# OATS: Opinion Aspect Target Sentiment Quadruple Extraction Dataset for
Aspect-Based Sentiment Analysis

Siva Uday Sampreeth Chebolu1, Franck Dernoncourt2,
Nedim Lipka3, Thamar Solorio1,4
1University of Houston, 4800 Calhoun Rd, Houston, TX, 77004, USA
2Adobe Research, 801 N 34th St, Seattle, WA 98103, USA
3Adobe Research, 321 Park Ave, San Jose, CA, 95110, USA
4MBZUAI, Masdar City, Abu Dhabi, UAE
<EMAIL_ADDRESS>
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Aspect-based sentiment Analysis (ABSA) delves into understanding sentiments
specific to distinct elements within textual content. It aims to analyze user-
generated reviews to determine a) the target entity being reviewed, b) the
high-level aspect to which it belongs, c) the sentiment words used to express
the opinion, and d) the sentiment expressed toward the targets and the
aspects. While various benchmark datasets have fostered advancements in ABSA,
they often come with domain limitations and data granularity challenges.
Addressing these, we introduce the OATS dataset, which encompasses three fresh
domains and consists of 20,000 sentence-level quadruples and 13,000 review-
level tuples. Our initiative seeks to bridge specific observed gaps: the
recurrent focus on familiar domains like restaurants and laptops, limited data
for intricate quadruple extraction tasks, and an occasional oversight of the
synergy between sentence and review-level sentiments. Moreover, to elucidate
OATS’s potential and shed light on various ABSA subtasks that OATS can solve,
we conducted in-domain and cross-domain experiments, establishing initial
baselines. We hope the OATS dataset augments current resources, paving the way
for an encompassing exploration of ABSA.

##  1 Introduction

User-generated reviews on e-commerce and other platforms are invaluable for
both consumers and product makers or service providers. These reviews provide
insights into past customer experiences, guiding potential consumers in their
decision-making process. On the other hand, producers and merchants benefit
from understanding user feedback on specific product features, allowing them
to strategize improvements. Given the explosive growth in the volume of online
reviews, discerning valuable insights from the vast amounts of data becomes
increasingly challenging. A local survey highlighted that a staggering 91% of
young consumers place significant trust in online reviews, often equating them
to personal recommendations Pitman (2022).

The trend in online reviews has shifted from a broad understanding of a
product’s overall performance to a more granular examination of individual
product aspects. This shift demands a different approach to analyzing reviews.
Aspect-based sentiment Analysis (ABSA) emerged as the answer to this nuanced
requirement, focusing on sentiment pertaining to specific aspects of an entity
Hu and Liu (2004a).

However, current datasets often fall short of capturing the complete spectrum
of ABSA. One of the significant limitations is the inability to perform joint
detection of all ABSA elements due to the absence of at least one critical
component in collected reviews. This limitation stunts the potential of ABSA
tasks. Furthermore, despite the SemEval datasets’ popularity, many of these
sentences often group multiple aspects under a single sentiment polarity. As
noted by Jiang et al. (2019), this approach simplifies the ABSA task,
effectively reducing it to mere sentence-level sentiment analysis. As a
solution, they proposed a new large-scale Multi-Aspect Multi-Sentiment (MAMS)
dataset, in which each phrase has at least two independent aspects with
different sentiment polarity. However, their approach of introducing
"miscellaneous" categories or neutral sentiments when a sentence contains only
one opinion tuple is impractical. While it adds complexity, it does not
necessarily reflect real-world sentiments in reviews.

Recently, ABSA datasets such as ACOS Cai et al. (2021) and ASQP Zhang et al.
(2021a) have been at the forefront, providing comprehensive annotations for
all four elements. However, they are largely limited to the long-standing
traditional domains of restaurants and laptops, a trend that has been observed
since 2014. In contrast, newer datasets like DM_ASTE and DE_ASTE (Domain
Expanded ASTE) have made strides by introducing annotations from fresh
domains, encompassing home appliances, fashion, groceries, and more, moving
beyond the typical restaurant and laptop reviews Xu et al. (2023); Chia et al.
(2023). Yet, these innovative datasets have their limitation: they lack the
critical aspect category annotations required for a holistic joint detection
of all elements.

Several crucial insights emerge in delving deep into the landscape of ABSA
datasets Chebolu et al. (2023). Firstly, there needs to be more consistency in
how ABSA components are defined and structured across various sources,
accentuating an urgent need for a universal standardized format. Secondly, the
challenge of accurately detecting opinion words, which are central to
discerning sentiment polarity and their corresponding aspect terms, still
needs to be solved. The current datasets, while extensive, do not always
reflect real-world complexities. It is imperative to have datasets
encompassing various domains or languages to achieve a truly comprehensive
ABSA. This need is further emphasized by recent advancements in unified models
using generative frameworks Chebolu et al. (2021); Zhang et al. (2021b),
highlighting the immense potential of models that jointly solve ABSA tasks.
These models necessitate datasets with richer annotations, similar to Zhang et
al. (2021a). Another stark observation is the overwhelming predominance of
English in ABSA datasets. This monopolization overlooks the vast linguistic
landscape, sidelining low-resource languages. Addressing this gap is vital for
a more inclusive and global ABSA application. Lastly, the prevailing ABSA
research seems overly fixated on a sentence-centric approach, often ignoring
that user reviews usually extend beyond single sentences. This calls for a
renewed focus on review-level ABSA datasets to capture sentiment in its entire
breadth and depth.

Given these challenges, the OATS dataset has been developed with a vision to
rectify existing gaps. The contributions of OATS are threefold:

  * •

Introduces both sentence-level quadruples and review-level tuples, capturing
sentiments in all their granularity.

  * •

The dataset spans multiple domains, ensuring its applicability across a
diverse range of ABSA tasks.

  * •

OATS will be publicly available in two formats: XML (as introduced by Pontiki
et al. (2014)), catering to detailed character-level annotations, and Text,
following the format set by Zhang et al. (2021a).

  * •

We provide a few baselines for our dataset, focusing on three primary tasks
(ASTE, TASD, and ASQP) from the 14 ABSA subtasks in two distinct settings: in-
domain ABSA and cross-domain ABSA.

##  2 OATS Dataset

In this section, we explore the resources behind the dataset and then discuss
the annotation process, its statistics, and its relevance to ABSA and the
wider NLP community.

###  2.1 Data Resources and Annotation Procedure

We first describe the sources from which we gathered the data for our corpus,
followed by a detailed account of the annotation procedure.

####  2.1.1 Data Resources

We derived three distinct datasets from multiple sources. Our corpus is
primarily built from text reviews available on Kaggle (consistent with the CCO
and ODbL licenses) and materials provided by Yin et al. (2017).

##### Amazon Fine Foods Dataset

We refer to this dataset as Amazon_FF, and is extracted from a Kaggle
competition 111https://www.kaggle.com/snap/amazon-fine-food-reviews containing
around 500k reviews of fine foods from Amazon. These reviews span topics such
as the quality of food or products, promptness of delivery, packaging
standards, and product availability, among others. We curated 1521 complete
reviews from this dataset, consisting of over 8200 sentences and approximately
5600 opinion quadruples.

##### Coursera Dataset

Originating from a Kaggle competition, this dataset contains reviews scraped
from the Coursera website, totaling close to 100k reviews. These reviews
reflect diverse perspectives on the course’s quality, content,
comprehensiveness, and the alignment of faculty lessons with course content.
For our work, we selected 1211 comprehensive reviews, which include roughly 6K
sentences and about 7K opinion quadruples.

##### TripAdvisor Dataset

This dataset is based on data from the Yin et al. (2017), featuring over 100k
hotel reviews. The reviews comment on various facets of the hotel experience,
such as pricing, design attributes, and more. We collated 1206 complete
reviews from this extensive collection, resulting in approximately 6.3K
sentences and nearly 8.3K opinion quadruples.

####  2.1.2 Annotation Procedure

After the data collection, we utilized the services of annotators from the
Upwork platform and employed the BRAT tool for annotation. Three annotators
were chosen for this task. We provided them with a guidelines document to
acquaint themselves with the topic, dataset, BRAT tool, and specific
annotation requirements. Initially, annotator (A) took the lead in annotating
a subset of the data, which was subsequently reviewed by annotator (B) for
corrections. This process was iterated twice, with pairs (A, C) and then (B,
C) for checks and balances. Any emerging disagreements were addressed and
resolved through consultations with one of our NLP experts. The remaining
dataset sentences were equally divided among the three annotators. They were
also given supplementary instructions derived from earlier discrepancies. When
consensus was elusive, collective decisions were made. When all three
annotators held differing views, a consensus was achieved in collaboration
with an expert annotator. Majority voting was the chosen mechanism to finalize
the annotations. For clarity, an illustrative example of an opinion quadruple
from the Coursera domain in BRAT format and the finalized XML format of
sentence-level and review-level annotations of a review are presented in
Figure 4 (located in the Appendix).

####  2.1.3 Inter-Annotator Agreement

According to Hripcsak and Rothschild (2005), the infamous Kappa metric Tomanek
and Hahn (2009); Becker et al. (2005) may not be the best fit for span-
extraction annotation in textual data to measure the inter-annotator
agreement. The limitation arises from the requirement of Kappa to compute the
number of negative cases, which is unidentifiable for spans as they constitute
sequences of words without a predetermined quantity of items for annotation in
a text. The F-measure is often more suitable for gauging inter-annotator
agreement in span extraction annotation tasks such as target and opinion
phrase extraction Deléger et al. (2012). We can calculate the F1-score by
considering one annotator’s annotations as the reference (or gold annotations
or groundtruth) and the other’s as a system’s responses (or predictions). We
computed the inter-annotator agreement scores, following this F1 metric, for
each domain using several combinations of the ABSA elements as shown in Table
1. The average quadruple extraction agreement F1 score for the three domains
is 69.39%, indicating high consistency among the annotators.

Dataset | Tgt. Span | Opi. Ph. Span | Asp-Sen | Quadruple
---|---|---|---|---
Amazon_FF | 72.57 | 69.72 | 85.43 | 65.78
Coursera | 78.26 | 71.26 | 79.63 | 68.56
Hotels | 74.78 | 72.05 | 87.32 | 73.84

Table 1: Inter-Annotator agreement F1-scores for the OATS Datasets. Tgt. Span:
Target span extraction. Opi. Ph. Span: Opinion Phrase span extraction. Asp-
Sent: Aspect Category and Sentiment combination categorization.

