# Do Language Models Understand Measurements?

Sungjin Park Seungwoo Ryu Edward Choi
KAIST
{zxznm, swryu<EMAIL_ADDRESS>
###### Abstract

Recent success of pre-trained language models (PLMs) has stimulated interest
in their ability to understand and work with numbers. Yet, the numerical
reasoning over measurements has not been formally studied despite their
importance. In this study, we show that PLMs lack the capability required for
reasoning over measurements. Furthermore, we find that a language model
trained on a measurement-rich corpus shows better performance on understanding
measurements. We propose a simple embedding strategy to better distinguish
between numbers and units, which leads to a significant improvement in the
probing tasks.

##  1 Introduction

The success of pre-trained language models (PLMs) has led to more research on
their ability to understand commonsense. In this context, numerical reasoning
over text (NRoT) is a NLP model’s ability to interpret and work with numbers
in either digit or word form Spithourakis and Riedel (2018). Recent studies on
NRoT test PLMs to answer questions on numeracy Wallace et al. (2019), scalar
magnitude comparison Zhang et al. (2020), numerical facts Lin et al. (2020),
and math word problems Wu et al. (2021).

Despite these efforts, existing works lack an analysis of the forms in which
numbers appear. In particular, we focus on the case where numbers appear as a
measurement in the context. In most scientific articles, measurements are an
integral part of the context for capturing its appropriate meaning. For
example, the two sentences "40g of Aspirin is lethal" and "40mg of Aspirin is
lethal" contain the same words except for the unit of measurement (UoM), but
the second sentence is incorrect because of the UoM.

In this work, we examine the measuring skill of PLMs: the ability to
understand the system of measurement and perform numerical reasoning over
measurements. We design three measuring skill tests (MSTs) and study how many
measuring skills can be acquired. Specifically, unit conversion, reference
range detection, and measurement comparison require understanding of the
system of measurement, the normal range of the biomedical entity, and the
ability to combine knowledge about the system of measurement and NRoT,
respectively. Table 1 shows an example of each of the measuring skill tests.

MST results showed that the models struggled to find the largest (or smallest)
value on the list of measurements and convert the measurement to another unit,
while they performed well on other tests. Compared to other PLMs, BioBERT Lee
et al. (2020) showed superior performance on unit conversion and reference
range detection, which implies that pre-training with measurement-rich text
helps the model understand the system of measurement. Finally, we speculate
that the lack of skills to distinguish numbers, units, and other words in the
context makes the models fail in some MSTs. To mitigate this, we introduce
scale embedding, which provides the model with the information regarding the
position and scale of the numbers in the input text. We show that scale
embedding significantly improves the MST performance of all PLMs.

Task | Example | Answer Candidates
---|---|---
Comparison | 1.59mg is $\mathsf{[MASK]}$ than 3.8g | larger, smaller
Argmin/max |  $\mathsf{[MASK]}$ value among 0.5mg, 3.4g, 2.8mg is 0.5mg | largest, smallest, middle
Sorting | sort 0.53mg, 32.54g, 2.8mg in $\mathsf{[MASK]}$ order is 0.53mg, 32.54g, 2.8mg | increasing, decreasing, random
Unit Conversion | 3.5g and 3500mg are $\mathsf{[MASK]}$ value |  same, different
Reference Range Detection | 85mg/dL of Glucose is $\mathsf{[MASK]}$ |  normal, abnormal

Table 1: Examples of measuring skill tests (MSTs). We underline the correct
answer for each example.

##  2 Measuring Skill Test

Task | Template
---|---
Comparison |  $\mathsf{[M]}$ is $\mathsf{[MASK]}$ than $\mathsf{[M]}$
Argmin/max |  $\mathsf{[MASK]}$ value among $\mathsf{[LoM]}$ is $\mathsf{[M]}$
Sorting | sort $\mathsf{[LoM]}$ in $\mathsf{[MASK]}$ order is $\mathsf{[LoM]}$
Unit Conversion |  $\mathsf{[M]}$ and $\mathsf{[M]}$ are $\mathsf{[MASK]}$ value
Reference Range Detection |  $\mathsf{[M]}$ of $\mathsf{[ENT]}$ is $\mathsf{[MASK]}$

Table 2: Templates which we used for data generation.
$\mathsf{[M]}$,$\mathsf{[LoM]}$, and $\mathsf{[ENT]}$ are the placeholder for
the measurement, the list of measurements, and the biomedical entity,
respectively.

In this section, we describe three MSTs to carefully study the ability of PLMs
to understand the system of measurement and perform numerical reasoning over
the measurements.

###  2.1 Unit Conversion

This task requires the model to decide whether the two measurements represent
the same quantity. For example, the model might correctly predict [MASK] in a
sentence, such as "3.5g and 3500mg are [MASK] value" to be filled with same if
it understands the conversion of units correctly. In general, it is a
convention to combine the unit (e.g., liter, meter) and its prefix (e.g.,
kilo, milli) to represent the numerical value of the measurement within a
range $[10^{-3},10^{3})$. Therefore, various unit prefixes can appear in a
single passage, even if the units are the same. To handle this, unit
conversion is essential for complex reasoning over measurements. To succeed in
unit conversion, we expect the model to handle the unit and numerical value
jointly, based on an understanding of the system of measurement.

###  2.2 Reference Range Detection

Given a biomedical entity and measurement, this task requires a model to
predict whether the measurement falls within the reference range. Knowledge of
the biomedical entity plays a crucial role in understanding measurements,
since the unit is determined by the biomedical entity. For example, we measure
the hemoglobin level in g/dL. In addition to understanding UoMs, PLMs must
rely on domain knowledge embedded in their parameters to solve this task, as
context alone does not provide sufficient clues as to what the reference range
is for the given biomedical entity.

###  2.3 Measurement Comparison

Given two measurements (or a series of n measurements), the task is to predict
the correct relationship between them. We created the synthetic dataset
following other well-known NRoT tasks. Here, we consider three numerical
reasoning tasks: Comparison Talmor et al. (2020), ArgMin/Max Wallace et al.
(2019), and Sorting Pal and Baral (2021), all requiring the model to compare
numbers. Note that each measurement in this task can have a different unit
prefix. For example, the sample "1.59mg is [MASK] than 3.8g" containing two
different units "mg" and "g" appears in the Comparison dataset. This task
assesses the model’s ability to combine an understanding of measurements and
numerical reasoning skills.

##  3 Experiments

Task | Measurement Comparison | Unit | Ref
---|---|---|---
Comp | Arg | Sort
Model | Notation | $in$ | $ex$ | $in$ | $ex$ | $in$ | $ex$ | $in$ | $ex$ | $in$ | $ex$
ALBERT | Sci | 81.2 | 77.3 | 60.4 | 58.0 | 78.2 | 76.5 | 48.6 | 49.9 | 71.9 | 59.9
Deci | 81.8 | 72.1 | 57.1 | 50.5 | 82.5 | 74.3 | 61.5 | 56.2 | 71.1 | 61.0
BERT | Sci | 73.3 | 72.4 | 55.1 | 52.2 | 45.6 | 45.0 | 52.7 | 51.2 | 73.5 | 64.3
Deci | 81.4 | 77.0 | 60.9 | 54.3 | 54.9 | 54.5 | 61.9 | 59.2 | 77.2 | 67.5
BioBERT | Sci | 82.7 | 82.3 | 55.0 | 54.4 | 68.2 | 69.1 | 58.7 | 57.3 | 81.3 | 63.7
Deci | 90.1 | 88.0 | 59.0 | 57.6 | 77.3 | 73.0 | 73.0 | 70.5 | 87.0 | 64.2
BlueBERT | Sci | 77.3 | 76.3 | 46.9 | 46.9 | 63.6 | 64.3 | 53.0 | 51.3 | 73.6 | 65.4
Deci | 74.6 | 73.2 | 57.0 | 55.5 | 73.0 | 68.0 | 59.2 | 57.1 | 77.1 | 69.0
Scratch | Sci | 50.9 | 50.8 | 40.2 | 37.1 | 33.3 | 33.8 | 52.5 | 50.7 | 66.3 | 60.8
Deci | 57.7 | 51.3 | 44.3 | 43.0 | 33.3 | 33.7 | 56.8 | 53.9 | 62.6 | 65.0

Table 3: Test-set results on MSTs. We report the classification accuracy on
interpolation ($in$) and extrapolation ($ex$) test dataset. comp, arg, sort,
unit, and ref are abbreviations of comparison, argmin/max, sorting, Unit
Conversion, and Reference Range Detection, respectively. Sci and Deci stand
for scientific and decimal notations, respectively.

Probing Setup We formulated MSTs as a Cloze test Talmor et al. (2020) to fully
utilize the knowledge captured by masked language modeling (MLM).
Specifically, a PLM received the masked inputs given in Table 1, and the MLM
head output the probability distribution of the answer candidates for [MASK].
Among the answer candidates, we chose the one with the highest probability as
the final prediction.

We probed four transformer-based PLMs. BERT Devlin et al. (2019) and ALBERT
Lan et al. (2020) were trained on Wikipedia articles and Book Corpus. BioBERT
Lee et al. (2020) was trained on biomedical articles from PubMed abstracts,
and BlueBERT Peng et al. (2020) used both clinical (MIMIC-III Johnson et al.
(2016)) and biomedical (PubMed abstracts) corpus for pre-training. We also
tested a randomly initialized transformer encoder (i.e. Scratch) to evaluate
the difficulty of our MSTs. For each model, we did not update the parameters
during training, except for the MLM head in the last transformer layer. In all
tasks, the models were trained with three random seeds and we report the mean
classification accuracy for all the probing tasks. Appendix A provides further
details on training and evaluation.

Data Preparation We manually crafted templates in Table 2 that contained at
most two slots for measurements and [MASK] token for an answer. We
instantiated [M] and [LoM] by sampling the measurement and the list of
measurements, respectively. For measurement sampling, we independently sampled
a number and a unit and then combined them. Specifically, we sampled units
from the predefined set in Table 7 which consists of SI units and some units
in MIMIC-III.

The numbers in the training dataset were sampled from $[10^{-2},10^{2})$. For
evaluation, we constructed two evaluation datasets: 1) Interpolation sampled
numbers from the same range as the training dataset; 2) Extrapolation sampled
numbers from $[10^{-3},10^{3})$. Note that we did not consider the numbers
outside the range $[10^{-3},10^{3})$, because many of the unit prefixes are in
the power of thousands. Zhang et al. (2020) reported that representing numbers
in scientific notation made it easier for the language model to capture the
scale of numbers. Following this observation, we tested two different number
notations: decimal and scientific. For example, 32.6 can be represented as
32.6 and 3.26E+01 in decimal and scientific notation, respectively. We
randomly varied the number of digits after the decimal point between zero and
three, and the significant digits were maintained after converting the number
notation.

For reference range detection, we collected biomedical entities from six
tables in MIMIC-III (INPUT, OUTPUT, LAB, PRESCRIPTION, PROCEDURE, and CHART)
and chose the subset.

We report the number of samples and the distribution of labels for each MST in
Table 8.

##  4 Results and Analysis

Task | Measurement Comparison | Unit | Ref
---|---|---|---
Comp | Arg | Sort
Prompt Set | Model | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$
Label | ALBERT | 78.7 (-3.1) | 70.8 (-1.3) | 40.9 (-16.2) | 33.1 (-17.4) | 73.6 (-8.9) | 67.2 (-7.1) | 55.5 (-6.0) | 56.0 (-0.2) | 51.1 (-20.0) | 36.0 (-25.0)
BERT | 73.1 (-8.3) | 70.8 (-6.2) | 54.0 (-6.9) | 50.7 (-3.6) | 54.0 (-0.9) | 54.3 (-0.2) | 56.6 (-5.3) | 55.0 (-4.2) | 40.3 (-36.9) | 13.7 (-53.8)
BioBERT | 82.8 (-7.3) | 80.2 (-7.8) | 56.7 (-2.3) | 55.7 (-1.9) | 66.4 (-10.9) | 62.6 (-10.4) | 61.9 (-11.1) | 60.4 (-10.1) | 69.1 (-17.9) | 59.6 (-4.6)
BlueBERT | 75.0 (0.4) | 69.7 (-3.5) | 56.9 (-0.1) | 55.3 (-0.2) | 70.1 (-2.9) | 66.6 (-1.4) | 56.4 (-2.8) | 54.9 (-2.2) | 76.4 (-0.7) | 70.6 (1.6)
Context | ALBERT | 67.8 (-14.0) | 61.8 (-10.3) | 49.1 (-8.0) | 43.5 (-7.0) | 72.3 (-10.2) | 68.2 (-6.1) | 50.4 (-11.1) | 50.5 (-5.7) | 65.8 (-5.3) | 56.9 (-4.1)
BERT | 70.2 (-11.2) | 67.9 (-9.1) | 52.4 (-8.5) | 47.1 (-7.2) | 51.8 (-3.1) | 50.6 (-3.9) | 56.1 (-5.8) | 55.2 (-4.0) | 66.4 (-10.8) | 63.2 (-4.3)
BioBERT | 80.7 (-9.4) | 78.4 (-9.6) | 58.1 (-0.9) | 55.9 (-1.7) | 73.6 (-3.7) | 69.7 (-3.3) | 60.4 (-12.6) | 59.3 (-11.2) | 75.2 (-11.8) | 64.8 (0.6)
BlueBERT | 71.5 (-3.1) | 65.6 (-7.6) | 51.9 (-5.1) | 48.5 (-7.0) | 60.6 (-12.4) | 56.4 (-11.6) | 50.7 (-8.5) | 50.7 (-6.4) | 67.4 (-9.7) | 64.9 (-4.1)
UoM | ALBERT | 87.9 (6.1) | 80.2 (8.1) | 71.0 (13.9) | 58.8 (8.3) | 90.5 (8.0) | 85.5 (11.2) | 64.2 (2.7) | 56.9 (0.7) | N/A | N/A
BERT | 90.0 (8.6) | 87.2 (10.2) | 69.4 (8.5) | 68.2 (13.9) | 67.1 (12.2) | 63.0 (8.5) | 67.2 (5.3) | 64.8 (5.6) | N/A | N/A
BioBERT | 96.5 (6.4) | 94.8 (6.8) | 72.1 (13.1) | 69.3 (11.7) | 84.0 (6.7) | 79.6 (6.6) | 82.5 (9.5) | 77.9 (7.4) | N/A | N/A
BlueBERT | 88.3 (13.7) | 83.8 (10.6) | 66.5 (9.5) | 63.8 (8.3) | 76.5 (3.5) | 71.7 (3.7) | 66.6 (7.4) | 62.4 (5.3) | N/A | N/A

Table 4: Test-set results on different sets of prompts. We report the
classification accuracy and the performance difference ($\Delta$). We obtain
$\Delta$ by subtracting the results in Table 3 from this table.

Task | Measurement Comparison | Ref
---|---|---
Comp | Arg | Sort
Model | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$
ALBERT | 87.5 (5.7) | 85.3 (13.2) | 71.8 (14.7) | 68.2 (17.7) | 87.3 (4.8) | 85.3 (11.0) | 65.6 (-5.5) | 53.8 (-7.2)
BERT | 88.3 (6.9) | 86.8 (9.8) | 59.3 (-1.6) | 60.6 (6.3) | 77.5 (22.6) | 77.5 (23.0) | 69.6 (-7.6) | 64.2 (-3.3)
BioBERT | 94.7 (4.6) | 93.5 (5.5) | 70.7 (11.7) | 68.3 (10.7) | 83.6 (6.3) | 82.9 (9.9) | 77.6 (-9.4) | 65.7 (1.5)
BlueBERT | 88.7 (14.1) | 86.1 (12.9) | 64.8 (7.8) | 60.9 (5.4) | 75.2 (2.2) | 74.2 (6.2) | 70.3 (-6.8) | 66.0 (-3.0)

Table 5: Test-set results on rule-based conversion experiments. We report the
classification accuracy and the performance difference ($\Delta$).

