# A fast algorithm for the detection of faint orbital debris tracks in optical
images

P. Hickson  Space science, Technologies and Astrophysics Research (STAR)
Institute,Université de Liège, Institut d′Astrophysique et de Géophysique,
Allée du 6 Août 19c, 4000 Liège, Belgium  National Astronomical Observatories,
Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing, China
<EMAIL_ADDRESS>
###### Abstract

Moving objects leave extended tracks in optical images acquired with a
telescope that is tracking stars or other targets. By searching images for
these tracks, one can obtain statistics on populations of space debris in
Earth orbit. The algorithm described here combines matched filtering with a
Fourier implementation of the discrete Radon transform and can detect long
linear tracks with high sensitivity and speed. Monte-Carlo simulations show
that such tracks, in a background of Poisson random noise, can be reliably
detected even if they are invisible to the eye. On a 2.2 GHz computer the
algorithm can process a $4096\times 4096$-pixel image in less than a minute.

###### keywords:

space debris , streak detection

††journal: Advances in Space Research††©2018. This manuscript version is made
available under the CC-BY-NC-ND 4.0 license
http://creativecommons.org/licenses/by-nc-nd/4.0/

##  1 Introduction

The detection of linear tracks in a two dimensional image is a common problem
in image processing. One important application is for the detection of orbital
debris. While known objects can be tracked with a telescope, increasing the
signal-to-noise ratio, unknown objects cannot. An appropriate observing
strategy is to track at the sidereal rate, thereby minimizing image
contamination by stars, and search for tracks in the image produced by objects
moving across the field of view during the exposure. Automated algorithms can
process large data sets and can reach detection limits fainter than can human
observers.

Many groups have developed algorithms to find streaked images, for the
detection of moving celestial objects (Sara & Cvrcek,, 2017; Waszczak et al.
,, 2017) as well as for satellite or debris detection (Zimmer et al. ,, 2013;
Ciurte & Danescu,, 2014; Vananti et al. ,, 2015; Virtanen et al. ,, 2017;
Vallduriola et al. ,, 2018). A variety of techniques have been employed. In
segmentation-based methods (Liu,, 1992; Virtanen et al. ,, 2017), pixels
having intensities above a threshold are analyzed. This is well-suited to
short, relatively-bright streaks. Stacking methods are useful when an object
is observed in multiple images (Yanagisawa et al. ,, 2012). Other methods
employ the Radon or Hough transform (Zimmer et al. ,, 2013; Ciurte & Danescu,,
2014) or matched filtering (Gural et al. ,, 2005; Schildknecht et al. ,, 2015;
Sara & Cvrcek,, 2017).

The classical Radon transform (Radon,, 1917; Radon & Parks,, 1986), and its
close relative the Hough transform (Hough,, 1959; Duda et al. ,, 1972), have
long been employed to identify linear features in images. These transforms map
lines to points in a two-dimensional “Hough space”, whose axes correspond to
position and angle. The angle is that between the normal to the line and the
reference axis, and the position is the perpendicular distance from the line
to the origin, usually taken to be the centre of the image. Thus the problem
of detecting long streaks becomes a simpler problem of detecting local maxima
in Hough space. A great advantage of this approach is that fast techniques
have been developed for the computation of the Radon transform (Beylkin,,
1987; Götz & Druckmüller,, 1995; Press,, 2006). These are fast in the sense
that for an $N\times N$ image, processing time grows roughly in proportion to
$N^{2}\ln N$, rather than $N^{3}$ for the classical Radon transform. For a
typical astronomical CCD image, $N\sim 10^{3}-10^{4}$, so the fast Radon
transform requires typically two to three orders of magnitude fewer
computations.

This paper describes a method that combines matched filtering with a fast
discrete Radon transform in order to achieve high sensitivity and speed. It is
best suited for the detection of long faint streaks in a single image, as
would be produced by fast-moving objects. The algorithm was tested by Monte
Carlo simulation of random linear tracks, having a Gaussian cross-section of
specified FWHM, superimposed on a constant background, to which was then added
random Poisson noise. It was found to be capable of reliably detecting faint
tracks that were invisible to the eye.

The sensitivity of the method comes from the use of matched filtering, which
provides the highest possible signal-to-noise ratio of any linear detection
technique, allowing the faintest possible detection limit. Direct application
of matched filtering, by integrating along all possible directions and
positions in the image, would be equally sensitive but very slow. The Radon
transform decreases the number of dimensions of the search, providing a large
increase in speed.

We begin by briefly reviewing the concepts of optimal detection and the Radon
transform. The algorithm is then described and results of the simulations are
presented. Our python source code implementing the Radon transform is
reproduced in Appendix A.

##  2 Method

###  2.1 Optimal detection

It has been known since 1953 that the optimal linear technique for the
detection and measurement of a signal in the presence of uniform uncorrelated
stochastic noise is that of matched filtering (Woodward,, 1953; Turin,, 1960).
It is optimal in the sense that no other linear filter can give a higher
signal-to-noise ratio.

In order to apply this method to the problem of detecting and measuring faint
tracks in noisy images, let us first suppose that we know a priori the angle
that the track makes with one of the axes of the image. Then, we can integrate
along this direction, summing the pixel values along lines parallel to the
track, in order to produce a one-dimensional mean profile of the cross-section
of the track. Effectively, this amounts to _projecting_ the image onto a line
that is orthogonal (transverse) to the track. In order to measure the position
of the track, and the total flux that it contains, we can search this one-
dimensional projection for a local maximum.

If we further know the shape and width of the track, in the transverse
direction, we can employ matched filtering. This will generally be true as
tracks left by orbital debris are typically unresolved, having a transverse
intensity profile that is well-approximated by the one-dimensional projected
profile of the point-spread function (PSF), found from images of stars in the
field. Denote the summed intensity along the projection line by $p(r)$, where
$r$ is a coordinate in the direction transverse to the track. Assume that any
constant intensity $I_{0}$, such as the sky background, has first been
subtracted. Let $f(r)$ be the expected profile, i.e. the projected PSF,
normalized so that its one-dimensional integral is unity,

$\int f(r)dr=1.$ (1)

For simplicity we show here the results for continuous functions and the
integral extends over the entire domain of the function. The extension to
discrete values can be made by replacing integrals by summations.

If the track has the expected profile, and is centred at $r=r_{0}$, we may
write

$p(r)=Ff(r-r_{0})+n(r),$ (2)

where $F$ is the total flux in the track (the integral along the projection
line of the summed intensity in the track) and $n(r)$ is a random variable
having zero mean and variance $\sigma^{2}_{n}(r)$, representing the noise.
Normally this will be the single-pixel noise variance $\sigma_{p}^{2}$
multiplied by the number of pixels that were summed in each line parallel to
the track.

To detect the track with optimal sensitivity, we cross-correlate the summed
intensity profile with a function $h(r)$ that is proportional to the expected
signal divided by the noise variance (King,, 1983),

$h(r)=\alpha\frac{f(r)}{\sigma^{2}_{n}(r)}.$ (3)

The constant $\alpha$ is chosen to make the integral of $h(r)$ unity,

$\int h(r)dr=1,$ (4)

The cross correlation will have a maximum at the location of the track, where
it takes the value

$\displaystyle g(r_{0})$ $\displaystyle=F\int h(r)f(r)dx+\int h(r)n(r)dr.$ (5)

This is a fluctuating quantity having an ensemble average

$\overline{g(r_{0})}=F\int f(r)h(r)dr.$ (6)

The second term has disappeared by virtue of $n(r)$ having zero mean. The best
estimate of the true flux $F$ is therefore

$\hat{F}=\frac{g(r_{0})}{Q},$ (7)

where

$Q=\int f(r)h(r)dr.$ (8)

The variance of this estimate is

$\displaystyle\text{Var }\hat{F}$ $\displaystyle=\frac{1}{Q^{2}}\text{Var
}g(r_{0}),$ $\displaystyle=\frac{1}{Q^{2}}\int h^{2}(r)\sigma^{2}_{n}(r)dr,$
$\displaystyle=\frac{\alpha}{Q},$ (9)

so the signal-to-noise ratio is

$s=F\sqrt{\frac{Q}{\alpha}}.$ (10)

If the noise variance can be assumed to be constant, independent of $r$, then
Eqns. (1), (3) and (4) require that $\alpha=\sigma_{n}^{2}$, and therefore
$h(r)=f(r)$. The matched filter is proportional to the expected signal. In
that case,

$Q=\int f^{2}(r)dr,$ (11)

and the signal-to-noise ratio (SNR) becomes

$s=\frac{F}{\sigma_{n}}\sqrt{Q}=\frac{\overline{g(r_{0})}}{\sigma_{n}\sqrt{Q}}.$
(12)

This shows the importance of a sharp PSF (small FWHM), which increases $Q$,
improving the SNR.

Although the noise in astronomical images arises from a number of different
sources (Newberry,, 1991; Howell et al. ,, 2003), it is often dominated by the
Poisson statistics of the detected photons. In that case, the noise variance
will be

$\sigma^{2}_{n}(r)=F_{0}+Ff(r),$ (13)

where $F_{0}$ represents the projected intensity of the background light,
before sky subtraction. Even though the mean background has been subtracted,
its noise remains.

