# Slice-by-slice deep learning aided oropharyngeal cancer segmentation with
adaptive thresholding for spatial uncertainty on FDG PET and CT images

Alessia De Biase
Department of Radiation Oncology
University Medical Center Groningen
Groningen, NL 9700RB
& Nanna Maria Sijtsema
Department of Radiation Oncology
University Medical Center Groningen
Groningen, NL 9700RB
& Lisanne van Dijk
Department of Radiation Oncology
University Medical Center Groningen
Groningen, NL 9700RB
Department of Radiation Oncology
University of Texas MD Anderson Cancer Center
Houston, TX 77030
& Johannes A. Langendijk
Department of Radiation Oncology
University Medical Center Groningen
Groningen, NL 9700RB
& Peter van Ooijen
Department of Radiation Oncology
Data Science Center in Health (DASH)
University Medical Center Groningen
Groningen, NL 9700RB
<EMAIL_ADDRESS>
###### Abstract

Tumor segmentation is a fundamental step for radiotherapy treatment planning.
To define an accurate segmentation of the primary tumor (GTVp) of
oropharyngeal cancer patients (OPC), simultaneous assessment of different
image modalities is needed, and each image volume is explored slice-by-slice
from different orientations. Moreover, the manual fixed boundary of
segmentation neglects the spatial uncertainty known to occur in tumor
delineation. This study proposes a novel automatic deep learning (DL) model to
assist radiation oncologists in a slice-by-slice adaptive GTVp segmentation on
registered FDG PET/CT images. We included 138 OPC patients treated with
(chemo)radiation in our institute. Our DL framework exploits both inter and
intra-slice context. Sequences of 3 consecutive 2D slices of concatenated FDG
PET/CT images and GTVp contours were used as input. A 3-fold cross validation
was performed three times, training on sequences extracted from the Axial (A),
Sagittal (S), and Coronal (C) plane of 113 patients. Since consecutive
sequences in a volume contain overlapping slices, each slice resulted in three
outcome predictions that were averaged. In the A, S, and C planes, the output
shows areas with different probabilities of predicting the tumor. The
performance of the models was assessed on 25 patients at different probability
thresholds using the mean Dice Score Coefficient (DSC). Predictions were the
closest to the ground truth at a probability threshold of 0.9 (DSC of 0.70 in
the A, 0.77 in the S, and 0.80 in the C plane). The promising results of the
proposed DL model show that the probability maps on registered FDG PET/CT
images could guide radiation oncologists in a slice-by-slice adaptive GTVp
segmentation.

_Keywords_ Automatic tumor segmentation $\cdot$ Adaptive Segmentation $\cdot$
Deep Learning

##  1 Introduction

Image-guided deep learning tools have shown enormous potential in the field of
radiation oncology. Two of the main goals are to speed up the radiotherapy
workflow and to automate error prone tasks. Thus, automatic segmentation of
organs at risk (OARs) and clinical target volumes (CTVs) is of great interest,
especially in cases where various imaging modalities are needed for
radiotherapy treatment planning. For the head and neck region, 3D deep
learning segmentation models on Positron Emission Tomography (PET) and
Computed Tomography (CT) images were designed and tested during the first and
second edition of the HECKTOR challenge Andrearczyk et al. (2021)Andrearczyk
et al. (2022a). The achieved results, in terms of Dice Similarity Coefficient
(DSC) and $95^{th}$ Hausdorff Distance, showed the potential of Artificial
Intelligence (AI) in the tumor segmentation task.

In recent years, analysis of the variability of target volume definitions
across different and same observers, documented as inter- and intra-observer
variability, have been published Sadeghi et al. (2021)Weiss and Hess (2003).
Several studies have investigated the extent of inter-observer variability in
target volume delineation, showing large discrepancies in the contoured
structures van der Veen et al. (2019). Automatic tumor segmentation has
demonstrated to achieve lower performance compared to automatic organs at risk
delineation, since tumor volumes highly depend on the clinical scenario and
clinical judgement of the treating physician Wong et al. (2020). During the
learning process of automatic segmentation models, manual annotations of the
volume of interest are used as ground truth. As contouring varies from
specialist to specialist, the use of manual contours as a reference may not be
reliable Sadeghi et al. (2021), thus these variations can cause lack of
generalization of the method.

Different strategies to improve tumor delineation and reduce inter-observer
variability have been proposed. Some examples are consensus guidelines,
reliable ground truth, the use of proper and complementary imaging modalities
Gudi et al. (2017)Sadeghi et al. (2021) or discuss the contours in a
multidisciplinary meeting. The use of FDG PET/CT for target volume
delineation, for example, showed to significantly reduce inter-observer
variability for head and neck cancers compared to using CT only Gudi et al.
(2017). In addition, the absence of a proper reference representing the
absolute gold standard makes it difficult to establish which contour can be
chosen for a certain clinical task and to quantify such variability. Despite
the above mentioned solutions may improve tumor delineation, they are not
always the most efficient. Adding more image modalities, in fact, is expected
to improve the contouring accuracy Rodríguez Outeiral et al. (2021), but makes
the contouring process more time consuming and more complex Anderson et al.
(2014).

In this paper we propose a 2D deep learning segmentation method that shows the
degree of uncertainty of each pixel to be classified as tumor or not on CT and
PET images of oropharyngeal cancer patients. The deep learning framework takes
sequences of three consecutive slices from the CT and PET images as input. The
corresponding sequences extracted from the gross tumor volume (GTV) manually
delineated by experts are used as ground truth. Each slice in a volume can be
either first, second, or third in a sequence, resulting in three different
predictions of the same slice. In the axial, coronal, and sagittal view, the
model outputs probability maps instead of binary masks. Bi-directional long
short term memory (Bi-LSTM) and spatial and channel attention mechanisms are
used to capture context information from the nearby slices and to enhance
inter-slice dependencies. With our method, we would like to assist radiation
oncologists in decision making, giving them a deep learning aided tool for
tumor segmentation in radiotherapy treatment planning, where the optimal
threshold for clinical acceptance could be customized and where the
uncertainty derived from the variability in contouring in the training set is
available and displayed.

##  2 Materials and methods

###  2.1 Data collection

Data used in this study were manually collected from the Picture Archiving and
Communication System (PACS) of the University Medical Center of Groningen
(UMCG) in the Netherlands retrospectively. We considered oropharyngeal cancer
patients (OPC) who were treated, in our institute, with (chemo)radiation
therapy between 2014 and 2017. 166 patients satisfied the eligibility
criteria. For each case, planning CT and FDG PET 3D images, and GTV primary
tumor delineations, that were used in the radiotherapy treatment planning
process, were collected. All imaging was acquired in treatment position using
fixation with a mask. The tumor delineations were reviewed by a specialized
head and neck nuclear medicine physician and by a head and neck radiologist in
cases where the MRI scan was also available. The tumor contours were approved
after a multidisciplinary second reading session attended by all head and neck
radiation oncologists.

Firstly, rigid image registration was performed. Radiation oncologists’ manual
annotations of primary tumors were extracted from the DICOM RTSTRUCT files.
Then, the imaging data were downloaded in DICOM format and transformed into
NIfTI file format M.D. et al. (1995) to facilitate data handling. PET images
were transformed from raw to Standardized Uptake Values (SUV) Ulaner (2019).
For 9 patients, the PET image could not be registered with the planning CT,
because either the PET image was missing or the images co-registration was not
accurate due to differences in patient posture between the PET and CT images.
Only patients having one primary tumor and who received radiation therapy as
the first cancer treatment were included in this study. The final group
satisfying the inclusion criteria consisted of 138 OPC patients (age: 61.95
(mean) ± 9.02 (std) years). Participants provided written informed consent for
their data to be used for research purposes. In Table 1, the dataset
description including gender, T and N stages, and HPV status is reported.

Table 1: Dataset description (n=138) Variables | Training | Test
---|---|---
(n=113) | (n=25)
_n (%)_ | _n (%)_
Gender | male | 69 (61) | 18 (72)
| female | 44 (39) | 7 (28)
T Stage | T1 | 16 (14) | 5 (20)
| T2 | 30 (27) | 4 (16)
| T3 | 13 (11) | 3 (12)
| T4 | 54 (48) | 13 (52)
N Stage | N0 | 18 (16) | 5 (20)
| N1 | 13 (11) | 0 (0)
| N2a | 6 (5) | 0 (0)
| N2b | 37 (33) | 8 (32)
| N2c | 36 (32) | 10 (40)
| N3 | 3 (3) | 2 (8)
HPV status | negative | 47 (42) | 13 (52)
| positive | 47 (42) | 9 (36)
| unknown | 19 (16) | 3 (12)

###  2.2 Region of Interest selection and pre-processing

Medical Imaging data can be considerably large in dimensions. Since this study
focused specifically on oropharyngeal cancer, a region of interest centered in
the oral cavity was automatically extracted for each patient. The method used
for the bounding box selection is inspired by Andrearczyk et al. (2020a). It
consists of a two step approach: first, brain segmentation is performed
identifying the brain as the largest connected component in the image after
applying a fixed threshold for the SUV values of three in the PET image;
second, a bounding box of fixed size (144x144x144) is determined with respect
to the brain position, as in Andrearczyk et al. (2020a). We observed that
using a fixed threshold of three to separate the brain from the rest of the
head did not work for some of the patients in our dataset. The main problem
occurred in cases where tumor activity on the PET image was included as it was
part of the brain. To avoid manual assessment of the quality of the bounding
boxes, we identified some metrics to guide us in manually correcting the
threshold for the SUV values. We considered the brain volume (expected to be
around 1260 $cm^{3}$ in men and 1130 $cm^{3}$ in women) and the maximum, the
mean and the standard deviation of the Hounsfield units (HU) in the
corresponding CT image of the segmented area of the brain. High values of
maximum HU suggest that the segmentation is too large because bones have a
much larger HU compared to soft tissues. Once the correct bounding boxes had
been identified, CT and PET images were reconstructed into 144x144x144 pixels
resulting in volumes of 144x144x144 $mm^{3}$.

To facilitate tumor segmentation, a level of 40 HU and a window of 350 HU were
chosen for the CT images. The PET images pixel values below 0 were set to 0.
Finally, for both PET and CT images z-normalization was performed.

###  2.3 Proposed Deep Learning framework

The proposed method is divided into two main sections: in the first part, a
deep learning network was trained on PET and CT images to identify tumor areas
based on image features; in the second part, the uncertainty of the
predictions of the first step was reconstructed for each patient using
probability maps.

Figure 1: The proposed deep learning framework. On the left side of the image
the input data is displayed as two concatenated sequences of three slices from
the PET [$pt_{i},pt_{i+1},pt_{i+2}$] and from the CT
[$ct_{i},ct_{i+1},ct_{i+2}$] 3D volumes of a patient in the training set. The
method is divided into two sections: I. An encoder decoder path, comprising
spatial and channel attention blocks; II. A bi-directional LSTM with spatial
and channel attention blocks. The output of the framework is a reconstructed
sequence of the three slices showing a probability map indicating the pixel
tumor probability.

####  2.3.1 Prediction model on 2D sequences

The proposed deep learning segmentation model is inspired by Pan et al.
(2021). As training data, we used sequences of three consecutive 2D slices of
PET [$pt_{i},pt_{i+1},pt_{i+2}$] and CT images [$ct_{i},ct_{i+1},ct_{i+2}$],
concatenated in the channel domain (as shown in figure 1), together with three
corresponding consecutive 2D slices of GTV primary tumor segmentation
[$gtv_{i},gtv_{i+1},gtv_{i+2}$], in the form of binary masks. The main reason
to use only three slices was computational costs.

Tumor segmentation, as many medical imaging problems, suffers of class
imbalance. In a 2D model, the problem is intensified, since negative slices
represent the majority of the volume. To train and validate the network,
sequence selection was needed. A sequence [$i,i+1,i+2$] was always selected if
its first slice $i$ contained tumor pixels for more than $5\%$ of the total
area. Only $10\%$ of the total number of sequences were selected if its first
slice was either a negative slice ($5\%$) or it contained tumor pixels for
less than $5\%$ of the total area (the other $5\%$). During the testing phase,
all sequences from a volume were selected, containing both positive and
negative slices. In order to recognize negative examples as such, we also
needed to include them in the training set.