###  2.2 Dataset Statistics

We provide the statistics for the ASQP dataset Zhang et al. (2021a), which is
the only dataset with all the elements of ABSA together in Table 2.2 for
comparison with the proposed dataset. We present several statistics in this
section that give an overview of the OATS dataset. We provide both the
sentence-level and review/text-level statistics for all the domains in Tables
2.2, 2.2, 2.2, and 4.

2  Dataset | #sents | #pos | #neg | #neu | #Total
---|---|---|---|---|---
2 Rest-15 | 1562 | 1710 | 701 | 85 | 2496
Rest-16 | 2024 | 2293 | 877 | 125 | 3295
Total | 3586 | 4003 | 1578 | 210 | 5791
2 |  |  |  |  |

Table 2: Current ASQP Dataset Statistics

To begin with, we present some basic statistics of each of the domains in the
dataset in Table 2.2. It provides information about the total number of
reviews, sentences, review-level opinion tuples, and sentence-level opinion
quadruples for every domain. Table 4 has the averaged statistics such as the
sentences per review, length of sentence (and review), and the number of
opinions per sentence (and review) with the domains as columns. We also
provide fine-grained statistics in Table 2.2, such as the number of sentences
and reviews with zero, one, two, three, and more than three opinions. The
numbers provided inside “()” for the number of opinions per sentence and per
review (in Table 4) are computed after excluding the zero opinion sentences
and reviews. The stats corresponding to the number of opinions with different
sentiment polarities, including positive, negative, neutral, and conflict at
both the sentence and review levels, is presented in Table 2.2.

2  Domain | #Revs. | #Sent. | #Rev.Op. | #Sent.Op. | #Total Op
---|---|---|---|---|---
2 Amazon_FF | 1616 | 8016 | 3879 | 7628 | 11507
Coursera | 1211 | 5852 | 3745 | 5360 | 9105
Hotels | 1204 | 6377 | 5874 | 8948 | 14882
Total | 4031 | 20245 | 13498 | 21936 | 35434
2 |  |  |  |  |

Table 3: OATS Dataset Overall Statistics

2  Stats/Domain | Amazon_FF | Coursera | Hotels
---|---|---|---
2 Avg. Sentences/Review | 4.98 | 4.9 | 5.2
Avg. Length of Sentence | 71.34 | 79.02 | 75.95
Avg. Length of Review | 359.9 | 391.55 | 405.29
Avg. Opinions/Sentence | 0.92 (1.25) | 0.94 (1.23) | 1.3 (1.69)
Avg. Opinions/Review | 2.4 (2.48) | 3.16 (3.18) | 4.59 (5.07)
2 |  |  |

Table 4: tab: OATS Average Statistics. The values inside the () are the
average opinions per sentence/review, excluding those with zero opinions.

The hotels domain has more number of opinions per sentence as well as per
review when compared to the other two domains. Despite having the most reviews
(from Table 2.2, the fine foods domain has the least number of opinions per
sentence and review among the three. One of the main reasons is that the
number of zero opinion sentences in the fine foods domain is higher than in
the hotels and Coursera (from Table 2.2). Furthermore, the ratio of two-
opinion sentences to one-opinion sentences for the Amazon_FF and the Coursera
domains ($\approx 1:5$) pushed the average numbers closer to one opinion per
sentence. The requirement of having a minimum of 2 distinct aspect categories
at the review level (as defined in Appendix Section A) is clearly evident from
the average number of opinions per review. However, it was lacking at the
sentence level.

2  | Sentence-Level ABSA |  Review-Level ABSA
---|---|---
Domain | 0-Op | 1-Op | 2-Op | 3-Op | >3-Op | 0-Op | 1-Op | 2-Op | 3-Op | >3-Op
2 Amazon_FF | 1957 | 4529 | 890 | 170 | 42 | 47 | 175 | 626 | 490 | 183
Coursera | 1395 | 3685 | 690 | 134 | 36 | 8 | 107 | 298 | 349 | 449
Hotels | 1454 | 2910 | 1167 | 499 | 337 | 116 | 11 | 58 | 138 | 883
Total | 4806 | 11124 | 2747 | 803 | 415 | 171 | 293 | 982 | 977 | 1515
2 |  |  |  |  |  |  |  |  |  |

Table 5: OATS Dataset Statistics for the total number of sentences/reviews
with the respective number of opinions

2  | Sentence-Level ABSA |  Review-Level ABSA
---|---|---
Domain | #pos | #neg | #neu | #pos | #neg | #neu | #conf
2 Amazon_FF | 5577 | 1187 | 234 | 2900 | 606 | 74 | 82
Coursera | 4403 | 1008 | 213 | 2910 | 721 | 129 | 71
Hotels | 6952 | 1207 | 169 | 4557 | 817 | 110 | 53
Total | 16932 | 3402 | 616 | 10367 | 2144 | 313 | 206
2 |  |  |  |  |  |  |

Table 6: OATS Dataset Statistics for the total number of positive, negative,
neutral, and conflict sentiment polarities observed for the aspect categories
at Review-Level and the aspect terms, aspect category, and opinion phrase
triplets at the Sentence-Level in each domain.

Figure 1: Distribution of Aspect Categories for Sentence-Level and Review-
Level Annotations for Amazon_FF Domain

Another important factor for ABSA datasets is the distribution of aspect
categories, which is presented in Figures 1, 2, and 3. The General attribute
for different entities in all the datasets has the highest number of opinions,
such as the FOOD#GENERAl in the Amazon_FF, COURSE# GENERAL and FACULTY#GENERAL
for the Coursera domain, and HOTEL#GENERAL, LOCATION#GENERAL, and
SERVICE#GENERAL in the hotel domain. The review-level and the sentence-level
opinions have similar distributions for the aspect categories.

Figure 2: Distribution of Aspect Categories for Sentence-Level and Review-
Level Annotations for Coursera Domain

Figure 3: Distribution of Aspect Categories for Sentence-Level and Review-
Level Annotations for Hotels Domain

##  3 Experimental Evaluation

This section will first list the tasks on which we conduct experiments using
our dataset. Then, we point to the baseline methods used for each task,
dividing them into task-specific and unified methods, followed by the
evaluation metrics and the experimental results.

###  3.1 Tasks

We selected three major joint detection tasks for our sentence-level ABSA
experiments, which are target-aspect-sentiment detection (TASD) Wan et al.
(2020), aspect sentiment triplet extraction (ASTE) Peng et al. (2020), and
aspect sentiment quadruple prediction (ASQP) Zhang et al. (2021a). The
rationale for choosing these tasks is that they cover all four elements of
ABSA in different combinations. Moreover, the emphasis was placed on tasks
involving joint extraction rather than single-element extraction, which stems
from findings in Chebolu et al. (2021). The authors have shown that joint
models, designed to handle multiple interrelated elements simultaneously,
consistently outperformed explicitly fine-tuned models for extracting a single
element.

For the review-level (or text-level) ABSA, where we have to extract aspect
category and sentiment tuples, we chose to use the aspect sentiment joint
detection task (ASD) Sun et al. (2019) for our experiments. The two main
challenges with the review-level task are that we have to use the entire
review context rather than simply using each sentence to detect the aspect
categories and the sentiment polarity of the review. Secondly, there can be
multiple opinions on the same aspect category with multiple polarities, from
which we must assign the dominating sentiment to each category.

###  3.2 Baseline Methods

We implement several representative models from various frameworks, including
MRC-based, generation-based, and BERT-based frameworks, for the task
evaluations.

##### Task-specific methods

Here, we present various approaches used to evaluate a specific task.

For ASTE, we use the following methods:

  * •

BMRC Liu et al. (2022): a MRC-based method. It extracts aspect-oriented
triplets and opinion-oriented triplets. Then, it obtains the final results by
merging the two directions.

  * •

GEN-NAT-SCL Peper and Wang (2022): a generative-based method. It combines a
new generative format with a supervised contrastive learning objective to
predict the ASTE triplets and quadruples.

  * •

BDTF (BERT) Zhang et al. (2022): a BERT-based method that uses table-filling
to solve the problem. It transforms the ASTE task into detecting and
classifying relation regions in the 2D table representing each triplet in
addition to an effective relation representation learning approach to
understand word and relation interactions.

These are the methods for ASQP:

  * •

Template-ILO Hu et al. (2022): a generative-based method. Similar to Zhang et
al. (2021a) with an additional step to identify the most proper orders at the
instance level, and further combine multiple proper templates as data
augmentation, instead of having a fixed order template, that is passed as
input to the generative model.

  * •

Template-ILO Hu et al. (2022): a generative-based method. Similar to Zhang et
al. (2021a), but has an additional step to identify the most proper orders at
the dataset level, and further combine multiple proper templates as data
augmentation, instead of having a fixed order template, that is passed as
input to the generative model.

For the TASD task, we use the TAS-BERT Wan et al. (2020), a BERT-based method.
It fine-tunes the pre-trained BERT model to solve the aspect-sentiment
detection task using the classification token and then detects the targets
corresponding to those tuples using the token classification with CRF/softmax
decoding.

##### Methods for review-level ASD

  * •

BERT-pair-NLI-B Sun et al. (2019): a BERT-based method. It generates an
auxiliary sentence from the aspect and sentiment to transform ABSA into a
sentence-pair classification task similar to natural language inference or
question-answering.

##### Unified methods

The following methods are unified generative frameworks that could be adapted
to any NLP problem. In our case, these are employed for all the sentence-level
and review-level tasks:

  * •

GEN-NAT-SCL Peper and Wang (2022): a generative-based method. It combines a
new generative format with a supervised contrastive learning objective to
predict the ASTE’s triplets and ASQP’s quadruples.

  * •

GAS Zhang et al. (2021c): a generation-based method. It transforms the ASTE,
ASQP, TASD, and ASD tasks into a text generation problem that inputs the
review and generates all the respective combinations of ABSA opinion elements
as output.

  * •

Paraphrase Zhang et al. (2021a): a generative-based method. It is similar to
GAS but also transforms output opinion elements into paraphrases that read as
natural sentences. We substituted the corresponding element for implicit
targets and opinions with the word "it."