Measuring Skills of PLMs Table 3 shows the results of MSTs stated in Section
2.

PLMs performed reasonably well on comparison, sorting, and reference range
detection, but struggled considerably on argmin/max and unit conversion tasks.
This shows that some measuring skills are difficult to learn from an LM
objective. Similar to previous NRoT studies Wallace et al. (2019); Pal and
Baral (2021), PLMs often failed to successfully extrapolate to values outside
the training range. Further, in most cases, MST results got worse when we
represented numbers in scientific notation.

We observed that BioBERT outperformed other PLMs in unit conversion, reference
range detection, and comparison, and showed comparable performance in the rest
of the MSTs. Compared to BioBERT, BlueBERT was pre-trained on a larger volume
of biomedical text, but showed worse performance. This shows that pre-training
on measurement-rich corpora assists the model in acquiring measuring skills,
but further training on the noisy clinical text could harm it when performing
reasoning over measurements. We also found that ALBERT outperformed its
competitors in Sorting even though it performed the same or worse on other
tasks. This may be because ALBERT benefits from its sentence order prediction
(SOP) objective, which predicts the ordering of two consecutive segments of
text.

Effect of using Different Prompts One can expect that the choice of prompt has
an impact on the results, and recent studies Jiang et al. (2020); Petroni et
al. (2019) support this. To see whether the results in Table 3 are maintained
as the prompt differs, we trained and evaluated PLMs on three distinct sets of
prompts: context, UoM, and label. Specifically, context, UoM, and label
examine how consistent MST results are against various linguistic expressions
of prompts, the set of unique UoMs in the dataset, and the choice of answer
candidates, respectively. Note that we considered answer candidates as part of
the prompt, since the prompt determines the set of correct answers.

For context, we manually created four additional templates that have the same
meaning as the original template in Table 2. For uom, we used only a subset of
units g, l, m, and s, which appear frequently in the general text. For label,
we included synonyms of the label as answer candidates. For example, "less",
"smaller", and "lower" are the answers for the prompt "1.59mg is [MASK] than
3.8mg.". More details of the experiments are in the Appendix B.

The results with the decimal notation are shown in Table 4. We can see that
the results vary with the choice of prompt, indicating that PLMs are indeed
sensitive to it. However, we found that MST performance maintains a similar
tendency in every experiment: BioBERT works well on comparsion, unit
conversion, and reference range detection, and ALBERT works well on sorting.

Figure 1: Our scale embedding.

Task | Measurement Comparison | Unit | Ref
---|---|---|---
Comp | Arg | Sort
Model | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$
ALBERT | 92.9 (11.1) | 78.9 (6.8) | 73.6 (16.5) | 63.5 (13.0) | 92.9 (10.4) | 86.8 (12.5) | 75.7 (14.2) | 68.0 (11.8) | 83.5 (12.4) | 63.9 (2.9)
BERT | 95.9 (14.5) | 89.0 (12.0) | 79.8 (18.9) | 75.6 (21.3) | 91.7 (36.8) | 90.9 (36.4) | 87.9 (26.0) | 80.2 (21.0) | 95.3 (18.1) | 62.6 (-4.9)
BioBERT | 98.4 (8.3) | 93.2 (5.2) | 85.9 (26.9) | 83.0 (25.4) | 94.0 (16.7) | 93.0 (20.0) | 90.1 (17.1) | 85.7 (15.2) | 98.4 (11.4) | 61.9 (-2.3)
BlueBERT | 97.4 (22.8) | 88.1 (14.9) | 75.9 (18.9) | 67.7 (12.2) | 91.8 (18.8) | 90.3 (22.3) | 80.0 (20.8) | 76.2 (19.1) | 94.3 (17.2) | 66.1 (-2.9)
Scratch | 70.2 (12.5) | 60.3 (9.0) | 45.5 (1.2) | 44.1 (1.1) | 33.2 (-0.1) | 33.8 (0.1) | 60.3 (3.5) | 56.1 (2.2) | 69.0 (6.4) | 66.3 (1.3)

Table 6: Effect of scale embedding on MSTs. We report the classification
accuracy and performance improvement ($\Delta$) after applying scale
embedding.

Rule-based Conversion of Measurements Measurements exhibit a certain pattern,
regardless of the domain, because of a global standard: the International
System of Units (SI). Thus, we can manually detect and convert all units in
the text without difficulty. Then, it is natural to wonder if converting all
units based on rules is easier than making the language model understand the
system of measurement. To answer this question, we tested the rule-based
conversion that detects measurements with the regular expression and converts
them into a prefix-free form. For example, the sentence "2.5mg is [MASK] than
3.8g" is converted to "0.0025g is [MASK] than 3.8g" after the rule-based
conversion. We examined the rule-based conversion on measurement comparison
and reference range detection.

The results with the decimal notation are shown in Table 5. The rule-based
conversion increased measurement comparison performance because the converted
measurement comparison does not require an understanding of unit conversion to
solve the problem. However, it can be seen that almost all models became worse
on reference range detection. This shows that the knowledge about the
reference range is highly correlated with the specific UoM. Thus, the rule-
based conversion is a suboptimal choice if we want to utilize the domain
knowledge embedded in PLMs.

Scale Embedding and its Effect In Section 4, we observed that none of the PLMs
showed a perfect understanding of each MST. We suspect that such a gap
originates in the deficiency of PLM’s ability to extract numerical values from
measurements and compare their magnitudes. To this end, we propose scale
embedding, an additional embedding that provides the model with the
information of the position and scale of numbers in the input text.

As described in Figure 1, we incrementally assigned the index to each token
from the end to the beginning of a sentence. If we encounter a token that is
not included in the numerical value, then we reset the index to zero and keep
assigning the index zero to tokens until another numerical value appears. We
distinguished between numerical and nonnumerical subwords using the regular
expression. Note that we trained only the scale embedding and MLM head while
freezing other pre-trained weights of the language model. This allows us to
adapt the model to any numerical reasoning tasks simply by plugging a
different scale embedding into them.

Table 6 shows the MST results after the scale embedding is applied to all
models, where we can see significantly improved test results, even for
Argmin/max and unit conversion.111The full set of experimental results are
shown in Table 13 in the Appendix. Note that the scale embedding is minimally
effective for Scratch, except for comparison. This shows that solving our MSTs
requires more than just simple embeddings, and a PLM that understands context
is an essential element.

##  5 Related Works

Over the years, numerical reasoning has been an active research area. Some
works investigate the numeracy of static word embeddings (Naik et al., 2019),
contextualized language embeddings (Wallace et al., 2019), and multilingual
words (Johnson et al., 2020). Wallace et al. (2019) shows that ELMo, BERT, and
GloVe embeddings are capable of capturing numeracy, but only within the range
of numbers seen during training. GenBERT (Geva et al., 2020), NumGPT (Jin et
al., 2021), and NT5 (Yang et al., 2021) focus on incorporating arithmetic
skills into pre-trained models. Another task that deals with numerical
quantities is measurement estimation. VerbPhysics (Forbes and Choi, 2017)
proposes the dataset to compare the relative scales between the physical
attributes of various objects. DoQ (Elazar et al., 2019) provides an empirical
distribution over possible values of quantitative attributes. Zhang et al.
(2020) tests that NLP models contain information about the scalar magnitudes
of physical objects. Although previous studies probed numerical reasoning over
numeral and physical attributes, no attempt has been made to investigate
reasoning over measurements.

##  6 Conclusion

To the best of our knowledge, our study is the first to investigate reasoning
over measurements. Our analysis shows that PLMs lack the capability required
for reasoning over measurements. We proposed a scale embedding approach that
provides information on the position and scale of numbers, and it
significantly increases the MST performance.

## Acknowledgements

Research supported with Cloud TPUs from Google’s TPU Research Cloud (TRC).
This work was supported by Institute of Information & Communications
Technology Planning & Evaluation (IITP) grant (No.2019-0-00075), the Korea
Health Industry Development Institute (KHIDI) grant (No.HI21C1138), and the
Korea Medical Device Development Fund grant (Project No.: 1711138160,
KMDF_PR_20200901_0097), funded by the Korea government (MSIT, MOTIE, MOHW,
MFDS).

## Limitations

Our scale embedding can make mistakes when the unit itself contains numbers
(e.g. mg/100ml). Therefore, scale embedding should not be applied to UoM
containing numbers through exception handling.

Our work will be largely affected by the created prompts. If the prompt is not
obvious for PLMs to understand, although they have such reasoning ability,
they may not give the correct answer. To mitigate this problem, we conducted
experiments with different sets of prompts in Section 4 and showed that the
results maintain their tendency across the prompts. Despite these efforts, it
is still unclear what the optimal choice of the prompt is. We remain this
problem as a future work.

## References

  * Devlin et al. (2019) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2019.  BERT: Pre-training of deep bidirectional transformers for language understanding.  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 4171–4186, Minneapolis, Minnesota. Association for Computational Linguistics.
  * Elazar et al. (2019) Yanai Elazar, Abhijit Mahabal, Deepak Ramachandran, Tania Bedrax-Weiss, and Dan Roth. 2019.  How large are lions? inducing distributions over quantitative attributes.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 3973–3983, Florence, Italy. Association for Computational Linguistics.
  * Forbes and Choi (2017) Maxwell Forbes and Yejin Choi. 2017.  Verb physics: Relative physical knowledge of actions and objects.  In _Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 266–276, Vancouver, Canada. Association for Computational Linguistics.
  * Geva et al. (2020) Mor Geva, Ankit Gupta, and Jonathan Berant. 2020.  Injecting numerical reasoning skills into language models.  In _Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics_ , pages 946–958, Online. Association for Computational Linguistics.
  * Jiang et al. (2020) Zhengbao Jiang, Frank F. Xu, Jun Araki, and Graham Neubig. 2020.  How Can We Know What Language Models Know? _Transactions of the Association for Computational Linguistics_ , 8:423–438.
  * Jin et al. (2021) Zhihua Jin, Xin Jiang, Xingbo Wang, Qun Liu, Yong Wang, Xiaozhe Ren, and Huamin Qu. 2021.  Numgpt: Improving numeracy ability of generative pre-trained models.  _arXiv preprint arXiv:2109.03137_.
  * Johnson et al. (2016) Alistair E W Johnson, Tom J Pollard, Lu Shen, Li-Wei H Lehman, Mengling Feng, Mohammad Ghassemi, Benjamin Moody, Peter Szolovits, Leo Anthony Celi, and Roger G Mark. 2016.  Mimic-iii, a freely accessible critical care database.  _Scientific data_ , 3(1):1–9.
  * Johnson et al. (2020) Devin Johnson, Denise Mak, Andrew Barker, and Lexi Loessberg-Zahl. 2020.  Probing for multilingual numerical understanding in transformer-based language models.  In _Proceedings of the Third BlackboxNLP Workshop on Analyzing and Interpreting Neural Networks for NLP_ , pages 184–192, Online. Association for Computational Linguistics.
  * Lan et al. (2020) Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut. 2020.  Albert: A lite bert for self-supervised learning of language representations.  In _International Conference on Learning Representations_.
  * Lee et al. (2020) Jinhyuk Lee, Wonjin Yoon, Sungdong Kim, Donghyeon Kim, Sunkyu Kim, Chan Ho So, and Jaewoo Kang. 2020.  Biobert: a pre-trained biomedical language representation model for biomedical text mining.  _Bioinformatics_ , 36(4):1234–1240.
  * Lin et al. (2020) Bill Yuchen Lin, Seyeon Lee, Rahul Khanna, and Xiang Ren. 2020.  Birds have four legs?! NumerSense: Probing Numerical Commonsense Knowledge of Pre-Trained Language Models.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP)_ , pages 6862–6868, Online. Association for Computational Linguistics.
  * Naik et al. (2019) Aakanksha Naik, Abhilasha Ravichander, Carolyn Rose, and Eduard Hovy. 2019.  Exploring numeracy in word embeddings.  In _Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics_ , pages 3374–3380, Florence, Italy. Association for Computational Linguistics.
  * Pal and Baral (2021) Kuntal Kumar Pal and Chitta Baral. 2021.  Investigating numeracy learning ability of a text-to-text transfer model.  In _Findings of the Association for Computational Linguistics: EMNLP 2021_ , pages 3095–3101, Punta Cana, Dominican Republic. Association for Computational Linguistics.
  * Paszke et al. (2019) Adam Paszke, Sam Gross, Francisco Massa, Adam Lerer, James Bradbury, Gregory Chanan, Trevor Killeen, Zeming Lin, Natalia Gimelshein, Luca Antiga, Alban Desmaison, Andreas Kopf, Edward Yang, Zachary DeVito, Martin Raison, Alykhan Tejani, Sasank Chilamkurthy, Benoit Steiner, Lu Fang, Junjie Bai, and Soumith Chintala. 2019.  Pytorch: An imperative style, high-performance deep learning library.  In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, _Advances in Neural Information Processing Systems 32_ , pages 8024–8035. Curran Associates, Inc.
  * Peng et al. (2020) Yifan Peng, Qingyu Chen, and Zhiyong Lu. 2020.  An empirical study of multi-task learning on BERT for biomedical text mining.  In _Proceedings of the 19th SIGBioMed Workshop on Biomedical Language Processing_ , pages 205–214, Online. Association for Computational Linguistics.
  * Petroni et al. (2019) Fabio Petroni, Tim Rocktäschel, Sebastian Riedel, Patrick Lewis, Anton Bakhtin, Yuxiang Wu, and Alexander Miller. 2019.  Language models as knowledge bases? 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 2463–2473, Hong Kong, China. Association for Computational Linguistics.
  * Spithourakis and Riedel (2018) Georgios Spithourakis and Sebastian Riedel. 2018.  Numeracy for language models: Evaluating and improving their ability to predict numbers.  In _Proceedings of the 56th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers)_ , pages 2104–2115, Melbourne, Australia. Association for Computational Linguistics.
  * Talmor et al. (2020) Alon Talmor, Yanai Elazar, Yoav Goldberg, and Jonathan Berant. 2020.  oLMpics-on what language model pre-training captures.  _Transactions of the Association for Computational Linguistics_ , 8:743–758.
  * Wallace et al. (2019) Eric Wallace, Yizhong Wang, Sujian Li, Sameer Singh, and Matt Gardner. 2019.  Do NLP models know numbers? probing numeracy in embeddings.  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 5307–5315, Hong Kong, China. Association for Computational Linguistics.
  * Wolf et al. (2020) Thomas Wolf, Lysandre Debut, Victor Sanh, Julien Chaumond, Clement Delangue, Anthony Moi, Pierric Cistac, Tim Rault, Remi Louf, Morgan Funtowicz, Joe Davison, Sam Shleifer, Patrick von Platen, Clara Ma, Yacine Jernite, Julien Plu, Canwen Xu, Teven Le Scao, Sylvain Gugger, Mariama Drame, Quentin Lhoest, and Alexander Rush. 2020.  Transformers: State-of-the-art natural language processing.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 38–45, Online. Association for Computational Linguistics.
  * Wu et al. (2021) Qinzhuo Wu, Qi Zhang, Zhongyu Wei, and Xuanjing Huang. 2021.  Math word problem solving with explicit numerical values.  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 5859–5869, Online. Association for Computational Linguistics.
  * Yang et al. (2021) Peng Jian Yang, Ying Ting Chen, Yuechan Chen, and Daniel Cer. 2021.  Nt5?! training T5 to perform numerical reasoning.  _arXiv preprint arXiv:2104.07307_.
  * Yuan et al. (2021) Weizhe Yuan, Graham Neubig, and Pengfei Liu. 2021.  Bartscore: Evaluating generated text as text generation.  In _Advances in Neural Information Processing Systems_ , volume 34, pages 27263–27277. Curran Associates, Inc.
  * Zhang et al. (2020) Xikun Zhang, Deepak Ramachandran, Ian Tenney, Yanai Elazar, and Dan Roth. 2020.  Do language embeddings capture scales? In _Findings of the Association for Computational Linguistics: EMNLP 2020_ , pages 4889–4896, Online. Association for Computational Linguistics.