The framework is divided into two main steps: first, an encoder decoder path,
comprising spatial and channel attention blocks, extracts the main image
features from each sequence; second, a bi-directional LSTM with spatial and
channel attention aims to capture context information from the nearby slices.
The network is described in all its parts in figure 1.

Loss functions were used as in Pan et al. (2021). Dice Score Coefficient (DSC)
was calculated on the validation set to monitor the training process. To
perform this task, a threshold of 0.5 was applied to the pixel probability
values of the predictions to transform segmentation results into binary masks.
We saved only three different checkpoints: one corresponding to the last
trained model after 150 epochs, the second one corresponding to the model with
the highest mean DSC on the validation set, and the final one corresponding to
the model which obtained the second highest value of mean DSC after the first
40 epochs, indicating the point where fluctuations in the loss functions
decreased.

####  2.3.2 Single slice probability prediction

As shown in figure 1, the output of the proposed network is a sequence
[$map_{i},map_{i+1},map_{i+2}$] obtained as result of a sigmoid function,
meaning that each pixel of the image corresponds to a probability indicating
the chance of it belonging to tumor or not. During the testing phase, from
each volume, a total of 142 sequences of three consecutive slices were
extracted and predictions were calculated. For each slice in a volume, a
minimum of one (for slice 1 and 144) and a maximum of three (from slice 3 to
142) different predictions were obtained. This is because slice number $k$,
with $k$ from 3 to 142, will be first, second or third in a sequence.

$\displaystyle seq_{1}=\Big{[}map_{1},map_{2},map_{3}\Big{]}$ (1)
$\displaystyle\dots$ $\displaystyle
seq_{k-1}=\Big{[}map_{k-2},map_{k-1},map_{k}\Big{]}$ $\displaystyle
seq_{k}=\Big{[}map_{k-1},map_{k},map_{k+1}\Big{]}$ $\displaystyle
seq_{k+1}=\Big{[}map_{k},map_{k+1},map_{k+2}\Big{]}$ $\displaystyle\dots$
$\displaystyle seq_{142}=\Big{[}map_{142},map_{143},map_{144}\Big{]}$

Lastly, each final predicted slice of a volume $out_{k}$ was calculated as the
average of the total number of predictions for that slice:

$out_{k}=\begin{cases}seq_{k}[1],&{k=1}\\\\[8.0pt]
\frac{1}{2}(seq_{k}[1]+seq_{k-1}[2]),&{k=2}\\\\[8.0pt]
\frac{1}{3}\sum_{i=1}^{3}seq_{k+1-i}[i],&{3\leqslant k\leqslant
142}\\\\[8.0pt] \frac{1}{2}(seq_{k-1}[2]+seq_{k-2}[3]),&{k=143}\\\\[8.0pt]
seq_{k-2}[3],&{k=144}\\\\[8.0pt] \end{cases}$ (2)

The idea behind this last step is showing the uncertainty of the network in
classifying each pixel as tumor to the end user. During testing on the hold-
out test set, model ensembling was performed, averaging predictions obtained
from the three models trained and validated during 3-fold cross validation.
Ensemble learning combines the predictions from multiple deep learning models
to reduce the variance of predictions and reduce generalization error Logan et
al. (2021).

###  2.4 Evaluation metrics and techniques

The first step of the evaluation procedure was based on the quality of the
predicted sequences. In order to evaluate the method, binary images were
needed. A threshold was chosen for the pixel probability values of the
predictions. For the second step of the evaluation procedure, we reconstructed
each single slice from the predicted sequences of probabilities. For each
patient we obtained 144 slices of probabilities for the axial, 144 for the
sagittal and 144 for the coronal view. We calculated the metrics slice by
slice and then we averaged over each patient at different probability
thresholds.

To evaluate the quality of the segmentation model, the mean DSC was used. Dice
Score coefficient is largely used in image segmentation tasks Gudi et al.
(2017), because it quantifies the overlap between the prediction result and
the ground truth image, which in our case is the GTV of the primary tumor. The
DSC was calculated with the following formula on 2D images:

$DSC=\frac{2\times
TP+\varepsilon_{smooth}}{(TP+FP)+(TP+FN)+\varepsilon_{smooth}}$ (3)

where TP indicated the number of true positives, FP the number of false
positives and FN the number of false negatives. A smoothing factor of
$\varepsilon_{smooth}=1\times 10^{-5}$ was used to avoid errors in cases of
well predicted sequences or slices not containing any tumor pixel.

Both precision and recall were used during the testing phase to point out the
precarious outcomes. Precision defines how correct is the model when
predicting that a specific area contains the tumor, while recall gives a
percentage of the amount of tumor that has been identified by the model for a
certain patient. Both these metrics are used to have an overall idea of the
quality of the predictions, but both of them are not completely reliable if
considered separately. After each final predicted slice $out_{k}$ of a volume
was calculated, precision and recall were obtained with the following
formulas:

$Precision=\frac{\sum_{i=1}^{144}TP_{out_{i}}}{\sum_{i=1}^{144}TP_{out_{i}}+\sum_{i=1}^{144}FP_{out_{i}}}$
(4)

$Recall=\frac{\sum_{i=1}^{144}TP_{out_{i}}}{\sum_{i=1}^{144}TP_{out_{i}}+\sum_{i=1}^{144}FN_{out_{i}}}$
(5)

##  3 Experimental results

###  3.1 Implementation details

The deep learning framework was implemented and trained using Python (v3.7.4),
PyTorch (v1.6.0), MONAI (v0.8.0), CUDA (v10.1), and cuDNN (v7.6.4) on GPUs of
the Peregrine Cluster of the University of Groningen. The training and
validation set consisted of 113 patients, while the test set consisted of 25
patients. The same network was trained in parallel three times using, each
time, different input data: sequences from the axial ($M_{a}$), sagittal
($M_{s}$) and coronal ($M_{c}$) view of the 3D volumes. Batch size of 1 was
used during training, consisting of one sequence of three consecutive slices
from the same patient, extracted from the same plane. To avoid overfitting,
3-fold cross validation was performed for $M_{a}$, $M_{s}$ and $M_{c}$,
resulting in a total of 9 final models. A fixed maximum number of epochs was
set to 150. Adam optimizer with a fixed learning rate equal to 0.0002 was used
for training.

###  3.2 Model performance analysis

Training and validation losses were monitored during training the network.
During training, the mean DSC was calculated over the 3 validation sets, made
of sequences selected using the criteria explained in section 2.3.1. The
highest value of mean DSC was achieved in the first 40 epochs when trained on
the selected sequences from the axial (0.72 (mean) ± 0.20 (std) DSC), coronal
(0.68 (mean) ± 0.25 (std) DSC) and sagittal (0.72 (mean) ± 0.21 (std) DSC)
planes of the validation set. Nevertheless, to obtain final predictions, we
selected the last trained model after 150 epochs because the validation loss
stabilized around a certain value and the fluctuations in both loss functions
decreased.

###  3.3 2D slices reconstruction results

As a final step, 142 consecutive sequences were extracted from each image
volume of each patient in the validation set and predictions were obtained.
Slices from each volume were then reconstructed as explained in section 2.3.2.
The performance of the method on each patient was assessed by comparing the
mean DSC of the reconstructed slices for $M_{a}$, $M_{s}$ and $M_{c}$. We used
different probability thresholds, when transforming images into binary masks,
and we recalculated the evaluation metrics. Table 2 shows the average results
over all patients in the validation set for the DSC using a probability
threshold of $th=0.9$. After the slice reconstruction, the mean DSC of the
$M_{s}$ achieved the highest score and the mean DSC of the $M_{a}$ the lowest
one.

When testing, in each direction, slices were obtained by ensembling
predictions of the three optimized models trained using 3-fold cross
validation. In figure 2 the distribution of the mean DSC obtained from the
test set is reported at different probability thresholds using boxplots. An
ascending pattern can be observed, meaning that when the threshold of the
predicted probability increases, the segmentation prediction overlaps better
with the ground truth delineation of the tumor. For all views we observed a
similar grouping pattern: boxplots behave similarly between threshold values
of 0.1-0.3, 0.4-0.6 and 0.7-0.9. In the coronal view boxplots show a more
distinctive difference in mean DSC distribution at $th=0.4$. In table 2
results on the test set are reported for the mean DSC. The values obtained on
the validation and test set are consistent. The results from the models
trained on sequences extracted from the coronal and sagittal view outperformed
the ones from the axial in both the validation and test results. The mean DSC
values in the test results confirmed the robustness of the ensemble technique.

Figure 2: Boxplots showing the mean DSC distributions of the patients in the test set at different probability thresholds. The plots report results obtained by ensembling the outputs of the models trained with sequences extracted from the axial, coronal and sagittal view. For the first and the third graph a similar ascending pattern can be observed from low to high thresholds of probability. For the coronal view, a bigger difference in results is shown in correspondence of $th=0.3$ and $th=0.4$. Table 2: Mean DSC for the validation and test set. The validation outputs were obtained using 3-fold cross validation, while the test outputs were obtained by ensembling predictions of the three models optimized on each fold. For each patient, slices were reconstructed from predicted sequences of slices extracted from the axial, coronal and sagittal plane of the 3D FDG PET/CT images. A threshold value of 0.9 was chosen to make the outputs binary. |  | mean $\pm$ std
---|---|---
Metric | model | (min - max)
|  | validation | test
DSC | $M_{a}$ | 0.61$\pm$0.11 | 0.70$\pm$0.14
|  | (0.22-0.82) | (0.35-0.86)
| $M_{c}$ | 0.66$\pm$0.18 |  0.80$\pm$0.11
|  | (0.21-0.93) | (0.48-0.91)
| $M_{s}$ |  0.68$\pm$0.11 | 0.77$\pm$0.10
|  | (0.34-0.87) | (0.50-0.94)
Figure 3: Examples of the output of the proposed method. A. Outcome
probabilities displayed on the axial view of slices extracted from the CT 3D
volume of a patient in the dataset. The red line represents the GTVp provided
as ground truth, while the different colors show different areas of tumor
probabilities. B. Four examples of customized outputs based on different
probability threshold settings, as shown by the color bars on the right side
of the images. In B.1 only the areas containing pixels with tumor
probabilities above 30% are displayed, above 50% in B.2, above 70% in B.3 and
above 90% in B.4.

In figures 5 and 6, in the Appendix, boxplots showing the mean DSC
distributions across probability thresholds for different T and N stages can
be found. An increasing pattern in mean DSC can be observed for T1, T2, T3 and
T4 stages in all three views. However, in the axial and coronal views,
predictions of T1 stages considerably improve when choosing probability
thresholds above 0.3. Predictions of T4 stages result always in higher mean
DSC compared to predictions of lower T stages, except for high probability
thresholds in the sagittal view. Low N stages show a similar increasing
pattern of low T stages across thresholds. For the N3 stages mean DSC seems
independent of the thresholds.

###  3.4 Qualitative results

In figure 3, we show how the probability map predictions of our network looks
on the CT image of a patient. On the left side, the total range of probability
is shown on few slices extracted from the axial plane. On the right side, four
cases of different probability threshold settings are displayed. The range of
probability shown on the image for each case corresponds to the one above the
selected threshold up to 100%. Increasing the probability percentage shrinks
the predicted area around the tumor.

After a qualitative assessment of the results from the axial view, we
acknowledged that the low performance of the models, in terms of mean DSC,
corresponds to cases where big metastatic lymph nodes were present and tumor
volume was larger. In figure 4 we show the relationship between precision and
recall per patient, at different probability thresholds, adding the tumor
volume information from the reference contours by the radiation oncologists.
For volumes reconstructed from sequences extracted from the coronal and
sagittal view a similar pattern was observed. For patients with larger tumor
volumes, precision is usually lower and recall is above 0.8. Having high
recall on the image translates into having a low false negative rate.
Precision in these cases is higher at low thresholds and decreases at higher
thresholds of probability. The first scatter plot on figure 4, which
represents results on the axial view, shows that for some of the patients with
larger tumor volumes, values of recall are around 0.6.

