# Linear Interpolation In Parameter Space is Good Enough for Fine-Tuned
Language Models

Mark Rofin, Nikita Balagansky, and Daniil Gavrilov
Tinkoff
{ext.mrofin, n.n.balaganskiy<EMAIL_ADDRESS>
###### Abstract

The simplest way to obtain continuous interpolation between two points in high
dimensional space is to draw a line between them111That applies if you are
operating in a Euclidean space.. While previous works focused on the general
connectivity between model parameters, we explored linear interpolation for
parameters of pre-trained models after fine-tuning. Surprisingly, we could
perform linear interpolation without a performance drop in intermediate points
for fine-tuned models. For controllable text generation, such interpolation
could be seen as moving a model towards or against the desired text attribute
(e.g., positive sentiment), which could be used as grounds for further methods
for controllable text generation without inference speed overhead.

##  1 Introduction

Currently, large pre-trained transformer models can be considered a default
choice for various NLP tasks. Training these models is a complex non-linear
task that is usually performed by feeding the model a large training corpus
and training it in a self-supervised manner (Devlin et al., 2019; Lan et al.,
2020; Liu et al., 2019; Radford et al., 2019). Weights obtained by this
process are used either for standard fine-tuning or other methods that can be
considered more effective in terms of trainable parameters (Liu et al., 2021c,
b; Li and Liang, 2021; Lester et al., 2021; Houlsby et al., 2019; Hu et al.,
2021).

Since initialization using pre-trained parameters is crucial for the final
model’s performance, it is fascinating to observe the changes in parameters
during the fine-tuning process on downstream tasks.

While recent works (Goodfellow et al. 2014, Lucas et al. 2021) explored
changes in parameter space during training, there is still little known about
the details of this process, specifically for model fine-tuning. In our work,
we are exploring properties of the fine-tuned language models obtained by
linear interpolation.

Figure 1: We experimented with linear interpolation for fine-tuned Language
Models. We observed that we could fine-tune a pre-trained model on two domains
(e.g., positive and negative movie reviews) and interpolate between trained
weights without loss in perplexity in between these models. Furthermore, we
could expand interpolation beyond trained models and get more positive or
negative models than fine-tuned ones.

Surprisingly, we observed compelling evidence of the linearity of some
parameter subspace of pre-trained models. The formula behind interpolation is
simply

$\alpha\theta^{-}+(1-\alpha)\theta^{+},$

where $\alpha\in\mathbb{R}$ is an interpolation weight222Note that $\alpha$
does not have to be restricted to be $\in[0;1]$, since we found that it could
exceed these boundaries during our experiments., and $\theta^{-}$,
$\theta^{+}$ are model parameters. If both $\theta^{-}$ and $\theta^{+}$ are
fine-tuned LMs (e.g., with negative and positive sentiment generations), the
model parameters obtained by applying this formula are well-behaved in terms
of perplexity. Therefore, the probability of positive sentiment occurring in
the generated text is smoothly growing with the interpolation coefficient
weight.

Figure 2: Schematic overview of interpolation schemas used in our experiments.
(Left) Interpolation between two fine-tuned model parameters $g_{1}(\alpha)$.
(Center) Direction obtained by two fine-tuned model parameters used to move
pre-trained model parameters to obtain $g_{2}(\alpha)$. (Right) Two directions
formed by fine-tuned parameters and pre-trained weight define the basis used
to define the direction in which the pre-trained parameter is moved to get
$g_{3}(\alpha,\beta)$. See Section 3.1 for more details.

We investigated the reasons for this phenomenon and found that the same
initialization from pre-trained models is crucial for the linear properties.

Utilizing the parameter space is interesting in terms of theoretical results
and insights, but what is more important is that it can be used for practical
tasks. E.g., linear interpolation makes it possible to apply two attributes in
a condition at the same time, or improve attribute presence with desired
weight without any computational overhead.

##  2 Related Work

Goodfellow et al. 2014 found that the loss landscape during interpolation
between initial weights and weights after training has no significant peaks
and decreases monotonically during interpolation. This is interesting since
training is a complex non-linear task, and model weights tend to fall into a
local optima point after training is complete. Continuing this line of
research Lucas et al. 2021 found a link between batch normalization Ioffe and
Szegedy (2015) and linearity of the logits’ path during training.