##  Appendix A Further Details of Measuring Skill Test

###  A.1 Data Statistics

Table 8 shows the statistics of MSTs we used for experiments.

###  A.2 Training and Evaluation

The BERT configuration of all models is the same as the base model (L=12,
H=768, A=12, Total Parameters=110M) in Devlin et al. (2019). Maximum sequence
length is 512.

We trained the model with batch size 256 for 30 epochs. We used the Adam
optimizer for training. The learning rate started from 5e-5 and linearly
decayed towards 1e-8. We early stopped the training when the validation
accuracy did not increase for 2 epochs. The batch size for evaluation is 128,
and other settings are the same as training. We found the optimal
hyperparameters using the grid search, where we evaluated the learning rate
[1e-5, 2e-5, 5e-5, 1e-4], batch size [16,32,64,128].

UoM
---
w/o Prefix | w/ Prefix
m | m, cm, mm, µm, nm
A | A, mA, µA, nA
K | K, mK, µK
M | M, mM, µM, nM
Eq/l | Eq/l, mEq/l, µEq/l, mEq/ml, mEq/µl
g/l | g/l, mg/l, µg/l, mg/dl, g/dl, µg/dl, ng/dl, g/ml, mg/ml
IU/l | IU/l, IU/ml, mIU/ml, µIU/ml, mIU/l, µIU/l, IU/µl, mIU/µl
U/l | U/l, U/ml, U/µl
l/min | l/min, dl/min, ml/min, µl/min
#/l | #/dl, #/ml, #/µl
k/l | k/dl, k/ml, k/µl
l | l, dl, ml, µl, nl, pl, fl
g | g, mg, µg, ng, pg, fg
s | s, ms, µs, ns
m/hr | m/hr, cm/hr, mm/hr, µm/hr
l/hr | l/hr, dl/hr, ml/hr, µl/hr

Table 7: List of units used for data generation.

Task | Split | Number Range | # Samples | Label Distribution
---|---|---|---|---
comparison | train | interpolation | 299,394 | smaller: 0.5, larger: 0.5
valid | interpolation | 29,986 | smaller: 0.498, larger: 0.502
extrapolation | 30,000 | smaller: 0.495, larger: 0.505
test | interpolation | 29,988 | smaller: 0.501, larger: 0.499
extrapolation | 30,000 | smaller: 0.501, larger: 0.499
argmin/max | train | interpolation | 300,000 | smallest:0.333, middle: 0.334, largest: 0.333
valid | interpolation | 30,000 | smallest:0.333, middle: 0.334, largest: 0.333
extrapolation | 30,000 | smallest:0.333, middle: 0.334, largest: 0.333
test | interpolation | 30,000 | smallest:0.332, middle: 0.335, largest: 0.333
extrapolation | 30,000 | smallest:0.335, middle: 0.332, largest: 0.333
sorting | train | interpolation | 300,000 | decreasing: 0.333, random: 0.332, increasing: 0.335
valid | extrapolation | 30,000 | decreasing: 0.333, random: 0.332, increasing: 0.335
extrapolation | 30,000 | decreasing: 0.337, random: 0.332, increasing: 0.331
test | interpolation | 30,000 | decreasing: 0.337, random: 0.332, increasing: 0.331
extrapolation | 30,000 | decreasing: 0.328, random: 0.339, increasing: 0.333
unit conversion | train | interpolation | 259,588 | same: 0.489, different: 0.511
valid | interpolation | 23,931 | same: 0.489, different: 0.511
extrapolation | 28,814 | same: 0.5, different: 0.5
test | interpolation | 23,538 | same: 0.483, different: 0.517
extrapolation | 28,696 | same: 0.498, different: 0.502
reference range detection | train | interpolation | 201,061 | normal: 0.575, abnormal: 0.425
valid | interpolation | 17,111 | normal: 0.593, abnormal: 0.407
extrapolation | 21,212 | normal: 0.618, abnormal: 0.382
test | interpolation | 16,948 | normal: 0.586, abnormal: 0.414
extrapolation | 18,429 | normal: 0.659, abnormal: 0.341

Table 8: Statistics of MSTs used for experiments.

##  Appendix B More Details of Prompt Sets

The results with both decimal and scientific notation are shown in Table 9.

###  B.1 label

Inspired by Yuan et al. (2021), we included synonyms as an answer to make the
prompt diverse. We used the website https://www.wordhippo.com/ to search for
synonyms. Among the search results, we chose two words that match the context.
We report the list of synonyms in Table 10.

###  B.2 context

If the context differs from what PLM saw during pre-training, then PLMs will
struggle to solve MSTs even if they understand the measuring skills. To
mitigate this, we prepared four additional prompts with the same meaning.
Additional prompts are listed in Table 11.

###  B.3 uom

In the general domain, some UoMs listed in Table 7 rarely appear in the
context. For example, international units per liter (IU/l) is frequently used
in pharmacology, but not in other scientific articles. Therefore, we can
wonder if some rare biomedical units disrupt the understanding of general
domain PLMs (e.g., BERT and ALBERT). To answer this question, we replaced all
UoMs in the dataset with the commonly used UoMs: g, l, m, and s.

##  Appendix C Additional Results on Rule-based Conversion

Table 12 describes the complete set of MST results after applying rule-based
conversion.

##  Appendix D Additional Results on Scale Embedding

Table 13 describes the MST results of scale embedding with decimal and
scientific notation.

##  Appendix E Experimental Environment

We trained the models with Google TPU v2-8 and v3-8. We used PyTorch 1.10.0
Paszke et al. (2019) and Huggingface Transformers Wolf et al. (2020) 4.3.3 for
experiments.

Task | Measurement Comparison | Unit | Ref
---|---|---|---
Comp | Arg | Sort
Prompt Set | Model | Notation | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$
Label | ALBERT | Sci | 77.1 (-4.1) | 73.8 (-3.5) | 40.5 (-19.9) | 34.2 (-23.8) | 71.3 (-6.9) | 65.6 (-10.9) | 54.5 (5.9) | 53.0 (3.1) | 51.4 (-20.5) | 34.6 (-25.3)
Deci | 78.7 (-3.1) | 70.8 (-1.3) | 40.9 (-16.2) | 33.1 (-17.4) | 73.6 (-8.9) | 67.2 (-7.1) | 55.5 (-6.0) | 56.0 (-0.2) | 51.1 (-20.0) | 36.0 (-25.0)
BERT | Sci | 68.9 (-4.4) | 68.7 (-3.7) | 42.9 (-12.2) | 42.9 (-9.3) | 47.3 (1.7) | 46.4 (1.4) | 53.4 (0.7) | 52.2 (1.0) | 47.4 (-26.1) | 23.0 (-41.3)
Deci | 73.1 (-8.3) | 70.8 (-6.2) | 54.0 (-6.9) | 50.7 (-3.6) | 54.0 (-0.9) | 54.3 (-0.2) | 56.6 (-5.3) | 55.0 (-4.2) | 40.3 (-36.9) | 13.7 (-53.8)
BioBERT | Sci | 74.0 (-8.7) | 73.3 (-9.0) | 50.9 (-4.1) | 50.8 (-3.6) | 61.8 (-6.4) | 61.7 (-7.4) | 55.8 (-2.9) | 54.1 (-3.2) | 59.3 (-22.0) | 54.3 (-9.4)
Deci | 82.8 (-7.3) | 80.2 (-7.8) | 56.7 (-2.3) | 55.7 (-1.9) | 66.4 (-10.9) | 62.6 (-10.4) | 61.9 (-11.1) | 60.4 (-10.1) | 69.1 (-17.9) | 59.6 (-4.6)
| BlueBERT | Sci | 77.0 (-0.3) | 76.3 (0.0) | 44.7 (-2.2) | 44.3 (-2.6) | 64.6 (1.0) | 65.1 (0.8) | 52.9 (-0.1) | 51.3 (0.0) | 70.0 (-3.6) | 63.8 (-1.6)
| Deci | 75.0 (0.4) | 69.7 (-3.5) | 56.9 (-0.1) | 55.3 (-0.2) | 70.1 (-2.9) | 66.6 (-1.4) | 56.4 (-2.8) | 54.9 (-2.2) | 76.4 (-0.7) | 70.6 (1.6)
Context | ALBERT | Sci | 68.4 (-12.8) | 65.4 (-11.9) | 49.3 (-11.1) | 48.2 (-9.8) | 71.4 (-6.8) | 69.2 (-7.3) | 49.7 (1.1) | 50.0 (0.1) | 65.7 (-6.2) | 57.2 (-2.7)
Deci | 67.8 (-14.0) | 61.8 (-10.3) | 49.1 (-8.0) | 43.5 (-7.0) | 72.3 (-10.2) | 68.2 (-6.1) | 50.4 (-11.1) | 50.5 (-5.7) | 65.8 (-5.3) | 56.9 (-4.1)
BERT | Sci | 65.4 (-7.9) | 64.5 (-7.9) | 46.8 (-8.3) | 44.6 (-7.6) | 44.6 (-1.0) | 43.9 (-1.1) | 52.7 (0.0) | 52.5 (1.3) | 65.4 (-8.1) | 58.9 (-5.4)
Deci | 70.2 (-11.2) | 67.9 (-9.1) | 52.4 (-8.5) | 47.1 (-7.2) | 51.8 (-3.1) | 50.6 (-3.9) | 56.1 (-5.8) | 55.2 (-4.0) | 66.4 (-10.8) | 63.2 (-4.3)
BioBERT | Sci | 74.6 (-8.1) | 74.2 (-8.1) | 53.3 (-1.7) | 52.9 (-1.5) | 68.4 (0.2) | 67.9 (-1.2) | 55.0 (-3.7) | 54.7 (-2.6) | 70.8 (-10.5) | 61.2 (-2.5)
Deci | 80.7 (-9.4) | 78.4 (-9.6) | 58.1 (-0.9) | 55.9 (-1.7) | 73.6 (-3.7) | 69.7 (-3.3) | 60.4 (-12.6) | 59.3 (-11.2) | 75.2 (-11.8) | 64.8 (0.6)
| BlueBERT | Sci | 68.4 (-8.9) | 67.1 (-9.2) | 40.4 (-6.5) | 40.6 (-6.3) | 56.8 (-6.8) | 56.5 (-7.8) | 50.3 (-2.7) | 50.2 (-1.1) | 66.5 (-7.1) | 60.3 (-5.1)
| Deci | 71.5 (-3.1) | 65.6 (-7.6) | 51.9 (-5.1) | 48.5 (-7.0) | 60.6 (-12.4) | 56.4 (-11.6) | 50.7 (-8.5) | 50.7 (-6.4) | 67.4 (-9.7) | 64.9 (-4.1)
UoM |  |  |  |  |  |  |  |  |  |  |  |
ALBERT | Sci | 92.8 (11.6) | 87.2 (9.9) | 73.0 (12.6) | 68.7 (10.7) | 89.0 (10.8) | 84.6 (8.1) | 60.5 (11.9) | 56.1 (6.2) | N/A | N/A
Deci | 87.9 (6.1) | 80.2 (8.1) | 71.0 (13.9) | 58.8 (8.3) | 90.5 (8.0) | 85.5 (11.2) | 64.2 (2.7) | 56.9 (0.7) | N/A | N/A
BERT | Sci | 81.5 (8.2) | 78.9 (6.5) | 60.6 (5.5) | 61.8 (9.6) | 50.7 (5.1) | 49.0 (4.0) | 56.5 (3.8) | 53.2 (2.0) | N/A | N/A
Deci | 90.0 (8.6) | 87.2 (10.2) | 69.4 (8.5) | 68.2 (13.9) | 67.1 (12.2) | 63.0 (8.5) | 67.2 (5.3) | 64.8 (5.6) | N/A | N/A
BioBERT | Sci | 91.3 (8.6) | 90.3 (8.0) | 53.6 (-1.4) | 52.2 (-2.2) | 72.7 (4.5) | 70.5 (1.4) | 66.1 (7.4) | 63.2 (5.9) | N/A | N/A
| Deci | 96.5 (6.4) | 94.8 (6.8) | 72.1 (13.1) | 69.3 (11.7) | 84.0 (6.7) | 79.6 (6.6) | 82.5 (9.5) | 77.9 (7.4) | N/A | N/A
| BlueBERT | Sci | 85.8 (8.5) | 83.0 (6.7) | 57.9 (11.0) | 57.3 (10.4) | 68.6 (5.0) | 69.4 (5.1) | 57.2 (4.2) | 53.5 (2.2) | N/A | N/A
| Deci | 88.3 (13.7) | 83.8 (10.6) | 66.5 (9.5) | 63.8 (8.3) | 76.5 (3.5) | 71.7 (3.7) | 66.6 (7.4) | 62.4 (5.3) | N/A | N/A

Table 9: Test-set results on different sets of prompts. We report the
classification accuracy and the performance difference ($\Delta$). We obtain
$\Delta$ by subtracting the results in Table 3 from this table.

Task | Answer Candidates | Synonyms
---|---|---
comparison | larger | higher, bigger
smaller | lower, less
argmin/max | largest | biggest, maximum
middle | medium, intermediate
smallest | lowest, minimum
sorting | increasing | growing, ascending
random | unclear, confusing
decreasing | reducing, descending
unit conversion | same | equal, identical
different | distinct, unlike
reference range detection | normal | regular, safe
abnormal | irregular, lethal

Table 10: Synonyms of the answer candidates we used for label.

Task | Template
---|---
comparison |  $\mathsf{[M]}$ is $\mathsf{[MASK]}$ than $\mathsf{[M]}$
compared to $\mathsf{[M]}$, $\mathsf{[M]}$ is $\mathsf{[MASK]}$ value
the measurement of control group ($\mathsf{[M]}$) is $\mathsf{[MASK]}$ than
$\mathsf{[M]}$
comparison: $\mathsf{[M]}$, $\mathsf{[M]}$, result: $\mathsf{[MASK]}$
$\mathsf{[M]}$ $\mathsf{[MASK]}$ $\mathsf{[M]}$
argmin/max | The $\mathsf{[MASK]}$ value among $\mathsf{[LoM]}$ is $\mathsf{[M]}$
$\mathsf{[M]}$ is the $\mathsf{[MASK]}$ value of $\mathsf{[LoM]}$
Among the list of measurements $\mathsf{[LoM]}$, the $\mathsf{[MASK]}$ value
is $\mathsf{[M]}$
argmin,argmax: $\mathsf{[LoM]}$, $\mathsf{[M]}$, result: $\mathsf{[MASK]}$
$\mathsf{[MASK]}$ $\mathsf{[LoM]}$ , $\mathsf{[M]}$
sorting | sort $\mathsf{[LoM]}$ in $\mathsf{[MASK]}$ order is $\mathsf{[LoM]}$
arranging $\mathsf{[LoM]}$ in $\mathsf{[MASK]}$ order is $\mathsf{[LoM]}$
$\mathsf{[LoM]}$ is obtained by sorting $\mathsf{[LoM]}$ in $\mathsf{[MASK]}$
order
sort: $\mathsf{[LoM]}$, $\mathsf{[LoM]}$, result: $\mathsf{[MASK]}$
$\mathsf{[LoM]}$ $\mathsf{[MASK]}$ $\mathsf{[LoM]}$
unit conversion |  $\mathsf{[M]}$ and $\mathsf{[M]}$ are the $\mathsf{[MASK]}$ value
convert $\mathsf{[M]}$ to $\mathsf{[MASK]}$ value, then the result is
$\mathsf{[M]}$
compare $\mathsf{[M]}$ to $\mathsf{[M]}$, the two are the $\mathsf{[MASK]}$
value
measurement comparison: $\mathsf{[M]}$, $\mathsf{[M]}$, result:
$\mathsf{[MASK]}$
$\mathsf{[M]}$ , $\mathsf{[M]}$ $\mathsf{[MASK]}$
reference range detection |  $\mathsf{[M]}$ of $\mathsf{[ENT]}$ is $\mathsf{[MASK]}$
$\mathsf{[M]}$ of $\mathsf{[ENT]}$ falls into $\mathsf{[MASK]}$ range
The physician decides $\mathsf{[M]}$ of $\mathsf{[ENT]}$ as $\mathsf{[MASK]}$
reference range: $\mathsf{[ENT]}$, $\mathsf{[M]}$, result: $\mathsf{[MASK]}$
$\mathsf{[ENT]}$ $\mathsf{[M]}$ $\mathsf{[MASK]}$

Table 11: Templates for context. $\mathsf{[M]}$ is the measurement and
$\mathsf{[LoM]}$ is the list of measurements.