Figure 4: Scatter plots of recall versus precision metric values for each
patient in the test set. For each patient precision and recall values are
calculated as the average of the metrics results obtained from the
reconstructed slices. The size of each data point represents the tumor volume,
while the color intensity represents the probability threshold chosen to make
the image binary.

##  4 Discussion

The experimental results show the potential of the proposed 2D deep learning
adaptive segmentation method for oropharyngeal cancer in PET-CT images. During
the second edition of the HECKTOR challenge Andrearczyk et al. (2022b), Xie
and Peng Xie and Peng (2022) ranked first in the task of tumor segmentation
using 3D PET-CT images, achieving a value of mean DSC of 0.7785 on a test set
of 101 cases from two different centers. The performance of our method in
terms of mean DSC is not comparable with the 3D segmentation results achieved
in the literature for the same task Oreiller et al. (2022), because in the
calculation of the mean DSC we also included no tumor slices present in the
selected bounding boxes. Values of precision and recall can be considered for
a fairer comparison of the results. As discussed in Andrearczyk et al.
(2020b), 2D methods result in higher precision and lower recall than 3D
methods. High values of precision and recall in figure 4 correspond to the top
right corner in the box plots where tumors with smaller volumes appear. Our
mean results in terms of precision range between 0.57 and 0.83 across
thresholds, in all three different views, against 0.6241 of the 2D PET/CT
baseline in Andrearczyk et al. (2021). In our study the mean recall ranges
between 0.5 and 0.67 at probability thresholds above 0.5, which is slightly
lower than the 2D PET/CT baseline in Andrearczyk et al. (2021).

In this paper we proposed a 2D deep learning adaptive segmentation method that
aims to assist radiation oncologists in segmenting primary tumors. The
framework mimics experts in their task of slice-by-slice tumor contouring on
PET and CT images of oropharyngeal cancer patients. The added value of the
method is providing probability maps which could be used in form of clinical
decision support as a starting point in the contouring process by the
radiation oncologist. Providing the radiation oncologist with contours of the
GTV primary tumor for different probabilities, makes it possible to choose the
most appropriate contour. This contour could than be optimized based on
clinical information and additional imaging. It is expected that by offering
the radiation oncologist a GTV contour to start with, the contouring process
can become faster and the inter observer variability could be reduced Vaassen
et al. (2020).

Despite several studies showing promising results in recognizing head and neck
tumors on multi-modal image data using deep learning Oreiller et al. (2022),
the limitations involved are still an obstacle for their clinical
implementation. In this paper, we suggest a solution that tackles the problem
of high variability in ground truth images used for tumor segmentation, and
that may enhance the utility of such a tool in clinical practice. The proposed
method, in fact, displays on different areas on the image a tumor probability
derived by the tumor segmentation learning process and supports the end user
towards a quicker and more attentive decision making. In tasks where the gold
standard is not represented by one and only one ground truth, like the one of
tumor segmentation, a large number of different annotations would be required
for training a DL network. In several deep learning studies, one adopted
solution, is to collect consistent tumor delineations obtained after
consultations of different radiation oncologists for the whole dataset. The
dataset used, in these cases, is consistent, and the deep learning framework
learns according to what has been decided to be "the truth". In our paper,
data has been collected between 2014 and 2017. In this time period, different
radiation oncologists have been working in our center and additional image
modalities became available for a more precise final tumor contour (e.g. MRI).
Despite the delineation guidelines being the same, the variation among tumor
delineations is large. Supervised learning methods reproduce human biases and
errors, thus the problem of inter-observer variability in target volume
delineation is a perpetuated error. This can be one of the reasons why our
results are sometimes lower than in other studies and hard to be compared
with.

However, relabeling a dataset is not practical and it is time consuming. New
data will always be collected and used to update existing models and similar
problems will again be encountered. In our method, in fact, we capture the
variation in contouring in the training set and we display the variation in
contours between the different probability threshold values. To widen the
output probability range and to obtain more robust predictions, ensembling
learning was performed. The ensambling technique provided more stable results,
as shown by the lower standard deviation of the test set results in table 2
compared to the validation set results. To evaluate the method presented in
this paper, we calculated mean DSC at the different probability thresholds to
establish whether the outcome predictions were overlapping the tumor area or
not. Increasing the probability threshold, predictions showed to become more
similar to what the oncologist defined as tumor delineation. Since the
segmentation task was performed on 2D images, the evaluation results for each
patient in each spatial direction were obtained as the average dice score
coefficient calculated between each 2D slice and the corresponding ground
truth. The same method was used also for precision and recall. When setting a
probability threshold of 0.9, only two out of 25 tumors have a corresponding
precision and recall below 0.03. In both cases the tumor dimension was small
and really difficult to identify on PET, as also mentioned in the patient
file.

In our work, we build on the fact that tumors are identified on a pixel level
on multi-modal images. The combination of PET and CT images gives
complementary information that makes the network assign a certain tumor
probability to a pixel. The channel attention blocks of the first part of the
framework (in figure 1) are intended to guarantee cross-modal learning. On one
hand, different imaging modalities have a positive impact on contouring
consistency, taking advantage of all the imaging information available for
segmentation Bird et al. (2015). On the other hand, contour variations can
result from differences in tumor visualization in various imaging modalities
and not accurate multi-modal image co-registrations Sadeghi et al. (2021). PET
imaging, in fact, does not allow a high delineation accuracy but is
fundamental in identifying head and neck cancer. The implemented deep learning
framework heavily relies on FDG uptake in PET images. Since the brain area is
characterized by a large amount of pixels with high SUV values, these areas
also appear in the outcome prediction, but with low probabilities. This
explains the large difference in mean DSC between probability thresholds below
and above 0.3 in the coronal view. When trained on sequences from the axial
plane, the network appears to rely on PET intensity and on tumor location. In
fact, in presence of primary tumors with larger volume that expand to
metastatic lymph nodes (higher N stages), the mean DSC values result lower.
For the purpose of this study, metrics need to be calculated at different
thresholds of probability to have an overall idea of the method performance. A
balance between precision and recall is what any model aims for, but in this
study a lower precision and an higher recall can indicate the volume at risk
for tumor extension, which should be further explored by the radiation
oncologist.

##  5 Conclusions

In conclusion, this study proposes a novel deep learning aided tool for
oropharyngeal tumor segmentation on FDG PET/CT images using probability maps.
The framework is designed to capture cross-modality information and inter-
slice context using spatial attention and bi-directional long short term
memory mechanisms. The main object is to display tumor probabilities on 2D
slices extracted from the axial, sagittal and coronal view of the 3D image
volumes, to assist radiation oncologists in decision making during
radiotherapy planning. The available wide range of probabilities is of
interest to minimise the risk of geographical misses and to make available the
uncertainty derived from the tumor segmentation task, leading cause of inter-
observer variability. Experimental results show the potential of the proposed
method in identifying the primary tumor.

## Availability of data and materials

The data used for this research comprise confidential patient health
information, which is protected and may not be released unless approved by the
Committee of Ethics of the UMCG.

## Declaration of competing interest

The authors declare no conflict of interest.

## Acknowledgment

This research was supported by the Hanarth Fonds. We would like to acknowledge
the contribution in data collection of the PhD students working on Head and
Neck cancer research in the Department of Radiation Oncology, at the
University Medical Hospital of Groningen, Netherlands. We would like to thank
the Center for Information Technology of the University of Groningen for their
support and for providing access to the Peregrine high performance computing
cluster.

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## Appendix

Figure 5: Mean DSC of patients in test set grouped by T stages at different
probability thresholds. Figure 6: Mean DSC of patients in test set grouped by
N stages at different probability thresholds.


# Endomorphism property of vertex operator algebras over arbitrary fields

Chao Yang  School of Mathematics, Sichuan University, Chengdu 610064 China
Jianqi Liu  Department of Mathematics, University of California, Santa Cruz,
CA 95064 USA

###### Abstract

In this paper, we study the endomorphism properties of vertex operator
algebras over an arbitrary field $\mathbb{F}$, with
$\text{Char}(\mathbb{F})\neq 2$. Let $V$ be a strongly finitely generated
vertex operator algebra over $\mathbb{F}$, and $M$ be an irreducible
admissible $V$-module. We prove that every element in $\text{End}_{V}(M)$ is
algebraic over $\mathbb{F}$ and that $\text{End}_{V}(M)$ is also finite-
dimensional. As an application, we prove Schur’s lemma for strongly finitely
generated vertex operator algebras over arbitrary algebraically closed fields,
and we give a test for absolute irreducibility of $V$-modules.

##  1 Introduction

The study of the endomorphism properties for associative algebras has a long
history and rich results(see [D, DS, MR, Q] etc.). If the associative algebra
$A$ is defined over an algebraically closed field and $S$ is a finite
dimensional simple $A$-module, the famous Schur’s Lemma tells us that
$\mathrm{End}_{A}(S)$ is one dimensional. However, when the base field
$\mathbb{F}$ is not algebraically closed, or $\text{dim}_{F}(S)$ is not
finite, $\mathrm{End}_{A}(S)$ is not so easy to characterize. One of the most
elegant results, in this case, is Quillen’s Theorem [Q], which says that each
element in $\text{End}_{A}(S)$ is algebraic over $\mathbb{F}$, provided that
$A$ is a filtered algebra whose associated graded algebra $\text{gr}(A)$ is
finitely generated and commutative.

In some special cases, the endomorphism ring $\text{End}_{A}(S)$ is finite-
dimensional. For example, if $A=U(\mathfrak{g})$ is the universal enveloping
algebra of a finite dimensional lie algebra $\mathfrak{g}$ over a field
$\mathbb{F}$ with $\text{char}(\mathbb{F})=0$, it is shown in [D] that
$\text{End}_{A}(S)$ is finite dimensional; if $A$ is a constructible algebra
over an arbitrary field $\mathbb{F}$, it is shown in [MR] that
$\text{End}_{A}(S)$ is also finite dimensional. Moreover, if $S$ is a
topologically irreducible admissible Banach space representation of a
$\mathbb{Q}_{p}$-analytic group $G$ over $\mathbb{F}$, where $\mathbb{F}$ is a
finite extension of $\mathbb{Q}_{p}$, then it is shown in [DS] that the
algebra $\rm{End}_{\mathbb{F}[G]}(S)$ of continuous $G$-equivariant
endomorphisms of $S$ is a finite dimensional division algebra over
$\mathbb{F}$. An associative algebra over $\mathbb{F}$ such that
$\mathrm{End}_{A}(S)$ is finite-dimensional for all simple modules $S$ is said
to admit the (strong) endomorphism property.

In the theory of vertex operator algebras (VOAs) ([DLM1, LL] etc.), when the
base field is ${\mathbb{C}}$, the “infinite dimensional Schur’s Lemma" due to
Dixmier [D] which says that $\mathrm{End}_{A}(M)={\mathbb{C}}\text{Id}_{M}$ if
$M$ is of countable dimension is commonly used in the proof of the
semisimplicity of the Hamiltonian $L(0)$ on an irreducible admissible module
$M$ over a vertex operator algebra $V$ (cf. [DLM, DLM1, Z].) In recent years,
the vertex operator algebras defined over an arbitrary field $\mathbb{F}$ has
brought much attention. A series of studies on vertex operator algebras over
arbitrary fields can be found in ([DR1, DR2, R, JLM, LM1, LM2] etc.) However,
since the usual Schur’s lemma no longer holds when $\mathbb{F}$ is not
uncountable, a natural question is what kind of endomorphism properties are
possessed by a vertex operator algebra over an arbitrary field $\mathbb{F}$.
The following Theorem is our main result.

######  Theorem 1.1.

Let $V$ be a strongly finitely generated vertex operator algebra over an
arbitrary field $\mathbb{F}$, and $M$ be an irreducible admissible $V$-module.
Then

  1. (1)

Every element in $\text{End}_{V}(M)$ is algebraic over $\mathbb{F}.$ Moreover,
$\text{End}_{V}(M)$ is also finite-dimensional over $\mathbb{F}$.

