$

Notice that an example of an infinite dimensional Lie coalgebra without finite
dimensional subcoagebras was constructed in [12] and [13].

##  5\. Non locally finite Jordan supecoalgebras

Let $G$ be the Grassman algebra with identity. Then $G=G_{0}+G_{1}$ is a
$Z_{2}$-graded algebra. Let $J=J_{0}+J_{1}$ be a $Z_{2}$-graded algebra. Then
$G(J)=J_{0}\otimes G_{0}+J_{1}\otimes G_{1}$ is a subalgebra of the algebra
$G\otimes J$. The subalgebra $G(J)$ is called Grassman envelope  of the
algebra $J$.

An algebra $J$ is called a Jordan superalgebra, if its Grassman envelope
$G(J)$ is a Jordan algebra, i.e., $G(J)$ satisfies the following identities:

$xy=yx,$ $(x^{2}y)x=x^{2}(yx).$

The Kantor construction [8]. Let $A$ be an associative commutative algebra
over $F$ with a derivation $D$. Denote by $\overline{A}$ an isomorphic copy of
the vector space $A$ with an isomorphism $a\mapsto\overline{a}$. On the direct
sum of the vector spaces

$J(A,D)=A\oplus\overline{A}$

define a product $(\cdot)$ by

$a\cdot b=ab,\,a\cdot\overline{b}=\overline{ab},\,\overline{a}\cdot
b=\overline{ab},\,\overline{a}\cdot\overline{b}=aD(b)-D(a)b,$

where $a,b\in A$ and $ab$ is the product of elements in $A$. Then $J(A,D)$ is
a Jordan superalgebra. The superalgebra $J(A,D)$ is called a superalgebra of
the vector type.

The Kantor construction for coalgebras. Let $(C,\Delta,d)$ be an associtive
and commutative differential coalgebra. Let $\overline{C}$ be an isomorphic
copy of the vector space $C$ with an isomorphism $c\mapsto\overline{c}$. On
the direct sum of vector spaces

$J(C,d)=C\oplus\overline{C}$

define a coproduct $\Delta_{J}$ by

$\displaystyle\Delta_{J}(c)=\sum_{(c)}c_{(1)}\otimes
c_{(2)}+\overline{c_{(1)}}\otimes\overline{d(c_{(2)})}-\overline{d(c_{(1)})}\otimes\overline{c_{(2)}},$
$\displaystyle\Delta_{J}(\overline{c})=\sum_{(c)}\overline{c_{(1)}}\otimes
c_{(2)}+c_{(1)}\otimes\overline{c_{(2)}},$

where $c\in C$ and $\Delta(c)=\sum_{(c)}c_{(1)}\otimes c_{(2)}$.

######  Proposition 5.

The coalgebra $(J(C,d),\Delta_{J})$ is a Jordan supercoalgebra and its dual
$J(C^{*},d^{*})$ is a Jordan superalgebra of the vector type.

Proof. By Corollary 1, $(C^{*},m_{\Delta},d^{*})$ is an associative and
commutative differential algebra. We have
$J(C,d)^{*}=C^{*}+(\overline{C})^{*}$. The isomorphism of $C$ and
$\overline{C}$ induces the isomorphism of $C^{*}$ and $(\overline{C})^{*}$.
Under this isomorphism, for any $\alpha\in C^{*}$ there corresponds
$\overline{\alpha}\in(\overline{C})^{*}$ such that
$\alpha(c)=\overline{\alpha}(\overline{c})$ for any $c\in C$. Therefore, we
can write that $J(C,d)^{*}=C^{*}+\overline{C^{*}}$.