###  3.3 Evaluation

Following Zhang et al. (2021a); Xu et al. (2023); Wan et al. (2020), we employ
the F1 score to measure the performance of different approaches on all the
tasks from Section 3.1. All experimental results are reported using the
average of 5 different runs using distinct random seeds. We divided each
domain dataset into train, validation, and test sets with 80%, 10%, and 10%
splits, respectively. A tuple, triplet, and quadruple is considered correct
only if all the corresponding prediction elements match the gold standard
labels. We consider any partial matches as wrong predictions following Zhang
et al. (2021a). We adopt the base versions of all the transformer models for
our experiments, including the BERT-base Devlin et al. (2019), T5-base Raffel
et al. (2020), RoBERTa-base Liu et al. (2019), and others.

###  3.4 Results

In this section, we present the results of different baseline systems on the
selected tasks, highlighting the benefits and challenges of current methods on
the OATS dataset and the specific task.

##### ASQP

Table 7 reports the performance on the ASQP task for the quadruple extraction
using five different baseline systems on the three datasets. On a high level,
the overall performance on the Hotels domain is better compared to the other
two domains in all the baseline systems.

For the Amazon_FF domain, even though Template-ILO gave the best performance,
all the other methods are not far from it. Similarly, for the Coursera domain,
GAS-T5 performed better than the Paraphrase and GEN-SCL-NAT, but the Template
method is close to its performance. Paraphrase outperformed all the other
methods in the Hotels domain with a 34.51% F1 score.

Prior works have shown that the Template method significantly outperformed
Paraphrase and GAS in the restaurant domain Hu et al. (2022). That trend only
translated to the Amazon_FF domain, leaving out Coursera and Hotels.
Similarly, GEN-SCL-NAT, which outperformed the Paraphrase method on the
restaurants and laptops domain, failed to carry it to any of the three
domains.

2  Method | Amazon_FF | Coursera | Hotels
---|---|---|---
2 GAS-T5 | 19.62 | 22.23 | 26.33
Paraphrase | 20.84 | 19.78 | 34.51
Template-ILO | 21.01 | 21.24 | 24.62
Template-DLO | 20.39 | 21.96 | 26.59
GEN-SCL-NAT | 20.36 | 20.20 | 28.58
2 |  |  |

Table 7: F1 scores of in-domain ASQP on OATS and the best results are
highlighted in bold font.

##### ASTE

Table 8 shows the performance on the ASTE task (the aspect term, opinion term,
and sentiment polarity) triplet extraction using five different baseline
systems on the three datasets. Unlike ASQP, a single method (BMRC)
outperformed all the other baselines on all domains for this ASTE task. The
GEN-SCL-NAT method consistently underperformed on all the domains.

2  Method | Amazon_FF | Coursera | Hotels
---|---|---|---
2 GEN-SCL-NAT | 20.36 | 20.20 | 28.58
GAS | 46.46 | 40.15 | 45.96
Paraphrase | 46.64 | 39.30 | 46.37
BDTF (BERT) | 27.81 | 43.98 | 42.02
BMRC | 49.48 | 50.85 | 50.71
2 |  |  |

Table 8: F1 scores of in-domain ASTE on OATS and the best results are
highlighted in bold font.

##### TASD

We present the results of the TASD task experiments on all the domains in
Table 9. The generative-based models outperformed the BERT-based methods on
the Amazon_FF and Hotels domains, while TAS-BERT performed best in the
Coursera reviews.

2  Method | Amazon_FF | Coursera | Hotels
---|---|---|---
2 TAS-BERT-BIO | 40.13 | 42.49 | 30.89
TAS-BERT-TO | 41.06 | 38.18 | 30.01
GAS | 43.04 | 41.53 | 50.69
Paraphrase | 44.89 | 40.24 | 49.81

Table 9: F1 scores of in-domain TASD on OATS and the best results are
highlighted in bold font.

##  4 Significance of OATS

Given that the ASQP task was introduced very recently, there is a need for a
benchmark dataset to evaluate the task effectively. Although Zhang et al.
(2021a) created a dataset when they proposed the task, they built it based on
several SemEval Shared Challenges Pontiki et al. (2015, 2016) that had
annotations for aspect terms, aspect categories, and their sentiment polarity.
The opinion term and aspect category annotations are derived from Fan et al.
(2019a) and Wan et al. (2020), respectively. Then, they align the samples from
these two sources, merge the annotations with the same aspect term in each
sentence as the anchor, and add the implicit target with their opinion terms
excluded by Fan et al. (2019a).

The main differences between the current ASQP datasets and OATS are:

  1. 1.

we annotate the data from scratch with all the elements together instead of
aligning from several sources

  2. 2.

we include all the implicit aspect terms and opinion phrases, unlike ASQP,
which excluded implicit opinion terms

  3. 3.

we provide both the review-level and sentence-level annotations facilitating
the analysis and experiments for the review/text-level aspect-based sentiment
analysis that differentiates Cai et al. (2023) from our dataset

  4. 4.

there are $\approx 5.8K$ opinion quadruples in total for the ASQP corpus,
which includes two restaurant datasets that are comparable to the $\approx
5.7K$ opinion quadruples of the Coursera domain itself, which is the smallest
in the OATS corpus. OATS has a total of $\approx 21K$ opinion quadruples,
almost four times larger than the ASQP corpus. Also, if we include the review-
level annotations, we have nearly $34K$ opinions in OATS.

The OATS corpus will help analyze and understand the inherent relationships
among all the elements of ABSA and exploit them to solve the ABSA task
holistically. The comprehensive nature of the OATS corpus not only allows the
evaluation of the ASQP task but also unlocks the potential analysis and
exploration of all the subtasks of ABSA, including TASD, ASTE, TOWE, TAD, and
many more. As pointed out by Zhang et al. (2021a), the characteristic of
tackling various ABSA tasks in a unified framework enables the knowledge to be
easily transferred across related subtasks, which is especially beneficial
under low-resource settings. It also allows cross-task transfer for subtasks
that underperform when trained using only a task-specific dataset. Many
researchers would like to explore ABSA for whole reviews rather than
individual sentences or single-sentence reviews. For such settings, OATS will
help push the research ahead with all the review-level annotations we provide
as a part of it.

##  5 Related Works

In this section, we explain the several tasks of ABSA and their related
datasets, followed by the task-specific methods for a few triplet and
quadruple extraction tasks.

###  5.1 Current Datasets

Traditionally, when ABSA was first introduced by Hu and Liu (2004b) as a task
in NLP, the main aim then was to extract different aspects given a sentence
followed by assigning polarities given those aspects. As a decade passed,
Pontiki et al. (2014) added another important element of ABSA: aspect
categories. In the SemEval-2014 shared task on ABSA, two sub-tasks were
drafted in addition to the aspect term extraction and its sentiment polarity.
They are the aspect category detection and their sentiment polarity. Two
datasets from the restaurant and laptop domains were released as part of this
shared task.

In the consecutive years, Pontiki et al. (2015, 2016) proposed two more ABSA
shared tasks at the SemEval, redefining the subtasks of ABSA and giving them a
more concrete structure. Aspect categories were a combination of entity and
attribute pairs, aspect terms were called targets, and the sentiment polarity
was assigned jointly for the targets and the aspect categories. Several
datasets with the three elements from multiple languages and domains were
published, including English, Dutch, Spanish, French, Russian, Turkish
restaurants, English laptops, Arabic hotels, and many others.

Later, Wan et al. (2020) utilized the data from the SemEval-2015 and
SemEval-2016 to define the target-aspect-sentiment joint detection task
(TASD). After that, researchers felt the need for opinion phrases to detect
the sentiment polarity and understand the relationship with targets and aspect
categories. This resulted in three new sub-tasks with new datasets that
stemmed from the SemEval challenges, namely aspect-opinion-pair extraction
(AOPE) Chen et al. (2020), target-opinion-word extraction (TOWE) Fan et al.
(2019b), and aspect-sentiment-triplet-extraction (ASTE) Peng et al. (2020).
More recently, the ABSA research has shifted towards the joint detection of
all four elements of ABSA, which proved to better identify the inter-
relationships among the elements, thereby enhancing the performance of other
subtasks. This task is called aspect sentiment quadruple prediction (ASQP)
Zhang et al. (2021a) or aspect-category-opinion-sentiment joint detection
(ACOS) Cai et al. (2021). The ASQP task is evaluated on the dataset with
quadruples created from the SemEval challenges, ASTE triplets, and TOWE and
AOPE tuples. The ACOS task introduced two datasets from the restaurant and the
laptop domain with implicit and explicit aspect terms and opinion phrases.

The current ABSA dataset landscape is notably skewed towards reviews and
particular domains such as restaurants and electronics. This stems from the
easy availability and volume of data on review websites and other online
platforms. Although these datasets have led to many successful applications,
this narrow focus restricts the wider application of ABSA, limiting its
ability to deliver insights across different sectors. Expanding the diversity
of ABSA datasets is essential to push research and applications beyond the
boundaries of mere reviews. We propose three new datasets for the quadruple
extraction task from three new domains, along with implicit and explicit
targets and opinion phrase annotations, to address this limitation. We also
include the review-level tuples for identifying the overall sentiment polarity
for different aspect categories in a review. This dataset could be used to
solve all the above-mentioned subtasks of ABSA.

###  5.2 Related Methods

We focus on a few tasks from the joint detection tasks arena: the TASD, ASTE,
and ASQP tasks for our research. Chebolu et al. (2023) have provided a
detailed survey of the other datasets, tasks, and their challenges.

##### Triplet Extraction Tasks

In ASTE research, three primary methodologies have emerged: MRC-based
techniques Liu et al. (2022), methods anchored on BERT and table-filling Zhang
et al. (2022), and generative approaches Peper and Wang (2022); Zhang et al.
(2021c, a). The MRC-based methods involve crafting a specific query for each
component in the triplet, subsequently extracting them based on the response
to this query. Generative methods, in contrast, frame the ASTE challenge as a
sequence generation task and employ sequence-to-sequence (seq2seq) models. The
triplets are then decoded using a specially tailored algorithm. In this study,
we employ representative techniques from these three categories and
investigate OATS using these methods.

For the TASD task, there are two paths: BERT-based methods Wan et al. (2020),
and generative-based approaches Chebolu et al. (2021); Zhang et al. (2021a).
The BERT-based methods extract the aspect sentiment tuple from a sentence
using the sentence-pair classification as a backbone Sun et al. (2019)
followed by extracting the targets for each pair using the token
classification with the BIO or a softmax classifier. The generative-based
approaches convert the task similar to an abstractive summarization and use
the sequence-to-sequence models like the T5, BART, and others Raffel et al.
(2020); Lewis et al. (2019) to predict the triplets.