For the Poisson case, the optimal filter, Eqn. (3), becomes

$h(r)=\frac{\alpha}{F_{0}}\frac{f(r)}{1+\beta f(r)},$ (14)

where

$\alpha=F_{0}\left[\int\frac{f(r)dr}{1+\beta f(r)}\right]^{-1}.$ (15)

Here $\beta=F/F_{0}$ is a measure of the relative brightness of the track
compared to the background. The constant $Q$ is now

$Q=\frac{\alpha}{F_{0}}\int\frac{f^{2}(r)dr}{1+\beta f(r)},$ (16)

so the SNR, Eqn. (10), becomes

$s=\frac{F}{\sigma_{0}}\left[\int\frac{f^{2}(r)dr}{1+\beta
f(r)}\right]^{1/2}.$ (17)

where $\sigma_{0}^{2}=F_{0}$ is the variance of the background.

For the Poisson case we see that the optimal filter depends on the flux of the
track, which is generally not known in advance. But, the problem considered in
this paper is the efficient detection of _faint_ tracks in a noisy image. In
that case, $\beta\lesssim 1$, and the Poisson equations approach those of the
constant-variance case, as expected. For bright streaks, it does not matter if
the matched filter that is used is somewhat less than optimal. They will be
detected in any case. This is primary justification for employing the
constant-variance matched filter when searching for faint tracks.

###  2.2 The fast discrete Radon transform

Of course if one knew in advance the orientation of the track, the detection
problem would be relatively simple. But in general the orientation is not
known. Thus, the algorithm must search all possible orientations. Directly
computing projections for $\sim N$ orientations is time consuming. However,
the speed of the process can be increased greatly by the use of the Fast Radon
Transform.

The algorithm that we employ to compute the Radon transform of the image is
based on the Fourier Slice Theorem (Bracewell,, 1960). This theorem, which is
easily proved, states that the values on a slice through the origin of the
two-dimensional Fourier transform of the image is equal to the one-dimensional
Fourier transform of the projection of the image onto a line parallel to the
slice. This allows one to employ fast Fourier transforms to compute the Radon
transform, by taking the inverse Fourier transform of each slice, for a
complete set of angles.

One way to compute the values on the slice would be to use two-dimensional
interpolation in the transformed image to estimate the values at integer
distances (in units of pixels) along the slice. However, a simpler and faster
method is employed here. If the angle $\theta$ between the slice and the $x$
axis is in the range $|\theta|<=45^{\circ}$ (for square images), a value on
the slice is determined for every $x$ pixel by taking the value of the image
at $(x,y)$, where $y=x\tan\theta$. Sinc interpolation is used to estimate the
value of the transformed image at fractional values of $y$. In this way, the
problem of interpolation in two dimensions is reduced to one-dimensional
interpolation. The resulting intensities along the slice have a spacing of
$x/\cos\theta$. To compensate, according to the Fourier scaling theorem the
intervals of $r$ after taking the inverse Fourier transform, are multiplied by
a factor of $\cos\theta$ and the intensity is multiplied by a factor of
$\sec\theta$. If $|\theta|>45^{\circ}$, the approach is similar but with $x$
and $y$ interchanged and $\tan\theta$ replaced by $\cot\theta$ and
$\cos\theta$ replaced by $\sin\theta$.

This differs from the standard Radon transform in that the position coordinate
no longer measures perpendicular distance from the track to the origin, but
distance along the $x$ or $y$ axis, depending on the value of $\theta$. The
matched filter is easily adapted to this by scaling the FWHM of the PSF by a
factor of $\sec\theta$ (or $\csc\theta$ if $|\theta|>45^{\circ}$) in order to
account for the oblique cut through the track.

The interpolation scheme that is used to compute values on the slice is
important. Simple linear interpolation, or even polynomial interpolation,
produces artifacts in the Radon transform, which degrade the photometric
accuracy. The correct approach is to use sinc interpolation. Here there is a
tradeoff between the order of the interpolation (the number of pixels that are
included in the summation) and the speed of the technique. A full $N-$point
sinc filter completely eliminates the artifacts, but significantly increases
processing time. On the other hand, linear interpolation, which is very fast,
results in systematic photometric errors that can be as great as 15%.
Employing a sinc interpolating filter encompassing 7 pixels reduces
photometric errors to less than 5%. Increasing this to $\sim 50$ pixels
reduces photometric errors to less than 1%, but increases execution time by
about a factor of 4. Even so, this is still more than an order of magnitude
faster than the corresponding two-dimensional interpolation.

###  2.3 The algorithm

Our method involves the following steps:

  * 1.

If the image is not square, divide it into overlapping square sub-images. Then
for each sub-image:

  * 2.

Mask stars and image defects.

  * 3.

Subtract the median background.

  * 4.

Compute the Radon transform.

  * 5.

Determine the RMS noise $\sigma_{n}$ in the Radon image and the threshold
value of $g$ corresponding to the desired SNR limit (from Eqn 12).

  * 6.

Find the highest value in the Radon image and record the corresponding
position, angle, flux and SNR.

  * 7.

Mask the region around the highest value bounded by specified tolerances in
position and angle.

  * 8.

Repeat this, finding the highest value in the masked Radon image, and continue
until the highest value falls below the threshold.

  * 9.

Combine the detections for all sub-images and reject duplicate detections.

The procedure requires some judgement about what constitutes a duplicate
detection. This is best found from experience, but typically, two detections
that have positions within a few FWHM of each other and angles within two or
three degrees are considered equivalent and the detection with the highest
flux is selected. In some cases, “ghost” detections may occur, which have the
same angle but differ in position by the number of pixels along an axis of the
image. This results from the periodicity of the fast Fourier transform.

##  3 Simulations and results

The algorithm was coded in Python 3 and uses the Numpy and Scipy libraries. In
order to test it, an image was created and filled with random values from a
standard normal distribution ($\sigma_{p}=1$). Then, a streak was constructed
having a random orientation and a transverse profile given by a Gaussian
having standard deviation $\sigma=w/\sqrt{8\ln 2}$, where $w$ is the desired
FWHM in pixels. The profile was normalized so that its integral, multiplied by
the number of pixels along the length of the track, equals $F$, the flux
required to achieve the desired signal-to-noise ratio according to Eqn. (12).
An example of a simulated image containing several tracks, and the
corresponding Radon transform, is shown in Figure 1.

Figure 1: Simulated image containing three orbital debris tracks (upper) and
its Radon transform (lower). The horizontal scale on the lower image is pixel
number, but it represents a range of angles from $-90^{\circ}$ to
$90^{\circ}$. The vertical scale is the $x$ position of the midpoint of the
streak. The tracks have a FWHM of 3.0 pixels and signal-to-noise ratios of
100, 50 and 25. Tracks having a signal-to-noise ratio of $\sim 20$ or less are
generally invisible to the eye in a 1K$\times$1K or larger image, but are
nevertheless detected by the algorithm.

A total of 2700 simulations were run using a range of SNR and FWHM. The
results are summarized in Table 1. Here FWHM is measured in pixels, and the
success rate is the fraction of runs for which the strongest detection found
by the algorithm matched the simulated track. The last column lists the
magnitude error, defined by
$-2.5\log_{10}(F_{\text{measured}}/F_{\text{true}})$.

The SNR values listed in Table 1 are computed using Eqn (12), where $F$ is the
modelled total flux of the track and $\sigma_{n}=\sqrt{F_{0}}$ is the
background Poisson noise variance.

Table 1: Monte-Carlo simulation summary

SNR | Size | No of trials | FWHM | Success rate | Mag error
---|---|---|---|---|---
2.0 | 1024 | 1000 | 2.0 | 0.001 | 1.15
|  |  | 3.0 | 0.002 | -1.02
|  |  | 4.0 | 0.001 | 1.01
3.0 | 1024 | 1000 | 2.0 | 0.008 | 0.76
|  |  | 3.0 | 0.011 | 0.66
|  |  | 4.0 | 0.027 | 0.66
4.0 | 1024 | 1000 | 2.0 | 0.075 | 0.39
|  |  | 3.0 | 0.094 | 0.42
|  |  | 4.0 | 0.125 | 0.37
5.0 | 1024 | 1000 | 2.0 | 0.255 | 0.20
|  |  | 3.0 | 0.328 | 0.22
|  |  | 4.0 | 0.370 | 0.22
6.0 | 1024 | 1000 | 2.0 | 0.620 | 0.13
|  |  | 3.0 | 0.642 | 0.14
|  |  | 4.0 | 0.687 | 0.15
7.0 | 1024 | 1000 | 2.0 | 0.859 | 0.14
|  |  | 3.0 | 0.879 | 0.13
|  |  | 4.0 | 0.914 | 0.14
8.0 | 1024 | 1000 | 2.0 | 0.964 | 0.15
|  |  | 3.0 | 0.971 | 0.14
|  |  | 4.0 | 0.979 | 0.13
9.0 | 1024 | 1000 | 2.0 | 0.997 | 0.15
|  |  | 3.0 | 0.999 | 0.14
|  |  | 4.0 | 0.997 | 0.13
10.0 | 1024 | 1000 | 2.0 | 0.999 | 0.14
|  |  | 3.0 | 1.000 | 0.13
|  |  | 4.0 | 1.000 | 0.12

Figure 2: Completeness and photometric error vs. signal-to-noise ratio, for
tracks having a FWHM of 3.0 pixels.