  2. (2)

(Schur’s lemma) If $\mathbb{F}$ is an algebraic closed field, then
$\text{End}_{V}(M)\cong\mathbb{F}$, and there exists a $\lambda\in\mathbb{F}$
such that $L(0)|_{M(n)}=\lambda+n$ for all $n$.

  3. (3)

$M$ is an absolutely irreducible $V$-module if and only if
$\text{End}_{V}(M)\cong\mathbb{F}.$

In this theorem, part (1) can be viewed as the analog of Quillen’s theorem in
the context of the vertex operator algebras. Schur’s lemma and the
characterization of the absolutely irreducible modules are natural corollaries
of Quillen’s theorem in (1). Although we imposed the condition that $V$ is
strongly finitely generated in Theorem 1.1, we note that this is not a strong
condition. In fact, almost all the classic examples of vertex operator
algebras over an arbitrary field, including the irrational ones, like vacuum
module VOA $V_{\hat{\mathfrak{g}}}(k,0)$ and the (universal) Virasoro VOA
$\bar{V}(c,0)$ [FZ], are strongly finitely generated [DR2, JLM, LM1].
Furthermore, for a strongly finitely generated VOA $V$, many examples indicate
that the orbifold $V^{G}$ of a reductive group $G\leq\mathrm{Aut}(V)$ is also
strongly finitely generated (see [LA, CL].) With Theorem 1.1, we can prove
that the conjecture proposed in [DR1], which states that the Hamiltonian
$L(0)$ acts semi-simply on an irreducible admissible $M$ without the
uncountable condition for $\mathbb{F}$ is indeed true.

To prove Theorem 1.1, we use the (generalized) Zhu’s algebra $A_{n}(V)$
defined in [DLM, R, Z] and transform the study of $\mathrm{End}_{V}(M)$ into
the study of the endomorphism property of an $A_{n}(V)$-modules. We will also
use the fact that there exists an epimorphism between Zhu’s $C_{2}$-algebra
$R(V)=V/C_{2}(V)$ and $\text{gr}A(V)$ [A, CJL, Liu], where $\mathrm{gr}A(V)$
is the associated graded algebra of the level filtration on $A(V)$, and
transform the strong generated property of $V$ into a finite generation
property of $\mathrm{gr}A(V)$. More precisely, we have the following results
toward the proof of Theorem 1.1:

######  Theorem 1.2.

  1. (1)

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$, and let
$M=\bigoplus_{i\geq 0}M(i)$ be an irreducible admissible $V$-module. Then
$\text{End}_{V}(M)\cong\text{End}_{A(V)}(M(0))$.

  2. (2)

Let $V$ be a strongly finitely generated vertex operator algebra over a field
$\mathbb{F}$. Then $A(V)$ is a filtered algebra such that the associated
graded algebra $\text{gr}A(V)$ is a finitely generated commutative algebra.

  3. (3)

Let $A$ be a filtered algebra such that the associated graded algebra
$\text{gr}(A)$ is a finitely generated commutative algebra. Then for each
simple $A$-module $S$, every element in $\text{End}_{A}(S)$ is algebraic over
$\mathbb{F}$. Moreover, $\text{End}_{V}(S)$ is finite dimensional.

This paper is organized as follows. Section 2 reviews the basics of vertex
operator algebras over an arbitrary field. We introduce some notations and
prove some basic results about scalar extensions of vertex operator algebra in
section 3. In section 4, we first recall the definition and properties of the
generalized Zhu’s algebra $A_{n}(V)$ in [DLM, R], and then prove an
intermediate result about the endomorphism property of the finite-type
filtered algebras. In section 5 and section 6, we prove the main results of
Theorem 1.1 and Theorem 1.2. Some applications of the main Theorems on the
scalar extensions are discussed in section 7.

The authors are grateful to our supervisor Professor Chongying Dong for
valuable advice.

Convention: All the fields in this paper have characterization not equal to
$2$.

##  2 Basics

In this section, we recall the basics of vertex operator algebras and modules
[DR1, LL, LM1, LM2] over an arbitrary field $\mathbb{F}$ with
$\text{char}(\mathbb{F})\neq 2$.

######  Definition 2.1.

A vertex algebra $(V,Y(-,z),{\bf 1})$ over a field $\mathbb{F}$ is a vector
space equipped with a linear map:

$\displaystyle Y(-,z):V$ $\displaystyle\to(\mbox{End}\,V)[[z,z^{-1}]],$
$\displaystyle v\mapsto Y(v,z)=\sum_{n\in{{\mathbb{Z}}}}v_{n}z^{-n-1}\ \ \ \
(v_{n}\in\mbox{End}\,(V)),$

and with a distinguished vector ${\bf 1}\in V$, satisfying the following
conditions for any $u,v\in V$ and $m,n\in{\mathbb{Z}}$:

$\displaystyle u_{n}v=0\ \ \ \ \ {\rm for}\ \ n\ \ {\rm sufficiently\ large};$
$\displaystyle Y({\bf 1},z)=Id_{V};$ $\displaystyle Y(v,z){\bf 1}\in V[[z]]\ \
\ {\rm and}\ \ \ \lim_{z\to 0}Y(v,z){\bf 1}=v;$

and the Jacobi identity:

$\displaystyle\displaystyle{z^{-1}_{0}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)Y(u,z_{1})Y(v,z_{2})-z^{-1}_{0}\delta\left(\frac{z_{2}-z_{1}}{-z_{0}}\right)Y(v,z_{2})Y(u,z_{1})}$
$\displaystyle\displaystyle{=z_{2}^{-1}\delta\left(\frac{z_{1}-z_{0}}{z_{2}}\right)Y(Y(u,z_{0})v,z_{2})}.$

We denote the set of non-negative integers by ${\mathbb{Z}}_{+}$.

######  Definition 2.2.

A vertex operator algebra $(V,Y(\ ,z),{\bf 1},\omega)$ over a field
$\mathbb{F}$ is a ${\mathbb{Z}}_{+}$-graded vertex algebra $(V,Y(-,z),{\bf
1})$:

$V=\bigoplus_{n\in{\mathbb{Z}}_{+}}V_{n},$

satisfying $\text{dim}V_{n}<\infty$ for all $n\in{\mathbb{Z}}_{+}$, ${\bf
1}\in V_{0}$, ${\omega}\in V_{2}$, and the following conditions holds for any
$u,v\in V$ and $m,n,s,t\in{\mathbb{Z}}$:

$\displaystyle u_{n}v\in V_{s+t-n-1}\ \text{for}\ u\in V_{s},v\in V_{t};$
$\displaystyle[L(m),L(n)]=(m-n)L(m+n)+\frac{1}{12}(m^{3}-m)\delta_{m+n,0}c;$
$\displaystyle\frac{d}{dz}Y(v,z)=Y(L(-1)v,z);$ $\displaystyle
L(0)|_{V_{n}}=n,$

where $L(m)=\omega_{m+1}$ for each $m\in{\mathbb{Z}}$, that is,
$Y(\omega,z)=\sum_{n\in{\mathbb{Z}}}L(n)z^{-n-2}$, and $c\in\mathbb{F}.$

We denote a vertex operator algebra $(V,Y(-,z),{\bf 1},\omega)$ over a field
$\mathbb{F}$ by $V$ for short. If $u\in V_{n}$, we call $n$ the degree of $u$
and write $\text{deg}u=n$.

######  Remark 2.3.

If $\text{char}(\mathbb{F})=0$, the assumption $u_{n}v\in V_{s+t-n-1}$ in
Definition 2.2 is a consequence of the other axioms. In the case that $F$ is a
field of finite characteristic $p$, $L(0)$ has only $p$ distinct eigenvalues
on $V$, and this assumption is necessary.

######  Definition 2.4.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$. A weak
$V$-module $M$ is a vector space equipped with a linear map:

$\begin{split}Y_{M}(-,z):V&\to{\rm End(M)}[[z,z^{-1}]]\\\
v&\mapsto\displaystyle{Y_{M}(v,z)=\sum_{n\in{\mathbb{Z}}}v_{n}z^{-n-1}\ \ \
(v_{n}\in{\rm End(M)}}),\end{split}$

which satisfies the following properties: for any $u,v\in V,$ and $w\in M$,

$\displaystyle u_{l}w=0\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{for}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ l\gg 0;$ $\displaystyle Y_{M}({\mathbf{1}},z)=Id_{M};$
$\displaystyle\displaystyle{z^{-1}_{0}\delta\left(\frac{z_{1}-z_{2}}{z_{0}}\right)Y_{M}(u,z_{1})Y_{M}(v,z_{2})-z^{-1}_{0}\delta\left(\frac{z_{2}-z_{1}}{-z_{0}}\right)Y_{M}(v,z_{2})Y_{M}(u,z_{1})}$
$\displaystyle\displaystyle{=z_{2}^{-1}\delta\left(\frac{z_{1}-z_{0}}{z_{2}}\right)Y_{M}(Y(u,z_{0})v,z_{2})}.$

######  Definition 2.5.

An admissible $V$-module is a ${\mathbb{Z}}_{+}$-graded weak $V$-module $M$:

$M=\bigoplus_{n\in{\mathbb{Z}}_{+}}M(n),$

satisfying

$u_{n}M(t)\in M(s+t-n-1)$

for $u\in V_{s}$, $n\in{\mathbb{Z}}$ and $t\in{\mathbb{Z}}_{+}$.

We also need the concept of strongly finitely generated vertex operator
algebra (cf. [L1]):

######  Definition 2.6.

Let $V$ be a vertex operator algebra over a field ${\mathbb{F}}$, and let
$U\subset V$ be a subset. $V$ is called strongly generated by $U$ if $V$ is
spanned by elements of the form:

$u^{1}_{-n_{1}}\dots u^{r}_{-n_{r}}{\bf 1},$

where $u^{1},\dots,u^{r}\in U$ for $r\geq 0$, and $n_{i}\geq 1$ for all $i$.
If $V$ is strongly generated by a finite dimensional subspace, then $V$ is
called strongly finitely generated.

######  Definition 2.7.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$.

  1. (1)

$V$ is called rational if the admissible $V$-module category is semi-simple.

  2. (2)

$V$ is called $C_{2}$-cofinite if $R(V)=V/C_{2}(V)$ is finite dimensional
[LM1], where

$C_{2}(V)=\text{span}\\{v_{-n-2}v\ |\ u,v\in V,n\in{\mathbb{Z}}_{+}\\}.$

######  Definition 2.8.

([MR]) Let $A$ be an associative algebra over a field $\mathbb{F}$, $A$ is
said to have the endomorphism property if for each simple $A$-module $S$,
every element in $\text{End}_{A}(S)$ is algebraic over $\mathbb{F}$. $A$ is
said to have the strong endomorphism property if for each simple $A$-module
$S$, the endomorphism algebra $\text{End}_{A}(S)$ is finite dimensional.

We introduce a similar definition of endomorphism properties for vertex
operator algebras over a field $\mathbb{F}$:

######  Definition 2.9.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$. $V$ is said to
have the endomorphism property if for each irreducible admissible $V$-module
$M$, every element in $\text{End}_{V}(M)$ is algebraic over $\mathbb{F}.$

$V$ is said to have the strong endomorphism property if for each irreducible
admissible $V$-module $M$, the endomorphism algebra $\text{End}_{V}(M)$ is
finite dimensional over $\mathbb{F}$.

##  3 Scalar extensions

In this section, we introduce some notations and prove some basic properties
of scalar extension of vertex operator algebras, which will be used later.

Let $A$ be an associative algebra over a field $\mathbb{F}$, and $M$ be a
(left) $A$-module. Let $\mathbb{K}/\mathbb{F}$ be a field extension. Define
$A^{\mathbb{K}}=\mathbb{K}\otimes_{\mathbb{F}}A$ and
$M^{\mathbb{K}}=\mathbb{K}\otimes_{\mathbb{F}}M$, then $A^{\mathbb{K}}$ is an
associative algebra over $\mathbb{K}$, and $M^{\mathbb{K}}$ is a (left)
$A^{\mathbb{K}}$-module.

######  Definition 3.1.

([D]) Let $A$ be an associative algebra over a field $\mathbb{F}$. An simple
$A$-module $S$ is called absolutely simple if $S^{\mathbb{K}}$ is an simple
$A^{\mathbb{K}}$-module for any field extension $\mathbb{K}$ of $\mathbb{F}$.