These observations raise a question about how we can interpolate between two
local optima without a loss in quality. Frankle et al. 2020 discovered
evidence showing that finding a winning ticket Frankle and Carbin (2018)
during iterative pruning is closely connected to finding linear connectivity
between optimal points in a weight space. In addition, Frankle et al. 2020
proposed the Loss Barrier metric for evaluating the connectivity between
parameters of two models.

Entezari et al. 2021 explored the impact of the width and depth of networks on
their connectivity. Their findings showed that the wider the network is, the
lower its loss barrier. Meanwhile, the deeper the network is, the higher its
barrier value. Furthermore, Entezari et al. 2021 proposed a conjecture about
weight permutations and solutions obtained by gradient descent. More
precisely, most SGD solutions belong to a set $S$, whose elements can be
permuted in such a way that there is no barrier to the linear interpolation
between any two permuted elements in $S$. Ainsworth et al. 2022 proposed
several methods for sufficient permutation in order to reduce the loss
barrier.

To further explore why a zero loss barrier is possible, the Lazy Training
theory Chizat et al. (2018) can be used. I.e., if a neural network has
sufficient width, the weights’ changes during training are small enough to use
the Taylor series expansion for the layer outputs. Therefore, inside some
small neighborhood of the initial point, $\theta^{0}$ in the weight space
model can be linearized in terms of the weights $\theta$.

##  3 Understanding Pre-Trained Weights’ Parameter Space

Having a model pre-trained on some general task (e.g., Language Modeling)
$\theta^{0}\in\mathbb{R}^{|\theta|}$, it is conventional to initialize a new
model with $\theta^{0}$ when solving a downstream task333We will refer to
superscription $0$, $-$ and $+$ as a sentiment characteristic, similar to how
superscription in particle physics refers to particle charge. $0$ refers to a
pre-trained model with a neutral sentiment, as we believe that pre-trained
models tend to generate texts with a neutral sentiment.. For example, GPT-2
could be used as $\theta^{0}$ when training an LM on some specific domain of
data (e.g., movie reviews). Doing so makes it possible to obtain faster
convergence of training procedures and better results than training from
scratch since $\theta^{0}$ is usually trained with a larger dataset than those
available for downstream tasks.

While many works explore the parameter space of models trained from scratch,
we are most interested in such a space for fine-tuned models. More
specifically, when a model is trained from scratch with different starting
points, there is evidence that different $\theta^{0}$ could be obtained.
Furthermore, if different random seeds are used to form mini-batches from the
training dataset, additional differences could occur in the resulting
parameters of trained models.

It is important to note that, if we train a model from a pre-trained state, we
eliminate the randomness caused by different starting points of optimization.
From such perspective, we should expect the parameter space of fine-tuned
models to be simpler than that of models trained from scratch. To explore the
limits of this simplicity, we experimented with linear interpolation between
weights of fine-tuned models described in the following sections.

###  3.1 Linear Interpolation

Consider two models with parameters $\theta^{+}\in\mathbb{R}^{|\theta|}$ and
$\theta^{-}\in\mathbb{R}^{|\theta|}$. Both $\theta^{+}$ and $\theta^{-}$ are
obtained after fine-tuning a pre-trained model $\theta^{0}$. For convenience,
let us consider that $\theta^{0}$ is Language Model trained on general domain
data (e.g., GPT-2), $\theta_{+}$ and $\theta_{-}$ are Language Models fine-
tuned on positive and negative sentiment data (e.g., SST dataset (Socher et
al., 2013)). We could linearly interpolate between them as

$g_{1}(\alpha)=\alpha\theta^{+}+(1-\alpha)\theta^{-},g_{1}:R\to R^{|\theta|}.$
(1)

We can also rewrite $g_{1}(\alpha)$ differently:

$g_{1}(\alpha)=\frac{1}{2}(\theta^{+}+\theta^{-})+\frac{1}{2}(2\alpha-1)(\theta^{+}-\theta^{-}),$
(2)

which could be seen as moving from starting point
$\frac{1}{2}(\theta^{+}+\theta^{-})$ in a direction $(\theta^{+}-\theta^{-})$.
Expressing interpolation with a starting point such as that in Equation 2
could be considered too verbose. However, it allows us to derive the second
possible formulation of interpolation, for which we replace the starting
moving point with $\theta_{0}$. To simplify the process even further, we use
$\alpha^{\prime}=2\alpha-1$ as an interpolation weight444Note that we have a
different scale for $\alpha^{\prime}$, since $\alpha=0$ implies
$\alpha^{\prime}=-\frac{1}{2}$, and $\alpha=1$ implies
$\alpha^{\prime}=\frac{1}{2}$..