##  4 Effect of track curvature

The algorithm is designed to find linear tracks. Tracks that are slightly
curved will still be detected, but with lower sensitivity. Tracks produced by
orbital debris are not perfectly straight, although the deviation from
linearity over the field of view of a typical astronomical camera is generally
quite small.

An exact analysis of the curvature of debris tracks is beyond the scope of
this paper, but a simple approximate treatment will suffice to provide an
estimate magnitude of the effect. Long tracks are made by fast-moving debris
in low or middle Earth orbit, which cross a typical imager in a few minutes or
less. For such objects, there is little error in ignoring the motion of the
observer due to the rotation of the Earth. Also, for simplicity, we shall
assume that the orbit is circular.

Figure 3: Geometry for estimating track curvature. A satellite at point $S$ in
a circular orbit is observed from point $O$ on the surface of the Earth. The
track appears elliptical when projected perpendicular to the line of sight.

The relevant geometry is shown in Figure 3. We choose a barycentric Cartesian
coordinate system in which the object orbits in the $x-y$ plane. Consider an
observer $O$ viewing the orbiting object when it is highest in the sky, which
is the point where it crosses the $x-z$ axis. The observer sees the orbit
projected on the sky, where it appears to be elliptical. The apparent
curvature of the track (the reciprocal of the angular radius of curvature in
radians) is

$\kappa=\frac{r}{R}\sin\alpha=\frac{R_{\oplus}}{R}\sin\theta,$ (18)

where $R$ is the orbital radius, $R_{\oplus}$ is the radius of the Earth, $r$
is the distance from the observer to the object, $\alpha$ is the angle between
the line of sight and the orbital plane, and $\theta$ is the angle between the
line connecting the observer to centre and the orbital plane. The second
equality follows by application of the sine rule for plane triangles.

From this we see that for a given angle $\theta$, the curvature is maximized
by making $R$ as small as possible. The smallest possible value is
$R=R_{\oplus}/\cos\theta$, which places the object on the observer’s horizon.
With this choice of $R$, Equation (18) becomes

$\kappa=\frac{1}{2}\sin 2\theta,$ (19)

which has a maximum value $\kappa_{\text{max}}=1/2$ which occurs when
$\theta=\pi/4$.

For a track segment of length $\beta$ radians, the maximum angular deviation
from the best-fit straight line is

$\epsilon=\frac{1}{16}\kappa\beta^{2}.$ (20)

In order for there to be no significant loss of sensitivity, this deviation
should be smaller than the half-width of the PSF. For example, if the PSF has
a FWHM of 1 arcsec, the maximum track length is 0.5 degrees. Longer tracks
will spread beyond the extend of the matched filter, lowering the detection
sensitivity. A lower limit on the sensitivity can be obtained by assuming that
the outer regions of the track, beyond this maximum length, contribute nothing
to the detection. In that case, the signal-to-noise ratio for the detection of
a maximally-curved 1-degree track would be reduced by a factor of two. In
practice, the loss would be smaller than this. Also, this is the worst-case
curvature. Most tracks should have much less deviation from linearity.

This analysis suggests that track curvature should not have a significant
impact for imagers having a field of view less than one degree. However, the
deviation increases quadratically with track length, so it is clear that
curvature could be an issue for wide-field cameras having a larger field of
view.

A second important source of nonlinearity is distortion within the telescope
and camera optics. This distortion needs to be corrected to a fairly tight
tolerance, on the order of 0.1% or less, in order to prevent significant
distortion of the tracks.

##  5 Discussion

It can be seen from Table 1 and Figure 1 that the completeness depends
primarily on the total signal-to-noise ratio. For a given SNR, There is no
significant variation with the width of the track or with the intensity or
flux of the track. The 50% completeness limit corresponds to $SNR\sim 5.5$,
and essentially all objects with $SNR\geq 8$ are detected. Larger images have
more distinguishable positions and angles, so more opportunities for noise
events to rise above the detection threshold. However this is compensated by
longer tracks, which give a higher SNR for true events. In our simulations,
tracks having a total $SNR\sim 20$ or less are generally invisible to the eye.
Yet they are readily detected by the algorithm described here.

The photometric errors found in these simulations are consistent with the
expected Poisson noise. As a check for systematic errors, several simulations
were run with very bright tracks, for which the Poisson noise was negligible.
These had photometric errors that were less than 0.01 magnitudes
(approximately 1%) when 51-point interpolaton was used and 0.05 magnitudes for
7-point interpolation.

No attempt was made to simulate stars, which need to be masked before running
this algorithm on astronomical images. Masking of stars can be done
automatically, and the effect on streaks is generally quite small unless the
field is very crowded (Zimmer et al. ,, 2013).

The method is most sensitive for the detection of tracks that completely cross
the image. Tracks that end within the image can also be detected but with
lower efficiency. The signal-to-noise ratio for such objects is proportional
to the track length. Tracks that cross the image are suboptimal for the
estimation of orbital and photometric parameters because the angular speed and
intrinsic luminosity of the object cannot be determined unless both endpoints
are contained within the image. Nevertheless, the relative brightness of the
track, its orientation, and the time of passage all provide useful
information.

The algorithm was implemented and tested on a computer having a 2.2 GHz 64-bit
processor. Execution time was $\sim 3$ s for a 1K$\times$1K image, $\sim 10$ s
for a 2K$\times$2K image and $\sim 30$ s for a 4K$\times$4K image. This speed
could be increased by parallelizing the code in order to take advantage of
multiple cores, however, it is already fast enough for many applications. For
example, it could be useful for surveys such as that planned for the
International Liquid Mirror Telescope, which will regularly scan the sky at
$29.36^{\circ}$ N latitude, acquiring a 16-Mpixel image every 102 seconds
(Surdej et al. ,, 2006). Such images can be scanned for streaks in near-real
time in order to acquire statistics on orbital debris populations.

## Acknowledgements

I am grateful to Prof. J. Surdej for many discussions and a careful reading of
the manuscript, and to the University of Liège for hospitality during a
sabbatical visit. This work was supported by grants from the Natural Sciences
and Engineering Research Council of Canada and the Fonds de la Recherche
Scientifique (FNRS) of Belgium, R.FNRS.4164-J-F-G. The hospitality of NAOC and
support from the Chinese Academy of Sciences, via the CAS President’s
International Fellowship Initiative, 2017VMA0013, is also gratefully
acknowledged.

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##  Appendix A Source code for the fast Radon transform employed here.

#—————————————————————————— # Fast 2-d Radon transform. # # Revision #
2018-04-20 # # Copyright (c) 2018 by P. Hickson. This code may be freely
copied and modified # for non-commercial applications only. This copyright
notice must be retained # in all copies and derivatives. # def
radon(img,theta=np.arange(-90,90,1),order=7): ””” The function computes the 2d
Radon transform of a square image. It uses the Fourier slice theorem. The 1-d
inverse Fourier transform of a slice through the 2-d Fourier transform of the
image is equal to the projection (sum) of the image along the slice axis. In
this implementation, r is the lesser of the distance along the x and y axes,
from the centre of the image to the streak. It is not the perpendicular
distance as with the standard Radon transform. This choice was made to
increase speed and reduce memory requirements. ”””

# Define the slice function. def im_slice(im,th,order): ””” Return a slice
through the origin along the specified direction. th is the angle between the
slice and the x axis, i.e. th = 0 is a horizontal slice y = 0). The origin is
located at (nx//2,ny//2). The range of th is -90 to +90 degrees.

The parameter order is the order of the sinc interpoltion. Increase it for
more accurate photometry, decrease it for faster speed. ””” ni = int(order)
ni2 = ni//2 ny,nx = im.shape x0 = nx//2 y0 = ny//2 tr = radians(th) ct =
cos(tr) st = sin(tr) sl = np.zeros(nx,dtype=complex) x = np.arange(nx) y =
np.arange(ny) if th ¡ -45 or th ¿ 45: xy = x0+(y-y0)*ct/st xmin = (xy-
ni2).astype(int) for i in range(ni): xv = np.clip(xmin+i,0,nx-1) sl +=
im[y,xv]*np.sinc(xy-xv) else: yx = y0+(x-x0)*st/ct ymin = (yx-ni2).astype(int)
for i in range(ni): yv = np.clip(ymin+i,0,ny-1) sl += im[yv,x]*np.sinc(yx-yv)
return sl

# Find the size of the image. ny,nx = img.shape if nx != ny: print(’image must
be square’) exit(0)

# Create an image to hold the output nt = len(theta) rt = np.zeros([nx,nt])

# Take the 2-d FFT of the image. fimg =
np.fft.fftshift(np.fft.fft2(np.fft.fftshift(img)))

# For each angle, extract the slice, then take the 1-d inverse FFT. for i in
range(nt): tr = radians(theta[i]) sl = im_slice(fimg,theta[i],order) rt[:,i] =
np.real(np.fft.ifftshift(np.fft.ifft(np.fft.ifftshift(sl)))) if theta[i] ¡
-45: rt[:,i] = rt[::-1,i]

return rt


25mm25mm30mm30mm

# Sweedler duality for Hom-algebras and Hom-modules

Jiacheng Sun  College of Sciences, Northeastern University, Shenyang, Liaoning
110004, China<EMAIL_ADDRESS>,  Shuanhong Wang  Shing-Tung Yau Center,
School of Mathematics, Southeast University, Nanjing 210096, China
<EMAIL_ADDRESS>,  Chi Zhang  College of Sciences, Northeastern
University, Shenyang, Liaoning 110004, China<EMAIL_ADDRESS>and
Haoran Zhu  College of Sciences, Northeastern University, Shenyang, Liaoning
110004, China<EMAIL_ADDRESS>
###### Abstract.