Let $\alpha,\beta\in C^{*}$, and $c\in C$. Denote by $(\cdot)$ the
multiplication of the algebra $(J(C,d),\Delta_{J})^{*}$. Then we have

$(\alpha\cdot\beta)(c)=\rho(\alpha\otimes\beta)\Delta_{J}(c),\,(\alpha\cdot\overline{\beta})(\overline{c})=\rho(\alpha\otimes\overline{\beta})\Delta_{J}(\overline{c}),\,$
$(\overline{\alpha}\cdot\beta)(\overline{c})=\rho(\overline{\alpha}\otimes\beta)\Delta_{J}(\overline{c}),\,(\overline{\alpha}\cdot\overline{\beta})(c)=\rho(\overline{\alpha}\otimes\overline{\beta})\Delta_{J}(c).$

From this we get

$\alpha\cdot\beta=\alpha\beta,\,\alpha\cdot\overline{\beta}=\overline{\alpha\beta},\,\overline{\alpha}\beta=\overline{\alpha\beta},\,\overline{\alpha}\cdot\overline{\beta}=\alpha
d^{*}(\beta)-d^{*}(\alpha)\beta,$

where $\alpha\beta$ is the product of elements in the dual algebra
$(C^{*},m_{\Delta},d^{*})$.

Consequently, $(J(C,d),\Delta_{J})^{*}=J(C^{*},d^{*})$. $\Box$

Example 7. Let $(C,\Delta,d)$ be the differential coalgebra from Example 1 and
let $(J(C,d),\Delta_{J})$ be the Jordan supercoalgebra obtained from
$(C,\Delta,d)$ by the Kantor construction for coalgebras. Notice that

$\Delta_{J}(e)=e\otimes e,\ \ \Delta_{J}(f_{i})=e\otimes f_{i}+f_{i}\otimes
e+\overline{e}\otimes\overline{f_{i+1}}-\overline{f_{i+1}}\otimes\overline{e},$
$\Delta_{J}(\overline{e})=e\otimes\overline{e}+\overline{e}\otimes e,\ \
\Delta_{J}(\overline{f_{i}})=e\otimes\overline{f_{i}}+\overline{f_{i}}\otimes
e+\overline{e}\otimes f_{i}+f_{i}\otimes\overline{e},$

for all $i\geq 1$.

######  Theorem 5.

The Jordan super-coalgebra $(J(C,d),\Delta_{J})$ is not locally finite and its
dual superalgebra satisfies the super identities

$\displaystyle xy=yx,xz=zx,(z_{1}z_{2})(z_{3}z_{4})=0$

for all even $x,y$ and odd $z,z_{1},z_{2},z_{3},z_{4}$.

Proof. Let $B$ be the subcoalgebra of $J(C,d)$ generated by
$\overline{f_{1}}$. If $\overline{f_{i}}\in B$, then
$\Delta_{J}(\overline{f_{i}})\in B\otimes B$ implies that
$e,\overline{e},f_{i}\in B$ by Lemma 2. If $f_{i}\in B$, then
$\Delta_{J}(f_{i})\in B\otimes B$ implies that $\overline{f_{i+1}}\in B$.
Consequently, $B=J(C,d)$ is infinite dimensional.

By Proposition 5, the dual of the supercoalgebra $(J(C,d),\Delta_{J})$ is the
Jordan superalgebra $J(C^{*},d^{*})$ obtained from the differential algebra
$(C^{*},m_{\Delta},d^{*})$ by the Kantor construction. The first two
identities of $J(C^{*},d^{*})$, mentioned in the lemma, directly follow from
the Kantor construction since $C^{*}$ is an associative and commutative
algebra. Notice that the product of two odd elements $z_{1},z_{2}$ from
$J(C^{*},d^{*})$ belongs to the ideal $d^{*}(C^{*})C^{*}$. By Proposition 3,
we have $d^{*}(C^{*})^{2}=0$. Consequently, $(z_{1}z_{2})(z_{3}z_{4})=0$ for
all odd elements $z_{1},z_{2},z_{3},z_{4}$. $\Box$

The following example is the graded dual of the simple Jordan superalgebra
$J(F[x],\partial)$ obtained from the simple differential algebra
$(F[x],\partial)$ by the Kantor construction.