##### Quadruple Extraction Task

Researchers have pointed out two promising directions. Cai et al. (2021)
propose a two-stage method by extracting the aspect and opinion terms first.
Then, these items are utilized to classify aspect categories and sentiment
polarity. Another method is based on the generation model Zhang et al.
(2021a). By paraphrasing the input sentence, the quadruplet can be extracted
end-to-end. In this work, we follow the generative direction and consider the
order-free property of the quadruplet. A few other studies also proposed
generative-based models with this paraphrasing as a backbone, including Hu et
al. (2022); Peper and Wang (2022).

##  6 Conclusion

ABSA still presents a lot of difficulties in the rapidly developing field of
generative artificial intelligence. We present a thoroughly curated ABSA
dataset including quadruples, both explicit and implicit attributes, and
views, spanning three unique domains. In addition, the dataset incorporates
review-level sentiment polarity for each aspect category, providing a
comprehensive perspective of the sentiments expressed in the reviews. Our
annotations outnumber those found in previous datasets. We do an in-depth
study, give comprehensive annotation guidelines, and provide dataset
statistics. We also enable evaluations of generating and non-generative
benchmarks on a range of common ABSA tasks.

## References

  * Becker et al. (2005) Markus Becker, Ben Hachey, Beatrice Alex, and Claire Grover. 2005.  Optimising selective sampling for bootstrapping named entity recognition.  In _ICML-2005 Workshop on Learning with Multiple Views_ , pages 5–11. Citeseer.
  * Cai et al. (2023) Hongjie Cai, Nan Song, Zengzhi Wang, Qiming Xie, Qiankun Zhao, Ke Li, Siwei Wu, Shijie Liu, Jianfei Yu, and Rui Xia. 2023.  Memd-absa: A multi-element multi-domain dataset for aspect-based sentiment analysis.  _ArXiv_ , abs/2306.16956.
  * Cai et al. (2021) Hongjie Cai, Rui Xia, and Jianfei Yu. 2021.  Aspect-category-opinion-sentiment quadruple extraction with implicit aspects and opinions.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 1: Long Papers)_ , pages 340–350, Online. Association for Computational Linguistics.
  * Chebolu et al. (2021) Siva Uday Sampreeth Chebolu, Franck Dernoncourt, Nedim Lipka, and Thamar Solorio. 2021.  Exploring conditional text generation for aspect-based sentiment analysis.  In _Proceedings of the 35th Pacific Asia Conference on Language, Information and Computation_ , pages 119–129, Shanghai, China. Association for Computational Lingustics.
  * Chebolu et al. (2023) Siva Uday Sampreeth Chebolu, Franck Dernoncourt, Nedim Lipka, and Thamar Solorio. 2023.  Survey of aspect-based sentiment analysis datasets.
  * Chen et al. (2020) Shaowei Chen, Jie Liu, Yu Wang, Wenzheng Zhang, and Ziming Chi. 2020.  Synchronous double-channel recurrent network for aspect-opinion pair extraction.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 6515–6524, Online. Association for Computational Linguistics.
  * Chia et al. (2023) Yew Ken Chia, Hui Chen, Wei Han, Guizhen Chen, Sharifah Mahani Aljunied, Soujanya Poria, and Lidong Bing. 2023.  Domain-expanded aste: Rethinking generalization in aspect sentiment triplet extraction.
  * Deléger et al. (2012) Louise Deléger, Qi Li, Todd Lingren, Megan Kaiser, Katalin Molnár, Laura Stoutenborough, Michal Kouril, Keith A. Marsolo, and Imre Solti. 2012.  Building gold standard corpora for medical natural language processing tasks.  _AMIA … Annual Symposium proceedings. AMIA Symposium_ , 2012:144–53.
  * Devlin et al. (2019) J. Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  Bert: Pre-training of deep bidirectional transformers for language understanding.  In _NAACL-HLT_.
  * Fan et al. (2019a) Fan, Zhifang and Wu, Zhen and Dai, Xin-Yu and Huang, Shujian and Chen, Jiajun. 2019a.  _Target-oriented Opinion Words Extraction with Target-fused Neural Sequence Labeling_.  Association for Computational Linguistics.  [link].
  * Fan et al. (2019b) Zhifang Fan, Zhen Wu, Xin-Yu Dai, Shujian Huang, and Jiajun Chen. 2019b.  Target-oriented opinion words extraction with target-fused neural sequence labeling.  In _Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers)_ , pages 2509–2518, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Hripcsak and Rothschild (2005) George Hripcsak and Adam S Rothschild. 2005.  Agreement, the f-measure, and reliability in information retrieval.  _Journal of the American medical informatics association_ , 12(3):296–298.
  * Hu et al. (2022) Mengting Hu, Yike Wu, Hang Gao, Yinhao Bai, and Shiwan Zhao. 2022.  Improving aspect sentiment quad prediction via template-order data augmentation.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 7889–7900, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.
  * Hu and Liu (2004a) Minqing Hu and Bing Liu. 2004a.  Mining and summarizing customer reviews.  In _Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , KDD ’04, page 168–177, New York, NY, USA. Association for Computing Machinery.
  * Hu and Liu (2004b) Minqing Hu and Bing Liu. 2004b.  Mining and summarizing customer reviews.  In _Proceedings of the Tenth ACM SIGKDD International Conference on Knowledge Discovery and Data Mining_ , KDD ’04, page 168–177, New York, NY, USA. Association for Computing Machinery.
  * Jiang et al. (2019) Qingnan Jiang, Lei Chen, Ruifeng Xu, Xiang Ao, and Min Yang. 2019.  A challenge dataset and effective models for aspect-based sentiment analysis.  pages 6281–6286.
  * Lewis et al. (2019) Mike Lewis, Yinhan Liu, Naman Goyal, Marjan Ghazvininejad, Abdelrahman Mohamed, Omer Levy, Veselin Stoyanov, and Luke Zettlemoyer. 2019.  BART: denoising sequence-to-sequence pre-training for natural language generation, translation, and comprehension.  _CoRR_ , abs/1910.13461.
  * Liu et al. (2022) Shu Liu, Kaiwen Li, and Zuhe Li. 2022.  A robustly optimized BMRC for aspect sentiment triplet extraction.  In _Proceedings of the 2022 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies_ , pages 272–278, Seattle, United States. Association for Computational Linguistics.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized bert pretraining approach.
  * Peng et al. (2020) Haiyun Peng, Lu Xu, Lidong Bing, Fei Huang, Wei Lu, and Luo Si. 2020.  Knowing what, how and why: A near complete solution for aspect-based sentiment analysis.  _Proceedings of the AAAI Conference on Artificial Intelligence_ , 34(05):8600–8607.
  * Peper and Wang (2022) Joseph J. Peper and Lu Wang. 2022.  Generative aspect-based sentiment analysis with contrastive learning and expressive structure.  In _Conference on Empirical Methods in Natural Language Processing_.
  * Pitman (2022) Jamie Pitman. 2022.  Local Consumer Review Survey 2022.  https://www.brightlocal.com/research/local-consumer-review-survey/.
  * Pontiki et al. (2016) Maria Pontiki, Dimitris Galanis, Haris Papageorgiou, Ion Androutsopoulos, Suresh Manandhar, Mohammad AL-Smadi, Mahmoud Al-Ayyoub, Yanyan Zhao, Bing Qin, Orphée De Clercq, Véronique Hoste, Marianna Apidianaki, Xavier Tannier, Natalia Loukachevitch, Evgeniy Kotelnikov, Nuria Bel, Salud María Jiménez-Zafra, and Gülşen Eryiğit. 2016.  SemEval-2016 task 5: Aspect based sentiment analysis.  In _Proceedings of the 10th International Workshop on Semantic Evaluation (SemEval-2016)_ , pages 19–30, San Diego, California. Association for Computational Linguistics.
  * Pontiki et al. (2015) Maria Pontiki, Dimitris Galanis, Haris Papageorgiou, Suresh Manandhar, and Ion Androutsopoulos. 2015.  SemEval-2015 task 12: Aspect based sentiment analysis.  In _Proceedings of the 9th International Workshop on Semantic Evaluation (SemEval 2015)_ , pages 486–495, Denver, Colorado. Association for Computational Linguistics.
  * Pontiki et al. (2014) Maria Pontiki, Dimitris Galanis, John Pavlopoulos, Harris Papageorgiou, Ion Androutsopoulos, and Suresh Manandhar. 2014.  SemEval-2014 task 4: Aspect based sentiment analysis.  In _Proceedings of the 8th International Workshop on Semantic Evaluation (SemEval 2014)_ , pages 27–35, Dublin, Ireland. Association for Computational Linguistics.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.
  * Reimers and Gurevych (2019) Nils Reimers and Iryna Gurevych. 2019.  Sentence-BERT: Sentence embeddings using Siamese BERT-networks.  In _Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Language Processing (EMNLP-IJCNLP)_ , pages 3982–3992, Hong Kong, China. Association for Computational Linguistics.
  * Sun et al. (2019) Chi Sun, Luyao Huang, and Xipeng Qiu. 2019.  Utilizing bert for aspect-based sentiment analysis via constructing auxiliary sentence.  _arXiv preprint arXiv:1903.09588_.
  * Tomanek and Hahn (2009) Katrin Tomanek and Udo Hahn. 2009.  Timed annotations—enhancing muc7 metadata by the time it takes to annotate named entities.  In _Proceedings of the Third Linguistic Annotation Workshop (LAW III)_ , pages 112–115.
  * Wan et al. (2020) Hai Wan, Y. Yang, Jianfeng Du, Y. Liu, Kunxun Qi, and J. Pan. 2020.  Target-aspect-sentiment joint detection for aspect-based sentiment analysis.  In _AAAI_.
  * Xu et al. (2023) Ting Xu, Huiyun Yang, Zhen Wu, Jiaze Chen, Fei Zhao, and Xinyu Dai. 2023.  Measuring your ASTE models in the wild: A diversified multi-domain dataset for aspect sentiment triplet extraction.  In _Findings of the Association for Computational Linguistics: ACL 2023_ , pages 2837–2853, Toronto, Canada. Association for Computational Linguistics.
  * Yin et al. (2017) Yichun Yin, Yangqiu Song, and Ming Zhang. 2017.  Document-level multi-aspect sentiment classification as machine comprehension.  In _Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing_ , pages 2044–2054, Copenhagen, Denmark. Association for Computational Linguistics.
  * Zhang et al. (2021a) Wenxuan Zhang, Yang Deng, Xin Li, Yifei Yuan, Lidong Bing, and Wai Lam. 2021a.  Aspect sentiment quad prediction as paraphrase generation.
  * Zhang et al. (2021b) Wenxuan Zhang, Xin Li, Yang Deng, Lidong Bing, and Wai Lam. 2021b.  Towards generative aspect-based sentiment analysis.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)_ , pages 504–510, Online. Association for Computational Linguistics.
  * Zhang et al. (2021c) Wenxuan Zhang, Xin Li, Yang Deng, Lidong Bing, and Wai Lam. 2021c.  Towards generative aspect-based sentiment analysis.  In _Proceedings of the 59th Annual Meeting of the Association for Computational Linguistics and the 11th International Joint Conference on Natural Language Processing (Volume 2: Short Papers)_ , pages 504–510, Online. Association for Computational Linguistics.
  * Zhang et al. (2022) Yice Zhang, Yifan Yang, Yihui Li, Bin Liang, Shiwei Chen, Yixue Dang, Min Yang, and Ruifeng Xu. 2022.  Boundary-driven table-filling for aspect sentiment triplet extraction.  In _Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing_ , pages 6485–6498, Abu Dhabi, United Arab Emirates. Association for Computational Linguistics.