The construction of Sweedler duality is an important tool in the theory of
Hopf algebras over a field, which is a right adjoint to the dual algebra
functor. In this paper, we study the Sweedler duality of Hom-algebras and
their Hom-modules. We delve into the structure of Hom-coalgebras and derive
the linear morphisms associated with them. Additionally, as an application, we
present the (right) Hom-(co)module morphisms under the Sweedler duality.

###### Key words and phrases:

Sweedler duality, Hom-algebra, Hom-coalgebra, Hom-module, Hom-comodule

###### 2020 Mathematics Subject Classification:

Primary 17A30; Secondary 17A60, 17D30

##  1\. Introduction

Algebraic deformation theory was first studied systematically and in depth by
Gerstenhaber [5] and is now one of the most important branches of algebra. In
recent years, the introduction of another type of algebraic deformation
algebra, namely Hom-algebra, has attracted the attention of many algebraists.
In 2006, Hartwing, Larsson and Silvestrov [7] introduced Hom-Lie algebras to
describe Q-transformations in Witt and Virasoro algebras. In 2008, Makhlouf
and Silvestrov [10] extended the structure of Hom-Lie algebras to quasi-Hom-
Lie algebras and quasi-Lie algebras where they provided a different method for
constructing Hom-Lie algebras, which is based on the process of generalising
the construction of Lie algebras from associative algebras by defining the Lie
bracket as a commutator. Building on this foundation, they [11][15]
generalized the general (co)algebraic structure to Hom-(co)algebras which
essentially extends the concept of associative algebras by deforming the
associativity law into an $\alpha$-associativity condition. Later, Yau
[17][18][19][21] proposed the concepts of Hom-(co)modules, Hom-Hopf modules,
and the duality of Hom-algebras, thus enriching the field of Hom-algebra
theory. As an emerging field within algebra, (Bi)Hom-algebra is intimately
connected to category theory (See,e.g.,[22][14]), combinatorics
(See,e.g.,[12]), braided group representations (See,e.g.,[9]), and quantum
algebras (See,e.g.,[13]), producing a wealth of research outcomes in recent
years.

The exponentials of infinitesimal characters and their convolutions are known
to produce characters with elegant combinatorial counterparts [2]. In a Hopf
algebra, the set of characters forms a group that acquires a natural structure
of an (infinite-dimensional) Lie group under specific conditions, which are
commonly met in practice, such as grading. All these elements pertain to the
largest dual one can consider for a Hopf algebra, known as Sweedler duality.
This duality is, in a sense, connected to finite state automaton theory [3]
and algebraic combinatorial theory [16].

In the finite dimensional picture, the structure of the Hom-coalgebra is
obtained from the Hom-algebra by an universal dual approach, and such a
conclusion is known. This naturally leads to the question that whether the
above structure can also be achieved in the infinite dimensional case? In
other words, is it possible to realize the induction of Hom-algebras to Hom-
coalgebras (resp. Hom-modules to Hom-comodules) by constructing a duality
under infinite dimensions (as we called it above the Sweedler duality)?

In this paper, we investigate the Sweedler duality of Hom-algebras and their
induction of Hom-coalgebras in general. We present the new structure of the
Hom-coalgebra that is induced by the Hom-algebra within the context of
Sweedler duality. Moreover, we extend our findings to Hom-modules and
establish the connection between Hom-modules and Hom-comodules.

The layout of this paper is as follows. In Section 2, we review some basic
concepts and lemmas that discuss the relationship between Hom-algebra and Hom-
coalgebra. The structure of the Hom-coalgebra under Sweedler duality is
obtained in Theorem 2. A morphism of Hom-coalgebra is derived using the
condition above in Corollary 2.10. In Section 3, we present a right Hom-
comodule of Hom-coalgebra, which is induced by the right-module of Hom-algebra
under Sweedler duality, as shown in Theorem 4. The (right) Hom-comodule
morphism is also provided in Theorem 5.

Throughout this paper, for the mapping role of elements, we will note them
like this $f(v):=\langle f,v\rangle$ for all $f\in G^{*}$, $v\in G$. More
details about notations can be found in [5]. Throughout, we will assume that
the characteristic of the field $\mathbb{K}$ is equal to zero.

##  2\. The Sweedler duality for Hom-algebras

In this section, we will first provide a brief overview of the fundamental
concepts related to the structures of Hom-(co)algebra and the
interrelationship between Hom-(co)algebras (see Definition 2.3 in [20]).
Subsequently, we introduce the concept of Sweedler duality, which differs from
the conventional duality (see Chapter 6 in [5]).

######  Definition 2.1.

A Hom-algebra is a triple $(G,\mu,\alpha)$, where $G$ is a $\mathbb{K}$-linear
space, $\mu:G\otimes G\rightarrow G$ is a bilinear multiplication with
$\mu(g\otimes g^{\prime})=gg^{\prime}$. For simplicity when there is no
confusion, we will omit the sign $\mu$. $\alpha:G\rightarrow G$ is a
$\mathbb{K}$-linear space homomorphism. Otherwise it satisfies the following
conditions

(2.1) $\alpha(g)(hk)=(gh)\alpha(k),\quad\text{ (Hom-associativity)}$ (2.2)
$\alpha(hk)=\alpha(h)\alpha(k).\quad\text{ (multiplicativity)}$

They can be represented by the following commutative diagrams:

$\textstyle{G\otimes G\otimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\otimes\mu}$$\textstyle{G\otimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu\otimes\alpha}$$\scriptstyle{\mu}$$\textstyle{G\otimes
G}$$\textstyle{G,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$
$\textstyle{G\otimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{G\otimes
G\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha\otimes\alpha}$$\scriptstyle{\mu}$$\textstyle{G}$$\textstyle{G.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$

In this context, we will refer to a Hom-algebra $(G,\mu,\alpha)$ as $G$. Let
us consider two Hom-algebras, $(G,\mu,\alpha)$ and
$(G^{\prime},\mu^{\prime},\alpha^{\prime})$, both defined over a field
$\mathbb{K}$. Let $f:G\to G^{\prime}$ be a $\mathbb{K}$-linear map. The linear
map $f$ is said to be a morphism of Hom-algebras if it satisfies the following
conditions:

(2.3) $\displaystyle\mu^{\prime}\circ(f\otimes f)=f\circ\mu,\quad\quad
f\circ\alpha=\alpha^{\prime}\circ f.$

In other words, $f$ is a morphism of Hom-algebras if it commutes with the
multiplication operations and respects the Hom-structures of $G$ and
$G^{\prime}$.

######  Definition 2.2.

Let $(H,\mu,\alpha)$ be a subspace of $G$. A Hom-subalgebra of G is defined as
a subspace of G that satisfies the following conditions:

$\displaystyle bc\in H,\quad\alpha(b)\in H,$

for any $b,c\in H$. In particular, if $ab\in H$ and $ba\in H$, for any $a\in
G$ and $b\in H$. The Hom-algebra $G$ is equipped with an ideal, which we
denote by $H$.

######  Definition 2.3.

A Hom-coalgebra is a triple $(C,\Delta,\beta)$, where $C$ is a
$\mathbb{K}$-vector space, $\Delta:C\rightarrow C\otimes C$ is a linear map
with notation $\Delta(c):=\sum_{(c)}c_{(1)}\otimes c_{(2)}$ for all $c\in C$.
$\beta:C\rightarrow C$ is a $\mathbb{K}$-linear space homomorphism. And it
satisfies

(2.4)
$(\beta\otimes\Delta)\circ\Delta=(\Delta\otimes\beta)\circ\Delta,\quad\text{
(Hom-coassociativity)}$ (2.5)
$\Delta\circ\beta=(\beta\otimes\beta)\circ\Delta.\quad\text{
(multiplicativity)}$

They can be expressed by the following commutative diagrams:

$\textstyle{C\otimes C\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta\otimes\beta}$$\textstyle{C\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\otimes\Delta}$$\scriptstyle{\Delta}$$\textstyle{C\otimes
C}$$\textstyle{C,\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$
$\textstyle{C\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Delta}$$\textstyle{C\otimes
C\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta\otimes\beta}$$\scriptstyle{\Delta}$$\textstyle{C}$$\textstyle{C.\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$

In a similar manner, we will refer to a Hom-coalgebra $(C,\Delta,\beta)$ as
$C$. A morphism of Hom-coalgebras is a linear map of the underlying
$\mathbb{K}$-linear spaces that commutes with the twisting maps and the
comultiplications.

Now, we will give some lemmas and describe their functions. Lemma 2.4 (see
[10]) describes the dual structure of the Hom-algebra. Lemma 2.5 (see [11])
introduces the relationship between the morphism of Hom-algebras and its
duality. Lemma 2.6 (see [4]) gives the quotient space properties induced by
subspaces under linear spaces. Lemma 2.7 (see [11]) explain the fundamental
isomorphism theorem of the Hom-algebra.