Example 8. Let $(C,\Delta,d)$ be the codifferential coalgebra from Example 4
and let $(J(C,d),\Delta_{J})$ be the Jordan super-coalgebra obtained from
$(C,\Delta,d)$ by the Kantor construction for coalgebras. Direct calculations
give that

$\Delta_{J}(x_{n})=\sum_{i=0}^{n}x_{i}\otimes
x_{n-i}+\sum_{i=0}^{n+1}(n+1-2i)\overline{x}_{i}\otimes\overline{x}_{n-i+1},$
$\Delta_{J}(\overline{x}_{n})=\sum_{i=0}^{n}(\overline{x}_{i}\otimes
x_{n-i}+x_{i}\otimes\overline{x}_{n-i}),$

for all $n\geq 0$.

######  Theorem 6.

The Jordan supercoalgebra $(J(C,d),\Delta_{J})$ is simple.

Proof. Let $B$ be a nonzero subcoalgebra of $(J(C,d),\Delta_{J})$ and let

$\displaystyle f=x_{n}+\sum_{i<n}\alpha_{i}x_{i}+\overline{c}\in B,$

where $\overline{c}\in\overline{C}$. Then

$\displaystyle\Delta_{J}(f)=\sum_{i\geq 0}a_{i}\otimes x_{i}+\sum_{i\geq
0}b_{i}\otimes\overline{x}_{i}\in B\otimes B$

and it is easy to check that $b_{n+1}\neq 0$. By Lemma 2,
$\overline{x}_{n+1}\in B$. Then $\Delta_{J}(\overline{x}_{n+1})\in B\otimes B$
implies that
$x_{0},\ldots,x_{n+1},\overline{x}_{0},\ldots,\overline{x}_{n+1}\in B$.
Consequently, $B=C$.

If

$\displaystyle f=\overline{x}_{n}+\sum_{i<n}\alpha_{i}\overline{x}_{i}\in B,$

then $\Delta_{J}(f)\in B\otimes B$ implies that $x_{n},\overline{x}_{n}\in B$
by Lemma 2. Continuing the same discussions, we get $B=C$. $\Box$

Notice that an example of a Jordan super-coalgebra without finite dimensional
subcoagebras was constructed in [23]

##  6\. Non locally finite right alternative coalgebra

In this section we give a corrected version of the example of a non locally
finite right-alternative coalgebra from [9]. This example was constructed on
the base of the example of a finitely generated metabelian right-alternative
algebra that is not residually finite [19].

An algebra $A$ is called right-alternative if it satisfies the following
identities:

(6) $\displaystyle(yx)x=yx^{2},$ (7) $\displaystyle((xy)z)y=x((yz)y).$

The identity (6) is called right-alternativity and can be written in terms of
the associators as

(8) $\displaystyle(y,x,x)=0.$

Over fields of characteristic $\neq 2$ this identity also implies the Moufang
identity (7) (see [25]).

######  Proposition 6.

Let $(A,\Delta)$ be a coalgebra over a field $F$ of characteristic $\neq 2$.
The coalgebra $(A,\Delta)$ is right-alternative if and only if the following
identity holds:

(9)
$\displaystyle(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta+(\mathrm{id}\otimes\tau)(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta=0.$

Proof. Let $(C,\Delta)$ be a coalgebra and let $\alpha,\beta,\gamma\in C^{*}$
and $c\in C$. Then

$(\alpha,\beta,\gamma)(c)=(\alpha\otimes\beta\otimes\gamma)((\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta(c))$

and

$(\alpha,\gamma,\beta)(c)=(\alpha\otimes\beta\otimes\gamma)((\mathrm{id}\otimes\tau)(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta(c)).$

Consequently,

$\displaystyle[(\alpha,\beta,\gamma)+(\alpha,\gamma,\beta)](c)$
$\displaystyle=(\alpha\otimes\beta\otimes\gamma)[(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta+(\mathrm{id}\otimes\tau)(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta](c).$