$g_{2}(\alpha^{\prime})=\theta^{0}+\alpha^{\prime}(\theta^{+}-\theta^{-}).$
(3)

Going even further, we can decompose the $(\theta^{+}-\theta^{-})$ direction
into $(\theta^{+}-\theta^{0})$ and $(\theta^{-}-\theta^{0})$, obtaining new
parametrization

$g_{3}(\alpha,\beta)=\theta^{0}+\alpha(\theta^{+}-\theta^{0})+\beta(\theta^{-}-\theta^{0}).$
(4)

Note that $\alpha+\beta=1$ reparametrizes $g_{1}$ and $\alpha+\beta=0$
reparametrizes $g_{2}$.

We discuss the limits of these reparametrizations in the Experiments section.

###  3.2 Ensembling

Another way to utilize several models at once is to combine them into an
ensemble. While linear interpolation is performed in the weight space,
ensembling can be seen as interpolating in the model output space. At every
step, language models yield logits $z$ for every token in vocabulary. As
proposed in DExperts Liu et al. (2021a), we could use these logits to obtain
the final tokens’ probability:

$\displaystyle P(x_{t}|x_{<t})=\operatorname{softmax(z)},$ $\displaystyle
z=z^{0}(x_{<t})+\alpha\cdot(z^{+}(x_{<t})-z^{-}(x_{<t}))$

However, this method requires significant computational time overhead compared
to interpolation in the parameter space since it requires evaluating several
models to get predictions.

((a))

((b))

((c))

Figure 3: Interpolation between two models fine-tuned on positive and negative
sentiment with $g_{1}(\alpha)$. We report the Mean probability of the positive
sentiment (a), the Perplexity of the generated text (b), and the Probability
of the grammatically correct text (c) for obtained interpolated models. See
Section 4.1 for more details.

##  4 Experiments

Figure 4: Words’ probability during interpolation with the text prompt "The
movie was". Words leading to positive sentiment are plotted in green, and
negative in red. While interpolating between $\theta^{-}$ and $\theta^{+}$,
probabilities of negative words decrease, and positive ones increase. See
Section 4.1 for more details.

###  4.1 Controllable Text Generation

Controllable text generation can be seen as the simplest way to explore the
parameter space of fine-tuned models. The performance of obtained $\theta$ can
be quickly evaluated with automatic metrics such as desired attribute
probability of generated texts, and text quality can be evaluated by
perplexity and grammar correctness.

Following the DExperts Liu et al. (2021a) setup, we took the SST dataset
containing texts with labels representing the sentiment of sequences. We
constructed a positive sentiment dataset containing texts with labels such as
"positive" or "very positive". In addition, we also created a negative dataset
with "negative" and "very negative" texts. We then fine-tuned two GPT-2 Large
models on the causal language modeling task on these datasets to obtain
$\theta^{+}$ and $\theta^{-}$, respectively.

We then evaluated the models obtained by $g_{1}$ and $g_{2}$ (See Equations 1,
3) to understand the limits of linear interpolation for fine-tuned models. To
do so, we used the same prompts as DExperts Liu et al. (2021a) for text
generation. For every prompt, we generated 25 continuations with their length
less or equal to 30 tokens.

We used three metrics to evaluate the generated texts’ sentiment and quality.
Positive text scores are evaluated using an external classifier and show the
mean probability of positive sentiment in the generated text. Grammar scores
are determined by a classifier trained on the CoLA Wang et al. (2018) dataset.
To evaluate the texts’ quality, we calculate perplexity using GPT-2 XL.

See Section A.1 of the Appendix for more details on training and evaluation
used in these experiments.

See Figure 3 for the results. We found that perplexity with $g_{1}(\alpha)$
remains stable in $\alpha\in[0;1]$, in which we have a zero perplexity
barrier. A wider interval of $\alpha$ also shows promising results, where the
positive sentiment probability increases with $\alpha>1$. Meanwhile,
perplexity and grammar remain stable. Based on this, we can assume that models
obtained by simple linear interpolation can still be considered language
models. Moreover, the original’s features, such as positive sentiment
probability, could be enhanced by $\alpha>1$ (and vice versa). In Section 5,
we hypothesise why linear interpolation in the weight space works even in
cases of complex non-linear models.