######  Lemma 2.4.

Let $(G,\mu,\alpha)$ is a finite-dimensional Hom-algebra, then
$(G^{*},\mu^{*},\alpha^{*})$ is a Hom-coalgebra where
$\mu^{*}:G^{*}\rightarrow G^{*}\otimes G^{*}$,
$\alpha^{*}(a^{*})=a^{*}\alpha$.

######  Lemma 2.5.

Let $(G,\mu,\alpha)$ and
$(G^{{}^{\prime}},\mu^{{}^{\prime}},\alpha^{{}^{\prime}})$ be two finite-
dimensional Hom-algebras, then $f:G\rightarrow G^{{}^{\prime}}$ is the
morphism of Hom-algebras if and only if $f^{*}:G^{{}^{\prime}*}\rightarrow
G^{*}$ is the morphism of Hom-coalgebras.

######  Lemma 2.6.

Let $A$ and $B$ be two $\mathbb{K}$-linear spaces, $I$ and $J$ be the
subspaces of $A$ and $B$ respectively. If there exists a map $\pi:A\otimes
B\rightarrow A/I\otimes B/J$, we have that $\operatorname{Ker}(\pi)=A\otimes
J+B\otimes I$ and $(A\otimes B)/(A\otimes J+B\otimes I)\cong A/I\otimes B/J$.

######  Lemma 2.7.

Let $(G,\mu,\alpha)$ and
$(G^{{}^{\prime}},\mu^{{}^{\prime}},\alpha^{{}^{\prime}})$ be two Hom-algebras
over $\mathbb{K}$, $f:G\rightarrow G^{{}^{\prime}}$ is a morphism of Hom-
algebras, $H$ is the ideal of Hom-algebra of $G$, the map $\pi:G\rightarrow
G/H$ is a regular full homomorphism. If
$\operatorname{Ker}\pi\subseteq\operatorname{Ker}f$, then there exists a
unique morphism of Hom-algebras $g:G/H\rightarrow G^{{}^{\prime}}$, so
$g\circ\pi=f$.

In order to investigate the structure of Hom-coalgebra induced by the infinite
dimensional Hom-algebra and the correspondence of the finite-codimensional
ideals in the morphism of Hom-algebras, we will give the definition of the
Sweedler duality in Definition 2.8 and the lemma 2.9 below.

######  Definition 2.8.

The Sweedler duality of $G$ is based on the universal duality $G^{*}$ as
follows:

$G^{\circ}:=\\{f\in G^{*}\mid\operatorname{Ker}f\supseteq J,\text{ $J$ is a
finite-codimensional ideal of }G\\}.$

######  Lemma 2.9.

Let $(G,\mu,\alpha)$ and
$(G^{{}^{\prime}},\mu^{{}^{\prime}},\alpha^{{}^{\prime}})$ be two Hom-algebras
over $\mathbb{K}$, $f:G\rightarrow G^{{}^{\prime}}$ is a morphism of Hom-
algebras. If $J$ is a finite-codimensional ideal of $G^{{}^{\prime}}$, then
$f^{-1}(J)$ is a finite-codimensional ideal of $G$.

###### Proof.

In accordance with the equation (2.3), we obtain the following which holds for
any $x\in f^{-1}(J)$ and $a\in G$:

$\displaystyle f(x)\in J\quad\text{and}\quad f(ax)=f(a)f(x)\in J.$

Then $ax\in f^{-1}(J)$. Similarly, we can show that $xa\in f^{-1}(J)$. For any
$x\in f^{-1}(J)$, it can be shown that $f(x)\in J$ and
$\alpha^{\prime}(f(x))\in J$.

Observe that $f$ is a morphism of Hom-algebras. Consequently, we have

$f\circ\alpha(x)=\alpha^{\prime}\circ f(x)\in J,$

there by establishing that $\alpha(x)\in f^{-1}(J)$. We then proceed with
$\alpha(f^{-1}(J))\subseteq f^{-1}(J)$. So $f^{-1}(J)$ is the ideal of Hom-
algebra $G$.

We now demonstrate that $f^{-1}(J)$ is finite-codimensional. Let $g:G\to
G^{\prime}/J$ be a map defined as follows: for any $a\in G$, $a=a_{1}+a_{2}$
where $a_{1}\in f^{-1}(J)$, we have $g(a)=f(a_{2})+J$. It is evident that
$\operatorname{Ker}(g)=f^{-1}(J)$. Then we can induce the injection
$\widetilde{g}:G/f^{-1}(J)\rightarrow G^{{}^{\prime}}/J$ ($b+f^{-1}(J)\mapsto
f(b)+J$), so we have

$\displaystyle dim(G/f^{-1}(J))\leq dim(G^{{}^{\prime}}/J)<+\infty$

which implies that the dimension of $G/f^{-1}(J)$ is finite. Consequently,
$f^{-1}(J)$ is a finite-codimensional ideal of $G$. ∎

Inspired by Liu’ construction [8] of basic algebras, it is sufficient for us
to construct the structure of Hom-coalgebras via Sweedler duality now.

######  Theorem 1.

$G^{\circ}$ is the subspace of $G^{*}$.

###### Proof.

It is straightforward to verify that $G^{\circ}$ is closed under
$\mathbb{K}$-multiplication. We proceed to show that $G^{\circ}$ is also
closed under addition.

For any $g,h\in G^{\circ}$, let $J$ and $H$ be two finite-codimensional ideals
of $G$ such that $\operatorname{Ker}g\supseteq J$ and
$\operatorname{Ker}h\supseteq H$. It is clear that $J\cap H$ is an ideal of
the Hom-algebra and that $\operatorname{Ker}(g+h)\supseteq J\cap H$.

The following proof explains that $J\cap H$ is finite-codimensional. Since
$H/(J\cap H)\cong(J+H)/J\subseteq G/J$, the quotient $H/(J\cap H)$ is a
finite-dimensional linear space. Furthermore, since $G/H\cong(G/(J\cap
H))/(H/(J\cap H))$, the finite dimension of $G/H$ implies that $dim(G/(J\cap
H))=dim(G/H)+dim(H/(J\cap H))$ is also finite. Consequently, $J\cap H$ is a
finite-codimensional ideal of $G$. ∎

######  Theorem 2.

Let $(G,\mu,\alpha)$ be a Hom-algebra. Then the Sweedler duality of $G$ is a
Hom-coalgebra $(G^{\circ},\Delta,\alpha^{\circ})$, where
$\Delta=\mu^{*}|_{G^{\circ}}$, $\alpha^{\circ}=\alpha^{*}|_{G^{\circ}}$ and
$\mu^{*}:G^{*}\rightarrow(G\otimes G)^{*}$.

###### Proof.

Let $f\in G^{\circ}$. We show that $\Delta(f)\in G^{\circ}\otimes G^{\circ}$.
There exists a finite-codimensional ideal $J$ of $G$ such that $f(J)=0$. Then
we do the formal arithmetic

$\displaystyle\Delta(f)(J\otimes G+G\otimes J)=f\circ\mu(J\otimes G+G\otimes
J)=0,$

since $\mu(J\otimes G)\subseteq J$ and $\mu(G\otimes J)\subseteq J$. Let a map
$\pi\colon G\otimes G\to G\otimes G/(J\otimes G+G\otimes J)$, then we have
$\operatorname{Ker}(\pi)=J\otimes G+G\otimes
J\subseteq\operatorname{Ker}(f\mu)$. By Lemma 2.7, this induces a unique
morphism $\overline{f\mu}\colon G\otimes G/(J\otimes G+G\otimes
J)\to\mathbb{K}$ satisfying $\overline{f\mu}\circ\pi=f\circ\mu$. That is to
say that $\overline{f\mu}\in(G\otimes G/(J\otimes G+G\otimes J))^{*}$.

According to Lemma 2.6, we have $G\otimes G/(J\otimes G+G\otimes
J)\cong(G/J)\otimes(G/J)$ and $G/J$ is finite, it is easy to see that

$\displaystyle\overline{f\mu}\in(G/J)^{*}\otimes(G/J)^{*}.$

Let
$\overline{f\mu}=\sum_{i,j=1}^{n}k_{ij}\overline{e_{i}}^{*}\otimes\overline{e_{j}}^{*}$,
where $\overline{e_{i}}^{*}\in(G/J)^{*}$ is the duality base elements
corresponding to a canonical map $\tilde{\pi}\colon G\to G/J$. So for any
$a,b\in G$, we have

$\displaystyle\langle f\circ\mu,a\otimes b\rangle$
$\displaystyle=\langle\overline{f\mu}\circ\pi,a\otimes b\rangle$
$\displaystyle=\langle\overline{f\mu},\pi(a\otimes b)\rangle$
$\displaystyle=\langle\overline{f\mu},\tilde{\pi}(a)\otimes\tilde{\pi}(b)\rangle$
$\displaystyle=\sum_{i,j=1}^{n}k_{ij}\langle\overline{e}_{i}^{*},\tilde{\pi}(a)\rangle\langle\overline{e}_{j}^{*},\tilde{\pi}(b)\rangle.$

Denote $e_{i}^{*}=\overline{e}_{i}^{*}\circ\tilde{\pi}$ Since
$\operatorname{Ker}(e_{i}^{*})\supseteq J$, we have $e_{i}^{*}\in G^{\circ}$.
Then it is straightforward to see

$\displaystyle\Delta(f)=f\circ\mu=\sum\limits_{i,j=1}^{n}k_{ij}e_{i}^{*}\otimes
e_{j}^{*}.$

Therefore, we conclude that $\Delta(f)\in G^{\circ}\otimes G^{\circ}$.