This implies that the identity (9) in $C$ is equivalent to linearized version
of the identity (8) in $C^{*}$. Consequently, (9) is equivalent to (8) over
fields of characteristic $\neq 2$. $\Box$

Example 9. Let $A$ be a vector space with a linear basis

$e_{1},e_{2},f_{1},f_{2},\ldots,f_{n},\ldots.$

Define a comultiplication $\Delta$ on the vector space $A$ by

$\Delta(e_{1})=0,\,\Delta(e_{2})=0,$ $\Delta(f_{3n-2})=e_{1}\otimes
f_{3n},\,\Delta(f_{3n-1})=e_{2}\otimes f_{3n},$ $\Delta(f_{3n})=e_{2}\otimes
f_{3n+1}-f_{3n+1}\otimes e_{2}-e_{1}\otimes f_{3n+2}+f_{3n+2}\otimes
e_{1},\,n\geq 1.$

######  Theorem 7.

The coalgebra $(A,\Delta)$ is a right-alternative non locally finite
coalgebra. Moreover, the dual algebra $A^{*}$ satisfies the identities
$(xy)(zt)=((xy)z)t=0$.

Proof. Let
$\Delta_{Ass}=(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)\Delta$. If
$f,g,h\in A^{*}$, then

$(f,g,h)(a)=(f\otimes g\otimes h)(\Delta_{ass}(a))$

for all $a\in A$. We have $(f,g,g)(e_{i})=0$ since $\Delta(e_{i})=0$ for
$i=1,2$.

Direct calculations give that

$\displaystyle\Delta_{Ass}(f_{3n-2})=(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)(e_{1}\otimes
f_{3n})=-e_{1}\otimes\Delta(f_{3n})$ $\displaystyle=-e_{1}\otimes e_{2}\otimes
f_{3n+1}+e_{1}\otimes f_{3n+1}\otimes e_{2}+e_{1}\otimes e_{1}\otimes
f_{3n+2}-e_{1}\otimes f_{3n+2}\otimes e_{1},$
$\displaystyle\Delta_{Ass}(f_{3n-1})=-e_{2}\otimes\Delta(f_{3n})=-e_{2}\otimes
e_{2}\otimes f_{3n+1}$ $\displaystyle+e_{2}\otimes f_{3n+1}\otimes
e_{2}+e_{2}\otimes e_{1}\otimes f_{3n+2}-e_{2}\otimes f_{3n+2}\otimes e_{1},$

and

$\displaystyle\Delta_{Ass}(f_{3n})=(\Delta\otimes\mathrm{id}-\mathrm{id}\otimes\Delta)(e_{2}\otimes
f_{3n+1}-f_{3n+1}\otimes e_{2}-e_{1}\otimes f_{3n+2}+f_{3n+2}\otimes e_{1})$
$\displaystyle=-\Delta(f_{3n+1})\otimes e_{2}+\Delta(f_{3n+2})\otimes
e_{1}-e_{2}\otimes\Delta(f_{3n+1})+e_{1}\otimes\Delta(f_{3n+2})$
$\displaystyle=-e_{1}\otimes f_{3(n+1)}\otimes e_{2}+e_{2}\otimes
f_{3(n+1)}\otimes e_{1}-e_{2}\otimes e_{1}\otimes f_{3(n+1)}+e_{1}\otimes
e_{2}\otimes f_{3(n+1)}.$

Consequently,

$\displaystyle(f,g,g)(f_{3n-2})=-f(e_{1})g(e_{2})g(f_{3n+1})+f(e_{1})g(f_{3n+1})g(e_{2})$
$\displaystyle+f(e_{1})g(e_{1})g(f_{3n+2})-f(e_{1})g(f_{3n+2})g(e_{1})=0,$
$\displaystyle(f,g,g)(f_{3n-1})=-f(e_{2})g(e_{2})g(f_{3n+1})+f(e_{2})g(f_{3n+1})g(e_{2})$
$\displaystyle+f(e_{2})g(e_{1})g(f_{3n+2})-f(e_{2})g(f_{3n+2})g(e_{1})=0,$

and

$\displaystyle(f,g,g)(f_{3n})=-f(e_{1})g(f_{3(n+1)})g(e_{2})+f(e_{2})g(f_{3(n+1)})g(e_{1})$
$\displaystyle-f(e_{2})g(e_{1})g(f_{3(n+1)})+f(e_{1})g(e_{2})g(f_{3(n+1)})=0.$