##  Appendix A Appendix: OATS Corpus Properties

We set the following properties or requirements for the corpus to make it
ideal for future works:

  * •

Each review should have both sentence-level and review/text-level annotations:
The goal of this dataset is not just to provide some sentence-level aspect-
related information but rather include review-level or text-level annotations
as well. That is, given a text, we should do the following two steps:

    1. 1.

Take each individual sentence in the text and extract all the aspect terms and
opinion phrases followed by categorizing them into aspect categories and
sentiment polarity, respectively, forming a quadruple. Each sentence can have
multiple such quadruples.

    2. 2.

After getting the quadruples from all the sentences, take the unique aspect
categories from them and assign the sentiment polarity (positive, negative,
neutral, or conflict) that is dominant from the sentiment assigned in those
quadruples. If the positive and negative sentiments are equally strong, assign
a conflict opinion polarity to that aspect category.

  * •

As little ambiguity as possible and easily understandable aspect categories:
To avoid ambiguity when assigning aspect categories to a review or a sentence,
one of the lessons learned from prior datasets is that those aspect categories
should be nearly mutually exclusive. Additionally, the aspect categories
should be relevant to the domain and easily comprehended by anyone.

  * •

Inclusion of both implicit targets and implicit opinions: As we mentioned in
Section 1, there may be cases where the opinion target or the opinion
expression could be implicit. We include both those cases and assign NULL to
that element in the quadruple. If both are implicit, both the elements will be
marked as NULL.

  * •

Start and End indices of the text spans should be provided: To facilitate
diverse formulations of ABSA, we aim to provide the start and end indices of
the opinion target and opinion phrase annotations for each quadruple. This
could be helpful for question-answering type formulations.

  * •

Reviews should have an adequate number of sentences: Each review in the
dataset should have a minimum number of sentences that will help differentiate
the sentence-level and review/text-level tasks of ABSA. If reviews are one to
two sentences long, there is a chance that review-level ABSA may degenerate to
sentence-level ABSA achieving good performance with the sentence-level models
itself. Therefore, we chose to keep only those reviews that are at least four
to eight sentences long.

##  Appendix B Appendix: Defining Fine-Grained Annotations

With the above requirements and properties in place, we began our process of
finding aspect categories for each domain. For the TripAdvisor Hotel reviews,
we chose to use the aspect categories defined in the shared task organized by
Pontiki et al. (2016), where there is a small Arabic dataset for hotel
reviews. Eight entities and seven attributes result in 26 Entity#Attribute
pairs as shown in Table 10. We followed the below steps to define the several
entities and attributes that form an aspect category for Coursera and amazon
fine foods domains:

  * •

From the entire review dataset, we generated a list of frequently occurring
words and phrases.

  * •

From that list, we filtered all the nouns and noun phrases. Then, we applied a
clustering algorithm on the nouns and noun phrases with the number of clusters
varying from three to eight.

  * •

From those clusters, we manually verified the intra-cluster word synonymity
and relationships that led to identifying the correct number of clusters.

  * •

From those clusters, we came up with the most generic entity-attribute pairs
that could sufficiently represent the cluster.

Then, we selected a pool of 2000 reviews from the entire dataset with the
following constraints:

  * •

There should be at least 4 sentences in each review

  * •

Each review should have at least 2 distinct aspect categories

We used a Siamese-BERT model Reimers and Gurevych (2019) to determine whether
a review has at least two of the defined aspect categories in the domain.
Specifically, we gave the review and the sentence “This review mentioned about
[aspect category]” as an input to the model. If the similarity score is
greater than 0.75 for more than two aspect categories, we take that review for
the annotation procedure. We present the entity-attribute pairs for each
domain in Tables 12, 11, and 10.

Entity/Attribute | GENERAL | PRICES | DESIGN&FEATURES | CLEANLINESS | COMFORT | QUALITY | STYLE&OPTIONS | MISCELLANEOUS
---|---|---|---|---|---|---|---|---
HOTEL | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
ROOMS | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
ROOM_AMENITIES | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
FACILITIES | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓
SERVICE | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
LOCATION | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
FOOD&DRINKS | ✗ | ✓ | ✗ | ✗ | ✗ | ✓ | ✓ | ✓

Table 10: Hotels domain possible Entity Attribute pairs

Entity/Attribute | GENERAL | QUALITY | COMPREHENSIVENESS | RELATABILITY | WORKLOAD | RESPONSE | VALUE | QUANTITY
---|---|---|---|---|---|---|---|---
PRESENTATION | ✗ | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | ✗
ASSIGNMENTS | ✗ | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓
MATERIAL | ✗ | ✓ | ✓ | ✓ | ✓ | ✗ | ✗ | ✓
GRADES | ✓ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗ | ✗
FACULTY | ✓ | ✗ | ✓ | ✓ | ✗ | ✓ | ✓ | ✗
COURSE | ✓ | ✓ | ✓ | ✓ | ✓ | ✗ | ✓ | ✗

Table 11: Coursera courses domain possible Entity Attribute pairs

Entity/Attribute | GENERAL | QUALITY | STYLE_OPTIONS | PRICES | DELIVERY | AVAILABILITY
---|---|---|---|---|---|---
FOOD | ✓ | ✓ | ✓ | ✓ | ✗ | ✗
SHIPMENT | ✗ | ✓ | ✗ | ✓ | ✓ | ✗
AMAZON | ✗ | ✗ | ✗ | ✓ | ✗ | ✓

Table 12: Amazon Fine Foods domain possible Entity Attribute pairs Figure 4:
Example of an opinion quadruple from a sentence in the Coursera domain in BRAT
format and the final XML format of the sentence-level and review-level
annotations of a review


# On polynomial symmetry algebras underlying superintegrable systems in
Darboux spaces

Ian Marquette<EMAIL_ADDRESS>Junze
<EMAIL_ADDRESS>and Yao-Zhong<EMAIL_ADDRESS>School of Mathematics and Physics, The University of Queensland, Brisbane, QLD
4072, Australia

###### Abstract

We review three different approaches to polynomial symmetry algebras
underlying superintegrable systems in Darboux spaces. The first method
consists of using deformed oscillator algebra to obtain finite-dimensional
representations of quadratic algebras. This allow one to gain information on
the spectrum of the superintegrable systems. The second method has
similarities with the induced module construction approach in the context of
Lie algebras and can be used to construct infinite dimensional representations
of the symmetry algebras. Explicit construction of these representations is a
non-trivial task due to the non-linearity of the polynomial algebras. This
method allows the construction of states of the superintegrable systems beyond
the reach of separation of variables. As a result, we are able to construct a
large number of states in terms of Airy, Bessel and Whittaker functions which
would be difficult to obtain in other ways. We also discuss the third approach
which is based on the notion of commutants of subalgebras in the enveloping
algebra of a Poisson algebra or a Lie algebra. This allows us to discover new
superintegrable models in the Darboux spaces and to construct their integrals
and symmetry algebras via polynomials in the enveloping algebras.

##  1 Introduction

Finite and Infinite-dimensional representations of symmetry algebras play a
significant role in determining the spectral properties of physical
Hamiltonians. In [1, 2] we focused on the representations of polynomial
symmetry algebras underlying superintegrable systems in 2D Darboux spaces. As
a result, we are able to construct a large number of states in terms of the
Airy, Bessel and Whittaker functions which would be difficult to obtain in
other ways.

Let $(\mathcal{M},g_{ij})$ be a smooth manifold with a metric tensor over
$\mathbb{C}$. Suppose that $\mathcal{M}$ admits local separable coordinates
$(x_{1},\ldots,x_{n})$. Let

$\displaystyle\hat{\mathcal{H}}=\sum_{i,j=1}^{n}\frac{1}{\sqrt{\det(g_{ij})}}\dfrac{\partial}{\partial
x_{j}}\left(\sqrt{\det(g_{ij})}g_{ij}\dfrac{\partial}{\partial
x_{j}}\right)+V(x_{1},\ldots,x_{n})\in\Gamma\left(T^{*}\mathcal{M}\right)$ (1)

be the Hamiltonian of a superintegrable system in $(\mathcal{M},g_{ij})$ and
$S_{m}=\\{\hat{\mathcal{H}},\hat{X}_{1},\ldots,\hat{X}_{m}\\}$ be a set of
integrals of motion. Let $\hat{\mathcal{Q}}(d)$ denote the polynomial algebra
of order $d$ over the polynomial ring $\mathbb{C}[\hat{\mathcal{H}}]$,
generated by the integrals from $S_{m}$ with the following non-trivial
commutators

$\displaystyle[\hat{X}_{s},\hat{X}_{t}]=\sum_{q}f_{q}(\hat{\mathcal{H}})\hat{X}_{q}+\sum_{p,q}f_{p,q}(\hat{\mathcal{H}})\hat{X}_{p}\hat{X}_{q}+\sum_{p,q,r}f_{p,q,r}(\hat{\mathcal{H}})\hat{X}_{p}\hat{X}_{q}\hat{X}_{r}+\ldots.$
(2)

Here $f_{p,q,r}(\hat{\mathcal{H}})$ is a polynomial function of the
Hamiltonian $\hat{\mathcal{H}}.$ Let $V$ be a vector space and
$\mathfrak{gl}(V)$ be the space of endomorphism of $V.$ Then representations
of the associative polynomial algebra $\hat{\mathcal{Q}}(d)$ are given by
$\rho:\hat{\mathcal{Q}}(d)\rightarrow\mathfrak{gl}(V)$.