For any $g\in G^{\circ}$, there exists a finite-codimensional ideal $J$ of $G$
such that $\operatorname{Ker}(g)\supseteq J$. Then, since $\alpha(J)\subseteq
J$ and $g(J)=0$, we have

$\displaystyle\langle\alpha^{\circ}(g),J\rangle=\langle\alpha^{*}(g),J\rangle=\langle
g,\alpha(J)\rangle=0.$

We therefore have $\operatorname{Ker}(\alpha^{\circ}(g))\supseteq J$, which
implies that $\alpha^{\circ}(g)\in G^{\circ}$. So we prove that
$\alpha^{\circ}(G^{\circ})\subseteq G^{\circ}$.

Then we will prove that
$\Delta\circ\alpha^{\circ}=(\alpha^{\circ}\otimes\alpha^{\circ})\circ\Delta$.
Using the equation (2.2), for any $c^{*}\in G^{\circ}$ and $x,y\in G$,

$\displaystyle\langle\Delta\circ\alpha^{\circ}(c^{*}),x\otimes y\rangle$
$\displaystyle=\langle c^{*},\alpha\circ\mu(x\otimes y)\rangle$
$\displaystyle=\langle c^{*},\mu\circ(\alpha\otimes\alpha)(x\otimes y)\rangle$
$\displaystyle=\langle\Delta(c^{*}),\alpha(x)\otimes\alpha(y)\rangle$
$\displaystyle=\sum_{(c^{*})}\langle c^{*}_{(1)}\otimes
c^{*}_{(2)},\alpha(x)\otimes\alpha(y)\rangle$
$\displaystyle=\sum_{(c^{*})}\langle\alpha^{\circ}(c^{*}_{(1)}),x\rangle\langle\alpha^{\circ}(c^{*}_{(2)}),y\rangle$
$\displaystyle=\langle(\alpha^{\circ}\otimes\alpha^{\circ})(\sum_{(c^{*})}c^{*}_{(1)}\otimes
c^{*}_{(2)}),x\otimes y\rangle$
$\displaystyle=\langle(\alpha^{\circ}\otimes\alpha^{\circ})\circ\Delta(c^{*}),x\otimes
y\rangle.$

Therefore we have
$\Delta\circ\alpha^{\circ}=(\alpha^{\circ}\otimes\alpha^{\circ})\circ\Delta$.

It is sufficient to verify that
$(\alpha^{\circ}\otimes\Delta)\circ\Delta=(\Delta\otimes\alpha^{\circ})\circ\Delta.$
For any $c^{*}\in G^{\circ}$ and $x,y,z\in G$, we have

$\displaystyle\langle(\alpha^{\circ}\otimes\Delta)\circ\Delta(c^{*}),x\otimes
y\otimes z\rangle$
$\displaystyle=\langle(\alpha^{\circ}\otimes\Delta)\circ(\sum_{(c^{*})}c^{*}_{(1)}\otimes
c^{*}_{(2)}),x\otimes y\otimes z\rangle$
$\displaystyle=\sum_{(c^{*})}\langle\alpha^{\circ}(c^{*}_{(1)})\otimes\Delta(c^{*}_{(2)}),x\otimes
y\otimes z\rangle$
$\displaystyle=\sum_{(c^{*})}\langle\alpha^{\circ}(c^{*}_{(1)}),x\rangle\langle\Delta(c^{*}_{(2)}),y\otimes
z\rangle$ $\displaystyle=\sum_{(c^{*})}\langle
c^{*}_{(1)},\alpha(x)\rangle\langle c^{*}_{(2)},yz\rangle$
$\displaystyle=\sum_{(c^{*})}\langle c^{*}_{(1)}\otimes
c^{*}_{(2)},\alpha(x)\otimes yz\rangle$
$\displaystyle=\langle\Delta(c^{*}),\alpha(x)\otimes yz\rangle$
$\displaystyle=\langle c^{*},\alpha(x)yz\rangle$

Similarly, we can get

$\displaystyle\langle(\Delta\otimes\alpha^{\circ})\circ\Delta(c^{*}),x\otimes
y\otimes z\rangle$ $\displaystyle=\langle c^{*},xy\alpha(z)\rangle.$

Due to the Hom-associativity (2.1) of $G$, we have $\alpha(x)yz=xy\alpha(z)$.
Thus
$(\alpha^{\circ}\otimes\Delta)\circ\Delta=(\Delta\otimes\alpha^{\circ})\circ\Delta$.
Therefore we prove this theorem. ∎

In the absence of specific instructions, we will examine the properties of
Hom-coalgebra $(G^{\circ},\Delta,\alpha^{\circ})$, the Sweedler duality of $G$
as outlined in Theorem 2. In light of the aforementioned considerations, we
proceed to derive the morphism of Hom-coalgebras as follows.

######  Corollary 2.10.

Let $(G,\mu,\alpha)$ and
$(G^{{}^{\prime}},\mu^{{}^{\prime}},\alpha^{{}^{\prime}})$ be two Hom-
algebras, $(G^{\circ},\Delta,\alpha^{\circ})$ and
$(G^{\prime\circ},\Delta^{\prime},\alpha^{\prime\circ})$ two corresponding
Sweedler dualities. The map $f:G\rightarrow G^{{}^{\prime}}$ is a morphism of
Hom-algebras. Note that the map
$f^{\circ}=f^{*}|_{G^{\prime\circ}}:G^{\prime\circ}\rightarrow G^{*}$. Thus,
$f^{\circ}$ is a morphism of Hom-coalgebras.

###### Proof.

Our primary objective is to demonstrate that
$f^{\circ}(G^{\prime\circ})\subseteq G^{\circ}$. For any $b^{*}\in
G^{\prime\circ}$, since $J$ is a finite-codimensional ideal of $G^{\prime}$
and is included in $\operatorname{Ker}(b^{*})$, we can utilize Lemma 2.9 to
deduce that $f^{-1}(J)$ constitutes a finite-codimensional ideal of the Hom-
algebra $G$.

Notice that

$\displaystyle\langle f^{\circ}(b^{*}),f^{-1}(J)\rangle$
$\displaystyle=\langle b^{*},f\circ f^{-1}(J)\rangle$ $\displaystyle=\langle
b^{*},J\rangle$ $\displaystyle=0,$

which shows that $\operatorname{Ker}(f^{\circ}(b^{*}))\supseteq f^{-1}(J)$. In
other words, $f^{\circ}(b^{*})\in G^{\circ}$. Consequently, it can be
concluded that $f^{\circ}(G^{\prime\circ})\subseteq G^{\circ}$.

Using the equation (2.3), for any $b^{*}\in G^{\prime\circ}$ and $a,c\in
G^{\circ}$, we have

$\displaystyle\langle\Delta\circ f^{\circ}(b^{*}),a\otimes c\rangle$
$\displaystyle=\langle b^{*},f\circ\mu(a\otimes c)\rangle$
$\displaystyle=\langle b^{*},\mu^{\prime}\circ(f\otimes f)(a\otimes c)\rangle$
$\displaystyle=\langle\Delta^{\prime}(b^{*}),f(a)\otimes f(c)\rangle$
$\displaystyle=\sum_{(b^{*})}\langle b^{*}_{(1)}\otimes
b^{*}_{(2)},f(a)\otimes f(c)\rangle$ $\displaystyle=\sum_{(b^{*})}\langle
b^{*}_{(1)},f(a)\rangle\langle b^{*}_{(2)},f(c)\rangle$
$\displaystyle=\sum_{(b^{*})}\langle f^{\circ}(b^{*}_{(1)}),a\rangle\langle
f^{\circ}(b^{*}_{(2)}),c\rangle$ $\displaystyle=\langle(f^{\circ}\otimes
f^{\circ})(\sum_{(b^{*})}b^{*}_{(1)}\otimes b^{*}_{(2)}),a\otimes c\rangle$
$\displaystyle=\langle(f^{\circ}\otimes
f^{\circ})\circ\Delta^{\prime}(b^{*}),a\otimes c\rangle.$

Thus, we obtain $\Delta\circ f^{\circ}=(f^{\circ}\otimes
f^{\circ})\circ\Delta^{\prime}$.

For any $b^{*}\in G^{\prime\circ}$ and $a\in G$, we have

$\displaystyle\langle\alpha^{\circ}\circ f^{\circ}(b^{*}),a\rangle$
$\displaystyle=\langle b^{*},f\circ\alpha(a)\rangle$ $\displaystyle=\langle
b^{*},\alpha^{{}^{\prime}}\circ f(a)\rangle$ $\displaystyle=\langle
f^{\circ}\circ\alpha^{\prime\circ}(b^{*}),a\rangle.$

Therefore, it follows that $\alpha^{\circ}\circ
f^{\circ}=f^{\circ}\circ\alpha^{\prime\circ}$, which is the desired result. ∎

##  3\. The Sweedler duality for Hom-module

In this section, we present the structure of Hom-(co)modules and the morphisms
of Hom-(co)modules (see Definition 4.1 in [20]). Subsequently, the Sweedler
duality of Hom-module is discussed (see [8]).