Consequently, $(f,g,g)(a)=0$ for all $f,g\in A^{*}$ and $a\in A$, i.e.,
$(f,g,g)=0$. This means that $A^{*}$ satisfies the identity (8).

Set
$\phi=(\Delta\otimes\mathrm{id}\otimes\mathrm{id})(\Delta\otimes\mathrm{id})\Delta$.
Direct calculations give that

$\phi(e_{1})=\phi(e_{2})=0,$
$\phi(f_{3n-2})=(\Delta\otimes\mathrm{id}\otimes\mathrm{id})(\Delta\otimes\mathrm{id})(e_{1}\otimes
f_{3n})=(\Delta\otimes\mathrm{id}\otimes\mathrm{id})(\Delta(e_{1})\otimes
f_{3n})=0,$
$\phi(f_{3n-1})=(\Delta\otimes\mathrm{id}\otimes\mathrm{id})(\Delta\otimes\mathrm{id})(e_{2}\otimes
f_{3n})=(\Delta\otimes\mathrm{id}\otimes\mathrm{id})(\Delta(e_{2})\otimes
f_{3n})=0,$

and

$\displaystyle\phi(f_{3n})=(\Delta\otimes\mathrm{id}\otimes\mathrm{id})(-\Delta(f_{3n+1})\otimes
e_{2}+\Delta(f_{3n+2})\otimes e_{1})=$
$\displaystyle(\Delta\otimes\mathrm{id}\otimes\mathrm{id})(-e_{1}\otimes
f_{3(n+1)}\otimes e_{2}+e_{2}\otimes f_{3(n+1)}\otimes e_{1})=0$

for all $n\geq 1$.

Consequently, $(((fg)h)e)(a)=0$ for all $f,g,h,e\in A^{*}$ and $a\in A$, i.
e., $((fg)h)e=0$. This means that $A^{*}$ satisfies the identity $((xy)z)t=0$.

Now set
$\psi=(\mathrm{id}\otimes\Delta\otimes\mathrm{id})(\mathrm{id}\otimes\Delta)\Delta$.
Then

$\psi(e_{1})=\psi(e_{2})=0,$
$\displaystyle\psi(f_{3n-2})=(\mathrm{id}\otimes\Delta\otimes\mathrm{id})(\mathrm{id}\otimes\Delta)(e_{1}\otimes
f_{3n})=(\mathrm{id}\otimes\Delta\otimes\mathrm{id})(e_{1}\otimes\Delta(f_{3n}))$
$\displaystyle=-e_{1}\otimes\Delta(f_{3n+1})\otimes
e_{2}+e_{1}\otimes\Delta(f_{3n+2})\otimes e_{1}$ $\displaystyle=-e_{1}\otimes
e_{1}\otimes f_{3(n+1)}\otimes e_{2}+e_{1}\otimes e_{2}\otimes
f_{3(n+1)}\otimes e_{1},$
$\displaystyle\psi(f_{3n-1})=(\mathrm{id}\otimes\Delta\otimes\mathrm{id})(\mathrm{id}\otimes\Delta)(e_{2}\otimes
f_{3n})=(\mathrm{id}\otimes\Delta\otimes\mathrm{id})(e_{2}\otimes\Delta(f_{3n}))$
$\displaystyle=-e_{2}\otimes\Delta(f_{3n+1})\otimes
e_{2}+e_{2}\otimes\Delta(f_{3n+2})\otimes e_{1}$ $\displaystyle=-e_{2}\otimes
e_{1}\otimes f_{3(n+1)}\otimes e_{2}+e_{2}\otimes e_{2}\otimes
f_{3(n+1)}\otimes e_{1},$