######  Definition 3.2.

([B]) A noetherian algebra over a field $\mathbb{F}$ is called stably
noetherian if $A^{\mathbb{K}}$ is also noetherian for any field extension
$\mathbb{K}$ of $\mathbb{F}$.

The following Lemma is well-known (see [Lang]):

######  Lemma 3.3.

Let $A$ be a finite dimensional semi-simple $\mathbb{F}$-algebra. Then for any
finite separable field extension $\mathbb{K}/\mathbb{F}$, the
$\mathbb{K}$-algebra $A^{\mathbb{K}}$ is also semi-simple.

Let $\mathbb{K}/\mathbb{F}$ be a field extension, and $(V,Y(-,z),{\bf
1},\omega)$ be a vertex operator algebra over $\mathbb{F}$. Let
$(M,Y_{M}(-,z))$ be an admissible $V$-module. We write:

$V^{\mathbb{K}}=\mathbb{K}\otimes_{\mathbb{F}}V,\leavevmode\nobreak\
\leavevmode\nobreak\ {\bf 1}^{\mathbb{K}}=1\otimes{\bf 1},\leavevmode\nobreak\
\leavevmode\nobreak\ \omega^{\mathbb{K}}=1\otimes\omega,$
$M^{\mathbb{K}}=\mathbb{K}\otimes_{\mathbb{F}}M=\bigoplus_{n=0}^{\infty}\mathbb{K}\otimes_{\mathbb{F}}M(n)=\bigoplus_{n=0}^{\infty}M(n)^{\mathbb{K}},$

then define
$Y^{\mathbb{K}}:V^{\mathbb{K}}\rightarrow\mathrm{End}{V^{\mathbb{K}}}[[z,z^{-1}]]$
and
$Y_{M^{\mathbb{K}}}:V^{\mathbb{K}}\rightarrow\mathrm{End}(M^{\mathbb{K}})[[z,z^{-1}]]$
by

$Y^{\mathbb{K}}(\lambda\otimes a,z)(\mu\otimes b):=\lambda\mu\otimes Y(a,z)b,$
(3.1) $Y_{M^{\mathbb{K}}}(\lambda\otimes a,z)(\mu\otimes u):=\lambda\mu\otimes
Y_{M}(a,z)u,$ (3.2)

for all $a,b\in V,$ $\lambda,\mu\in\mathbb{K}$, and $u\in M$. Then it is easy
to check that $(V^{\mathbb{K}},Y^{\mathbb{K}}(-,z),{\bf
1}^{\mathbb{K}},\omega^{\mathbb{K}})$ is a vertex operator algebra over
$\mathbb{K}$, and $(M^{\mathbb{K}},Y_{M^{\mathbb{K}}}(-,z))$ is a
$V^{\mathbb{K}}$-module.

######  Proposition 3.4.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$, and let
$\mathbb{K}/\mathbb{F}$ be a field extension. Then we have:

  1. (1)

$V$ is $C_{2}$-cofinite if and only if $V^{\mathbb{K}}$ is $C_{2}$-cofinite.

  2. (2)

$V$ is strongly finitely generated if and only if $V^{\mathbb{K}}$ is strongly
finitely generated.

###### Proof.

The proof is straightforward from the definition. ∎

We can similarly define the absolute irreducibility for irreducible admissible
modules of vertex operator algebras over arbitrary field.

######  Definition 3.5.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$. An irreducible
admissible $V$-module $M$ is called absolutely irreducible if $M^{\mathbb{K}}$
is an irreducible admissible $V^{\mathbb{K}}$-module for any field extension
$\mathbb{K}/\mathbb{F}$.

######  Lemma 3.6.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$, and $M$ be an
irreducible admissible $V$-module. Let $\mathbb{K}/\mathbb{F}$ be a field
extension. Then $M^{\mathbb{K}}$ is an irreducible admissible
$V^{\mathbb{K}}$-module if and only if $M(0)^{\mathbb{K}}$ is a simple
$A(V)^{\mathbb{K}}$-module.

###### Proof.

Clearly, if $M^{\mathbb{K}}$ is irreducible then $M(0)^{\mathbb{K}}$ is a
simple $A(V)^{\mathbb{K}}$-module.

Conversely, suppose $M(0)^{\mathbb{K}}$ is a simple
$A(V)^{\mathbb{K}}$-module, and $N$ is an admissible submodule of
$M^{\mathbb{K}}$ such that $0\lneq N\lneq M^{\mathbb{K}}$. Since
$M(0)^{\mathbb{K}}$ is simple and $M^{\mathbb{K}}$ is generated by
$M(0)^{\mathbb{K}}$, we have $N\cap M(0)^{\mathbb{K}}=0.$ Hence there exists
some $s>0$ such that the bottom level $N(0)$ of $N$ is contained in
$M^{\mathbb{K}}(s)$. Take a nonzero element $k_{1}\otimes
m_{1}+\cdots+k_{p}\otimes m_{p}\in N(0)\ (k_{i}\in\mathbb{K},\ 0\neq m_{i}\in
M(s),\ p>0)$ such that $k_{1},\cdots,k_{p}$ are linearly independence over
$\mathbb{F}$. For any $v\in V$ and $i\geq 0$, by (3.2) we have:

$\displaystyle 0$ $\displaystyle=(1\otimes v)_{\mathrm{wt}(v)+i}(k_{1}\otimes
m_{1}+\cdots+k_{p}\otimes m_{p})$
$\displaystyle=k_{1}\otimes(v_{\mathrm{wt}(v)+i}m_{1})+\cdots+k_{p}\otimes(v_{\mathrm{wt}(v)+i}m_{p}).$

Since $k_{1},\dots,k_{p}$ are linearly independence over $\mathbb{F}$, we have
$v_{\mathrm{wt}(v)+i}m_{1}=0$ for any $v\in V$ and $i\geq 0$. Let $\langle
m_{1}\rangle$ be the $V$-submodule of $M$ generated by $m_{1}$. Then it
follows from an easy induction that $\langle m_{1}\rangle\cap M(0)=0,$ and so
$\langle m_{1}\rangle$ is a proper submodule of $M$. This contradicts the
assumption that $M$ is irreducible. Hence $M^{\mathbb{K}}$ is irreducible, and
the proof is complete. ∎

The following corollaries are immediate consequences of Lemma 3.6.

######  Corollary 3.7.

Let $V$ be a simple vertex operator algebra over a field $\mathbb{F}$, with
$V_{0}=\mathbb{F}{\bf 1}$, and let $\mathbb{K}/\mathbb{F}$ be a field
extension. Then $V^{\mathbb{K}}$ is also a simple vertex operator algebra over
a field $\mathbb{K}$.

######  Corollary 3.8.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$. If $M$ is an
irreducible admissible $V$-module such that $M(0)\cong\mathbb{F}$, then $M$ is
absolutely irreducible.

##  4 Associative algebras $A_{n}(V)$ and Quillen’s Theorem

Our goal in this section is to give a characterization of
$\mathrm{End}_{V}(M)$, using the $A_{n}(V)$-theory of vertex operator algebras
(over an arbitrary field) in [DLM, R], as a preparation of the main theorem of
the paper. We will also recall Quillen’s theorem [Q] for finite type filtered
algebras and prove some preparation Lemmas for the following sections.

We first review the $A_{n}(V)$-theory of vertex operator algebras over
arbitrary fields.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$, and $n$ be a
nonnegative integer. For any homogeneous elements $u\in V$, $v\in V$, and
$s,t\in{\mathbb{Z}}_{+}$ with $t\geq s$, define

$u\circ_{n,t}^{s}v=\text{Res}_{z}Y(u,z)v\frac{(1+z)^{\mathrm{wt}u+n+s}}{z^{2n+2+t}},$

and

$u*_{n}v=\sum_{m=0}^{\infty}(-1)^{m}{m+n\choose
n}\text{Res}_{z}Y(u,z)v\frac{(1+z)^{\mathrm{wt}u+n}}{z^{n+m+1}}.$

Extend $\circ_{n,t}^{s}$ and $*_{n}$ bilinearly onto $V$. Let $O_{n}(V)$ be
the span of elements $u\circ_{n,t}^{s}v$ and $L(-1)u+L(0)u$, for all $u,v\in
V$, and $s,t\in{\mathbb{Z}}_{+}$ with $t\geq s$. Define $A_{n}(V)$ to be the
quotient space $V/O_{n}(V)$. In the case $n=0$, $A_{0}(V)$ is just the usual
Zhu’s algebra $A(V)$ (over field $\mathbb{F}$) studied in [DR1, Z].

######  Remark 4.1.

In the case $\text{Char}(\mathbb{F})=0$, it is proved in [DLM] that $O_{n}(V)$
is spanned by $u\circ_{n,0}^{0}v$ and $L(-1)u+L(0)u$, for all $u,v\in V$. But
for an arbitrary field $\mathbb{F}$, it is necessary (cf.[R]) to include the
elements $u\circ_{n,t}^{s}v$ in the definition of $O_{n}(V)$. The notion
$u\circ_{n,t}^{s}v$ comes from [DJ].

For $u\in V$, we write $[u]=u+O_{n}(V)\in A_{n}(V)$. For a homogeneous element
$u\in V$, we write $o(u)$ for the operator $u_{\mathrm{wt}u-1}$ on any weak
$V$-module $M$.

######  Theorem 4.2.

([DLM, R]) Let $V$ be a vertex operator algebra over a field $\mathbb{F}$, and
let $M$ be an admissible $V$-module. We have:

  1. (1)

For any $n\geq 0$, $(A_{n}(V),*_{n},{\bf 1}+O_{n}(V))$ is an associative
algebra.

  2. (2)

For any $n\geq s\geq 0$, the identity map on $V$ induces an onto algebra
homomorphism from $A_{n}(V)$ to $A_{s}(V)$.

  3. (3)

For any $n\geq s\geq 0$, $M(s)$ is an $A_{n}(V)$-module such that $[u]\in
A_{n}(V)$ acts as $o(u)$ for homogeneous $u\in V$.

  4. (4)

If $V$ is rational, then $A_{n}(V)$ is a semi-simple associative algebra for
any $n\geq 0$.

  5. (5)

If $V$ is simple, then $V$ is rational if and only if $A_{n}(V)$ is a semi-
simple associative algebra for any $n\geq 0$.

######  Remark 4.3.

In [DR1, R], the field $\mathbb{F}$ is assumed to be uncountable and
algebraically closed, then by using the infinite-dimensional Schur’s Lemma,
one can conclude that $A_{n}(V)$ is not only a semi-simple algebra over
$\mathbb{F}$ but finite-dimensional as well.

However, since we do not assume that F is uncountable, it is not obvious
whether or not $A_{n}(V)$ is finite-dimensional in parts (4) and (5) of
theorem 4.2.

######  Theorem 4.4.

([DLM, R]) Let $V$ be a vertex operator algebra over a field $\mathbb{F}$, and
let $U$ be an $A_{n}(V)$-module. Then there exists an admissible $V$-module
$\mathcal{M}_{n}(U)$ such that

  1. (1)

$\mathcal{M}_{n}(U)$ is generated by $\mathcal{M}_{n}(U)(n)=U$, and it has the
following universal property: for any admissible $V$-module $M$ and any
$A_{n}(V)$-morphism $\phi:U\rightarrow M(n)$, there exists a unique
$V$-morphism $\overline{\phi}:\mathcal{M}_{n}(U)\rightarrow M$ such that
$\overline{\phi}|_{U}=\phi$.

  2. (2)

$\mathcal{M}_{n}(U)$ has a unique maximal admissible $V$-submodule
$\mathcal{J}_{n}(U)$, with the property that $\mathcal{J}_{n}(U)\cap U=0.$ Set
$L_{n}(U)=\mathcal{M}_{n}(U)/\mathcal{J}_{n}(U)$. Then $L_{n}(U)$ is an
admissible $V$-module generated by $L_{n}(U)(n)=U$.

  3. (3)

If $U$ is a simple $A_{n}(V)$-module, then $L_{n}(U)$ is an irreducible
$V$-module.