##  2 Construction of representations of polynomial algebras

Superintegrable systems in 2D Darboux spaces were classified [3, 4] and it was
found that there exist 12 distinct classes of second order superintegrable
systems in the Darboux spaces. In [1] we presented exact solutions via purely
algebraic means for the energies of all the 12 classes of superintegrable
systems in four different 2D Darboux spaces. This was achieved by constructing
the deformed oscillator realization and finite-dimensional irreducible
representation of the underlying quadratic symmetry algebra generated by
quadratic integrals respectively for each of the 12 superintegrable systems.

###  2.1 Deformed oscillator algebra realizations

One way to construct representations of quadratic symmetry algebras is through
their realizations in terms of the deformed oscillator algebra.

Consider the generic quadratic algebra $\hat{\mathcal{Q}}(3)$ with three
generators, proposed in [5, 6],

$\displaystyle\begin{matrix}[A,B]=C,\\\ [A,C]=\alpha
A^{2}+\gamma\\{A,B\\}+\delta A+\epsilon B+\zeta,\\\ [B,C]=aA^{2}-\gamma
B^{2}-\alpha\\{A,B\\}+dA-\delta B+z.\end{matrix}$ (3)

Its Casimir operator is given by

$\displaystyle K=$ $\displaystyle
C^{2}-\alpha\\{A^{2},B\\}-\gamma\\{A,B^{2}\\}+(\alpha\gamma-\delta)\\{A,B\\}+(\gamma^{2}-\epsilon)B^{2}+(\gamma\delta-2\zeta)B$
$\displaystyle+\frac{2a}{3}A^{3}+(d+\frac{a\gamma}{3}+\alpha^{2})A^{2}+(\frac{a\epsilon}{3}+\alpha\delta+2z)A.$

Notice that the generic form $\eqref{eq:comm}$ was proposed for the case of
two quadratic integrals $A,B$ and is quite formal because in general there is
no guarantee that integrals of degree 2 would close to form a quadratic
algebra.

Finite-dimensional representations of $\eqref{eq:comm}$ can be conveniently
constructed via the deformed oscillator algebra with generators
$\\{b^{\dagger},b,\mathcal{N}\\}$, which denotes by $\mathfrak{o}(3),$
satisfying the following commutation relations [5]

$\displaystyle[\mathcal{N},b^{\dagger}]=b^{\dagger},\quad\text{
}[\mathcal{N},b]=-b,\quad\text{ }bb^{\dagger}=\Phi(\mathcal{N}+1),\quad\text{
}b^{\dagger}b=\Phi(\mathcal{N}).$ (4)

Here $\mathcal{N}$ is the number operator and $\Phi$ is a well defined real
function satisfying

$\displaystyle\Phi(0)=0,\text{ }\Phi(x)>0,\text{ for all }x\in\mathbb{R}^{+}.$
(5)

Then representations of $\mathfrak{o}(3)$ are given by

$\displaystyle\mathcal{N}\psi_{n}=n\psi_{n},\quad\text{
}b^{\dagger}\psi_{n}=\sqrt{\Phi(n+1)}\psi_{n+1},\quad
b\psi_{n}=\sqrt{\Phi(n)}\psi_{n-1}.$ (6)

where $\psi_{n}$ are eigenvectors that form the Fock basis of the oscillator
algebra. Imposing $\Phi(p+1)=0$ for all $p\in\mathbb{N}$, one obtains
$p+1$-dimensional unitary representations of $\mathfrak{o}(3)$.

Applying the deformed oscillator technique, in [1] we gave algebraic
derivations of the spectra for the 12 superintegrable systems in the 2D
Darboux spaces. As an example, we here review some of the results for the
superintegrable system in the Darboux space II with the Hamiltonian
$\hat{\mathcal{H}}=\frac{x^{2}}{x^{2}+1}\left(\partial_{x}^{2}+\partial_{y}^{2}\right)+\frac{x^{2}}{x^{2}+1}\left(a_{1}\left(\frac{x^{2}}{4}+y^{2}\right)+a_{2}y+\frac{a_{3}}{x^{2}}\right)$.
The constants of motion are $A=\partial_{y}^{2}+a_{1}y^{2}+a_{2}y$ and

$\displaystyle
B=\frac{2y}{x^{2}+1}\left(\partial_{y}^{2}-x^{2}\partial_{x}^{2}\right)+2x\partial_{x}\partial_{y}+\partial_{y}+\frac{a_{1}}{2}y\left(x^{2}+\frac{x^{2}+4y^{2}}{x^{2}+1}\right)+\frac{a_{2}}{2}\left(x^{2}+\frac{4y^{2}}{x^{2}+1}\right)-\frac{2a_{3}y}{x^{2}+1}.$

These integrals satisfy the quadratic algebra $\mathcal{Q}(3)$ [3]

$\displaystyle[A,B]=C,\quad[A,C]=-4a_{1}B-4a_{2}A,$
$\displaystyle[B,C]=-24A^{2}+4a_{2}B+32\hat{\mathcal{H}}A-8\hat{\mathcal{H}}^{2}-8a_{1}\hat{\mathcal{H}}+6a_{1}+8a_{1}a_{3}.$

with the Casimir operator given by

$\displaystyle
K=C^{2}-16A^{3}+4a_{1}B^{2}+4a_{2}\\{A,B\\}+\left(4a_{1}(4a_{3}-11)-(16a_{1}\hat{\mathcal{H}}+16\hat{\mathcal{H}}^{2})\right)A+32\hat{\mathcal{H}}A^{2}.$

In term of the differential realization of $A,B$, the Casimir $K$ takes the
simple form $K=(32a_{1}+4a_{2}^{2})\hat{\mathcal{H}}-a_{2}^{2}(3+4a_{3})$. It
was shown in [1] that the transformation
$A=2\sqrt{-a_{1}}(\mathcal{N}+\eta),\quad\text{
}B=\frac{2a_{2}}{\sqrt{-a_{1}}}(\mathcal{N}+\eta)+\frac{a_{2}\hat{\mathcal{H}}_{2,1}}{a_{1}}+b^{\dagger}+b,$
maps $\hat{\mathcal{Q}}(3)$ is to the deformed oscillator algebra
$\eqref{eq:alg}$ with the structure function cubic in $(\mathcal{N}+\eta)$

Here $\eta$ is a constant determined from the constraints of the structure
function. In general it is very difficult to obtain analytical solutions to
$\eta$ for arbitrary coefficients $a_{i}$. We consider the case where
$-a_{1}=a_{2}=a_{3}=a$, $a\in\mathbb{R}$. For such model parameters, the
energies $E$ and their corresponding structure functions for distinct $\eta$
are

$\displaystyle
E_{\epsilon}=\frac{1}{4}\left(8\sqrt{a}(p+1)+3a+2\epsilon\sqrt{8a^{3/2}(p+1)-2a^{2}+a}\right),$
(7)

where $\epsilon=\pm 1$, with the associated structure function
$\Phi^{(II)}_{E_{\epsilon}}(z)=z(p+1-z)^{2}.$ The energy spectrum
$E_{\epsilon}$ is real for $0<a\leq 1/2$. The corresponding energy spectrum of
the system and structure function for the $(p+1)$-dimensional unirreps of the
deformed oscillator algebra are given respectively by $E=p(p+2)+a+\frac{3}{4}$
and

$\displaystyle\Phi_{E}(z)=z(z-p-1)\left(z+\frac{1}{8a}\left(3a^{3/2}-4a(p+1)+\sqrt{a}(4p^{2}+8p+3)\right)\right).$
(8)

###  2.2 Verma module constructions on $\hat{\mathcal{Q}}(d)$

In this subsection, we review our method in [2] of determining the
representations of $\hat{\mathcal{Q}}(d)$ without relying on the deformed
oscillator algebra realizations. In what follows, we always assume that the
Schrödinger equation $\hat{\mathcal{H}}\Psi=E\Psi$ has solutions of the
separable form $\Psi(x_{1},\ldots,x_{n})=F_{1}(x_{1})\ldots F_{n}(x_{n})$,
where $\hat{\mathcal{H}}$ is the Hamiltonian defined in $\eqref{eq:Ham}$ and
$F_{j}$, $j=1,2,\ldots,$ are functions of the coordinates $x_{j}$. Notice that
the solution space of the Schrödinger equation will form an infinite-
dimensional vector space. As $\hat{\mathcal{Q}}(d)$ is the symmetry algebra of
the Hamiltonian $\hat{\mathcal{H}}$, infinite dimensional representations of
$\hat{\mathcal{Q}}(d)$ can be obtained through actions of its generators on
eigenstates of $\hat{\mathcal{H}}$.

Since solutions of the Schrödinger equation are separable, we assume that
$S_{p}=\\{\hat{\mathcal{H}},\hat{X}_{1},\ldots,\hat{X}_{p}\\}$ is a subset of
$S_{m}$ such that $\hat{X}_{j}\Psi=\lambda_{j}\Psi$, with
$\lambda_{j}\in\mathbb{R}$ for all $1\leq j\leq p.$ It is clear that the
existence of $S_{p}$ depends on the explicit form of the integrals. Moreover,
superintegrable systems with scalar potentials and $n$ degrees of freedom
admit up to $n$ integrals (including the Hamiltonian) forming a set of
commuting operators (then $p\leq n$). Then for some $n_{j}\in\mathbb{N}$
define the following reiterated vector

$\displaystyle\prod_{p+1\leq j\leq
m}\hat{X}_{j}^{n_{j}}\Psi=\Psi_{n_{p+1},\ldots,n_{m}},$ (9)

in the eigenspace $V_{E}$ of $\hat{\mathcal{H}}$, i.e.
$\Psi_{n_{p+1},\ldots,n_{m}}\in
V_{E}=\\{\textbf{v}:\hat{\mathcal{H}}\textbf{v}=E\textbf{v}\\}.$ Due to the
complexities of the commutation relations $\eqref{eq:aba}$ and the form of
$\Psi$, analytic computation of $\prod_{p+1\leq j\leq m}\hat{X}_{j}^{n_{j}}$
on $\Psi$ is not in general feasible.