((a))

((b))

((c))

Figure 5: Comparison of $g_{1}(\alpha)$ and $g_{2}(\alpha)$ interpolations
with neutral prompts. We evaluated the positive text score (a), perplexity
(b), and Grammar Correctness (c) for interpolated models. Note that these
interpolation methods differ in the scale of $\alpha$ (See Section 3.1 for
details). Therefore, we used different scales to report these results.
$\alpha$ values for $g_{1}(\alpha)$ are shown below the plot, while those for
$g_{2}(\alpha)$ are above. While the positiveness of both approaches is
comparable, $g_{2}(\alpha)$ obtained better perplexity and grammar correctness
through utilizing $\theta^{0}$ parameters. See Section 4.2 for more details.

((a))

((b))

Figure 6: Perplexity (a) and Positive Text Score (b) with respect to
interpolation weight in the logits space (DExperts) and the weight space. Note
that the text’s positiveness increases smoothly in booth cases, while
perplexity remains low. See Section 4 for more details.

We also measured the next token probability on our "The movie was" prompt
using parameters obtained from $g_{1}$ interpolation. See Figure 4 for the
results. The probabilities of the words leading to positive sentiment are
monotonically increasing, while the probabilities of the negative sentiment
words are monotonically decreasing with $\alpha$.

###  4.2 Which Parametrization Is Best?

In Section 3.1, we discussed different ways to parametrize interpolation and
move the model’s weights in the desired direction. However, note that it is
not fully clear what the differences are between them. To compare the proposed
interpolation schemes, we conducted experiments with the pre-trained model
GPT-2 Large as $\theta^{0}$ and Fine-Tuned models from the previous section as
$\theta^{+}$ and $\theta^{-}$. Note that parametrization $g_{2}$ takes three
models as input.

((a))

((b))

Figure 7: Postive text score (a) and perplexity (b) for $g_{3}(\alpha,\beta)$
interpolation. Positive text score has a clear growing direction from upper
left to lower right. We also found plateau of perplexity near $\alpha=-\beta$
line. See Section 4.3 for more details.

In this subsection, all experiments were conducted with neutral prompts only.
See Figure 5 for the results. We observed that $g_{1}(\alpha)$ obtained a
positiveness score comparable with $g_{2}(\alpha)$, while the latter showed
better perplexity and grammar correctness. However, we would like to note
that, in this case, perplexity should not be considered fully representative
of the generated texts’ quality. Since $g_{2}(\alpha)$ utilizes $\theta^{0}$,
its samplings are more likely to produce texts which would be treated as more
probable by GPT-2, while $g_{1}(\alpha)$ has a stronger shift towards movie
reviews.

###  4.3 Interpolation Space

To further analyze the interpolation points, we conduct experiments with
interpolation $\theta=g_{3}(\alpha,\beta)$. Note that
$g_{3}:\mathbb{R}^{2}\to\mathbb{R}^{|\theta|}$ maps point $(0,1)$ to
$g(0,1)=\theta^{0}-\theta^{0}+\theta^{+}-0\cdot(\theta^{-}-\theta^{0})=\theta^{+}$,
analogously $(0,0)\to\theta^{0}$ and $(1,0)\to\theta^{-}$. Then we can choose
any point in $\mathbb{R}^{2}=\operatorname{dom}g$ and obtain a model
$\theta=g(\alpha,\beta)$. We use a 2d uniform grid with values from -4 to +4
and 20 points in every dimension
($G=\\{i/2.5\\}_{i=-10}^{10}\times\\{j/2.5\\}_{j=-10}^{10}$) to obtain 400
models, and measure the properties of these models. As a result, we get 400
points of perplexity and a positive sentiment score shown in Figure 7. In
addition, we also count the models’ mean negative log-likelihood loss value on
a test set of SST positive and negative subsets. See Figure 8 for the results.

((a))

((b))

Figure 8: Positive (a) and negative (b) test set losses for the models
obtained by $\theta=g_{3}(\alpha,\beta)$ interpolation. See Section 4.3 for
more details.

We found a plateau of perplexity near the $\alpha=-\beta$ line. Furthermore,
we also found that the loss values of models near $\alpha=-\beta+1$ (equals to
$\theta=g_{2}(\cdot)$ parametrization) are significantly lower. These results
can be explained as follows. The first parametrization $\theta=g_{1}(\cdot)$
does not utilize a pre-trained $\theta^{0}$ model. Therefore, the obtained
models remain within the SST dataset domain (movie reviews). We can assume
that it is because of lower loss on both test subsets. The perplexity of the
specific texts is higher due to them becoming biased toward the movie review
style. The model we used to measure perplexity (pre-trained GPT-2 XL) is not a
domain-specific model and therefore measures information contained in the
generated text. As the domain of the generated texts shifted, we observed
consistently higher perplexity compared to $\theta=g_{2}(\cdot)$. However,
this perplexity remained stable and in a meaningful value below 50. On the
other hand, the parametrization $\theta=g_{2}(\cdot)$ did not shift toward the
movie reviews domain because of the constant persistence of the $\theta^{0}$
term. Lower perplexity, in this case, did not indicate better quality of the
generated texts.