######  Definition 3.1.

Let $(G,\mu,\alpha)$ be a Hom-algebra. A right Hom-module of $G$ is a triple
$(M,\psi,\gamma)$ where $M$ is a $\mathbb{K}$-linear space, a map
$\psi:M\otimes G\rightarrow M$ with $\psi(m\otimes g)=m\cdot g$ and a map
$\gamma:M\rightarrow M$ satisfying:

$\psi\circ(\psi\otimes\alpha)=\psi\circ(\gamma\otimes\mu),\quad\text{(Hom-
associativity)}$
$\psi\circ(\gamma\otimes\alpha)=\gamma\circ\psi.\quad\quad\text{(multiplicativity)}$

We will refer to a right Hom-module $(M,\psi,\gamma)$ as $M$. Let
$(M,\psi,\gamma)$ and $(N,\tilde{\psi},\tilde{\gamma})$ be two right Hom-
modules of $G$. A map $\sigma:M\rightarrow N$ is said to be a morphism of
right Hom-modules when the following conditions hold:

(3.1)
$\displaystyle\sigma\circ\psi=\tilde{\psi}\circ(\sigma\otimes\alpha),\quad\quad\tilde{\gamma}\circ\sigma=\sigma\circ\gamma.$

######  Definition 3.2.

Let $(C,\Delta,\beta)$ be a Hom-coalgebra. A right Hom-comodule of $C$ is a
triple $(A,\phi,\epsilon)$ where $A$ is a $\mathbb{K}$-linear space, a map
$\phi:A\rightarrow A\otimes C$ with $\psi(a)=\sum_{(a)}a_{(0)}\otimes a_{(1)}$
and a map $\gamma:M\rightarrow M$ satisfying:

$(\phi\otimes\beta)\circ\phi=(\epsilon\otimes\Delta)\circ\phi,\quad\text{(Hom-
coassociativity)}$
$(\epsilon\otimes\beta)\circ\phi=\phi\circ\epsilon.\quad\text{(multiplicativity)}$

We will refer to a right Hom-comodule $(A,\phi,\epsilon)$ as $A$. Let
$(A,\phi,\epsilon)$ and $(A^{\prime},\phi^{\prime},\epsilon^{\prime})$ be two
right Hom-comodule of $C$. A map $f:A\rightarrow A^{\prime}$ is called a
morphism of right Hom-comodules if it satisfies:

$\phi^{\prime}\circ
f=(f\otimes\beta)\circ\phi,\quad\quad\epsilon^{\prime}\circ f=f\circ\epsilon.$

######  Definition 3.3.

The Sweedler duality of the right Hom-module $M$ based on the universal
duality $M^{*}$ is defined by

$M^{\circ}:=\\{f\in M^{*}\mid f(M\cdot I)=0,I\text{ is a finite-codimensional
ideal of G }\\}.$

Then we will give the isomorphism in dual spaces (see [8]) and the structure
of Hom-comodule induced by the universal duality as follows.

######  Lemma 3.4.

Let $U$ and $V$ be two $\mathbb{K}$-linear space. Denote $\rho:U^{*}\otimes
V^{*}\rightarrow(U\otimes V)^{*}$ with corresponding rule $\rho(f\otimes
g)(x\otimes y)=f(x)g(y)$, for any $f\in U^{*}$, $g\in V^{*}$, $x\in U$ and
$y\in V$. If $\operatorname{dim}U<+\infty$ or $\operatorname{dim}V<+\infty$,
$\rho$ is the isomorphism of the linear spaces.

######  Lemma 3.5.

Let $(G,\mu,\alpha)$ be a finite-dimensional Hom-algebra and $(M,\psi,\gamma)$
be a right Hom-module of $G$. Then, $(M^{*},\psi^{*},\gamma^{*})$ is the right
Hom-comodule of the Hom-coalgebra $(G^{*},\mu^{*},\alpha^{*})$, where
$\psi^{*}:M^{*}\rightarrow M^{*}\otimes G^{*}$ and
$\gamma^{*}:M^{*}\rightarrow M^{*}$.

###### Proof.

By applying Lemma 2.4, we can get $(G^{*},\mu^{*},\alpha^{*})$ is a Hom-
coalgebra. Then we want to show that
$\psi^{*}\circ\gamma^{*}=(\gamma^{*}\otimes\alpha^{*})\circ\psi^{*}$. For any
$m^{*}\in M$, $m\in M$ and $g\in G$, we have

$\displaystyle\langle\psi^{*}\circ\gamma^{*}(m^{*}),m\otimes g\rangle$
$\displaystyle=\langle m^{*},\gamma\circ\psi(m\otimes g)\rangle$
$\displaystyle=\langle m^{*},\psi\circ(\gamma\otimes\alpha)(m\otimes
g)\rangle$
$\displaystyle=\langle\psi^{*}(m^{*}),\gamma(m)\otimes\alpha(g)\rangle$
$\displaystyle=\sum_{(m^{*})}\langle m^{*}_{(0)}\otimes
m^{*}_{(1)},\gamma(m)\otimes\alpha(g)\rangle$
$\displaystyle=\sum_{(m^{*})}\langle\gamma^{*}(m^{*}_{(0)}),m\rangle\langle\alpha^{*}(m^{*}_{(1)}),g\rangle$
$\displaystyle=\sum_{(m^{*})}\langle(\gamma^{*}\otimes\alpha^{*})(m^{*}_{(0)}\otimes
m^{*}_{(1)}),m\otimes g\rangle$
$\displaystyle=\langle(\gamma^{*}\otimes\alpha^{*})\circ\psi^{*}(m^{*}),m\otimes
g\rangle.$

Therefore we have
$\psi^{*}\circ\gamma^{*}=(\gamma^{*}\otimes\alpha^{*})\circ\psi^{*}$.

Then it is sufficient for us to verify that
$(\psi^{*}\otimes\alpha^{*})\circ\psi^{*}=(\gamma^{*}\otimes\mu^{*})\circ\psi^{*}$.
For any $a^{*}\in M^{*}$ and $x\in M,y,z\in G$,

$\displaystyle\langle(\psi^{*}\otimes\alpha^{*})\circ\psi^{*}(a^{*}),x\otimes
y\otimes z\rangle$
$\displaystyle=\langle(\psi^{*}\otimes\alpha^{*})(\sum_{(a^{*})}a^{*}_{(0)}\otimes
a^{*}_{(1)}),x\otimes y\otimes z\rangle$
$\displaystyle=\sum_{(a^{*})}\langle\psi^{*}(a^{*}_{(0)}),x\otimes
y\rangle\langle\alpha^{*}(a^{*}_{(1)}),z\rangle$
$\displaystyle=\sum_{(a^{*})}\langle a^{*}_{(0)},\psi(x\otimes
y)\rangle\langle a^{*}_{(1)},\alpha(z)\rangle$
$\displaystyle=\langle\psi^{*}(a^{*}),(\psi\otimes\alpha)(x\otimes y\otimes
z)\rangle$ $\displaystyle=\langle a^{*},\psi\circ(\psi\otimes\alpha)(x\otimes
y\otimes z)\rangle.$

Similarly, we have
$\langle(\gamma^{*}\otimes\mu^{*})\circ\psi^{*}(a^{*}),x\otimes y\otimes
z\rangle=\langle a^{*},\psi\circ(\gamma\otimes\mu)(x\otimes y\otimes
z)\rangle$. According to the equation (3.1), It can be shown that
$(\psi^{*}\otimes\alpha^{*})\circ\psi^{*}=(\gamma^{*}\otimes\mu^{*})\circ\psi^{*}$.
This completes the proof of the lemma. ∎

Now, we will construct the structure of Hom-comodule via the Sweedler duality.

######  Theorem 3.

$M^{\circ}$ is the subspace of $M^{*}$.

###### Proof.

Firstly, it can be observed that $M^{\circ}$ is closed under
$\mathbb{K}$-multiplication. So it is sufficient for us to show that
$M^{\circ}$ is closed under addition.

Let $f,g\in M^{\circ}$, there exist finite-codimensional ideals $I$ and $J$ of
$G$ such that $f(M\cdot I)=0$ and $g(M\cdot J)=0$. By Theorem 1, we can show
that $I\cap J$ is an ideal of $G$ and $G/(I\cap J)$ is finite-dimensional.
Then, we have $(f+g)(M\cdot(I\cap J))=0$. And we may conclude that $f+g\in
M^{\circ}$, thereby completing the proof. ∎

######  Theorem 4.

Let $(M,\psi,\gamma)$ be a right Hom-module of a Hom-algebra $(G,\mu,\alpha)$
where $\alpha$ is an automorphism map of $G$, then its Sweedler duality
$(M^{\circ},\psi^{\circ},\gamma^{\circ})$ is a right Hom-comodule of
$G^{\circ}$ where $\psi^{\circ}=\psi^{*}|_{M^{\circ}}$,
$\gamma^{\circ}=\gamma^{*}|_{M^{\circ}}$ and
$\psi^{*}:M^{*}\rightarrow(M\otimes G)^{*}$.