Task | Measurement Comparison | Ref
---|---|---
Comp | Arg | Sort
Model | Notation | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$
ALBERT | Sci | 73.8 (-7.4) | 73.0 (-4.3) | 54.7 (-5.7) | 51.0 (-7.0) | 80.4 (2.2) | 77.7 (1.2) | 71.8 (-0.1) | 62.3 (2.4)
Deci | 87.5 (5.7) | 85.3 (13.2) | 71.8 (14.7) | 68.2 (17.7) | 87.3 (4.8) | 85.3 (11.0) | 65.6 (-5.5) | 53.8 (-7.2)
BERT | Sci | 69.2 (-4.1) | 68.6 (-3.8) | 53.7 (-1.4) | 52.4 (0.2) | 53.4 (7.8) | 53.4 (8.4) | 73.2 (-0.3) | 63.5 (-0.8)
Deci | 88.3 (6.9) | 86.8 (9.8) | 59.3 (-1.6) | 60.6 (6.3) | 77.5 (22.6) | 77.5 (23.0) | 69.6 (-7.6) | 64.2 (-3.3)
BioBERT | Sci | 80.6 (-2.1) | 80.1 (-2.2) | 50.4 (-4.6) | 47.0 (-7.4) | 69.0 (0.8) | 68.1 (-1.0) | 79.0 (-2.3) | 64.3 (0.6)
Deci | 94.7 (4.6) | 93.5 (5.5) | 70.7 (11.7) | 68.3 (10.7) | 83.6 (6.3) | 82.9 (9.9) | 77.6 (-9.4) | 65.7 (1.5)
BlueBERT | Sci | 68.3 (-9.0) | 65.8 (-10.5) | 40.2 (-6.7) | 40.1 (-6.8) | 66.9 (3.3) | 67.0 (2.7) | 74.0 (0.4) | 65.0 (-0.4)
Deci | 88.7 (14.1) | 86.1 (12.9) | 64.8 (7.8) | 60.9 (5.4) | 75.2 (2.2) | 74.2 (6.2) | 70.3 (-6.8) | 66.0 (-3.0)
Scratch | Sci | 58.2 (7.3) | 54.6 (3.8) | 43.0 (2.8) | 41.0 (3.9) | 33.3 (0.0) | 33.7 (-0.1) | 64.7 (-1.6) | 62.8 (2.0)
Deci | 78.8 (21.1) | 73.9 (22.6) | 43.5 (-0.8) | 42.9 (-0.1) | 33.2 (-0.1) | 33.9 (0.2) | 63.6 (1.0) | 64.3 (-0.7)

Table 12: Test-set results on rule-based conversion experiments. We report the
classification accuracy and the performance difference.

Task | Measurement Comparison | Unit | Ref
---|---|---|---
Comp | Arg | Sort
Model | Notation | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$ | $in(\Delta)$ | $ex(\Delta)$
ALBERT | Sci | 93.4 (12.2) | 86.3 (9.0) | 73.2 (12.8) | 66.0 (8.0) | 92.7 (14.5) | 90.1 (13.6) | 74.8 (26.2) | 61.6 (11.7) | 87.0 (15.1) | 63.4 (3.5)
Deci | 92.9 (11.1) | 78.9 (6.8) | 73.6 (16.5) | 63.5 (13.0) | 92.9 (10.4) | 86.8 (12.5) | 75.7 (14.2) | 68.0 (11.8) | 83.5 (12.4) | 63.9 (2.9)
BERT | Sci | 96.4 (23.1) | 95.0 (22.6) | 80.9 (25.8) | 80.5 (28.3) | 89.8 (44.2) | 89.5 (44.5) | 79.9 (27.2) | 67.5 (16.3) | 92.4 (18.9) | 61.9 (-2.4)
Deci | 95.9 (14.5) | 89.0 (12.0) | 79.8 (18.9) | 75.6 (21.3) | 91.7 (36.8) | 90.9 (36.4) | 87.9 (26.0) | 80.2 (21.0) | 95.3 (18.1) | 62.6 (-4.9)
BioBERT | Sci | 98.3 (15.6) | 96.3 (14.0) | 81.3 (26.3) | 80.7 (26.3) | 94.0 (25.8) | 93.6 (24.5) | 89.3 (30.6) | 66.7 (9.4) | 96.0 (14.7) | 64.7 (1.0)
Deci | 98.4 (8.3) | 93.2 (5.2) | 85.9 (26.9) | 83.0 (25.4) | 94.0 (16.7) | 93.0 (20.0) | 90.1 (17.1) | 85.7 (15.2) | 98.4 (11.4) | 61.9 (-2.3)
BlueBERT | Sci | 96.0 (18.7) | 93.1 (16.8) | 76.0 (29.1) | 74.9 (28.0) | 86.2 (22.6) | 85.8 (21.5) | 77.4 (24.4) | 63.3 (12.0) | 91.1 (17.5) | 66.7 (1.3)
Deci | 97.4 (22.8) | 88.1 (14.9) | 75.9 (18.9) | 67.7 (12.2) | 91.8 (18.8) | 90.3 (22.3) | 80.0 (20.8) | 76.2 (19.1) | 94.3 (17.2) | 66.1 (-2.9)
Scratch | Sci | 59.5 (8.6) | 57.4 (6.6) | 41.4 (1.2) | 39.9 (2.8) | 33.4 (0.1) | 33.8 (0.0) | 52.5 (0.0) | 50.6 (-0.1) | 80.0 (13.7) | 61.8 (1.0)
Deci | 70.2 (12.5) | 60.3 (9.0) | 45.5 (1.2) | 44.1 (1.1) | 33.2 (-0.1) | 33.8 (0.1) | 60.3 (3.5) | 56.1 (2.2) | 69.0 (6.4) | 66.3 (1.3)

Table 13: Effect of scale embedding on MSTs. We report the classification
accuracy and performance improvement ($\Delta$) after applying scale
embedding.


# Weight modules over split Lie algebras

Antonio J. Calderón Martín
José M. Sánchez Delgado
Departamento de Matemáticas.
Universidad de Cádiz. 11510 Puerto Real, Cádiz, Spain.
e-mail<EMAIL_ADDRESS>
e-mail<EMAIL_ADDRESS>Supported by the PCI of the UCA
‘Teoría de Lie y Teoría de Espacios de Banach’, by the PAI with project
numbers FQM298, FQM2467, FQM3737 and by the project of the Spanish Ministerio
de Educación y Ciencia MTM2007-60333.

###### Abstract

We study the structure of weight modules $V$ with restrictions neither on the
dimension nor on the base field, over split Lie algebras $L$. We show that if
$L$ is perfect and $V$ satisfies $LV=V$ and ${\mathcal{Z}}(V)=0$, then

$L=\bigoplus\limits_{i\in I}I_{i}$ and $V=\bigoplus\limits_{j\in J}V_{j}$

with any $I_{i}$ an ideal of $L$ satisfying $[I_{i},I_{k}]=0$ if $i\neq k$,
and any $V_{j}$ a (weight) submodule of $V$ in such a way that for any $j\in
J$ there exists a unique $i\in I$ such that $I_{i}V_{j}\neq 0,$ being $V_{j}$
a weight module over $I_{i}$. Under certain conditions, it is shown that the
above decomposition of $V$ is by means of the family of its minimal
submodules, each one being a simple (weight) submodule.

Keywords: Infinite dimensional Lie module, infinite dimensional split Lie
algebra, structure theory.

##  1 Introduction and previous definitions

Throughout this paper, weight modules $V$ and split Lie algebras $L$ are
considered of arbitrary dimensions and over an arbitrary base field 𝕂. It is
worth to mention that, unless otherwise stated, there is not any restriction
on $\dim{V}_{\gamma}$, $\dim{L}_{\alpha}$ or the products
$L_{\alpha}V_{\gamma}$ where $V_{\gamma}$ denotes the weight space associated
to the weight $\gamma$ of $V$ and $L_{\alpha}$ the root space associated to
the root $\alpha$ of $L$.

In $\S 2$ we develop connection of weights techniques in the framework of
weight modules $V$ over split Lie algebras $L$ so as to show that in case
$LV=V$ and ${\mathcal{Z}}(V)=0$ then $V$ decomposes as the direct sum of an
adequate family of nonzero (weight) submodules of $V$,
$V=\bigoplus\limits_{j\in J}V_{j}$. In $\S 3$ we slightly modify the above
mentioned connection techniques of weights in order to apply them to the set
of nonzero roots of $L$, as consequence we show that if $L$ is perfect then
$L=\bigoplus\limits_{i\in I}I_{i}$ where any $I_{i}$ is a nonzero ideal of $L$
satisfying $[I_{i},I_{k}]=0$ if $i\neq k$. In $\S 4$ we relate the
decompositions of $V$ and $L$ obtained in the previous sections so as to get
as main result that in case $LV=V$, ${\mathcal{Z}}(V)=0$ and $L$ is perfect
then $L=\bigoplus\limits_{i\in I}I_{i}$ and $V=\bigoplus\limits_{j\in J}V_{j}$
as above, in such a way that for any $j\in J$ there exists one and only one
$i\in I$ such that $I_{i}V_{j}\neq 0$, being any $V_{j}$ a weight module over
$I_{i}$. In the last section, $\S 5$, we study if any of the $V_{j}$, $j\in
J$, in the above decomposition of $V$ is simple. Under certain conditions we
give an affirmative answer. Finally, we would like to note the increasing
interest in the study of weight modules over split Lie algebras, (and
superalgebras), specially motivated by their relation with mathematical
physics (see [12, 14, 17, 20, 23, 30, 31]).

Given an element $x$ of a Lie algebra $L,$ we denote by ${\rm ad}_{x}$ the
adjoint mapping ${\rm ad}_{x}$ defined as ${\rm ad}_{x}(y):=[x,y]$ for any
$y\in L$. A splitting Cartan subalgebra $H$ of ${L}$ is defined as a maximal
abelian subalgebra of $L$, satisfying that the adjoint mappings ${\rm
ad}_{h}$, for $h\in H$, are simultaneously diagonalizable. If $L$ contains a
splitting Cartan subalgebra $H$, then $L$ is called a split Lie algebra, (see
for instance [28]). This means that we have a root spaces decomposition

$L=H\oplus(\bigoplus\limits_{\alpha\in\Lambda}{L}_{\alpha})$

where $L_{\alpha}=\\{v_{\alpha}\in L:[v_{\alpha},h]=\alpha(h)v_{\alpha}$ for
any $h\in H\\}$ for a linear functional $\alpha\in H^{*}$ and
$\Lambda:=\\{\alpha\in H^{*}\backslash\\{0\\}:L_{\alpha}\neq 0\\}$. The
subspaces $L_{\alpha}$ for $\alpha\in H^{*}$ are called root spaces of $L$,
(respect to $H$), and the elements $\alpha\in\Lambda\cup\\{0\\}$ are called
roots of $L$, (respect to $H$). Clearly $L_{0}=H$ and, as consequence of
Jacobi identity, $[L_{\alpha},L_{\beta}]\subset L_{\alpha+\beta}$ for any
$\alpha,\beta\in\Lambda\cup\\{0\\}$. We also say that $\Lambda$ is symmetric
if for any $\alpha\in\Lambda$ we have that $-\alpha\in\Lambda$. Here we note
that there are many interesting examples of split Lie algebras with a
symmetric root system. For instance, we have the separable semisimple
$L^{*}$-algebras ([27]), the semisimple locally finite split Lie algebras over
a field of characteristic zero ([24]), the generalized oscilator algebras
([28, Example 1.4, (a)]) or the Virasoro algebras ([28, Example 1.4, (b)]).

We will denote by ${\mathcal{Z}}(L)=\\{e\in L:[e,L]=0\\}$ the center of a Lie
algebra $L$. We also recall that $L$ is called perfect if ${\mathcal{Z}}(L)=0$
and $[L,L]=L$.

######  Definition 1.1.

Let $V$ be a module over a Lie algebra $L$ with splitting Cartan subalgebra
$H$. For a linear functional $\gamma:H\longrightarrow\hbox{\hueca K},$ the
weight space of $V$, (respect to $H$), associated with $\gamma$ is the
subspace

$V_{\gamma}=\\{v_{\gamma}\in V:hv_{\gamma}=\gamma(h)v_{\gamma}\hskip
5.69046pt{\it for}\hskip 5.69046pt{\it any}\hskip 5.69046pth\in H\\}.$

The elements $\gamma\in H^{*}$ satisfying $V_{\gamma}\neq 0$ are called
weights of $V$ respect to $H$ and we denote $\mathcal{P}:=\\{\gamma\in
H^{*}\backslash\\{0\\}:V_{\gamma}\neq 0\\}$. We say that $V$ is a weight
module, respect to $H$, if

$V=V_{0}\oplus(\bigoplus\limits_{\gamma\in\mathcal{P}}{V}_{\gamma}).$

We also say that $\mathcal{P}$ is the weight system of $V$.

The weight system $\mathcal{P}$ is called symmetric if for any
$\gamma\in\mathcal{P}$ we have that $-\gamma\in\mathcal{P}$.

Split Lie algebras are examples of weight modules over themselves, where
$\mathcal{P}=\Lambda$ and $V_{\gamma}=L_{\gamma}$ for
$\gamma\in\mathcal{P}\cup\\{0\\}$. Since the even part $L^{0}$ of the standard
embedding of a split Lie triple system $T$ and of a split twisted inner
derivation triple system $M$ is a split Lie algebra, the natural actions of
$L^{0}$ over $T$ and $M$ make of $T$ and $M$ weight modules over the split Lie
algebra $L^{0}$. So the present paper extend the results in [6, 8, 9]. We also
have that any split Lie superalgebra, see [10], $L=L^{\bar{0}}\oplus
L^{\bar{1}}$ is a weight module over the split Lie algebra $L^{\bar{0}}$.
These examples provide us of interesting applications of the results in the
paper which can be interpreted from a common viewpoint, the one of weight
modules over a split Lie algebra. We devote Section 6 to develop these
examples in detail. We also remark in $\S 6$ future perspectives of the
matter.

##  2 Connections of weights. Decompositions of $V$

From now on, (throughout the paper),

$V=V_{0}\oplus(\bigoplus\limits_{\gamma\in\mathcal{P}}{V}_{\gamma})$

denotes a weight module with a symmetric weight system $\mathcal{P}$, respect
to a split Lie algebra

$L=H\oplus(\bigoplus\limits_{\alpha\in\Lambda}L_{\alpha})$

with a symmetric root system $\Lambda$.