###  4.4 Interpolation vs. Ensembling

In this section, we compare two methods of utilizing several models for
controllable text generation tasks.

As discussed in 3.2, DExperts could be seen as a linear interpolation in the
model outputs space.

We generated texts with several values of the $\alpha$ parameter. Then, we
used $\theta^{+}$ model as an expert and $\theta^{-}$ model as an anti-expert.
We compared this setup with $\theta=g_{2}(\alpha)$ parametrization. The
results are presented in Figure 6.

We found that the curves are almost identical for $\alpha\in[0;1]$. Similar
results could be obtained if all model backbones were linear. Surprisingly for
us, we also discovered that linear interpolation in weight space is highly
competitive to ensembling and does not damage the internal knowledge of the
model.

##  5 Results Analysis

((a))

((b))

Figure 9: The norm of the weight difference for $\theta^{+}$ and $\theta^{0}$
(a), as well as DistilGPT-2 trained on C4 Raffel et al. (2020) from scratch
and pre-trained DistilGPT-2 (b). Each row represents one layer, and the
parameter names can be found on the x-axis ticks. See Section 5.3 for more
details.

To explain the results of the above observations, we will now try to establish
some intuition on why linear interpolation works so well for pre-trained
language models.

###  5.1 Lazy Training

Lazy training, introduced by Chizat et al. 2018, has a solid connection to
linear interpolation in the weight space.

Let us say that we have some function
$f(\theta)=R(h(\theta)):\mathbb{R}^{|\theta|}\to\mathbb{R}_{+}$, where
$h(\theta):\mathbb{R}^{|\theta|}\to\mathcal{F}$ is our model. Now, let us
define a linearized model
$\overline{h}(\theta)=h(\theta^{0})-Dh(\theta^{0})(\theta^{0}-\theta)$.
$\overline{h}(\theta)$ could be seen as a Taylor’s series expansion to the
first order of $h(\theta)$. If we can accurately approximate $h(\theta)$ with
$\overline{h}(\theta)$, then
$\overline{f}(\theta)=R\left(\overline{h}(\theta)\right)$ becomes a good
approximation of $f$ if we are using gradient descent for optimizing $R$.

###  5.2 Pre-Trained Models Fine-Tuning and Lazy Traing

Chizat et al. 2018 considers function $f$ to be a loss function optimized by
gradient descent. In our work, we will talk about a proxy function that can be
seen as a differentiable interpolation of the positive text score or other
desired attributes for controllable text generation. For example, we can start
our optimization process at $\theta^{0}$ and train models $\theta_{+}$ and
$\theta_{-}$ using stochastic gradient descent on the NLL target function. We
believe that after minimization, some function $f$ (in our case, the
probability of positive sentiment in the generated text) will have a lower
value on $\theta^{-}$ and a higher value on $\theta^{+}$. In other words,
during the training procedure, we are trying to find weights $\theta^{-}$ and
$\theta^{+}$ such that $f(\theta^{-})<f(\theta^{0})<f(\theta^{+})$.

Conjecture. Point $\hat{\theta}$ with the lowest value of the loss function
$L$, such as NLL, does not imply optimal value of the truly desired $f$
function. In other words, we can find a value of $\theta^{*}$ with
$L(\theta^{*})>L(\hat{\theta})$, but $f(\theta^{*})>f(\hat{\theta})$.

Function $f$ can be a composition of other functions such as a weighted sum of
grammar scores, desired and present attributes, and perplexity.

Assumption. If the weights $\theta$ obtained after the fine-tuning procedure
are close to pre-trained initialization $\theta^{0}$, we can linearize the
function $f$ as $\overline{f}(\theta)=f(\theta^{0})+\nabla
f(\theta^{0})^{T}(\theta-\theta_{0})$ in some neighbourhood of $\theta^{0}$.