From [7, Theorem 1], actions by any elements in $S_{p}$ on
$\Psi_{n_{p+1},\ldots,n_{m}}$ are still in the eigenspace space. For integrals
$\hat{X}_{i}\in S_{m}/S_{p},$ we aim to construct the recurrence relations

$\displaystyle\hat{X}_{i}\Psi_{n_{p+1},\ldots,n_{m}}=\sum_{i_{1}\in
W_{1},\,\ldots,\,i_{m}\in W_{m}}\Psi_{i_{1},...,i_{m}},$ (10)

where $W_{j}$ are the sets of integer tuples and $S_{m}/S_{p}$ defines a
subset of $S_{m}$ with all the elements in $S_{p}$ excluded. This allows us to
understand how the operators in $S_{m}/S_{p}$ acts on the eigenspace of
$S_{p}$. Let

$V_{n_{j}}:=\\{\Psi,X_{p+1}\Psi,\ldots,\Psi_{n_{p+1}},\ldots,X_{p+2}\Psi_{n_{p+2}},\ldots,\Psi_{n_{p+1},\ldots,n_{m}}\\}.$

In general it is not easy to show that the elements in $V_{n_{j}}$ form a
basis. However, if $V_{n_{j}}$ is a finite-dimensional vector space of
homogeneous polynomials, then the infinite-dimensional representations of
$\hat{\mathcal{Q}}(d)$ are given by
$\pi:\hat{\mathcal{Q}}(d)\rightarrow\mathfrak{gl}(V)$ with
$V=\bigoplus_{n_{j}\in\mathbb{N}}V_{n_{j}}.$ The polynomial commutation
relations $\eqref{eq:aba}$ can be used to simplify the vectors in
$\eqref{eq:a1}$ for explicit construction of the recurrence relations
$\eqref{eq:recur}$.

As illustrations, we apply the above construction to the cubic algebra
underlying the superintegrable integrable system
$S_{3}=\\{\hat{\mathcal{H}},\hat{X}_{1},\hat{X}_{2}\\}$ in the 2D Darboux
space with separable local coordinates $(x,y)$. Here
$\hat{\mathcal{H}}=\varphi(x)\left(\partial_{x}^{2}+\partial_{y}^{2}+c\right)$,
where $c$ is a constant, is the Hamiltonian and $\hat{X}_{1},\hat{X}_{2}$ are
integrals. Solution of the Schrödinger equation has the separable form
$\Psi(x,y)=X(x)Y(y)$, where $X(x)$ and $Y(y)$ satisfy the second order ODE
$X^{\prime\prime}+\lambda\varphi(x)X=0$ and $Y^{\prime\prime}-\lambda Y=0$
with separation constant $\lambda.$ As shown in [1], the integrals form cubic
algebra $\hat{\mathcal{Q}}(3)$ with the following commutation relations

$\displaystyle[\hat{X}_{1},\hat{X}_{2}]=$ $\displaystyle\hat{F},$
$\displaystyle[\hat{X}_{1},\hat{F}]=$ $\displaystyle
u_{1}\hat{X}_{1}^{2}+u_{2}\hat{X}_{1}+u_{3}\hat{X}_{2}+u,$ (11)
$\displaystyle[\hat{X}_{2},\hat{F}]=$ $\displaystyle
v_{1}\hat{X}_{1}^{3}+v_{2}\hat{X}_{1}^{2}+v_{3}\hat{X}_{1}-u_{2}\hat{X}_{2}-u_{1}\\{\hat{X}_{1},\hat{X}_{2}\\}+v,$

where $u_{j},v_{j},\ldots,u,v$ are polynomials of the Hamiltonian
$\hat{\mathcal{H}}$ and $v_{1}$ in (11) is non-zero coefficient. The Casimir
operator $C_{(3)}$ for the cubic algebra is given by

$\displaystyle C_{(3)}=$
$\displaystyle\hat{F}^{2}-u_{1}\\{\hat{X}_{1}^{2},\hat{X}_{2}\\}-u_{2}\\{\hat{X}_{1},\hat{X}_{2}\\}+\frac{v_{1}}{2}\hat{X}_{1}^{4}+\frac{2}{3}v_{2}\hat{X}_{1}^{3}$
$\displaystyle+\left(v_{3}+u_{1}^{2}\right)\hat{X}_{1}^{2}+\left(u_{1}u_{2}+2v\right)\hat{X}_{1}-2u\hat{X}_{2}-u_{3}\hat{X}_{2}^{2}.$

For the current case, the subset
$S_{1}=\\{\hat{\mathcal{H}},\hat{X}_{1}\\}\subset S_{3}$ contains the
simultaneously diagonalizable operators. We can show that the set $V_{m,n}$
formed by the eigenvectors $\hat{X}_{2}^{n}\hat{F}^{m}$ is a vector space such
that $\rho:\hat{\mathcal{Q}}(3)\rightarrow\mathfrak{gl}(V)$ provides infinite
dimensional representations, where $V=\oplus_{m,n\in\mathbb{N}}V_{m,n}$ and
$V_{m,n}=\\{\Psi,\ldots,\hat{F}^{m}\Psi,\ldots,\hat{X}_{2}^{n}\hat{F}^{m}\Psi\\}.$
Extending results proved in [2, Proposition 3.1, Proposition 3.4], we state

######  Proposition 2.1.

Let $\hat{\mathcal{Q}}(3)$ be the cubic algebra $\eqref{eq:a},$ and let $W$ be
the space cyclically generated by $\hat{F}.$ Let $\psi_{m}=\hat{F}\Psi^{m}$
for all $m\in\mathbb{N}$. Suppose that $u_{3}=0$. Then $V\cong W$ and $V$ form
a $\hat{\mathcal{Q}}(3)$-module. In particular, if $u_{1}=u_{2}=u_{3}=0,$ then
the representation of $\rho:\hat{\mathcal{Q}}(3)\rightarrow\mathfrak{gl}(W)$
has the form of

$\displaystyle\hat{F}\psi_{m}=$ $\displaystyle E\psi_{m+1},$
$\displaystyle\hat{X}_{1}\psi_{m}=$
$\displaystyle(m-2)u\psi_{m}+\lambda\psi_{m},$
$\displaystyle\hat{X}_{2}\psi_{m}=$ $\displaystyle
f(\psi_{m-3},\psi_{m-2},\psi_{m-1},\psi_{m},\psi_{m+2})$

where $f$ has coefficient depending on certain polynomials of $m$.

For further details, we refer the reader to the paper.

##  3 Polynomial algebras with Poisson-Lie brackets under PBW basis

Let’s first define the commutants relative to subalgebras of the universal
enveloping algebra and the symmetry algebra of a Lie algebra $\mathfrak{g}.$

######  Definition 3.1.

Let $\mathfrak{g}$ be a $n$-dimensional Lie algebra with a basis
$\beta_{\mathfrak{g}}=\\{X_{1},\ldots,X_{n}\\}$, and let $\mathfrak{g}^{*}$ be
its dual with a basis $\beta_{\mathfrak{g}^{*}}=\\{x_{1},\ldots,x_{n}\\}$. Let
$(\mathcal{U}(\mathfrak{g}),[\cdot,\cdot]$ and
$(\mathcal{S}(\mathfrak{g}^{*}),\\{\cdot,\cdot\\})$ be the universal
enveloping algebra of $\mathfrak{g}$ and the symmetric algebra of
$\mathfrak{g^{*}}$, respectively. Then the commutants relative to subalgebras
$\mathfrak{a}\subset\mathfrak{g}$ and
$\mathfrak{a}^{*}\subset\mathfrak{g}^{*}$ in $\mathcal{U}(\mathfrak{g})$ and
$\mathcal{S}(\mathfrak{g}^{*})$, denoted respectively as
$\mathcal{C}_{\mathcal{U}(\mathfrak{g})}(\mathfrak{a})$ and
$\mathcal{C}_{\mathcal{S}(\mathfrak{g}^{*})}(\mathfrak{a}^{*})$, are defined
as follows

$\displaystyle\mathcal{C}_{\mathcal{U}(\mathfrak{g})}(\mathfrak{a})$
$\displaystyle=\left\\{Y\in\mathcal{U}(\mathfrak{g}):[X,Y]=0,\quad\forall
X\in\mathfrak{a}\right\\},$
$\displaystyle\mathcal{C}_{\mathcal{S}(\mathfrak{g}^{*})}(\mathfrak{a}^{*})$
$\displaystyle=\\{y\in\mathcal{S}(\mathfrak{g}^{*}):\\{x,y\\}=0,\quad\forall
x\in\mathfrak{a}^{*}\\}.$

The adjoint action of $\mathfrak{g}$ and co-adjoint action of
$\mathfrak{g}^{*}$ on the universal enveloping algebra
$\mathcal{U}(\mathfrak{g})$ and the symmetric algebra
$\mathcal{S}(\mathfrak{g}^{*})$ are given by

$\displaystyle P\left(X_{1},\ldots,X_{n}\right)\in\mathcal{U}(\mathfrak{g})$
$\displaystyle\mapsto[X_{j},P]\in\mathcal{U}(\mathfrak{g}),$ (12)
$\displaystyle p(x_{1},\ldots,x_{n})\in\mathcal{S}(\mathfrak{g}^{*})$
$\displaystyle\mapsto\\{x_{j},p\\}=\tilde{X}_{j}(p(\textbf{x}))=\sum_{l}C_{jk}^{l}x_{l}\dfrac{\partial
p}{\partial x_{j}}\in\mathcal{S}(\mathfrak{g}^{*}),$ (13)

respectively, where
$\tilde{X}_{j}=\sum_{l}C_{jk}^{l}x_{l}\dfrac{\partial}{\partial x_{j}}$ are
vector field realizations of the generators of the Lie algebra $\mathfrak{g}$
and $\\{\cdot,\cdot\\}:C^{\infty}(\mathfrak{g}^{*})\times
C^{\infty}(\mathfrak{g}^{*})\rightarrow C^{\infty}(\mathfrak{g}^{*})$ is a
Poisson-Lie structure induced by the co-adjoint action of $\mathfrak{g}^{*}$.