and

$\displaystyle\psi(f_{3n})=(\mathrm{id}\otimes\Delta\otimes\mathrm{id})(\mathrm{id}\otimes\Delta)(e_{2}\otimes
f_{3n+1}-f_{3n+1}\otimes e_{2}-e_{1}\otimes f_{3n+2}+f_{3n+2}\otimes e_{1})$
$\displaystyle=(\mathrm{id}\otimes\Delta\otimes\mathrm{id})(e_{2}\otimes\Delta(f_{3n+1})-e_{1}\otimes\Delta(f_{3n+2}))$
$\displaystyle=(\mathrm{id}\otimes\Delta\otimes\mathrm{id})(e_{2}\otimes
e_{1}\otimes f_{3(n+1)}-e_{1}\otimes e_{2}\otimes f_{3(n+1)})$
$\displaystyle=e_{2}\otimes\Delta(e_{1})\otimes
f_{3(n+1)}-e_{1}\otimes\Delta(e_{2})\otimes f_{3(n+1)}=0$

for all $n\geq 1$.

Therefore,

$\displaystyle(f((gh)g))(f_{3n-2})=-f(e_{1})g(e_{1})h(f_{3(n+1)})g(e_{2})+f(e_{1})g(e_{2})h(f_{3(n+1)})g(e_{1})=0,$
$\displaystyle(f((gh)g))(f_{3n-1})=0$

for all $f,g,h\in A^{*}$.

This means $f((gh)g)=0$ for all $f,g,h\in A^{*}$. Together with the identity
$((xy)z)t=0$, this proves that the Moufang identity (7) holds in $A^{*}$.

It is easy to check that $(\Delta\otimes\Delta)\Delta(a)=0$ for all $a\in A$.
Therefore $A^{*}$ satisfies the identity $(xy)(zt)=0$.

Now we show that$(A,\Delta)$ is not locally finite. Let $B$ a subalgebra of
$(A,\Delta)$ generated by $f_{1},f_{2}$.

Suppose that $f_{i}\in B$. If $i=3n-1$ or $i=3n-2$, then $\Delta(f_{i})\in
B\otimes B$ implies that $e_{1},e_{2},f_{3n}\in B$ by Lemma 2. If $i=3n$, then
we get $e_{1},e_{2},f_{3n+1},f_{3n+2}\in B$. This implies that $B=C$ is
infinite-dimensional. $\Box$

Acknowledgments

The first author is supported by the grant of the Ministry of Education and
Science of the Republic of Kazakhstan (project AP09261086) and the other two
authors are supported by the Russian Science Foundation (project 21-11-00286).