  4. (4)

If $M=\bigoplus_{i=0}^{\infty}M(i)$ is an irreducible admissible $V$-module,
then $L_{n}(M(n))$ is irreducible, and $L_{n}(M(n))\cong M$.

######  Remark 4.5.

We note that $L_{n}(U)(0)$ might be zero if $U$ is also a module over
$A_{n-1}(V)$.

Similar to Proposition 3.4, $A_{n}(V)$ satisfies the following properties:

######  Proposition 4.6.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$, and let
$\mathbb{K}/\mathbb{F}$ be a field extension. Then for any
$n\in{\mathbb{Z}}_{+}$, we have
$A_{n}(V^{\mathbb{K}})\cong(A_{n}(V))^{\mathbb{K}}.$

######  Corollary 4.7.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$. If
$M=\bigoplus_{i\geq 0}M(i)$ is an irreducible admissible $V$-module, then
$\text{End}_{V}(M)\cong\text{End}_{A_{n}(V)}(M(n))$ for any
$n\in{\mathbb{Z}}_{+}.$

###### Proof.

Define a linear map
$\varphi:\text{End}_{V}(M)\rightarrow\text{End}_{A_{n}(V)}(M(n)),$
$\varphi(F):=F|_{M(n)}$, for any $F\in\text{End}_{V}(M)$. Since $M$ is
irreducible, it is easy to see that $\varphi$ is injective.

Let $f\in\text{End}_{A_{n}(V)}(M(n))$, by Theorem 4.4 (1), there exists a
$V$-homomorphism:

$\overline{f}:\mathcal{M}_{n}(M(n))\rightarrow M$

such that $\overline{f}|_{M(n)}=f$. Since $M$ is irreducible, we have
$\text{Ker}\overline{f}=\mathcal{J}(M(n))$ by the definition of
$\mathcal{J}(M(n))$. Hence the $\overline{f}$ induces a $V$-homomorphism
$\widetilde{f}:M\cong L_{n}(M(n))\rightarrow M$ such that
$\varphi(\widetilde{f})=\widetilde{f}|_{M(n)}=f$. This shows that $\varphi$ is
also surjective, and the proof is complete. ∎

Next, we recall Quillen’s theorem for finite-type filtered algebras and prove
some related results under field extensions.

######  Definition 4.8.

An associative algebra $A$ over a field $\mathbb{F}$ is called filtered if it
is endowed with an increasing filtration: $0=F_{-1}A\subset F_{0}A\subset
F_{1}A\subset\cdots$ by $\mathbb{F}$-subspaces such that

  1. (1)

$1_{A}\in F_{0}A,$

  2. (2)

$\ F_{p}A\cdot F_{q}A\subset F_{p+q}A$ for all $p,q\geq 0,$

  3. (3)

$A=\bigcup_{p\geq 0}F_{p}A.$

Let $A$ be a filtered algebra. Set
$\text{gr}A=\oplus_{n=0}^{\infty}F_{n}A/F_{n-1}A=\bigoplus_{n=0}^{\infty}(\text{gr}A)_{n}$.
The multiplication in $A$ gives rise to a well-defined product:

$(F_{p}A/F_{p-1}A)\times(F_{q}A/F_{q-1}A)\rightarrow F_{p+q}A/F_{p+q-1}A,\ \
(\bar{a},\bar{b})\mapsto\overline{a\cdot b},$

for all $a,b\in V$, making $\text{gr}A$ a graded associative algebra.

######  Definition 4.9.

A filtered algebra $A$ over a field $\mathbb{F}$ is called finite-type if
$\text{gr}A$ is a finitely generated commutative algebra.

######  Lemma 4.10.

(Quillen’s theorem) Let $A$ be a finite-type filtered algebra over a field
$\mathbb{F}$. Then $A$ has the endomorphism property. i.e., if $M$ is a simple
$A$-module, then every element in $\text{End}_{A}(M)$ is algebraic over
$\mathbb{F}$.

The following Proposition, which is a stronger version of Quillen’s theorem,
might be well-known, but we cannot find any reference to this version. So we
present the proof for it in the rest of this section.

######  Proposition 4.11.

Let $A$ be a finite-type filtered algebra over $\mathbb{F}$. Then $A$ has the
strong endomorphism property. i.e., if $M$ is a simple $A$-module, then
$\text{dim}_{\mathbb{F}}\text{End}_{A}(M)<\infty.$

######  Remark 4.12.

In the case when $\text{Char}(\mathbb{F})=0$ and $A=U(\mathfrak{g})$ the
universal enveloping algebra of a finite dimensional Lie algebra
$\mathfrak{g}$ over $\mathbb{F}$, Proposition 4.11 is proved in [D,
Proposition 2.6.9]. On the other hand, when $\mathbb{F}$ is of arbitrary
characteristic and $A$ is a constructible algebra over $\mathbb{F}$,
Proposition 4.11 is proved in [MR, Theorem 9.5.5]. Note that $U(\mathfrak{g})$
is a constructible algebra when $\mathfrak{g}$ is a finite-dimensional Lie
algebra over $\mathbb{F}$.

In our case, we cannot assume that $\text{gr}A$ is generated by
$F_{1}A/F_{0}A$, since we will let $A$ be Zhu’s algebra $A(V)$ later. So $A$
might not be a constructible algebra. But we can still prove Proposition 4.11
using some techniques in [MR].

######  Lemma 4.13.

([MR]) Let $D$ be a division ring whose center is $\mathbb{K}$, and let
$\mathbb{E}$ be a maximal subfield of $D$. If
$\text{dim}_{\mathbb{K}}(\mathbb{E})<\infty$, then
$\text{dim}_{\mathbb{E}}(D)<\infty$ and $\text{dim}_{\mathbb{K}}(D)<\infty.$

The proof of the following Lemma is fairly standard (see [MR]).

######  Lemma 4.14.

If $A$ is a filtered algebra such that $\text{gr}(A)$ is left noetherian, then
$A$ is left noteherian.

The next Lemma shows that the finite-type property and the notherianess are
preserved under field extensions.

######  Lemma 4.15.

Let $A$ be a finite-type filtered algebra over a field $\mathbb{F}$. Let
$\mathbb{K}/\mathbb{F}$ be a field extension. Then

  1. (1)

$A^{\mathbb{K}}$ is also a finite-type filtered algebra.

  2. (2)

$A^{\mathbb{K}}$ is noetherian. In particular, $A$ is noetherian.

###### Proof.

Let $0=F_{-1}A\subset F_{0}A\subset F_{1}A\subset\cdots$ be a filtration of
$A$. Then

$0=\mathbb{K}\otimes F_{-1}A\subset\mathbb{K}\otimes
F_{0}A\subset\mathbb{K}\otimes F_{1}A\subset\cdots$

is a filtration of $A^{\mathbb{K}}$, here $\otimes$ means
$\otimes_{\mathbb{F}}$. It is clear that
$\text{gr}(A^{\mathbb{K}})\cong\mathbb{K}\otimes_{\mathbb{F}}\text{gr}(A)$.
Hence $\text{gr}(A^{\mathbb{K}})$ is a finitely generated commutative algebra.
Consequently, $A^{\mathbb{K}}$ is a finite-type filtered algebra and
$\text{gr}A^{\mathbb{K}}$ is left noetherian. By Lemma 4.14, $A^{\mathbb{K}}$
is left noetherian.

To prove $A^{\mathbb{K}}$ is right noetherian, we consider the opposite ring
$(A^{\mathbb{K}})^{\text{op}}$ of $A^{\mathbb{K}}$. Note that $A^{\mathbb{K}}$
is right noetherian if and only if $(A^{\mathbb{K}})^{\text{op}}$ is left
noetherian. It is easy to verify that $(F_{n}A^{\mathbb{K}})_{n=0}^{\infty}$
is also a filtration of $(A^{\mathbb{K}})^{\text{op}}$, where
$F_{n}A^{\mathbb{K}}=K\otimes_{\mathbb{F}}F_{n}(A)$ for each $n\geq 0$. Since
$\text{gr}A^{\mathbb{K}}$ is commutative,
$\text{gr}((A^{\mathbb{K}})^{\text{op}})\cong\text{gr}A^{\mathbb{K}}$ is a
finitely generated commutative algebra. Therefore,
$(A^{\mathbb{K}})^{\text{op}}$ is left noetherian, and so $A^{\mathbb{K}}$ is
right noetherian. The proof is complete. ∎

The following lemma is the key to prove Proposition 4.11.

######  Lemma 4.16.

([MR]) Let $A$ be a $\mathbb{F}$-algebra, $M$ be a finite generated
$A$-module, and $\mathbb{K}$ be a subfield of $\text{End}_{A}(M)$ containing
$\mathbb{F}$. If $A^{\mathbb{K}}$ is left noetherian, then $\mathbb{K}$ is a
finitely generated field extension of $\mathbb{F}$.

We now return to the proof of Proposition 4.11.

######  Proof of Proposition 4.11.

Let $M$ be a simple $A$-module and $D=\text{End}_{A}(M)$. Then $D$ is a
division ring. Let $\mathbb{K}$ be the center of $D$, and let $\mathbb{E}$ be
a maximal subfield of $D$.

Note that $\mathbb{F}\leq\mathbb{K}\leq\mathbb{E}.$ By Lemma 4.15,
$A^{\mathbb{E}}$ is left noetherian. Since $M$ is a simple $A$-module, it is
finitely generated. Then by Lemma 4.16, $\mathbb{E}$ is a finitely-generated
field extension of $\mathbb{F}$. Moreover, every element in $D$ is algebraic
over $\mathbb{F}$ by Quillen’s theorem, then $\mathbb{E}/\mathbb{F}$ is a
finitely generated algebraic extension, hence $\mathbb{E}/\mathbb{F}$ is a
finite extension. In particular, $\text{dim}_{\mathbb{K}}(\mathbb{E})<\infty$
and $\text{dim}_{\mathbb{F}}(\mathbb{K})<\infty$. By Lemma 4.13, we have
$\text{dim}_{\mathbb{K}}(D)<\infty$. Consequently,
$\text{dim}_{\mathbb{F}}(D)<\infty$, and the proof is complete. ∎

##  5 Strong endomorphism property of $A(V)$

Using the results from previous sections, we can prove that Zhu’s algebra
$A(V)$ of a strongly finitely generated vertex operator algebra $V$ over an
arbitrary field $\mathbb{F}$ satisfies the strong endomorphism property.

To proof this theorem, we first review the relations between the commutative
algebras $R(V)$ and $\text{gr}A(V)$ studied in [A, L1, L2, Liu, Z].

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$. Recall that
$R(V)=V/C_{2}(V)$ is a commutative Poisson algebra (cf. [Z]), with the
identity element $\bar{{\bf 1}}$, under the product and Lie bracket:

$\overline{u}\cdot\overline{v}=\overline{u_{-1}v},\qquad\\{\bar{u},\bar{v}\\}=\overline{u_{0}v},\quad\text{for}\
u,v\in V,$ (5.1)

where $\overline{u}$ represents the image of $u$ in $R(V)$.

For any $n\in{\mathbb{Z}}_{+}$, we define
$F_{n}(A(V))=A(V)_{n}=(V_{0}+\cdots+V_{n}+O(V))/O(V)\subseteq A(V)$(cf. [Z]).
It is easy to verify that $(A(V)_{n})_{n=0}^{\infty}$ is a filtration of
$A(V)$. Hence $A(V)$ is a filtrated algebra. Moreover, by the following
property of $A(V)$:

$u\ast v-v\ast u\equiv\sum_{i\geq 0}\binom{\mathrm{wt}u}{i}u_{i}v\pmod{O(V)},$

we can show that $\text{gr}A(V)$ is also a commutative Poisson algebra [A,
Liu]. Define a linear map $\phi:R(V)\rightarrow\text{gr}A(V)$ by letting:

$\phi(\bar{u}):=u+A(V)_{n-1}\in A(V)_{n}/A(V)_{n-1},$

for $n\in{\mathbb{Z}}_{+}$ and $u\in V_{n}$. Then $\phi$ is a well-defined
epimorphism between the commutative Poisson algebras $R(V)$ and
$\text{gr}A(V)$ (see [CJL, Liu]).