Let us provide a short review on the construction of symmetry algebras from
the subalgebras $\mathfrak{a}$ of $\mathfrak{g}$ (see also [8, 9]). Using
$\eqref{eq:dual},$ the commutant
$\mathcal{C}_{\mathcal{S}(\mathfrak{g}^{*})}(\mathfrak{a}^{*})$ is generated
by the linearly independent solutions of the systems of PDEs

$\displaystyle\tilde{X}_{j}(p_{h})(\textbf{x})=\\{x_{j},p_{h}\\}(\textbf{x})=\sum_{l}C_{jk}^{l}x_{l}\dfrac{\partial
p_{h}}{\partial x_{j}}=0,\text{ }\quad 1\leq j\leq\dim\mathfrak{a}\equiv s,$
(14)

where $p_{h}(\textbf{x})$ is a homogeneous polynomial of degree $h\geq 1$ with
the generic form

$\displaystyle p_{h}(\textbf{x})=\sum_{i_{1}+\ldots+i_{s}\leq
h}\Gamma_{i_{1},\ldots,i_{s}}\,x_{1}^{i_{1}}\ldots
x_{s}^{i_{s}}\in\mathcal{S}(\mathfrak{g}^{*}).$ (15)

Notice that, depending on the structure of the Lie algebra $\mathfrak{g},$
solutions of $\eqref{eq:func}$ may not be polynomials ( see [10] and the
reference therein).

By systematically analyzing the polynomial solutions of $\eqref{eq:func}$ up
to certain degrees, we obtain polynomials that can be decomposed in terms of
products of polynomials of lower degrees. Using the symmetrization map,
commutants in the enveloping algebra are obtained as
$\mathcal{C}_{\mathcal{U}(\mathfrak{g})}(\mathfrak{a})=\Lambda\left(\mathcal{C}_{\mathcal{S}(\mathfrak{g}^{*})}(\mathfrak{a}^{*})\right)$.
Then the linearly independent monomials form a finite-generated Poisson
algebra. From the construction above, one can define algebraic Hamiltonians as
follows:

######  Definition 3.2.

[11]. Let $\mathfrak{a}\subset\mathfrak{g}$ be a Lie subalgebra and
$\mathcal{C}_{\mathcal{U}(\mathfrak{g})}(\mathfrak{a})$ be its commutant,
where $\mathfrak{a}$ admits a basis
$\beta_{\mathfrak{a}}=\\{X_{1},\ldots,X_{s}\\}$ with $\dim\mathfrak{a}=s.$ An
algebraic Hamiltonian with respect to $\mathfrak{a}$ is given by

$\displaystyle\hat{\mathcal{H}}=\sum_{1\leq i_{1},\ldots,i_{k}\leq
h}^{s}\Gamma_{i_{1},\ldots,i_{k}}X_{i_{1}}\ldots
X_{i_{k}}+\sum_{t}C_{t}K_{t},$

where $\Gamma_{i_{1},\ldots,i_{k}}$ is the constant coefficient and $K_{t}$
are the Casimir invariants of $\mathfrak{g}$.

In the following we obtain polynomial algebras from the 2D conformal algebra
$\mathfrak{c}(2)$ and their representations. In [12], the connection of
$\mathfrak{c}(2)$ with 2D Darboux spaces was established. As an example we
here present a derivation of an algebraic Hamiltonian and the underlying
symmetry algebra from the conformal algebra.

We consider the subalgebra $\mathfrak{a}_{1}$ and construct the commutant
$\mathcal{C}_{\mathcal{S}(\mathfrak{c}^{*}(2))}(\mathfrak{a}_{1}^{*})$. By
solving $\left\\{p(x_{1},\ldots,x_{6}),x_{1}\right\\}=0,$ we find $6$ linearly
independent polynomials as follows $A_{1}=x_{1},$ $A_{2}=x_{2},$
$A_{3}=x_{1}x_{3}+x_{2}x_{4},$ $A_{4}=x_{2}x_{6}+x_{3}^{2},$
$A_{5}=x_{1}x_{6}-2x_{3}x_{4}-x_{2}x_{5},$ $A_{6}=-x_{1}x_{5}+x_{4}^{2}.$ Then
possible algebraic Hamiltonian is given by

$\mathcal{H}=\alpha
x_{1}+\gamma_{1}\mathcal{C}_{1}+\gamma_{2}\mathcal{C}_{2}.$

It is easy to check that $\\{\mathcal{H},A_{j}\\}=0$ for $1\leq j\leq 6.$ The
linearly independent polynomial integrals form a quadratic Poisson algebra
$\mathcal{Q}_{1}(2)=\mathcal{Q}_{1}\oplus\mathcal{Q}_{2}$, where
$\mathcal{Q}_{1}=\mathrm{Span}\\{A_{1},A_{2}\\}$ and
$\mathcal{Q}_{2}=\mathrm{Span}\\{A_{3},\ldots,A_{6}\\},$ with the following
non-zero brackets

$\displaystyle\\{A_{2},A_{3}\\}$
$\displaystyle=A_{1}^{2}+A_{2}^{2},\quad\text{
}\\{A_{2},A_{4}\\}=-\\{A_{2},A_{6}\\}=2A_{3},$
$\displaystyle\\{A_{3},A_{4}\\}$
$\displaystyle=-A_{1}A_{5}-2A_{2}A_{6}=-\\{A_{3},A_{6}\\}.$ (16)

Notice that
$\mathcal{Z}\left(\mathcal{Q}_{1}(2)\right)=\mathrm{Span}\\{A_{1},A_{5}\\}.$
The universal enveloping Poisson algebra has the form [13]

$K_{\mathcal{Q}_{1}(2)}^{h}=\sum_{\alpha_{1}+\ldots+\alpha_{6}\leq
h}\Gamma_{\alpha_{1},\ldots,\alpha_{6}}A_{1}^{\alpha_{1}}\ldots
A_{6}^{\alpha_{6}}\in\mathcal{S}_{h}\left(\mathcal{Q}_{1}(2)\right),$

where $\mathcal{S}_{h}(\cdot)$ is the symmetry algebra of the finitely-
generated quadratic Poisson algebra $\mathcal{Q}_{1}(2).$ The functionally
independent Casimir operators are

$\displaystyle K_{\mathcal{Q}_{1}(2)}^{1,1}=A_{1},\quad\text{
}K_{\mathcal{Q}_{1}(2)}^{1,2}=A_{5},\quad\text{
}K_{\mathcal{Q}_{1}(2)}^{1,3}=A_{4}+A_{6},$ $\displaystyle
K_{\mathcal{Q}_{1}(2)}^{3,1}=(A_{1}^{2}+A_{2}^{2})A_{6}+A_{2}A_{5}A_{1}+A_{3}^{2}.$

The corresponding commutator algebra generated by the integrals has the
following commutation relations

$\displaystyle[\hat{A}_{2},\hat{A}_{3}]$
$\displaystyle=\hat{A}_{1}^{2}+\hat{A}_{2}^{2},\quad\text{
}[\hat{A}_{2},\hat{A}_{4}]=-[\hat{A}_{2},\hat{A}_{6}]=2\hat{A}_{3},$
$\displaystyle[\hat{A}_{3},\hat{A}_{4}]$
$\displaystyle=2\left(\hat{A}_{2}\hat{A}_{6}-\hat{A}_{3}\right)+2\hat{A}_{1}\hat{A}_{5}=[\hat{A}_{3},\hat{A}_{6}].$

It is clear that this is a quadratic algebra.

##  4 conclusion

In this short note, we have reviewed three different approaches to the
construction of polynomial symmetry algebras of superintegrable systems and
their representations. We have first described the application of the deformed
oscillator algebra technique to the superintegrable system in the Darboux
space. Then we have described the method for constructing infinite-dimensional
representations and commented its relevance to solving superintegrable systems
without the use separation of variables. Finally, we have described the
algebraic method based on commutants and as an example have applied the
construction to obtain the algebraic Hamiltonian and underlying quadratic
symmetric algebra from the 2D conformal algebra.

## Acknowledgement

IM was supported by the Australian Research Council Future Fellowship
FT180100099. YZZ was supported by the Australian Research Council Discovery
Project DP190101529.

##  5 Bibliography

## References

  * [1] Ian Marquette, Junze Zhang, and Yao-Zhong Zhang.  Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces.  J. Phys. A, 56(35):355201, 2023.
  * [2] I. Marquette, J. Zhang, and Y.-Z. Zhang.  Infinite-dimensional representations of cubic and quintic algebras and special functions.  The European Physical Journal Plus, 138(6):1–17, 2023.
  * [3] E. G. Kalnins, J. M. Kress, W. Miller Jr, and P. Winternitz.  Superintegrable systems in Darboux spaces.  J. Math. Phys., 44(12):5811–5848, 2003.
  * [4] E. G. Kalnins, J. M. Kress, and P. Winternitz.  Superintegrability in a two-dimensional space of nonconstant curvature.  J. Math. Phys., 43(2):970–983, 2002.
  * [5] C. Daskaloyannis.  Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems.  J. Math. Phys., 42(3):1100–1119, 2001.
  * [6] C. Daskaloyannis and K. Ypsilantis.  Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold.  J. Math. Phys., 47(4):042904, 38, 2006.
  * [7] W. Miller Jr.  Symmetry and separation of variables.  Encyclopedia of Mathematics and its Applications, Vol. 4. Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1977.  With a foreword by Richard Askey.
  * [8] L. E. Dickson.  Recent Publications: Reviews: Vorlesungen uber die Theorie der Algebraischen Zahlen.  Amer. Math. Monthly, 31(1):45–46, 1924.
  * [9] E. G. Beltrametti and A. Blasi.  On the number of Casimir operators associated with any Lie group.  Phys. Lett., 20:62–64, 1966.
  * [10] R. Campoamor-Stursberg, D. Latini, I. Marquette, and Y.-Z. Zhang.  Polynomial algebras from lie algebra reduction chains $\mathfrak{g}^{\prime}\subset\mathfrak{g}$.  arXiv preprint arXiv:2303.00975, 2023.
  * [11] L. J. Boya and R. Campoamor-Stursberg.  Commutativity of missing label operators in terms of Berezin brackets.  J. Phys. A, 42(23):235203, 12, 2009.
  * [12] A. P. Fordy.  First integrals from conformal symmetries: Darboux-Koenigs metrics and beyond.  J. Geom. Phys., 145:103475, 13, 2019.
  * [13] Thierry Lambre, Cyrille Ospel, and Pol Vanhaecke.  Poisson enveloping algebras and the Poincaré-Birkhoff-Witt theorem.  J. Algebra, 485:166–198, 2017.