If we then parametrize $\theta$ with the general parametrization
$\theta=g_{3}(\alpha,\beta)=\theta^{0}+\alpha(\theta^{+}-\theta^{0})+\beta(\theta^{-}-\theta^{0})$
and pass it to $\overline{f}$, we obtain

$\displaystyle\overline{f}\circ g(\alpha,\beta)=f(\theta^{0})$ (5)
$\displaystyle+\alpha\cdot\nabla f(\theta^{0})^{T}(\theta^{+}-\theta^{0})$
$\displaystyle+\beta\cdot\nabla f(\theta^{0})^{T}(\theta^{-}-\theta^{0})$
$\displaystyle=\alpha\cdot C^{+}+\beta\cdot C^{-},$

where $C^{+}$ and $C^{-}$ are constants.

Note that $C^{+}\approx\partial(f\circ g)/\partial\alpha$ and
$C^{-}\approx\partial(f\circ g)/\partial\beta$ in some $\theta^{0}$
neighbourhood.

The scheme with linear interpolation works even if $C^{+}$ and $C^{-}$ are not
constants, since $C^{+}(\alpha)$ > 0 and $C^{-}(\beta)<0$ is a sufficient
condition.

This model clarifies the similarity between DExperts and linear weight
interpolation in Figure 6. If Assumption 5.2 holds, then the interpolation
between weights will be approximately equal to the interpolation between
outputs in a small enough region around $\theta^{0}$

((a))

((b))

((c))

Figure 10:  Linear interpolation between DistilGPT-2 and GPT-C4 weights. We
observed that points between original weights performed poorly with increased
perplexity (a), the reduced fraction of unique n-grams (b), and grammar score
(c). See Section 5.3 for more details.

###  5.3 Interpolating Between Two Different Decorrelated Language Models

The small difference between weights is the main factor for the above-
mentioned theory. We hypothesize that we obtained such a low difference since
we performed fine-tuning of the same pre-trained model $\theta^{0}$. To
support this, we conduct experiments with two different language models.

For the simplicity of our experiment, we chose a small DistilGPT-2 model Sanh
et al. (2019) with 6 hidden layers. We trained a GPT model from scratch on C4
Raffel et al. (2019), namely GPT-C4 with architecture identical to
DistilGPT-2. For more details, refer to Section A.2.

Firstly, we measured the norm of the weight differences between two pairs of
models.

  1. (a)

Pre-trained original DistilGPT-2 $\longleftrightarrow$ Fine-tuned original
DistilGPT-2 on positive sentiment.

  2. (b)

Pre-trained original DistilGPT-2 $\longleftrightarrow$ Pre-trained on C4
dataset GPT-C4.

To evaluate the difference between 1-d tensors (biases), we use the scaled
$\ell^{2}$-norm:

$\Delta_{b}=\frac{\lVert
b_{1}-b_{2}\rVert}{\sqrt{d}}=\frac{\sqrt{\sum_{i}\left(b_{1}^{i}-b_{2}^{i}\right)^{2}}}{\sqrt{d}}.$

For matrices with sizes $n\times m$, we use:

$\Delta_{w}=\frac{\lVert w_{1}-w_{2}\rVert}{\sqrt{n\cdot
m}}=\frac{\sqrt{\sum_{i}^{n}\sum_{j}^{m}\left(w_{1}^{ij}-w_{2}^{ij}\right)^{2}}}{\sqrt{n\cdot
m}}.$

Results showed in Figure 9. Note that two comparisons are plotted on the same
scale. While the differences between the (a) pair are small, the differences
between (b) can be observed.

The second experiment interpolates between two language models: DistilGPT-2
and GPT-C4. See Figure 10 for the results. We found that models obtained at
every interpolation step completely forget the knowledge obtained during the
training procedure. We additionally estimate the fraction of the distinct
n-grams. At every point where perplexity becomes lower than initial values (0
and 1), we observe a significant drop in unique n-grams. The grammar score has
two major peaks at points 0 and 1.

Models obtained by interpolating between different pre-trained models were
found to fail at the basic language model tasks. This experiment confirms the
importance of initializing fine-tuned models in the same way.

##  6 Conclusion

In our paper, we looked into simple linear weight interpolation between pre-
trained and fine-tuned models, and concluded that this method performs
surprisingly well. We found that different types of interpolation have
different strengths and flaws, which we discuss in detail in the Experiments
section. We have researched this phenomenon and provided intuition on why
large language models, highly non-linear complex functions, are capable of
generating texts with good metrics even after simple linear interpolation.