In order to clarify the results along the paper let us introduce a concrete
example which is of potential interest in the modelling of physical problems
involving split Lie algebras and some involutive characteristic. After any
main result, we will refer to this example to illustrate it.

######  Example 2.1.

Let
${\mathfrak{L}}=H\oplus\bigoplus\limits_{\alpha\in\Lambda}{\mathfrak{L}}_{\alpha}$
be a split Lie algebra over a base field of characteristic distinct to two
with a symmetric root system, and $\xi$ an involutive automorphism of
${\mathfrak{L}}$. Hence we can write

${\mathfrak{L}}=Sym({\mathfrak{L}},\xi)\oplus Skw({\mathfrak{L}},\xi),$

where $Sym({\mathfrak{L}},\xi)=\\{e\in{\mathfrak{L}}:\xi(e)=e\\}$ and
$Skw({\mathfrak{L}},\xi)=\\{e\in{\mathfrak{L}}:\xi(e)=-e\\}.$ Denote by
$\Pi_{0}:{\mathfrak{L}}\to Sym({\mathfrak{L}},\xi)$ and
$\Pi_{1}:{\mathfrak{L}}\to Skw({\mathfrak{L}},\xi)$ the projection maps
$\Pi_{i}(x_{0}+x_{1})=x_{i}$. Suppose $\xi(H)\subset H$ and
$\Pi_{i}({\mathfrak{L}}_{\alpha})\neq 0$ for any
$\alpha\in\Lambda\cup\\{0\\}$, $i\in\\{0,1\\}$. Then it is straightforward to
verify that

$L:=Sym({\mathfrak{L}},\xi)$

is a split Lie algebra respect to the splitting Cartan subalgebra $Sym(H,\xi)$
with set of nonzero roots
$\Lambda|_{Sym(H,\xi)}=\\{{\alpha}|_{Sym(H,\xi)}:\alpha\in\Lambda\\}$ and with
nonzero root spaces

$L_{{\alpha}|_{Sym(H,\xi)}}=\Pi_{0}({\mathfrak{L}}_{\alpha})$

and that

$V:=Skw({\mathfrak{L}},\xi)$

is a weight module respect to the split Lie algebra $L$ under the natural
action

$Sym({\mathfrak{L}},\xi)\times Skw({\mathfrak{L}},\xi)\to
Skw({\mathfrak{L}},\xi)$ $(x,y)\mapsto[x,y],$

with set of nonzero weights
$\mathcal{P}=\Lambda=\\{{\alpha}|_{Sym(H,\xi)}:\alpha\in\Lambda\\}$, with
nonzero weight spaces

$V_{{\alpha}|_{Sym(H,\xi)}}=\Pi_{1}({\mathfrak{L}}_{\alpha})$

and with $V_{0}=Skw(H,\xi)$.

Let us return to our study of weight modules over split Lie algebras by
developing connections of weights techniques in this framework.

######  Definition 2.1.

Let $\gamma$ and $\delta$ be two nonzero weights. We say that $\gamma$ is
connected to $\delta$ if there exist $\alpha_{1},...,\alpha_{n}\in\Lambda$
such that

  1. 1.

$\\{\gamma+\alpha_{1},\gamma+\alpha_{1}+\alpha_{2},...,\gamma+\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n-1}\\}\subset\mathcal{P}$,

  2. 2.

$\gamma+\alpha_{1}+\alpha_{2}+\dots+\alpha_{n}\in\\{\delta,-\delta\\}$,

where the sums are considered in $H^{*}$.

We also say that $\\{\gamma,\alpha_{1},...,\alpha_{n}\\}$ is a connection from
$\gamma$ to $\delta$.

######  Note 2.1.

For an easier notation, we will understand that $\\{\gamma\\}$ is a connection
from $\gamma$ to itself and to $-\gamma$.

The next result shows the connection relation is of equivalence.

######  Proposition 2.1.

The relation $\sim$ in $\mathcal{P}$ defined by $\gamma\sim\delta$ if and only
if $\gamma$ is connected to $\delta$ is an equivalence relation.

###### Proof.

By Note 2.1, $\\{\gamma\\}$ is a connection from $\gamma$ to itself and
therefore $\gamma\sim\gamma$.

Let us see the symmetric character of $\sim$: If $\gamma\sim\delta$, there
exists a connection $\\{\gamma,\alpha_{1},...,\alpha_{n}\\}$ from $\gamma$ to
$\delta$, being so

$\\{\gamma+\alpha_{1},\gamma+\alpha_{1}+\alpha_{2},...,\gamma+\alpha_{1}+\cdots+\alpha_{n-1}\\}\subset\mathcal{P}$

and $\gamma+\alpha_{1}+\cdots+\alpha_{n}\in\\{\delta,-\delta\\}.$ Hence, we
can distinguish two possibilities. In the first one
$\gamma+\alpha_{1}+\cdots+\alpha_{n}=\delta,$ and in the second one
$\gamma+\alpha_{1}+\cdots+\alpha_{n}=-\delta.$ Now observe that the set
$\\{\delta,-\alpha_{n},-\alpha_{n-1},...,-\alpha_{1}\\}$ gives us a connection
from $\delta$ to $\gamma$ if we have the first possibility and
$\\{\delta,\alpha_{n},\alpha_{n-1},...,\alpha_{1}\\}$ if we have the second
one. Hence $\sim$ is symmetric.

Finally, suppose $\gamma\sim\delta$ and $\delta\sim\eta$, and write
$\\{\gamma,\alpha_{1},...,\alpha_{n}\\}$ for a connection from $\gamma$ to
$\delta$ and $\\{\delta,\beta_{1},...,\beta_{m}\\}$ for a connection from
$\delta$ to $\eta$. If $\delta\neq\pm\eta$, then $m\geq 1$ and so
$\\{\gamma,\alpha_{1},...,\alpha_{n},\beta_{1},...,\beta_{m}\\}$ is a
connection from $\gamma$ to $\eta$ in case
$\gamma+\alpha_{1}+\cdots+\alpha_{n}=\delta$, and
$\\{\gamma,\alpha_{1},...,\alpha_{n},-\beta_{1},...,-\beta_{m}\\}$ in case
$\gamma+\alpha_{1}+\cdots+\alpha_{n}=-\delta$. If $\delta\in\\{\eta,-\eta\\}$
then $\\{\gamma,\alpha_{1},...,\alpha_{n}\\}$ is a connection from $\gamma$ to
$\eta$. Therefore $\gamma\sim\eta$ and $\sim$ is of equivalence. ∎

Given $\gamma\in\mathcal{P}$, we denote by

$\mathcal{P}_{\gamma}:=\\{\delta\in\mathcal{P}:\delta\sim\gamma\\}.$

Clearly if $\delta\in\mathcal{P}_{\gamma}$ then
$-\delta\in\mathcal{P}_{\gamma}$ and, by Proposition 2.1, if
$\eta\notin\mathcal{P}_{\gamma}$ then
$\mathcal{P}_{\gamma}\cap\mathcal{P}_{\eta}=\emptyset$.

Our next goal is to associate an (adequate) weight submodule
$V_{\mathcal{P}_{\gamma}}$ to any $\mathcal{P}_{\gamma}$. For
$\mathcal{P}_{\gamma},\gamma\in\mathcal{P},$ we define the following linear
subspace of $V$:

$V_{\mathcal{P}_{\gamma}}:=(\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}_{\gamma}}L_{-\alpha}V_{\alpha})\oplus(\bigoplus\limits_{\delta\in\mathcal{P}_{\gamma}}V_{\delta}).$

(We also denote by
$V_{0,\mathcal{P}_{\gamma}}:=\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}_{\gamma}}L_{-\alpha}V_{\alpha}\subset
V_{0}$).

######  Lemma 2.1.

The following assertions hold:

  1. 1.

For any $\alpha\in\Lambda$ and $\gamma\in\mathcal{P}$ with
$\alpha\neq-\gamma$, if $L_{\alpha}V_{\gamma}\neq 0$ then
$\alpha+\gamma\sim\gamma$.

  2. 2.

For any $\alpha,\beta\in\Lambda\cap\mathcal{P}$, if
$L_{\beta}(L_{-\alpha}V_{\alpha})\neq 0$ then $\alpha\sim\beta$.

###### Proof.

1\. The fact $L_{\alpha}V_{\gamma}\neq 0$ with $\alpha\neq-\gamma$ ensures
$\alpha+\gamma\in\mathcal{P}$. Hence, just consider the connection
$\\{\gamma,\alpha\\}$.

2\. If $\beta=\pm\alpha$ it is clear. Hence suppose $\beta\neq\pm\alpha$. From
$L_{\beta}(L_{-\alpha}V_{\alpha})\neq 0$ we have either
$[L_{\beta},L_{-\alpha}]V_{\alpha}\neq 0$ or
$L_{-\alpha}(L_{\beta}V_{\alpha})\neq 0$. In the first case
$\beta-\alpha\in\Lambda$ and $\\{\alpha,\beta-\alpha\\}$ is a connection form
$\alpha$ to $\beta$. In the second case we have $\alpha+\beta\in\mathcal{P}$
and then $\\{\alpha,\beta,-\alpha\\}$ is a connection form $\alpha$ to
$\beta$. ∎

We recall that a Lie module $V$ is said to be simple if its only submodules
are $\\{0\\}$ and $V$.

######  Theorem 2.1.

Let $\gamma\in\mathcal{P}$. Then the following assertions hold.

  1. 1.

$V_{\mathcal{P}_{\gamma}}$ is a weight submodule of $V$.

  2. 2.

If $V$ is simple, then there exists a connection from $\gamma$ to $\delta$ for
any $\gamma,\delta\in\mathcal{P}$ and
$V_{0}=\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}_{\gamma}}L_{-\alpha}V_{\alpha}$.

###### Proof.

1\. For any $e\in L$ and for any $v\in V_{\mathcal{P}_{\gamma}}$ we can write
$e=h+\sum\limits_{i=1}^{n}e_{\alpha_{i}}$, with $h\in H$, $e_{\alpha_{i}}\in
L_{\alpha_{i}}$ and $\alpha_{i}\in\Lambda$; and
$v=v_{0}+\sum\limits_{j=1}^{m}v_{\gamma_{j}}$ with
$v_{0}\in\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}_{\gamma}}L_{-\alpha}V_{\alpha}$,
$v_{\gamma_{j}}\in V_{\gamma_{j}},$ and $\gamma_{j}\in\mathcal{P}_{\gamma}.$

From here

$ev\in(\bigoplus\limits_{j=1}^{m}V_{\gamma_{j}})+(\sum\limits_{i=1}^{n}L_{\alpha_{i}}(L_{-\alpha}V_{\alpha}))+(\bigoplus\limits_{i=1,j=1}^{n,m}L_{\alpha_{i}}V_{\gamma_{j}}).$
(1)

If $L_{\alpha_{i}}(L_{-\alpha}V_{\alpha})\neq 0$, then
$\alpha_{i}\in\mathcal{P}$ and so Lemma 2.1-2 gives us $\alpha\sim\alpha_{i}$.
We have $\alpha_{i}\in\mathcal{P}_{\gamma}$ and we get

$\sum\limits_{i=1}^{n}L_{\alpha_{i}}(L_{-\alpha}V_{\alpha})\subset\sum\limits_{i=1}^{n}V_{\alpha_{i}}\subset
V_{\mathcal{P}_{\gamma}}.$

Consider now the third summand in (1). If $\alpha_{i}=-\gamma_{j}$ then

$L_{\alpha_{i}}V_{\gamma_{j}}=L_{-\gamma_{j}}V_{\gamma_{j}}\subset
V_{\mathcal{P}_{\gamma}}.$ (2)

If $\alpha_{i}\neq-\gamma_{j}$, Lemma 2.1-1 gives us
$\alpha_{i}+\gamma_{j}\sim\gamma_{j}$. Hence
$\alpha_{i}+\gamma_{j}\in\mathcal{P}_{\gamma}$ and so

$L_{\alpha_{i}}V_{\gamma_{j}}\subset V_{\alpha_{i}+\gamma_{j}}\subset
V_{\mathcal{P}_{\gamma}}.$

From here, and taking into account equation (2), we get
$\bigoplus\limits_{i=1,j=1}^{n,m}L_{\alpha_{i}}V_{\gamma_{j}}\subset
V_{\mathcal{P}_{\gamma}}$ which completes the proof of 1.

2\. The simplicity of $V$ implies $V_{\mathcal{P}_{\gamma}}=V$ for any
$\gamma\in\mathcal{P}$. Therefore $\mathcal{P}_{\gamma}=\mathcal{P}$, and so
$V$ has all its nonzero weights connected, and
$V_{0}=\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}_{\gamma}}L_{-\alpha}V_{\alpha}$.
∎

Theorem 2.1-1 let us assert that for any $\gamma\in\mathcal{P}$,
$V_{\mathcal{P}_{\gamma}}$ is a weight submodule of $V$ that we call the
submodule of $V$ associated to $\mathcal{P}_{\gamma}$.

######  Proposition 2.2.

For a linear complement $\mathcal{U}$ of $span_{\hbox{\hueca
K}}\\{L_{-\alpha}V_{\alpha}:\alpha\in\Lambda\cap\mathcal{P}\\}$ in $V_{0}$, we
have the sum of 𝕂-vector subspaces

$V=\mathcal{U}+(\sum\limits_{[\gamma]\in\mathcal{P}/\sim}V_{[\gamma]}),$

where any $V_{[\gamma]}$ is one of the weight submodules described in Theorem
2.1-1.

###### Proof.

By Proposition 2.1, we can consider the quotient set

$\mathcal{P}/\sim:=\\{[\gamma]:\gamma\in\mathcal{P}\\}.$

Let us denote by $V_{[\gamma]}:=V_{\mathcal{P}_{\gamma}}$. We have
$V_{[\gamma]}$ is well defined and by Theorem 2.1-1 a weight submodule of $V$.
Therefore
$V=\mathcal{U}+(\sum\limits_{[\gamma]\in\mathcal{P}/\sim}V_{[\gamma]}).$ ∎

Recall that the center of $V$ is defined as the set ${\mathcal{Z}}(V)=\\{v\in
V:Lv=0\\}$.

######  Theorem 2.2.

If $LV=V$ and ${\mathcal{Z}}(V)=0$, then $V$ is the direct sum of the weight
submodules given in Proposition 2.2,

$V=\bigoplus\limits_{[\gamma]\in\mathcal{P}/\sim}V_{[\gamma]}.$

###### Proof.

Taking into account Proposition 2.2, it is clear that the fact $LV=V$ gives us
$V=\sum\limits_{[\gamma]\in\mathcal{P}/\sim}V_{[\gamma]}.$ To verify the
direct character of this sum, we just have to prove that the sum
$V_{0}=\sum\limits_{[\gamma]\in\mathcal{P}/\sim}V_{0,[{\gamma}]}$, where
$V_{0,[{\gamma}]}=\sum\limits_{\beta\in\Lambda\cap[{\gamma}]}L_{-\beta}V_{\beta}$
is direct. So, take $v_{0}\in
V_{0,[{\gamma}]}\cap(\sum\limits_{[\gamma_{i}]\in\mathcal{P}/\sim\setminus[\gamma]}V_{0,[{\gamma_{i}}]})$.
Since ${\mathcal{Z}}(V)=0$, $L$ is split and $HV_{0}=0$, there exists $0\neq
e_{\alpha}\in L_{\alpha}$, $\alpha\in\Lambda$, such that $e_{\alpha}v_{0}\neq
0$, being then $\alpha\in\Lambda\cap\mathcal{P}$. By Lemma 2.1-2, we have
$\alpha\sim\gamma$ and $\alpha\sim\gamma_{i}$, for some $i$, with
$[\gamma]\neq[\gamma_{i}]$ which is a contradiction. Therefore we can write
$V=\bigoplus\limits_{[\gamma]\in\mathcal{P}/\sim}V_{[\gamma]}.$ ∎

Let us concrete this result in Example 2.1. We have that in case

$[Sym({\mathfrak{L}},\xi),Skw({\mathfrak{L}},\xi)]=Skw({\mathfrak{L}},\xi)$

and $\\{v\in Skw({\mathfrak{L}},\xi):[Sym({\mathfrak{L}},\xi),v]\\}=0$, we can
decompose $Skw({\mathfrak{L}},\xi)$ by means of an adequate family of weight
submodules as

$Skw({\mathfrak{L}},\xi)=\bigoplus\limits_{[\alpha]\in\Lambda|_{Sym(H,\xi)/\sim}}Skw({\mathfrak{L}},\xi)_{[\alpha]}.$

##  3 Connections of roots. Decompositions of $L$

We begin this section by introducing a concept of connections of roots for $L$
in a slightly different way to the one of connection of weights for
$\mathcal{P}$ developed in the previous section. To do that, we will connect
the nonzero roots of $L$ through nonzero roots of $L$ and nonzero weights of
$\mathcal{P}$ considered both as elements in $H^{*}$.