###### Proof.

Noticing $f\in M^{\circ}$, we verify that $\psi^{\circ}(f)\in M^{\circ}\otimes
G^{\circ}$. There exists a finite-codimensional ideal $J$ of $G$ such that
$f(M\cdot J)=0$. Since $\alpha$ is the automorphism map of $G$,
$\gamma(M)\subseteq M$ and $JG\subseteq J$, we have

$\displaystyle\psi^{\circ}(f)(M\cdot J\otimes G+M\otimes J)$
$\displaystyle=\psi^{\circ}(f)(M\cdot J\otimes\alpha(G)+M\otimes J)$
$\displaystyle=f\circ\psi(M\cdot J\otimes\alpha(G)+M\otimes J)$
$\displaystyle=f(\psi(M\cdot J\otimes\alpha(G))+\psi(M\otimes J))$
$\displaystyle=f(\psi(\gamma(M)\otimes JG)+\psi(M\otimes J))$
$\displaystyle=0.$

Then we denote $\pi:M\otimes G\rightarrow M\otimes G/(M\cdot J\otimes
G+M\otimes J)$ and $\operatorname{Ker}(\pi)=M\cdot J\otimes G+M\otimes
J\subseteq\operatorname{Ker}(f\circ\psi).$

By Lemma 2.6, we can induce a unique morphism $\overline{f\psi}:M\otimes
G/(M\cdot J\otimes G+M\otimes J)\rightarrow\mathbb{K}$ satisfying
$\overline{f\psi}\circ\pi=f\circ\psi$, that is to say

$\displaystyle\overline{f\psi}\in\left(M\otimes G/(M\cdot J\otimes G+M\otimes
J)\right)^{*}.$

According to the Lemma 3.4, we can get $M\otimes G/(M\cdot J\otimes G+M\otimes
J)\cong(M/M\cdot J)\otimes(G/J)$ with finite-dimensional linear space $G/J$,
so we have

$\displaystyle\overline{f\psi}\in(M/M\cdot J)^{*}\otimes(G/J)^{*}.$

Denote two canonical maps $\pi_{1}:M\rightarrow M/M\cdot J$ and
$\pi_{2}:G\rightarrow G/J$. And then we denote
$\overline{f\psi}=\sum^{n}_{i,j=1}k_{ij}\overline{d_{i}}^{*}\otimes\overline{e_{j}}^{*}$
where $\overline{d_{i}}^{*}\in(M/M\cdot J)^{*}$,
$\overline{e_{j}}^{*}\in(G/J)^{*}$ are the dual base elements. After a similar
discussion as in Theorem 2, for any $x\in M$ and $y\in G$, we can get that

$\langle f\circ\psi,x\otimes
y\rangle=\sum_{i,j=1}^{n}k_{ij}\langle\overline{d_{i}}^{*},\pi_{1}(x)\rangle\langle\overline{e_{j}}^{*},\pi_{2}(y)\rangle.$

Denote $d_{i}^{*}=\overline{d_{i}}^{*}\circ\pi_{1}$ and
$e_{j}^{*}=\overline{e_{j}}^{*}\circ\pi_{2}$. Since $d_{i}^{*}(M\cdot J)=0$
and $\operatorname{Ker}(e_{j}^{*})\supseteq J$, we have $d_{i}^{*}\in
M^{\circ}$ and $e_{j}^{*}\in G^{\circ}$. Thus we can show that

$\psi^{\circ}(f)=f\circ\psi=\sum_{i,j=1}^{n}k_{ij}d_{i}^{*}\otimes e_{j}^{*}.$

Consequently, $\psi^{\circ}(f)\in M^{\circ}\otimes G^{\circ}$.

For any $m\in M^{\circ}$, there exists a finite-codimensional ideal $J$ of $G$
such that $m(M\cdot J)=0$. Then we have

$\displaystyle\langle\gamma^{\circ}(m),M\cdot J\rangle=\langle m,\gamma(M\cdot
J)\rangle=0,$

since $\gamma(M\cdot J)=\gamma(M)\cdot\alpha(J)\subseteq M\cdot J$. Then we
have $\gamma^{\circ}(m)(M\cdot J)=0$, which implies $\gamma^{\circ}(m)\in
M^{\circ}$. Thus, $\gamma^{\circ}(M^{\circ})\subseteq M^{\circ}$. Then by
Lemma 3.5 and the relationship between the Sweedler duality and the universal
duality we can show that
$\psi^{\circ}\circ\gamma^{\circ}=(\gamma^{\circ}\otimes\alpha^{\circ})\circ\psi^{\circ}$
and
$(\psi^{\circ}\otimes\alpha^{\circ})\circ\psi^{\circ}=(\gamma^{\circ}\otimes\Delta)\circ\psi^{\circ}$.
Hence, $M^{\circ}$ is the right Hom-comodule of $G^{\circ}$. ∎

In the subsequent proof, we will study the properties of Hom-comodule
$(M^{\circ},\psi^{\circ},\gamma^{\circ})$ proposed in Theorem 4. Based on the
above considerations, we continue to derive the morphism of Hom-comodules as
follows.

######  Theorem 5.

Let $(M,\psi,\gamma)$ and $(N,\psi^{\prime},\gamma^{\prime})$ be two right
Hom-modules over Hom-algebra $(G,\mu,\alpha)$ where $\alpha$ is an
automorphism map of $G$, $(M^{\circ},\psi^{\circ},\gamma^{\circ})$ and
$(N^{\circ},\psi^{\prime\circ},\gamma^{\prime\circ})$ be two corresponding
Sweedler dualities over $G^{\circ}$. Let $\sigma:M\rightarrow N$ be a morphism
of right Hom-modules. Note that
$\sigma^{\circ}:=\sigma^{*}|_{N^{\circ}}:N^{\circ}\rightarrow M^{*}$. Then
$\sigma^{\circ}$ is a morphism of Hom-comodules over $G^{\circ}$

###### Proof.

For all $f\in N^{\circ}$, let $J$ be a finite-codimensional ideal of $G$ such
that $f(N\cdot J)=0$. By the properties of the morphism of the right Hom-
modules, we have

$\displaystyle\sigma^{\circ}(f)(M\cdot J)$
$\displaystyle=\sigma^{\circ}(f)\circ\psi(M\otimes J)$
$\displaystyle=f(\sigma\circ\psi(M\otimes J))$
$\displaystyle=f(\psi^{\prime}\circ(\sigma\otimes\alpha)(M\otimes J))$
$\displaystyle=f(\sigma(M)\cdot\alpha(J))$ $\displaystyle=0,$

since $\sigma(M)\subseteq N$ and $\alpha(J)\subseteq J$. Hence
$\sigma^{\circ}(N^{\circ})\subseteq M^{\circ}$. Then we will prove that
$(\sigma^{\circ}\otimes\alpha^{\circ})\circ\psi^{\prime\circ}=\psi^{\circ}\circ\sigma^{\circ}$.
For any $b^{*}\in N^{\circ}$ and $x\in M,y\in G$, we have

$\displaystyle\langle(\sigma^{\circ}\otimes\alpha^{\circ})\circ\psi^{\prime\circ}(b^{*}),x\otimes
y\rangle$
$\displaystyle=\sum_{(b^{*})}\langle(\sigma^{\circ}\otimes\alpha^{\circ})(b^{*}_{(0)}\otimes
b^{*}_{(1)}),x\otimes y\rangle$
$\displaystyle=\sum_{(b^{*})}\langle\sigma^{\circ}(b^{*}_{(0)}),x\rangle\langle\alpha^{\circ}(b^{*}_{(1)}),y\rangle$
$\displaystyle=\langle\psi^{\prime\circ}(b^{*}),(\sigma\otimes\alpha)(x\otimes
y)\rangle$ $\displaystyle=\langle
b^{*},\psi^{\prime}\circ(\sigma\otimes\alpha)(x\otimes y)\rangle$
$\displaystyle=\langle b^{*},\sigma\circ\psi(x\otimes y)\rangle$
$\displaystyle=\langle\psi^{\circ}\circ\sigma^{\circ}(b^{*}),x\otimes
y\rangle.$

Thus we obtain
$(\sigma^{\circ}\otimes\alpha^{\circ})\circ\psi^{\prime\circ}=\psi^{\circ}\circ\sigma^{\circ}$.

Then we vertify
$\gamma^{\circ}\circ\sigma^{\circ}=\sigma^{\circ}\circ\gamma^{\prime\circ}$.
For all $c^{*}\in N^{\circ}$ and $x\in M$,

$\displaystyle\langle\gamma^{\circ}\circ\sigma^{\circ}(c^{*}),x\rangle$
$\displaystyle=\langle c^{*},\sigma\circ\gamma(x)\rangle$
$\displaystyle=\langle c^{*},\gamma^{\prime}\circ\sigma(x)\rangle$
$\displaystyle=\langle\sigma^{\circ}\circ\gamma^{\prime\circ}(c^{*}),x\rangle.$

Thus it follows that
$\gamma^{\circ}\circ\sigma^{\circ}=\sigma^{\circ}\circ\gamma^{\prime\circ}$.
Therefore this theorem is proved. ∎

In the special case when $M=G$, we have $M\cdot J=GJ\subseteq J$. Then the
Sweedler duality of the Hom-module $M$ reduces to the general Sweedler duality
of the Hom-algebra.

Acknowledgements S. Wang was partially supported by NSFC grant No. 12271089,
and C. Zhang was partially supported by NSFC grant No. 12101111.

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