## References

  * Ainsworth et al. (2022) Samuel K. Ainsworth, Jonathan Hayase, and Siddhartha Srinivasa. 2022.  Git re-basin: Merging models modulo permutation symmetries.
  * Burgess et al. (2019) Neil Burgess, Jelena Milanovic, Nigel Stephens, Konstantinos Monachopoulos, and David Mansell. 2019.  Bfloat16 processing for neural networks.  In _2019 IEEE 26th Symposium on Computer Arithmetic (ARITH)_ , pages 88–91.
  * Chizat et al. (2018) Lenaic Chizat, Edouard Oyallon, and Francis Bach. 2018.  On lazy training in differentiable programming.
  * 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.
  * Entezari et al. (2021) Rahim Entezari, Hanie Sedghi, Olga Saukh, and Behnam Neyshabur. 2021.  The role of permutation invariance in linear mode connectivity of neural networks.
  * Frankle and Carbin (2018) Jonathan Frankle and Michael Carbin. 2018.  The lottery ticket hypothesis: Training pruned neural networks.  _CoRR_ , abs/1803.03635.
  * Frankle et al. (2020) Jonathan Frankle, Gintare Karolina Dziugaite, Daniel Roy, and Michael Carbin. 2020\.  Linear mode connectivity and the lottery ticket hypothesis.  In _Proceedings of the 37th International Conference on Machine Learning_ , volume 119 of _Proceedings of Machine Learning Research_ , pages 3259–3269. PMLR.
  * Goodfellow et al. (2014) Ian J. Goodfellow, Oriol Vinyals, and Andrew M. Saxe. 2014.  Qualitatively characterizing neural network optimization problems.
  * Houlsby et al. (2019) Neil Houlsby, Andrei Giurgiu, Stanislaw Jastrzebski, Bruna Morrone, Quentin de Laroussilhe, Andrea Gesmundo, Mona Attariyan, and Sylvain Gelly. 2019.  Parameter-efficient transfer learning for nlp.  In _ICML_.
  * Hu et al. (2021) Edward J. Hu, Yelong Shen, Phillip Wallis, Zeyuan Allen-Zhu, Yuanzhi Li, Shean Wang, Lu Wang, and Weizhu Chen. 2021.  Lora: Low-rank adaptation of large language models.
  * Ioffe and Szegedy (2015) Sergey Ioffe and Christian Szegedy. 2015.  Batch normalization: Accelerating deep network training by reducing internal covariate shift.  In _Proceedings of the 32nd International Conference on Machine Learning_ , volume 37 of _Proceedings of Machine Learning Research_ , pages 448–456, Lille, France. PMLR.
  * 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_.
  * Lester et al. (2021) Brian Lester, Rami Al-Rfou, and Noah Constant. 2021.  The power of scale for parameter-efficient prompt tuning.
  * Li and Liang (2021) Xiang Lisa Li and Percy Liang. 2021.  Prefix-tuning: Optimizing continuous prompts for generation.  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 4582–4597, Online. Association for Computational Linguistics.
  * Liu et al. (2021a) Alisa Liu, Maarten Sap, Ximing Lu, Swabha Swayamdipta, Chandra Bhagavatula, Noah A. Smith, and Yejin Choi. 2021a.  DExperts: Decoding-time controlled text generation with experts and anti-experts.  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 6691–6706, Online. Association for Computational Linguistics.
  * Liu et al. (2021b) Xiao Liu, Kaixuan Ji, Yicheng Fu, Zhengxiao Du, Zhilin Yang, and Jie Tang. 2021b.  P-tuning v2: Prompt tuning can be comparable to fine-tuning universally across scales and tasks.  _CoRR_ , abs/2110.07602.
  * Liu et al. (2021c) Xiao Liu, Yanan Zheng, Zhengxiao Du, Ming Ding, Yujie Qian, Zhilin Yang, and Jie Tang. 2021c.  Gpt understands, too.  _arXiv:2103.10385_.
  * Liu et al. (2019) Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Mandar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov. 2019.  Roberta: A robustly optimized bert pretraining approach.  Cite arxiv:1907.11692.
  * Loshchilov and Hutter (2019) Ilya Loshchilov and Frank Hutter. 2019.  Decoupled weight decay regularization.
  * Lucas et al. (2021) James Lucas, Juhan Bae, Michael R. Zhang, Stanislav Fort, Richard Zemel, and Roger Grosse. 2021.  Analyzing monotonic linear interpolation in neural network loss landscapes.
  * Morris et al. (2020) John Morris, Eli Lifland, Jin Yong Yoo, Jake Grigsby, Di Jin, and Yanjun Qi. 2020\.  Textattack: A framework for adversarial attacks, data augmentation, and adversarial training in nlp.  In _Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations_ , pages 119–126.
  * Radford et al. (2019) Alec Radford, Jeff Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019.  Language models are unsupervised multitask learners.
  * Raffel et al. (2019) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2019.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _arXiv e-prints_.
  * Raffel et al. (2020) Colin Raffel, Noam Shazeer, Adam Roberts, Katherine Lee, Sharan Narang, Michael Matena, Yanqi Zhou, Wei Li, and Peter J. Liu. 2020.  Exploring the limits of transfer learning with a unified text-to-text transformer.  _Journal of Machine Learning Research_ , 21(140):1–67.
  * Rosenthal et al. (2017) Sara Rosenthal, Noura Farra, and Preslav Nakov. 2017.  Semeval-2017 task 4: Sentiment analysis in twitter.  In _Proceedings of the 11th international workshop on semantic evaluation (SemEval-2017)_ , pages 502–518.
  * Sanh et al. (2019) Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf. 2019.  Distilbert, a distilled version of bert: smaller, faster, cheaper and lighter.  In _NeurIPS EMC Workshop_.
  * Socher et al. (2013) Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D. Manning, Andrew Ng, and Christopher Potts. 2013.  Recursive deep models for semantic compositionality over a sentiment treebank.  In _Proceedings of the 2013 Conference on Empirical Methods in Natural Language Processing_ , pages 1631–1642, Seattle, Washington, USA. Association for Computational Linguistics.
  * Wang et al. (2018) Alex Wang, Amapreet Singh, Julian Michael, Felix Hill, Omer Levy, and Samuel R. Bowman. 2018.  Glue: A multi-task benchmark and analysis platform for natural language understanding.  Cite arxiv:1804.07461Comment: https://gluebenchmark.com/.