######  Definition 3.1.

Let $\alpha,\beta$ be two nonzero roots of $L$. We say that $\alpha$ is
connected to $\beta$ if there exist
$\zeta_{1},...,\zeta_{n}\in{\Lambda}\cup\mathcal{P}$ such that

  1. 1.

$\alpha=\zeta_{1}$,

  2. 1.

$\\{\zeta_{1},\zeta_{1}+\zeta_{2},...,\zeta_{1}+\cdots+\zeta_{n-1}\\}\subset\Lambda\cup\mathcal{P}$,

  3. 2.

$\zeta_{1}+\cdots+\zeta_{n}\in\\{\beta,-\beta\\}$,

where the sums are considered in $H^{*}$.

We also say that $\\{\zeta_{1},...,\zeta_{n}\\}$ is a connection from $\alpha$
to $\beta$.

We can prove that the connection relation is of equivalence in $\Lambda$ by
arguing as in the proof of Proposition 2.1. So we can assert.

######  Proposition 3.1.

The relation $\approx$ in $\Lambda$ defined by $\alpha\approx\beta$ if and
only if $\alpha$ is connected to $\beta$ is an equivalence relation.

Given $\alpha\in\Lambda$, we denote by

$\Lambda_{\alpha}:=\\{\beta\in\Lambda:\beta\approx\alpha\\}.$

We also have that if $\beta\in\Lambda_{\alpha}$ then
$-\beta\in\Lambda_{\alpha}$ and, by Proposition 3.1, if
$\mu\notin\Lambda_{\alpha}$ then
$\Lambda_{\alpha}\cap\Lambda_{\mu}=\emptyset$.

Our next aim is to associate an adequate ideal of $L$ to any
$\Lambda_{\alpha}$. For $\Lambda_{\alpha},\alpha\in\Lambda$, we define

$H_{\Lambda_{\alpha}}:=span_{\hbox{\hueca
K}}\\{[L_{\beta},L_{-\beta}]:\beta\in\Lambda_{\alpha}\\}\subset H,$

and

$N_{\Lambda_{\alpha}}:=\bigoplus\limits_{\beta\in\Lambda_{\alpha}}L_{\beta}.$

We denote by $L_{\Lambda_{\alpha}}$ the following linear subspace of $L$,

$L_{\Lambda_{\alpha}}:=H_{\Lambda_{\alpha}}\oplus N_{\Lambda_{\alpha}}.$

######  Proposition 3.2.

Let $\alpha\in\Lambda$. Then the following assertions hold.

  1. 1.

$[L_{\Lambda_{\alpha}},L_{\Lambda_{\alpha}}]\subset L_{\Lambda_{\alpha}}$.

  2. 2.

If $\mu\notin\Lambda_{\alpha}$ then
$[L_{\Lambda_{\alpha}},L_{\Lambda_{\mu}}]=0$.

###### Proof.

1\. Taking into account $H=L_{0}$ and $[L_{\alpha},L_{\beta}]\subset
L_{\alpha+\beta}$, we have

$[L_{\Lambda_{\alpha}},L_{\Lambda_{\alpha}}]=[H_{\Lambda_{\alpha}}\oplus
N_{\Lambda_{\alpha}},H_{\Lambda_{\alpha}}\oplus N_{\Lambda_{\alpha}}]\subset
N_{\Lambda_{\alpha}}+\sum\limits_{\beta,\lambda\in\Lambda_{\alpha}}[L_{\beta},L_{\lambda}].$
(3)

If $\lambda=-\beta$ then

$[L_{\beta},L_{\lambda}]\subset H_{\Lambda_{\alpha}}.$ (4)

If $\lambda\neq-\beta$ and $[L_{\beta},L_{\lambda}]\neq 0$, then
$\beta+\lambda\in\Lambda$. From here, if $\\{\zeta_{1},...,\zeta_{n}\\}$ is a
connection from $\alpha$ to $\beta$ then
$\\{\zeta_{1},...,\zeta_{n},\lambda\\}$ is a connection from $\alpha$ to
$\beta+\lambda$ in case $\zeta_{1}+\cdots+\zeta_{n}=\beta$ and
$\\{\zeta_{1},.....,\zeta_{n},-\lambda\\}$ in case
$\zeta_{1}+\cdots+\zeta_{n}=-\beta$. Hence $\beta+\lambda\in\Lambda_{\alpha}$
and so

$[L_{\beta},L_{\lambda}]\subset N_{\Lambda_{\alpha}}.$ (5)

From equations (3), (4) and (5) we conclude
$[L_{\Lambda_{\alpha}},L_{\Lambda_{\alpha}}]\subset L_{\Lambda_{\alpha}}$.

2\. We have

$[L_{\Lambda_{\alpha}},L_{\Lambda_{\mu}}]=[H_{\Lambda_{\alpha}}\oplus
N_{\Lambda_{\alpha}},H_{\Lambda_{\mu}}\oplus N_{\Lambda_{\mu}}]\subset$
$\subset[H_{\Lambda_{\alpha}},N_{\Lambda_{\mu}}]+[N_{\Lambda_{\alpha}},H_{\Lambda_{\mu}}]+[N_{\Lambda_{\alpha}},N_{\Lambda_{\mu}}].$
(6)

Consider in (6) the third summand $[N_{\Lambda_{\alpha}},N_{\Lambda_{\mu}}]$
and suppose there exist $\beta\in\Lambda_{\alpha}$ and $\eta\in\Lambda_{\mu}$
such that $[L_{\beta},L_{\eta}]\neq 0$. As necessarily $\beta\neq-\eta$, then
$\beta+\eta\in\Lambda$. So $\\{\beta,\eta,-\beta\\}$ is a connection between
$\beta$ and $\eta$. By the transitivity of the connection relation we have
$\mu\in\Lambda_{\alpha}$, a contradiction. Hence $[L_{\beta},L_{\eta}]=0$ and
so

$[N_{\Lambda_{\alpha}},N_{\Lambda_{\mu}}]=0.$ (7)

Consider now the first summand $[H_{\Lambda_{\alpha}},N_{\Lambda_{\mu}}]$ in
equation (6) and suppose there exist $\beta\in\Lambda_{\alpha}$ and
$\eta\in\Lambda_{\mu}$ such that $[[L_{\beta},L_{-\beta}],L_{\eta}]\neq 0$. By
applying Jacobi identity, either $[[L_{-\beta},L_{\eta}],L_{\beta}]\neq 0$ or
$[[L_{\eta},L_{\beta}],L_{-\beta}]\neq 0$ and so either
$[L_{-\beta},L_{\eta}]\neq 0$ or $[L_{\eta},L_{\beta}]\neq 0$, what
contradicts equation (7). Hence

$[H_{\Lambda_{\alpha}},N_{\Lambda_{\mu}}]=0.$

Finally, we note that the same above argument shows

$[N_{\Lambda_{\alpha}},H_{\Lambda_{\mu}}]=0.$

By equation (6) we conclude $[L_{\Lambda_{\alpha}},L_{\Lambda_{\mu}}]=0$. ∎

By Proposition 3.2-1 we can assert that for any $\alpha\in\Lambda$,
$L_{\Lambda_{\alpha}}$ is a subalgebra of $L$ that we call the subalgebra of
$L$ associated to $\Lambda_{\alpha}$.

######  Theorem 3.1.

The following assertions hold.

  1. 1.

For any $\alpha\in\Lambda$, the subalgebra

$L_{\Lambda_{\alpha}}=H_{\Lambda_{\alpha}}\oplus N_{\Lambda_{\alpha}}$

of $L$ associated to $\Lambda_{\alpha}$ is an ideal of $L$.

  2. 2.

If $L$ is simple, then there exists a connection from $\alpha$ to $\beta$ for
any $\alpha,\beta\in\Lambda$ and
$H=\sum\limits_{\alpha\in\Lambda}[L_{\alpha},L_{-\alpha}]$.

###### Proof.

1\. Since $[L_{\Lambda_{\alpha}},H]\subset N_{\Lambda_{\alpha}}$, taking into
account Proposition 3.2, we have

$[L_{\Lambda_{\alpha}},L]=[L_{\Lambda_{\alpha}},H\oplus(\bigoplus\limits_{\beta\in\Lambda_{\alpha}}L_{\beta})\oplus(\bigoplus\limits_{\mu\notin\Lambda_{\alpha}}L_{\mu})]\subset
L_{\Lambda_{\alpha}}.$

2\. The simplicity of $L$ implies $L_{\Lambda_{\alpha}}=L$. Therefore
$\Lambda_{\alpha}=\Lambda$ and
$H=\sum\limits_{\alpha\in\Lambda}[L_{\alpha},L_{-\alpha}]$. ∎

######  Proposition 3.3.

For a linear complement $\mathcal{U}$ of $span_{\hbox{\hueca
K}}\\{[L_{\alpha},L_{-\alpha}]:\alpha\in\Lambda\\}$ in $H$, we have

$L=\mathcal{U}+\sum\limits_{[\alpha]\in\Lambda/\approx}I_{[\alpha]},$

where any $I_{[\alpha]}$ is one of the ideals $L_{\Lambda_{\alpha}}$ of $L$
described in Theorem 3.1-1, satisfying $[I_{[\alpha]},I_{[\beta]}]=0$ if
$[\alpha]\neq[\beta].$

###### Proof.

The decomposition of $L$ can be obtained as the one of $V$ in Proposition 2.2.
Finally, by applying Proposition 3.2-2 we also obtain
$[I_{[\alpha]},I_{[\beta]}]=0$ if $[\alpha]\neq[\beta].$ ∎

######  Theorem 3.2.

If ${\mathcal{Z}}(L)=0$ and
$H=\sum\limits_{\alpha\in\Lambda}[L_{\alpha},L_{-\alpha}]$, then $L$ is the
direct sum of the ideals given in Theorem 3.1,

$L=\bigoplus\limits_{[\alpha]\in\Lambda/\approx}I_{[\alpha]}.$

###### Proof.

By Proposition 3.3,
$L=\mathcal{U}+\sum\limits_{[\alpha]\in\Lambda/\approx}I_{[\alpha]}$. From
$H=\sum\limits_{\alpha\in\Lambda}[L_{\alpha},L_{-\alpha}]$ it is clear that
$\mathcal{U}=0$. Finally, the direct character of the sum now follows from the
facts $[I_{[\alpha]},I_{[\beta]}]=0$, if $[\alpha]\neq[\beta]$, and
${\mathcal{Z}}(L)=0$. ∎

Let us also concrete this result in Example 2.1. We have that in case $\\{e\in
Sym({\mathfrak{L}},\xi):[e,Sym({\mathfrak{L}},\xi)]\\}=0$ and

$Sym(H,\xi)=\sum\limits_{{\alpha}\in\Lambda}[\Pi_{0}({\mathfrak{L}}_{\alpha}),\Pi_{0}({\mathfrak{L}}_{-\alpha})],$

we can decompose $Sym({\mathfrak{L}},\xi)$ by means of an adequate family of
ideals as

$Sym({\mathfrak{L}},\xi)=\bigoplus\limits_{[\alpha]\in\Lambda|_{Sym(H,\xi)/\approx}}Sym({\mathfrak{L}},\xi)_{[\alpha]}.$

##  4 Relating the decompositions of $V$ and $L$

Suppose that $V$ satisfies $LV=V$ and ${\mathcal{Z}}(V)=0$, and that $L$ is
perfect. Since the fact $[L,L]=L$ implies
$H=\sum\limits_{\alpha\in\Lambda}[L_{\alpha},L_{-\alpha}]$, Theorems 2.2 and
3.2 give us

$V=\bigoplus\limits_{[\gamma]\in\mathcal{P}/\sim}V_{[\gamma]}$ and
$L=\bigoplus\limits_{[\alpha]\in\Lambda/\approx}I_{[\alpha]}$

with any $V_{[\gamma]}$ a weight submodule of $V$ and any $I_{[\alpha]}$ an
ideal of $L$ satisfying $[I_{[\alpha]},I_{[\beta]}]=0$ if
$[\alpha]\neq[\beta]$.

For any $[\gamma]\in\mathcal{P}/\sim$, since ${\mathcal{Z}}(V)=0$, there
exists $[\alpha]\in\Lambda/\approx$ such that $I_{[\alpha]}V_{[\gamma]}\neq
0$. Let us show that this $[\alpha]$ is unique. To do that, consider
$\beta\in\Lambda$ satisfying $I_{[\beta]}V_{[\gamma]}\neq 0$. From the facts
$I_{[\alpha]}V_{[\gamma]}\neq 0$ and $I_{[\beta]}V_{[\gamma]}\neq 0$ we can
take $\alpha^{\prime}\in{[\alpha]}$, $\beta^{\prime}\in{[\beta]}$ and
$\gamma^{\prime},\gamma^{\prime\prime}\in{[\gamma]}$ such that
$L_{\alpha^{\prime}}V_{\gamma^{\prime}}\neq 0$ and
$L_{\beta^{\prime}}V_{\gamma^{\prime\prime}}\neq 0$. Since
$\gamma^{\prime},\gamma^{\prime\prime}\in{[\gamma]}$, we can fix a connection

$\\{\gamma^{\prime},\alpha_{1},...,\alpha_{n}\\},$

($\alpha_{1},...,\alpha_{n}\in\Lambda$), from the weight $\gamma^{\prime}$ to
the weight $\gamma^{\prime\prime}$.