The following lemma is well-known (see [A, L1, L2]). We include the proof of
it for the completeness.

######  Lemma 5.1.

Let $V$ be a strongly finitely generated vertex operator algebra over
$\mathbb{F}$. Then:

  1. (1)

$R(V)$ is a finitely generated commutative algebra. In particular, $R(V)$ is
noetherian.

  2. (2)

$A(V)$ is a finite-type filtered algebra.

###### Proof.

(1) Let $U$ be a finite set that strongly generates $V$. Since

$a_{-n}b_{-2}c=b_{-2}a_{-n}c+\sum_{j\geq
0}\binom{-n}{j}(a_{j}b)_{-n-2-j}c\equiv b_{-2}a_{-n}c\pmod{C_{2}(V)},$

for any $a,b,c\in V$ and $n\geq 0$. It follows that for any spanning element
$u^{1}_{-n_{1}}\dots u^{r}_{-n_{r}}{\bf 1}$ of $V$, where $u^{i}\in U$ and
$n_{i}\geq 1$ for all $i$, if there exists some $i$ such that $n_{i}\geq 2$,
then $u^{1}_{-n_{1}}\dots u^{r}_{-n_{r}}{\bf 1}\in C_{2}(V)$. Then by (5.1),
we have $R(V)$ is generated by $(U+C_{2}(V))/C_{2}(V)$, and so $R(V)$ is
noetherian by Hilbert’ basis theorem.

(2) Since $\phi$ is an epimorphism between the commutative algebras $R(V)$ and
$\text{gr}A(V)$, it follows that $\text{gr}A(V)$ as a commutative algebra is
also finitely generated, and so $A(V)$ is a finite-type filtered algebra. ∎

Now we prove the main theorem of this section:

######  Theorem 5.2.

If $V$ is a strongly finitely generated vertex operator algebra over a field
$\mathbb{F}$, and $\mathbb{K}/\mathbb{F}$ is a field extension, then
$A(V)^{\mathbb{K}}$ has the strong endomorphism property. In particular,
$A(V)$ has the strong endomorphism property.

###### Proof.

Because $V$ is strongly finitely generated, $V^{\mathbb{K}}$ is also strongly
finitely generated by Proposition 3.4. This implies that
$A(V)^{\mathbb{K}}\cong A(V^{\mathbb{K}})$ is a finite-type filtered algebra
by Proposition 4.6 and Lemma 5.1. Now it follows from Proposition 4.11 that
$A(V)^{\mathbb{K}}$ has the strong endomorphism property. ∎

##  6 The endomorphism property of vertex operator algebras

In this section, we will prove the main results of the paper: Theorem 6.4 and
Corollary 6.10.

######  Lemma 6.1.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$. Then the
following conditions are equivalent:

  1. (1)

$A(V)$ has the endomorphism property.

  2. (2)

$V$ has the endomorphism property.

  3. (3)

For any $n\in{\mathbb{Z}}_{+}$, $A_{n}(V)$ has the endomorphism property.

###### Proof.

(1) $\Rightarrow$ (2). Let $M$ be any irreducible admissible $V$-module over
$\mathbb{F}$. Note that $M(0)$ is a simple $A(V)$-module. By Corollary 4.7, we
have $\text{End}_{V}(M)\cong\text{End}_{A(V)}(M(0))$. Since $A(V)$ has the
endomorphism property, every element in $\text{End}_{V}(M)$ is algebraic over
$\mathbb{F}$. Thus, $V$ has the endomorphism property as well.

(2) $\Rightarrow$ (3). Let $U$ be a simple $A_{n}(V)$-module. By (3) of
Theorem 4.4, $L_{n}(U)$ is an irreducible admissible $V$-module. By Corollary
4.7 again, we have $\text{End}_{V}(L_{n}(U))\cong\text{End}_{A_{n}(V)}(U)$.
Hence each element in $\text{End}_{A_{n}(V)}(U)$ is algebraic over
$\mathbb{F}$.

(3) $\Rightarrow$ (1). Let $U$ be a simple $A(V)$-module. By Theorem 4.2, we
know that $A(V)$ is a quotient algebra of $A_{n}(V)$, hence $U$ is also a
simple $A_{n}(V)$-module and
$\text{End}_{A(V)}(U)\cong\text{End}_{A_{n}(V)}(U)$. Hence each element in
$\text{End}_{A(V)}(U)$ is algebraic over $\mathbb{F}$. ∎

By adopting a similar argument, we can prove the equivalency of the strong
endomorphism properties:

######  Lemma 6.2.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$. Then the
following conditions are equivalent:

  1. (1)

$A(V)$ has the strong endomorphism property.

  2. (2)

$V$ has the strong endomorphism property.

  3. (3)

For any $n\in{\mathbb{Z}}_{+}$, $A_{n}(V)$ has the strong endomorphism
property.

Before we prove the main theorem, we give some examples of vertex operator
algebras with the strong endomorphism property. Note that the vertex operator
algebras in the following example are all strongly finitely generated, but
none of them are $C_{2}$-cofinite.

######  Example 6.3.

Let $\mathbb{F}$ be an arbitrary field with $\text{Char}(\mathbb{F})=0$.

(1) If $V=\bar{V}_{vir}(c,0)$ is the (universal) Virasoro vertex operator
algebra over $\mathbb{F}$ with the central charge $c$. It is well-known that
$A(V)\cong\mathbb{F}[x]$ (see [W]). Since $\mathbb{F}[x]$ is a principal ideal
domain, every simple $\mathbb{F}[x]$-module is finite-dimensional, then it
follows that $A(V)$ has the strong endomorphism property, and so
$\bar{V}_{vir}(c,0)$ also has the strong endomorphism property by Lemma 6.2.

(2) If $V=M_{\mathfrak{\hat{h}}}(1,0)$ is the level one Heisenberg vertex
operator algebra ([FZ]) over $\mathbb{F}$ associated with the $n$-dimensional
vector space $\mathfrak{h}$, then $A(V)\cong\mathbb{F}[x_{1},\cdots,x_{n}]$.
Let $S$ be any simple $\mathbb{F}[x_{1},\cdots,x_{n}]$-module. Then
$S\cong\mathbb{F}[x_{1},\cdots,x_{n}]/J$, for some maximal ideal of
$\mathbb{F}[x_{1},\cdots,x_{n}]$. Clearly, $\mathbb{F}[x_{1},\cdots,x_{n}]/J$
is a field extension of $\mathbb{F}$, and it is a finitely generated
$\mathbb{F}$-algebra. Hence $\mathbb{F}[x_{1},\cdots,x_{n}]/J$ is a finte
algebraic extension of $\mathbb{F}$ by Hilbert’s Nullstellensatz.
Consequently, every simple $A(V)$-module is finite-dimensional. Hence,
$M_{\mathfrak{\hat{h}}}(1,0)$ has the strong endomorphism property by Lemma
6.2.

(3) Let $\mathfrak{g}$ be a finite-dimensional simple Lie algebra over a field
$\mathbb{F}$, equipped with a nondegenerate symmetric invariant bilinear form
$\langle\cdot,\cdot\rangle$. Assume that the Casimir operator $\Omega$ acts on
$\mathfrak{g}$ as a scalar $2h\in\mathbb{F}$, with $k\neq-h$. Let
$V=V_{\mathfrak{\hat{g}}}(k,0)$ be the level $k$ vacuum module vertex operator
algebra over $\mathbb{F}$ associated with the finite-dimensional simple Lie
algebra $\mathfrak{g}$ ([FZ, LL]). Then $A(V)\cong U(\mathfrak{g})$, the
universal enveloping algebra of $\mathfrak{g}$ (see [FZ]). It is well-known
that $U(\mathfrak{g})$ has the strong endomorphism property ([D, MR]). Hence
$V_{\mathfrak{\hat{g}}}(k,0)$ has the strong endomorphism property by Lemma
6.2.

######  Theorem 6.4.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$.

  1. (1)

If $\mathbb{F}$ is an uncountable field, then $V$ has the endomorphism
property.

  2. (2)

If $V$ is $C_{2}$-cofinite, then $V$ has the strong endomorphism property.

  3. (3)

If $\mathbb{F}={\mathbb{R}}$ the real number field, then $V$ has the strong
endomorphism property.

  4. (4)

(Quillen’s theorem) If $V$ is strongly finitely generated over $\mathbb{F}$,
then $V$ has the strong endomorphism property.

###### Proof.

(1) By Lemma 6.1, it is enough to prove that $A(V)$ has the endomorphism
property. Let $S$ be a simple $A(V)$-module. Recall that $V$ is of countable
dimension over $\mathbb{F}$. Then $A(V)$, as a quotient space of $V$, also has
countable dimension. Thus $S$ has countable dimension over $\mathbb{F}$ as
well.

Let $\theta\in\text{End}_{A(V)}(S)$, we adopt a similar argument in [Q] to
show that $\theta$ is algebraic. Suppose $\theta$ is not algebraic, then $S$
is a nonzero vector space over $\mathbb{F}(\theta)$, and
$\text{dim}_{\mathbb{F}}(S)\geq\text{dim}_{\mathbb{F}}(\mathbb{F}(\theta))\geq\text{card}\leavevmode\nobreak\
\mathbb{F}$, where the last inequality follows from the fact that the elements
of the form $(\theta-\lambda)^{-1}\ (\lambda\in\mathbb{F})$ in
$\mathbb{F}(\theta)$ are linearly independent. This contradicts the assumption
that $\mathbb{F}$ is uncountable. Hence $\theta$ must be algebraic.

(2) If $V$ is $C_{2}$-cofinite, then by the epimorphism
$\phi:R(V)\rightarrow\mathrm{gr}(A(V))$ in Section 5 we know that
$\mathrm{gr}(A(V))$, as well as $A(V)$, are finite-dimensional. Naturally,
$A(V)$ has the strong endomorphism property. Then by Lemma 6.2, $V$ also has
the strong endomorphism property.

(3) By Lemma 6.2 again, it is enough to prove that $A(V)$ has the strong
endomorphism property. Let $S$ be any simple $A(V)$-module, and let
$D=\text{End}_{A(V)}(S)$. Since ${\mathbb{R}}$ is an uncountable field, every
element in $D$ is algebraic over ${\mathbb{R}}$ by (1). Consequently, $D$ is a
division algebra over ${\mathbb{R}}$ such that every element in $D$ is
algebraic over ${\mathbb{R}}$. Hence $D$ must be isomorphic to
${\mathbb{R}},{\mathbb{C}}$ or $\mathbb{H}$, where $\mathbb{H}$ is the
quaternion algebra (see [H]). This implies that
$\text{dim}_{{\mathbb{R}}}(D)<\infty$, as desired.

(4) Since $V$ is strongly finitely generated, it follows from Theorem 5.2 that
$A(V)$ has the strong endomorphism property. So does $V$ by Lemma 6.2. ∎

######  Remark 6.5.

The $C_{2}$-cofinite CFT-type vertex operator algebras over the complex number
field ${\mathbb{C}}$ are all strongly finitely generated (see [L1]). So part
(2) in Theorem 6.4 is a special case of part (4) when
$\mathbb{F}={\mathbb{C}}$ and $V$ is of the CFT-type.

The following corollary is an immediate consequence of Lemma 6.2 and Theorem
6.4:

######  Corollary 6.6.

(1) If $V$ is a vertex operator algebra over the real number field
${\mathbb{R}}$, then $A_{n}(V)$ has the strong endomorphism property.

(2) If $V$ is a strongly finitely generated vertex operator algebra over an
arbitrary field $\mathbb{F}$, then $A_{n}(V)$ has the strong endomorphism
property.

######  Remark 6.7.

If $V$ is a strongly finitely generated vertex operator algebra over a field
$\mathbb{F}$, by Lemma 4.15 and Lemma 5.1(2), we know that $A(V)$ is stably
noetherian. Moreover, by Corollary 6.6, $A_{n}(V)$ has the strong endomorphism
property for any $n\in{\mathbb{Z}}_{+}$. A natural question is whether
$A_{n}(V)$ is also stably noetherian.