## References

  * [1] J. Anquela, T. Cortes, F. Montaner, Nonassociative coalgebras. Comm. Algebra 22 (1994), no. 12, 4693–4716.
  * [2] L.A. Bokut, Y. Chen, Z. Zhang, Gröbner-Shirshov bases method for Gelfand-Dorfman-Novikov algebras. J. Algebra Appl. 16 (2017), no. 1, 1750001, 22 pp.
  * [3] V.G. Drinfeld, Quantum groups. Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 798–820, Amer. Math. Soc., Providence, RI, 1987.
  * [4] V.G. Drinfeld, Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations. (Russian) Dokl. Akad. Nauk SSSR 268 (1983), no. 2, 285–287.
  * [5] I.M. Gelfand, I.Ya. Dorfman, Hamiltonian operators and algebraic structures associated with them. (Russian) Funktsional. Anal. i Prilozhen. 13 (1979), no. 4, 13–30
  * [6] M.E. Goncharov, V.N. Zhelyabin, Mikheev’s construction for Mal’tsev coalgebras. (Russian) Algebra Logika 51 (2012), no. 5, 668–671; translation in Algebra Logic 51 (2012), no. 5, 445–447.
  * [7] M.E. Goncharov, V.N. Zhelyabin, Embedding of Mal’tsev coalgebras into Lie coalgebras with triality. (Russian) Algebra Logika 52 (2013), no. 1, 34–56; translation in Algebra Logic 52 (2013), no. 1, 24–40.
  * [8] I.L. Kantor, Jordan and Lie superalgebras determined by a Poisson algebra. Algebra and Analysis (Tomsk, 1989), 55–80, Amer. Math. Soc. Transl. Ser. 2, 151, Amer. Math. Soc., Providence, RI, 1992.
  * [9] D.Kh. Kozybaev, Right-alterenative and right-symmertric coalgebras, South-Kazakhstan Science and Education. 19 (2000), No. 12, 155–163.
  * [10] D. Kozybaev, U. Umirbaev, Identities of the left-symmetric Witt algebras. Internat. J. Algebra Comput. 26 (2016), no. 2, 435–450.
  * [11] W. Michaelis, Lie coalgebras. Adv. in Math. 38 (1980), no. 1, 1–54.
  * [12] W. Michaelis, An example of a non-zero Lie coalgebra $M$ for which $\mathrm{Loc}(M)=0$. J. Pure Appl. Algebra 68 (1990), no. 3, 341–348.
  * [13] W.D. Nichols, The structure of the dual Lie coalgebra of the Witt algebra. J. Pure Appl. Algebra 68 (1990), no. 3, 359–364.
  * [14] D.E. Radford, Hopf algebras. Series on Knots and Everything, 49. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. xxii+559 pp.
  * [15] G. Santos Filho, L. Murakami, I. Shestakov, Locally finite coalgebras and the locally nilpotent radical I. Linear Algebra Appl. 621 (2021), 235–253.
  * [16] G. Santos Filho, L. Murakami, I. Shestakov. Locally finite coalgebras and the locally nilpotent radical II. Comm. Algebra 49 (2021), no. 12, 5472–5482.
  * [17] A.M. Slinko, Local finiteness of coalgebraic Lie coalgebras. Comm. Algebra 23 (1995), no. 3, 1165–1170.
  * [18] M. Sweedler, Hopf algebras. Mathematics Lecture Note Series W. A. Benjamin, Inc., New York 1969 vii+336 pp.
  * [19] U.U. Umirbaev, The word problem for metabelian right-alternative algebras. (Russian) Model-theoretic algebra, 130–139, Kazakh. Gos. Univ., Alma Ata, 1989.
  * [20] V.N. Zhelyabin, Structurable coalgebras. (Russian) Algebra i Logika 35 (1996), no. 5, 529–542; translation in Algebra and Logic 35 (1996), no. 5, 296–304.
  * [21] V.N. Zhelyabin, The Kantor-Koecher-Tits construction for Jordan coalgebras. (Russian) Algebra i Logika 35 (1996), no. 2, 173–189; translation in Algebra and Logic 35 (1996), no. 2, 96–104.
  * [22] V.N. Zhelyabin, Jordan (super)coalgebras and Lie (super)coalgebras. (Russian) Sibirsk. Mat. Zh. 44 (2003), no. 1, 87–111; translation in Siberian Math. J. 44 (2003), no. 1, 73–92.
  * [23] V.N. Zhelyabin, Dual coalgebras of Jordan bialgebras and superalgebras. (Russian) Sibirsk. Mat. Zh. 46 (2005), no. 6, 1302–1315; translation in Siberian Math. J. 46 (2005), no. 6, 1050–1061.
  * [24] V.N. Zhelyabin, Embedding of Jordan copairs into Lie coalgebras. Comm. Algebra 35 (2007), no. 2, 561–576.
  * [25] K.A. Zhevlakov, A. M. Slinko, I. P. Shestakov, A. I. Shirshov, Rings that are nearly associative. Pure and Applied Mathematics, 104. Academic Press, New York-London, 1982. xi+371 pp