Let us distinguish four cases. First, if $\alpha^{\prime}+\gamma^{\prime}\neq
0$, $\beta^{\prime}+\gamma^{\prime\prime}\neq 0$ and so
$\alpha^{\prime}+\gamma^{\prime}$,
$\beta^{\prime}+\gamma^{\prime\prime}\in\mathcal{P}$, we have that

$\\{\alpha^{\prime},\gamma^{\prime},-\alpha^{\prime},\alpha_{1},...,\alpha_{n},\beta^{\prime},-\gamma^{\prime\prime}\\}\subset\Lambda\cup\mathcal{P}$

is a connection from the root $\alpha^{\prime}$ to the root $\beta^{\prime}$
in the case
$\gamma^{\prime}+\alpha_{1}+\cdots+\alpha_{n}=\gamma^{\prime\prime}$ while
$\\{\alpha^{\prime},\gamma^{\prime},-\alpha^{\prime},\alpha_{1},...,\alpha_{n},-\beta^{\prime},\gamma^{\prime\prime}\\}$
gives us the same connection of roots in the case
$\gamma^{\prime}+\alpha_{1}+\cdots+\alpha_{n}=-\gamma^{\prime\prime}$. From
here $\alpha^{\prime}\approx\beta^{\prime}$ and so $[\alpha]=[\beta]$. Second,
if $\alpha^{\prime}+\gamma^{\prime}=0$,
$\beta^{\prime}+\gamma^{\prime\prime}\neq 0$ and so
$\alpha^{\prime}=-\gamma^{\prime}$,
$\beta^{\prime}+\gamma^{\prime\prime}\in\mathcal{P}$, we have that

$\\{-\gamma^{\prime},-\alpha_{1},...,-\alpha_{n},-\beta^{\prime},\gamma^{\prime\prime}\\}$

is a connection of roots between $\alpha^{\prime}$ and $\beta^{\prime}$ in the
case $\gamma^{\prime}+\alpha_{1}+\cdots+\alpha_{n}=\gamma^{\prime\prime}$
while
$\\{-\gamma^{\prime},-\alpha_{1},...,-\alpha_{n},\beta^{\prime},-\gamma^{\prime\prime}\\}$
it is in the case
$\gamma^{\prime}+\alpha_{1}+\cdots+\alpha_{n}=-\gamma^{\prime\prime}$. From
here, $[\alpha]=[\beta]$. Third, if $\alpha^{\prime}+\gamma^{\prime}\neq 0$,
$\beta^{\prime}+\gamma^{\prime\prime}=0$ we can argue as in the previous case
to get $[\alpha]=[\beta]$. Finally, in the fourth case we suppose
$\alpha^{\prime}+\gamma^{\prime}=0$, $\beta^{\prime}+\gamma^{\prime\prime}=0$
and so $\alpha^{\prime}=-\gamma^{\prime}$,
$\beta^{\prime}=-\gamma^{\prime\prime}.$ Then

$\\{-\gamma^{\prime},-\alpha_{1},...,-\alpha_{n}\\}$

is a connection of roots between $\alpha^{\prime}$ and $\beta^{\prime}$ which
implies $[\alpha]=[\beta]$. We conclude that the element
$[\alpha]\in\Lambda/\approx$ such that $I_{[\alpha]}V_{[\gamma]}\neq 0$ is
unique. Also, let us observe that Lemma 2.1 gives us that $V_{[\gamma]}$ is a
weight module over $I_{[\alpha]}$. Hence we can assert.

######  Theorem 4.1.

Let $V$ be weight module respect to a perfect split Lie algebra $L$ such that
$LV=V$ and ${\mathcal{Z}}(V)=0$. Then

$L=\bigoplus\limits_{i\in I}I_{i}$ and $V=\bigoplus\limits_{j\in J}V_{j}$

with any $I_{i}$ a nonzero ideal of $L$ satisfying $[I_{i},I_{k}]=0$ if $i\neq
k$, and any $V_{j}$ a nonzero weight submodule of $V$ in such a way that for
any $j\in J$ there exists a unique $i\in I$ such that

$I_{i}V_{j}\neq 0.$

Furthermore $V_{j}$ is a weight module over $I_{i}$.

Finally, let us concrete this result in Example 2.1 as always. Observe that in
this case ${\mathcal{P}}=\Lambda$ and so the equivalence relations $\sim$ and
$\approx$ in Definitions 2.1 and 3.1 agree. From here, we have that in case
$Sym({\mathfrak{L}},\xi)$ is perfect,
$[Sym({\mathfrak{L}},\xi),Skw({\mathfrak{L}},\xi)]=Skw({\mathfrak{L}},\xi)$
and $\\{v\in Skw({\mathfrak{L}},\xi):[Sym({\mathfrak{L}},\xi),v]\\}=0$, we can
decompose $Sym({\mathfrak{L}},\xi)$ and $Skw({\mathfrak{L}},\xi)$ by means of
adequate families of ideals and weight submodules respectively

$Sym({\mathfrak{L}},\xi)=\bigoplus\limits_{[\alpha]\in\Lambda|_{Sym(H,\xi)/\sim}}Sym({\mathfrak{L}},\xi)_{[\alpha]}$

and

$Skw({\mathfrak{L}},\xi)=\bigoplus\limits_{[\alpha]\in\Lambda|_{Sym(H,\xi)/\sim}}Skw({\mathfrak{L}},\xi)_{[\alpha]}$

in such a way that for any $Skw({\mathfrak{L}},\xi)_{[\alpha]}$, precisely
$Sym({\mathfrak{L}},\xi)_{[\alpha]}$ is the unique ideal in the decomposition
of $Sym({\mathfrak{L}},\xi)$ such that
$[Sym({\mathfrak{L}},\xi)_{[\alpha]},Skw({\mathfrak{L}},\xi)_{[\alpha]}]\neq
0$, being $Skw({\mathfrak{L}},\xi)_{[\alpha]}$ a weight module over
$Sym({\mathfrak{L}},\xi)_{[\alpha]}$.

##  5 The simple components

In this section we are showing that, under certain conditions, the
decomposition of $V$ given in Theorem 4.1 can be given by means of the family
of its minimal submodules, each one being a simple (weight) submodule.

######  Lemma 5.1.

Suppose ${\mathcal{Z}}(V)=0$. If $W$ is a submodule of $V$ such that $W\subset
V_{0}$, then $W=\\{0\\}$.

###### Proof.

On the one hand $LW\subset W\subset V_{0}$, and on the other hand

$LW=(H\oplus(\bigoplus_{\alpha\in\Lambda}{L_{\alpha}}))W\subset\bigoplus\limits_{\beta\in\Lambda\cap\mathcal{P}}{V_{\beta}}.$

Hence, $LW\subset
V_{0}\cap(\bigoplus\limits_{\beta\in\Lambda\cap\mathcal{P}}{V_{\beta}})=0$.
From here $W=\\{0\\}$. ∎

We recall the following definition (see [2, 4]).

######  Definition 5.1.

A weight module $V$ it is said completely pointed if $\dim{V}_{\gamma}=1$ for
any $\gamma\in\mathcal{P}$.

Let us note that completely pointed weight modules appears in a natural way in
several contexts. See for instance [22], [29] and [32] for the cases over
Virasoro, generalized Virasolo and Witt algebras respectively.

Given any submodule $W$ of $V$, it is well known that any submodule of a
weight module is again a weight module. So we have

$W=(W\cap V_{0})\oplus(\bigoplus\limits_{\gamma\in\mathcal{P}}(W\cap
V_{\gamma})).$

Observe that if $V$ is completely pointed then we can write

$\hbox{$W=(W\cap
V_{0})\oplus(\bigoplus\limits_{\gamma\in\mathcal{P}^{W}}V_{\gamma})$ where
$\mathcal{P}^{W}:=\\{\gamma\in\mathcal{P}:W\cap V_{\gamma}\neq 0\\}$}.$ (8)

Let us also introduce the concept of weight-multiplicativity in the framework
of weight modules over spit Lie algebras, in a similar way to the analogous
one for split Lie algebras, split Lie triple systems and split twisted inner
derivation triple systems, (see [6, 8, 9] for these notions and examples).

######  Definition 5.2.

We say that a weight module $V$ over a split Lie algebra $L$ is weight-
multiplicative if given $\alpha\in\Lambda$ and $\gamma\in\mathcal{P}$ such
that $\alpha+\gamma\in\mathcal{P}$, then $L_{\alpha}V_{\gamma}\neq 0.$

Here we note that if $V$ satisfies
$V_{0}=\sum\limits_{\beta\in\Lambda\cap\mathcal{P}}{L_{-\beta}V_{\beta}}$ we
will understand the weight-multiplicativity of $V$ by supposing also that if
$L_{\beta}(L_{-\beta}V_{\beta})\neq 0$ then
$L_{-\beta}(L_{\beta}V_{-\beta})\neq 0$.

######  Theorem 5.1.

Let $V$ be a completely pointed weight module, weight-multiplicative and with
${\mathcal{Z}}(V)=0$ over a split Lie algebra $L$. If $V$ has all its nonzero
weights connected and
$V_{0}=\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}}{L_{-\alpha}V_{\alpha}}$
then either $V$ is simple or $V=W\oplus W^{\prime}$ with $W$ and $W^{\prime}$
simple (weight) submodules of $V$.

###### Proof.

Consider $W$ a nonzero submodule of $V$. By Lemma 5.1 and equation (8) we can
write $W=(W\cap
V_{0})\oplus(\bigoplus\limits_{\gamma\in\mathcal{P}^{W}}V_{\gamma})$ with
$\mathcal{P}^{W}\subset\mathcal{P}$ and $\mathcal{P}^{W}\neq\emptyset$. Hence,
we can take $\gamma_{0}\in\mathcal{P}^{W}$ being so

$0\neq V_{\gamma_{0}}\subset W.$ (9)

For $\delta\in\mathcal{P}\setminus\\{\pm\gamma_{0}\\}$ we have $\gamma_{0}$ is
connected to $\delta.$ Therefore, there exists a connection
$\\{\gamma_{0},\alpha_{1},...,\alpha_{n}\\}$ from $\gamma_{0}$ to $\delta$
such that

$\gamma_{0},\gamma_{0}+\alpha_{1},...,\gamma_{0}+\alpha_{1}+\cdots+\alpha_{n-1}\in\mathcal{P}$

and

$\gamma_{0}+\alpha_{1}+\cdots+\alpha_{n}=\epsilon_{\tiny{\delta}}\delta,$

for some $\epsilon_{\tiny{\delta}}\in\pm 1$. Consider $\gamma_{0}$,
$\alpha_{1}$ and $\gamma_{0}+\alpha_{1}$. The weight-multiplicativity of $W$
gives us $0\neq L_{\alpha_{1}}V_{\gamma_{0}}$. Hence, the completely pointed
character of $W$ implies $0\neq
L_{\alpha_{1}}V_{\gamma_{0}}=V_{\gamma_{0}+\alpha_{1}}$ and equation (9) let
us conclude

$0\neq V_{\gamma_{0}+\alpha_{1}}\subset W.$

We can argue in a similar way from $\gamma_{0}+\alpha_{1}$, $\alpha_{2}$ and
$\gamma_{0}+\alpha_{1}+\alpha_{2}$ to get

$0\neq V_{\gamma_{0}+\alpha_{1}+\alpha_{2}}\subset W.$

Following this process we obtain that

$0\neq V_{\gamma_{0}+\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}}\subset W$

and so

$\hbox{$0\neq V_{\epsilon_{\tiny{\delta}}\delta}\subset W$ for any
$\delta\in\mathcal{P}$ and some $\epsilon_{\tiny{\delta}}\in\pm 1$}.$ (10)

Let us distinguish two possibilities. In the first one
$-\gamma_{0}\in\mathcal{P}^{W}$. Then

$\\{-\gamma_{0},-\alpha_{1},...,-\alpha_{n}\\}$

is a connection from $-\gamma_{0}$ to $\delta$ satisfying

$-\gamma_{0}-\alpha_{1}-\cdots-\alpha_{n}=-\epsilon_{\tiny{\delta}}\delta.$

By arguing as above we get

$0\neq V_{-\epsilon_{\tiny{\delta}}\delta}\subset W$

and so $\mathcal{P}^{W}=\mathcal{P}.$ From here, we also have
$V_{0}=\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}}{L_{-\alpha}V_{\alpha}}\subset
W$ and so $W=V$. Hence $V$ is simple.

In the second possibility, there is not any $\gamma_{0}\in\mathcal{P}^{W}$
such that $-\gamma_{0}\in\mathcal{P}^{W}$. By equation (10) we can write

$\mathcal{P}=\mathcal{P}^{W}\dot{\cup}-\mathcal{P}^{W}$ (11)

where $-\mathcal{P}^{W}:=\\{-\gamma:\gamma\in\mathcal{P}^{W}\\}.$ Let us
denote by

$W^{\prime}:=(\sum\limits_{\beta\in\Lambda\cap-\mathcal{P}^{W}}L_{-\beta}V_{\beta})\oplus(\bigoplus\limits_{\delta\in-\mathcal{P}^{W}}{V_{\delta}}).$

We have that $W^{\prime}$ is a submodule of $V$. Indeed,

$LW^{\prime}=(H\oplus(\bigoplus\limits_{\alpha\in\Lambda}L_{\alpha}))((\sum\limits_{\beta\in\Lambda\cap-\mathcal{P}^{W}}L_{-\beta}V_{\beta})\oplus(\bigoplus\limits_{\delta\in-\mathcal{P}^{W}}{V_{\delta}}))\subset$
$(\bigoplus\limits_{\delta\in-\mathcal{P}^{W}}{V_{\delta}})+(\bigoplus\limits_{\alpha\in\Lambda}L_{\alpha})(\sum\limits_{\beta\in\Lambda\cap-\mathcal{P}^{W}}L_{-\beta}V_{\beta})+(\bigoplus\limits_{\alpha\in\Lambda}L_{\alpha})(\bigoplus\limits_{\delta\in-\mathcal{P}^{W}}{V_{\delta}})).$
(12)

Consider the second summand in equation (12). If some
$L_{\alpha}(L_{-\beta}V_{\beta})\neq 0$ we have that in case $\alpha=\beta$,
clearly $L_{\alpha}(L_{-\beta}V_{\beta})=L_{\beta}(L_{-\beta}V_{\beta})\subset
V_{\beta}\subset W^{\prime},$ and in case $\alpha=-\beta$, since the facts $W$
is a submodule of $V$ and $\beta\notin{\mathcal{P}}^{W}$ imply
$L_{\beta}(L_{\beta}V_{-\beta})=0$, we get by weight-multiplicativity
$L_{-\beta}(L_{-\beta}V_{\beta})=0$. Suppose then $\alpha\neq\pm\beta$. Then
either $[L_{\alpha}L_{-\beta}]V_{\beta}\neq 0$ or
$L_{-\beta}(L_{\alpha}V_{\beta})\neq 0$ and, by the completely pointed
character of $V$, either $[L_{\alpha}L_{-\beta}]V_{\beta}=V_{\alpha}$ or
$L_{-\beta}(L_{\alpha}V_{\beta})=V_{\alpha}.$ In the first case, since
$-\beta\in\mathcal{P}^{W}$, we have by weight-multiplicativity that
$L_{-\alpha+\beta}V_{-\beta}=V_{-\alpha}\subset W.$ That is,
$-\alpha\in\mathcal{P}^{W}$. From here $\alpha\in-\mathcal{P}^{W}$ and then
$V_{\alpha}\subset W^{\prime}$. In the second case we have in a similar way
that $L_{\beta}(L_{-\alpha}V_{-\beta})=V_{-\alpha}\subset W$ and so
$V_{\alpha}\subset W^{\prime}$. Therefore

$(\bigoplus\limits_{\alpha\in\Lambda}L_{\alpha})(\sum\limits_{\beta\in\Lambda\cap-\mathcal{P}^{W}}L_{-\beta}V_{\beta})\subset
W^{\prime}.$

Finally, if we consider the third summand in equation (12) and some
$L_{\alpha}V_{\delta}\neq 0$, we have that in case $\alpha=-\delta$, clearly
$L_{\alpha}V_{\delta}=L_{-\delta}V_{\delta}\subset W^{\prime}$. Suppose then
$\alpha\neq-\delta$ and so $L_{\alpha}V_{\delta}=V_{\alpha+\delta}$. Since
$-\delta\in\mathcal{P}^{W}$, the weight-multiplicativity of $V$ gives us
$L_{-\alpha}V_{-\delta}=V_{-\alpha-\delta}\subset W$. Hence
$\alpha+\delta\in-\mathcal{P}^{W}$ and then $V_{\alpha+\delta}\subset
W^{\prime}$. Consequently
$(\bigoplus\limits_{\alpha\in\Lambda}L_{\alpha})(\bigoplus\limits_{\delta\in-\mathcal{P}^{W}}{V_{\delta}})\subset
W^{\prime}$ and $W^{\prime}$ is a submodule of $V$.