######  Proposition 6.8.

Let $V$ be a vertex operator algebra over a field $\mathbb{F}$, with the
strong endomorphism property. Then

  1. (1)

If $V$ is rational, then $A_{n}(V)$ is a finite dimensional semi-simple
associative algebra for each $n\in{\mathbb{Z}}_{+}$.

  2. (2)

If $V$ is simple, then $V$ is rational if and only if $A_{n}(V)$ is finite-
dimensional semi-simple for each $n\in{\mathbb{Z}}_{+}$.

###### Proof.

(1) If $V$ is rational, then by Theorem 4.2 (4), $A_{n}(V)$ is a semi-simple
left $A_{n}(V)$-module for each $n\in{\mathbb{Z}}_{+}$. Thus $A_{n}(V)$ is an
aritinian algebra. Let $\\{S_{1},\cdots,S_{k}\\}$ be the set of the
isomorphism classes of simple $A_{n}(V)$-modules, then by Artin-Wedderburn
theorem,

$A_{n}(V)^{op}\cong\bigoplus_{i=1}^{k}\text{M}_{n_{i}}(D_{i})\ \text{for
some}\ n_{i}\in{\mathbb{Z}}_{+},\ \text{where}\
D_{i}=\text{End}_{A_{n}(V)}(S_{i}),\ \text{for\ all\ }i.$

Since $V$ has the strong endomorphism property, we have
$\text{dim}_{\mathbb{F}}(D_{i})<\infty$ for all $i$. Consequently, $A_{n}(V)$
is a finite dimensional semi-simple associative algebra for each
$n\in{\mathbb{Z}}_{+}$.

(2) follows immediately from (1) and Theorem 4.2 (5). ∎

######  Proposition 6.9.

Let $\mathbb{F}$ be an algebraically closed field, and let $V$ be a vertex
operator algebra over $\mathbb{F}$, with the endomorphism property. Let
$M=\bigoplus_{n\in{\mathbb{Z}}_{+}}M(n)$ be an irreducible admissible
$V$-module. Then

  1. (1)

$\text{End}_{V}(M)\cong\mathbb{F}$. Moreover, there exists a
$\lambda\in\mathbb{F}$ such that $L(0)|_{M(n)}=\lambda+n$ for all $n$.

  2. (2)

If $V$ is rational, then $A_{n}(V)$ is a finite dimensional semi-simple
associative algebra for every $n\in{\mathbb{Z}}_{+}$.

###### Proof.

The proof of (1) is straightforward (see also [LL, Proposition 4.5.5]). It
follows from (1) that $V$ has the strong endomorphism property. Hence (2) is
an immediate consequence of Proposition 6.8. ∎

By the previous results, Theorem 6.4 (4), Proposition 6.8, and Proposition
6.9, we have the following:

######  Corollary 6.10.

Let $V$ be a strongly finitely generated vertex operator algebra over a field
$\mathbb{F}$, and let $M=\bigoplus_{n\in{\mathbb{Z}}_{+}}M(n)$ be an
irreducible admissible $V$-module. Then we have:

  1. (1)

(Quillen’s theorem) $\text{End}_{V}(M)<\infty$.

  2. (2)

(Schur’s lemma) If $\mathbb{F}$ is an algebraic closed field, then
$\text{End}_{V}(M)\cong\mathbb{F}$. Moreover, there exists a
$\lambda\in\mathbb{F}$ such that $L(0)|_{M(n)}=\lambda+n$ for all $n$.

  3. (3)

If $V$ is rational, then $A_{n}(V)$ is a finite-dimensional semi-simple
associative algebra for any $n\geq 0$.

######  Remark 6.11.

Let $V$ be a vertex operator algebra over an arbitrary field $\mathbb{F}$. It
is well-known that if $\mathbb{F}$ is uncountable and algebraically closed,
then Schur’s Lemma in Corollary 6.10 (2) is satisfied. In [DR1], the authors
made the conjecture that the assumption $\mathbb{F}$ is uncountable in Schur’s
Lemma is unnecessary.

Now by Corollary 6.10, we can drop the uncountable assumption for $\mathbb{F}$
in Schur’s Lemma for strongly finitely generated vertex operator algebra $V$.

##  7 Applications: Scalar Extensions

In [M], the author studied the vertex operator algebras $V$ over
${\mathbb{R}}$, and proved that
$V^{\mathbb{C}}={\mathbb{C}}\otimes_{\mathbb{R}}V$ is rational if $V$ is
rational over ${\mathbb{R}}$. As an application of Proposition 6.8, we will
prove that the inverse is also true.

######  Theorem 7.1.

Let $\mathbb{K}/\mathbb{F}$ be a finite separable field extension, and $V$ be
a simple vertex operator algebra over $\mathbb{F}$ s.t. $V_{0}=\mathbb{F}{\bf
1}$. Assume that $V$ and $V^{\mathbb{K}}$ both have the strong endomorphism
property. Then $V$ is rational if and only if $V^{\mathbb{K}}$ is rational.

###### Proof.

The “if" part: Since $V^{\mathbb{K}}$ is rational and has the strong
endomorphism property, then by Proposition 6.8,
$A_{n}(V^{\mathbb{K}})\cong\mathbb{K}\otimes A_{n}(V)$ is a finite-dimensional
semi-simple associative algebra for any $n\in{\mathbb{Z}}_{+}$. Since
$\text{dim}_{\mathbb{F}}(A_{n}(V))=\text{dim}_{\mathbb{K}}(\mathbb{K}\otimes
A_{n}(V))$, we have $\text{dim}_{\mathbb{F}}(A_{n}(V))<\infty$. We claim that
$A_{n}(V)$ is also semi-simple for each $n\in{\mathbb{Z}}_{+}$. Indeed, we
just need to show that $A_{n}(V)$ has no nonzero nilpotent ideal since it is
already finite-dimensional. Suppose that $A_{n}(V)$ has a nonzero nilpotent
ideal $I$, then $\mathbb{K}\otimes I$ is a nonzero nilpotent ideal of
$A_{n}(V^{\mathbb{K}})$, which contradicts the fact that
$A_{n}(V^{\mathbb{K}})$ is semi-simple. Hence each $A_{n}(V)$ is semi-simple,
and so $V$ is rational by Theorem 4.2(5).

The “only if" part: If $V$ is rational, then by Proposition 6.8 again,
$A_{n}(V)$ is a finite-dimensional semi-simple associative algebra for each
$n\in{\mathbb{Z}}_{+}$. Since $\mathbb{K}/\mathbb{F}$ is a finite separable
extension, it follows from Lemma 3.3 that
$A_{n}(V^{\mathbb{K}})\cong\mathbb{K}\otimes A_{n}(V)$ is also finite
dimensional semi-simple over $\mathbb{K}$. Moreover, $V^{\mathbb{K}}$ is a
simple vertex operator algebra over $\mathbb{K}$ by Corollary 3.7. Hence
$V^{\mathbb{K}}$ is rational by Theorem 4.2(5). ∎

######  Remark 7.2.

When $\mathbb{K}/\mathbb{F}$ is a finite separable extension, we believe that
$V$ has the strong endomorphism property if and only if $V^{\mathbb{K}}$ has
the strong endomorphism property. But we do not know how to prove it.

######  Corollary 7.3.

Let $V$ be a simple vertex operator algebra over the real number field
${\mathbb{R}}$, with $V_{0}={\mathbb{R}}{\bf 1}$. Then $V$ is rational if and
only if $V^{{\mathbb{C}}}$ is rational.

###### Proof.

By Theorem 6.4 (3), any vertex operator algebra $V$ over ${\mathbb{R}}$ has
the strong endomorphism property, and by Proposition 6.9 (1), $V^{\mathbb{C}}$
also has the strong endomorphism property. Moreover, the extension
${\mathbb{C}}/{\mathbb{R}}$ is clearly finite separable. Then by Theorem 7.1,
we have $V$ is rational if and only if $V^{\mathbb{C}}$ is rational. ∎

######  Corollary 7.4.

Let $\mathbb{K}/\mathbb{F}$ be a finite separable extension, and $V$ be a
simple strongly finitely generated vertex operator algebra over a field
$\mathbb{F}$ such that $V_{0}=\mathbb{F}{\bf 1}$. Then $V$ is rational if and
only if $V^{\mathbb{K}}$ is rational.

###### Proof.

Note that $V^{\mathbb{K}}$ is also strongly finitely generated by Proposition
3.4. By Corollary 6.10 (1), both $V$ and $V^{\mathbb{K}}$ have the strong
endomorphism property. Now the conclusion follows from Theorem 7.1. ∎

######  Theorem 7.5.

Let $V$ be a strongly finitely generated vertex operator algebra over a field
$\mathbb{F}$, and $M$ be an irreducible admissible $V$-module. Then the
following conditions are equivalent:

  1. (1)

$M$ is an absolutely irreducible $V$-module.

  2. (2)

$M(0)$ is an absolutely simple $A(V)$-module.

  3. (3)

$\text{End}_{V}(M)=\text{End}_{A(V)}(M(0))=\mathbb{F}.$

###### Proof.

By Lemma 3.6 and Corollary 4.7, it is enough to prove that $M(0)$ is
absolutely simple if and only if $\text{End}_{A(V)}(M(0))=\mathbb{F}.$ The
following proof is similar to [D, Proposition 2.6.5].

Assume that $\text{End}_{A(V)}(M(0))=\mathbb{F}$. We claim that
$M(0)^{\mathbb{K}}$ is a simple $A(V)^{\mathbb{K}}$-module for any field
extension $\mathbb{K}/\mathbb{F}$. Indeed, let $N$ be a nonzero submodule of
$M(0)^{\mathbb{K}}$. Take a nonzero element $k_{1}\otimes
m_{1}+\cdots+k_{p}\otimes m_{p}\in N\ (k_{i}\in\mathbb{K},m_{i}\in M(0),p>0)$
such that $m_{1},\dots,m_{p}$ are linearly independent over
$\mathbb{F}=\mathrm{End}_{A(V)}(M(0))$. Since $M(0)$ is a simple
$A(V)$-module, by the Jacobson’s density theorem, there exists some $v\in
A(V)$ such that $[v]\cdot m_{1}=o(v)m_{1}=m_{1}$ and $[v]\cdot
m_{i}=o(v)m_{i}=0$ for any $i\neq 1$. Hence $o(k_{1}^{-1}\otimes
v)(k_{1}\otimes m_{1}+\cdots+k_{p}\otimes m_{p})=1\otimes m_{1}\in N.$ Since
$M(0)$ is a simple $V$-module, we have $M(0)=\\{o(v)m_{1}\leavevmode\nobreak\
|\leavevmode\nobreak\ v\in V\\}.$ Therefore, we have $M(0)^{\mathbb{K}}=N$,
and $M(0)^{\mathbb{K}}$ is a simple $A(V)^{\mathbb{K}}$-module.

Now we assume that $M(0)$ is absolutely simple. For any
$f\in\text{End}_{A(V)}(M(0)),$ we need to show that $f=\lambda Id_{M(0)}$ for
some $\lambda\in F$. Let $\overline{\mathbb{F}}$ be an algebraic closure of
$\mathbb{F}$. Then
$\text{End}_{A(V)^{\mathbb{K}}}(M(0)^{\mathbb{K}})=\overline{\mathbb{F}}$ by
Corollary 6.10. On the other hand, $1\otimes
f\in\text{End}_{A(V)^{\mathbb{K}}}(M(0)^{\mathbb{K}}),$ so $1\otimes
f=\lambda\otimes Id_{M(0)}$ for some $\lambda\in\overline{\mathbb{F}}.$ This
implies that $\lambda\in\mathbb{F}$ and $f=\lambda Id_{M(0)}.$ The proof is
complete. ∎

######  Remark 7.6.

The Theorem 7.5 is useful, especially because it provides an intrinsic test
for a module to be absolutely simple, which involves only the computation of
$\text{End}_{V}(M)$ or $\text{End}_{A(V)}(M(0))$, rather than the
consideration of extension fields.

######  Corollary 7.7.

Let $V$ be a strongly finitely generated vertex operator algebra over an
algebraically closed field $\mathbb{F}$, and let $M$ be a simple admissible
$V$-module. Then $M$ is absolutely simple.

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