Now observe that we can write the direct sum

$(\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}^{W}}L_{-\alpha}V_{\alpha})\oplus(\sum\limits_{\beta\in\Lambda\cap-\mathcal{P}^{W}}L_{-\beta}V_{\beta}).$

In fact, if there exists $0\neq
v_{0}\in(\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}^{W}}L_{-\alpha}V_{\alpha})\cap(\sum\limits_{\beta\in\Lambda\cap-\mathcal{P}^{W}}L_{-\beta}V_{\beta})$,
and taking into account ${\mathcal{Z}}(V)=0$, $L$ is split and $HV_{0}=0$,
there exists $0\neq e_{\alpha^{\prime}}\in L_{\alpha^{\prime}}$,
$\alpha^{\prime}\in\Lambda$, such that $e_{\alpha^{\prime}}v_{0}\neq 0$, being
then $V_{\alpha^{\prime}}\in W\cap W^{\prime}$, a contradiction. Hence
$v_{0}=0$ and the sum is direct.

Taking into account the above observation, the fact
$V_{0}=\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}}{L_{-\alpha}V_{\alpha}}$
and equation (11) we have

$V=W\oplus W^{\prime}.$

Finally, we note that can argue with $W$ and $W^{\prime}$ as we did at the
first of the proof with $V$ to conclude that $W$ and $W^{\prime}$ are simple
submodules, which completes the proof of the theorem. ∎

######  Theorem 5.2.

Let $V$ be a completely pointed weight module, weight-multiplicative and with
$LV=V$, ${\mathcal{Z}}(V)=0$ over a split Lie algebra $L$. Then
$L=\bigoplus\limits_{i\in I}I_{i}$ with any $I_{i}$ a nonzero ideal of $L$
satisfying $[I_{i},I_{j}]=0$ if $i\neq j$, and $V=\bigoplus\limits_{k\in
K}V_{k}$ is the direct sum of the family of its minimal submodules, each one
being a simple weight submodule of $V$ in such a way that for any $k\in K$
there exists a unique $i\in I$ such that $I_{i}V_{k}\neq 0.$ Furthermore
$V_{k}$ is a weight module over $I_{i}$.

###### Proof.

By Theorem 4.1,

$V=\bigoplus\limits_{j\in J}V_{j}$ (13)

is the direct sum of the weight submodules

$V_{j}=V_{[\gamma]}=(\sum\limits_{\alpha\in\Lambda\cap\mathcal{P}_{\gamma}}L_{-\alpha}V_{\alpha})\oplus(\bigoplus\limits_{\delta\in\mathcal{P}_{\gamma}}V_{\delta}),$

$[\gamma]\in\mathcal{P}/\sim$, having any $V_{[\gamma]}$ its weight system,
${\mathcal{P}}_{\gamma}$, with all of its weights connected. Furthermore
$L=\bigoplus\limits_{i\in I}I_{i}$ with any $I_{i}$ an ideal of $L$ satisfying
that $[I_{i},I_{k}]=0$ if $i\neq k$, that for any $j\in J$ there exists a
unique $i\in I$ such that $I_{i}V_{j}\neq 0$ and that $V_{j}$ is a weight
module over $I_{i}$. We also have that any of the $V_{[\gamma]}$ is weight-
multiplicative as consequence of the weight-multiplicativity of $V$. Clearly
$V_{[\gamma]}$ is completely pointed, and finally
${\mathcal{Z}}(V_{[\gamma]})=0$ as consequence of ${\mathcal{Z}}(V)=0$. We can
apply Theorem 5.1 to any $V_{[\gamma]}$ so as to conclude that either
$V_{[\gamma]}$ is simple or $V_{[\gamma]}=W_{[\gamma]}\oplus
W_{[\gamma]}^{\prime}$ with $W_{[\gamma]}$ and $W_{[\gamma]}^{\prime}$ simple
submodules of $V$. From here, it is clear that by writing $V_{j}=W_{j}\oplus
W_{j}^{\prime}$ in equation (13) if $V_{j}$ is not simple, we get that the
resulting decomposition satisfies the assertions of the theorem. ∎

##  6 Some applications and final remarks

We recall that split Lie algebras plays an important role in mathematical
physics. As it is explained in [13], the split decomposition of a Lie algebra
is one of the most interesting fine gradings of a Lie algebra, with a heavy
influence in the field of particle physics via the usual identification of
observables with generators in a splitting Cartan subalgebra, and particles
living comfortably in the root spaces. Also, this split decomposition appears
as a common feature in many attempts of describing strong interactions in
nature. Other independent role comes from the theory of contraction of Lie
algebras. As claimed in [15], contractions are important in physics because
they explain in terms of Lie algebras why some theories arise as a limit of
more exact theories. Consider a split Lie algebra

$L=H\oplus(\bigoplus\limits_{\alpha\in\Lambda}{L}_{\alpha}).$

We have that split Lie algebras are examples of weight modules over
themselves, where $\mathcal{P}=\Lambda$ and $V_{\alpha}=L_{\alpha}$ for
$\alpha\in\mathcal{P}\cup\\{0\\}$. From here, Definition 3.1 and Theorem 5.2
let us assert that if $L$ is completely pointed, weight-multiplicative and
perfect, then $L$ is the direct sum of the family of its minimal ideals,
$L=\bigoplus\limits_{[\alpha]\in\Lambda/\approx}I_{[\alpha]}$, each one being
a simple split Lie algebra satisfying $[I_{[\alpha]},I_{[\beta]}]=0$ if
$[\alpha]\neq[\beta]$.

Let us now center our attention on split Lie superalgebras
$L=L^{\bar{0}}\oplus L^{\bar{1}}$, see [10]. Lie superalgebras plays an
important role in theoretical physics, specially in conformal field theory and
supersymmetries (see [3, 21, 25] for recent references). The notion of
supersymmetry reflects the known symmetry between bosons and fermions, being
the mathematical structure formalizing this idea the one of supergroup, or
${\hbox{\hueca Z}}_{2}$-graded Lie group. As mentioned in [26], its job is
that of modelling continuous supersymmetry transformations between bosons and
fermions. As Lie algebras consist of generators of Lie groups, the
infinitesimal Lie group elements tangent to the identity, so ${\hbox{\hueca
Z}}_{2}$–graded Lie algebras, otherwise known as Lie superalgebras, consist of
generators of (or infinitesimal) supersymmetry transformations. We also refers
to [1] and [16] for more interesting applications of Lie superalgebras. We
have that $L=L^{\bar{0}}\oplus L^{\bar{1}}$ is a weight module over the split
Lie algebra $L^{\bar{0}}$. So Theorem 5.2 let us also assert that if $L$ is
completely pointed, weight-multiplicative and $L^{\bar{0}}$ is perfect, then
$L$ is the direct sum of the family of its minimal (graded) ideals, each one
being a simple split Lie superalgebra.

Consider now a Lie triple system $T$ with triple product
$[\cdot,\cdot,\cdot]$. As it is pointed out in [19], Lie triple systems are
well related to the theory of quantum mechanics with $PT$-symmetric
Hamiltonians and Krein space-related models in general, by identifying this
underlying structure in the recognizing of $PT$-like involutory structures in
physical models. If we consider a split Lie triple system $T$, see [8], it is
well known that the even part ${\mathfrak{L}}$ of its standard embedding is a
split Lie algebra in such a way that the natural action

${\mathfrak{L}}\times T\to T$

makes $T$ a weight module over the split Lie algebra ${\mathfrak{L}}$. From
here the results in the present paper apply and we can also conclude that if
$T$ is completely pointed, weight-multiplicative and ${\mathfrak{L}}$ is
perfect, then $T$ is the direct sum of the family of its minimal ternary
ideals, $T=\bigoplus\limits_{[\alpha]\in\Lambda/\approx}I_{[\alpha]}$, each
one being a simple split Lie triple system satisfying
$[I_{[\alpha]},T,I_{[\beta]}]=0$ if $[\alpha]\neq[\beta]$.

Finally, we would like to centering on Jordan triple systems
$(J,\\{\cdot,\cdot,\cdot\\})$. Jordan triple systems also plays a sensible
role in the study of four-dimensional superstring or heterotic string
theories, and in the study of orthosymmetric ortholattices which are
fundamental to approximate to Hilbert lattices (see [11, 18]). Split Jordan
triple systems are a particular case of split twisted inner derivation triple
systems, see [9]. The even part ${\mathfrak{L}}$ of the standard embedding of
$J$ is also a split Lie algebra in such a way that the natural action
${\mathfrak{L}}\times J\to J$ also makes $J$ a weight module over the split
Lie algebra ${\mathfrak{L}}$. From here, we have as in the previous example
that the results in the present paper apply and we get that if $J$ is
completely pointed, weight-multiplicative and ${\mathfrak{L}}$ is perfect,
then $J$ is the direct sum of the family of its minimal ternary ideals,
$J=\bigoplus\limits_{[\alpha]\in\Lambda/\approx}I_{[\alpha]}$, each one being
a simple split Jordan triple system satisfying
$\\{I_{[\alpha]},T,I_{[\beta]}\\}+\\{I_{[\alpha]},I_{[\beta]},T\\}+\\{T,I_{[\alpha]},I_{[\beta]}\\}=0$
if $[\alpha]\neq[\beta]$.

######  Remark 6.1.

An interesting open question is to describe simple weight modules (of
arbitrary dimension) $V$ over split Lie algebras. Motivated by the results in
[5] and [7] on simple split Lie algebras and simple split Lie triple systems
respectively, we conjecture $V$ could be described as a direct limit of well-
known finite dimensional weight modules. Also an interesting question is that
of studying weight modules over split Lie algebras with non-necessarily
symmetric weight and root systems. These will be the topics of a future
research of the authors.

## References

  * [1] Benamor, H. and Pinczon, G.: The graded Lie algebra structure of Lie superalgebra deformation theory. Lett. Math. Phys. 18 no. 4, 307–313, (1989).
  * [2] Benkart, G., Britten, D., Lemire, F.: Modules with bounded weight multiplicities for simple Lie algebras, Math. Z. 225(2), 333–353, (1997).
  * [3] Boyallian, C., Meinardi, V.: Quasifinite representations of the Lie superalgebra of quantum pseudodifferential operators. J. Math. Phys. 49, no. 2, 023505, 13 pp. (2008).
  * [4] Britten, D. J.; Lemire, F.: On modules of bounded multiplicities for the symplectic algebras. Trans. Amer. Math. Soc. 351, n. 8, 3413–3431, (1999).
  * [5] Calderón, A.J.: On involutive Lie algebras having a Cartan decomposition. Bull. Austral. Math. Soc. 69, 191-202, (2004).
  * [6] Calderón, A.J.: On split Lie algebras with symmetric root systems. Proc. Indian. Acad. Sci, Math. Sci. 118, 351-356, (2008).
  * [7] Calderón, A.J.: On simple split Lie triple systems. Algebr. Represent. Theory 12, 401-415, (2009).
  * [8] Calderón, A.J.: On split Lie triple systems II. Proc. Indian. Acad. Sci, Math. Sci. 120, 185-198, (2010).
  * [9] Calderón, A.J., Forero, M.: Split Twisted inner derivation triple systems. Comm. Alg. 38, n.1, 28-45, (2010).
  * [10] Calderón, A.J., Sánchez, J.M.: On split Lie superalgebras. J. Math. Phys. 51, 073511, 9 pp, (2010).
  * [11] Chevalier, G.: Orthosymmetries and Jordan triples. Int. J. of Theoretical Physics 37, n.1, 577-583, (1998).
  * [12] Chu, Y.J., Huang, F., Zheng, Z.J.: A commutant of $\beta\gamma$-system associated to the highest weight module $V_{4}$ of $sl(2,\mathbb{C})$ . J. Math. Phys. 51(9), 092301, 32 pp, (2010).
  * [13] Draper, C.; Martín, C. and Viruel, A.: Fine Gradings on the Lie algebra $d_{4}$. Forum Math. 22, 863–877, (2010).
  * [14] Dimitrov, I., Futorny, V., Penkov, I.: A reduction theorem for highest weight modules over toroidal Lie algebras. Comm. Math. Phys. 250(1), 47–63, (2004).
  * [15] Fialowski, A. and De Montigny, M.: On Deformations and Contractions of Lie Algebras. Symmetry, Integrability and Geometry: Methods and Applications Vol. 2, paper 048, 10 pp, (2006).
  * [16] Gié, P.A.; Pinczon, G. and Ushirobira, R.: Back to the Amitsur-Levitzki theorem: a super version for the orthosymplectic Lie superalgebra ${\mathfrak{osp}}(1,2n)$. Lett. Math. Phys. 66 no. 1-2, 141–155, (2003).
  * [17] Grantcharov, D., Jung, J.H., Kang, S.J., Kim, M.: Highest weight modules over quantum queer superalgebra $U_{q}({{\mathfrak{q}}(n)})$. Comm. Math. Phys. 296(3), 827–860, (2010).
  * [18] Gunaydin, M.: $N=2$ superconformal algebras and Jordan triple systems. Physics Letters B. 255 Vol. 1, 46–50, (1991).
  * [19] Gunther, U. and Kuzhel, S.: $PT$-symmetry, Cartan decompositions, Lie triple systems and Krein space-related Clifford algebras. J. Phys. A: Math. Theor. 43, 392002, 10 pp, (2010).
  * [20] Iohara, K.: Unitarizable highest weight modules of the $N=2$ super Virasoro algebras: untwisted sectors. Lett. Math. Phys. 91(3), 289–305, (2010).
  * [21] Iohara, K., Koga, Y.: Note on spin modules associated to $\mathbb{Z}$-graded Lie superalgebras. J. Math. Phys. 50, no. 10, 103508, 9 pp. (2009).
  * [22] Kaplansky I.: Virasoro algebras. Comm. Math. Phys. 86, 49–54, (1982).
  * [23] Liu, D., Gao, S., Zhu, L.: Classification of irreducible weight modules over $W$-algebra $W(2,2)$. J. Math. Phys. 49(1), 113503, 6 pp, (2008).
  * [24] Neeb, K.-H.: Integrable Roots in Split Graded Lie Algebras. J. Algebra. 225, 534–580, 2000.
  * [25] Poletaeva, E.: Embedding of the Lie superalgebra $D(2,1;\alpha)$ into the Lie superalgebra of pseudodifferential symbols on $S^{1|2}$. J. Math. Phys. 48, no. 10, 103504, 17 pp. (2007).
  * [26] Raptis, I.: Mixed-Jordan-Lie superalgebra I. arXiv:math-ph/0110030v4, 14 Nov. 2001.
  * [27] Schue, J.R.: Hilbert Space methods in the theory of Lie algebras. Trans. Amer. Math. Soc. 95, 69–80, (1960).
  * [28] Stumme, N.: The structure of locally finite split Lie algebras. J. Algebra 220, 664–693, (1999).
  * [29] Su, Y.C., Zhao K.: Generalized Virasoro and super-Virasoro algebras and modules of intermediate series. J. Algebra 252, 1-19, (2002).
  * [30] Takemura, K.: The decomposition of level-$1$ irreducible highest-weight modules with respect to the level-$0$ actions of the quantum affine algebra. J. Phys. A 31, n. 5., 1467–1485, (1998).
  * [31] Zapletal, A. Difference equations and highest-weight modules of $U_{q}[{\rm sl}(n)]$. J. Phys. A 31, n. 47, 9593–9600, (1998).
  * [32] Zhao, K.: Weight modules over generalized Witt algebras with 1-dimensional weight spaces. Forum Math. 16, no. 5, 725-748, (2004).