##  Appendix A Experiment Details

###  A.1 Details of Controllable Text Generation Experiments

We fine-tuned two GPT-2 Large models on the SST dataset and ran a
hyperparameter search using the grid from Table 1.

Parameter | Values range
---|---
Learning rate | [1e-4, 1e-5, 1e-6]
Batch size | [32, 64, 128, 256]
Steps | [500, 1000, 2000]
Table 1: Hyperparameter search ranges used in fine-tuning.

After the training, we proceeded with the best model in terms of perplexity on
the corresponding validation sets. The best parameters are reported in Table
2.

Parameter | Positive ($\theta_{+}$) | Negative ($\theta_{-}$)
---|---|---
Learning rate | 1e-6 | 1e-6
Batch size | 64 | 64
Steps | 1000 | 1000
Table 2: Best hyperparameters.

As a Positive Text Score metric, we use outputs of the RoBERTa-base model
trained by CardiffNLP555https://huggingface.co/cardiffnlp/twitter-roberta-
base-sentiment Rosenthal et al. (2017). The model outputs consist of three
probabilities: negative, neutral and positive sentiment. For the final score,
we use the expectation of positive sentiment (see Equation 6).

$score=0\cdot P(neg)+0.5\cdot P(neutral)+1\cdot P(pos)$ (6)

For perplexity, we use the GPT-2 XL model and count the perplexity of all
generated texts.

We also evaluate the Grammar Score using the RoBERTa-base model fine-tuned on
the CoLA dataset by TextAttack666https://huggingface.co/textattack/roberta-
base-CoLA Morris et al. (2020). The final score is the mean probability of the
text being grammatically correct.

Text generation parameters can be found in Table 3:

Parameter | Value
---|---
top-p | 0.9
max new tokens | 30
Table 3: Parameters used for text generation.

###  A.2 LM Training Details

We trained a GPT-Like language model on the C4 Raffel et al. (2019) dataset.
This model’s architecture is identical to the DistilGPT-2 model Sanh et al.
(2019). We used 8x NVidia A100-SXM-80GB GPUs with bf16 mixed precision Burgess
et al. (2019). We trained our model for 37K steps with the AdamW Loshchilov
and Hutter (2019) optimizer and a cosine scheduler with warmup. Parameters for
the training procedure can be found in the table 4. Model was trained until
convergence, and the loss dynamic can be found in Figure 11.

Parameter | Value
---|---
Max LR | 3e-4
Weight decay | 0.01
$\beta_{1}$ | 0.9
$\beta_{2}$ | 0.95
$\varepsilon$ | 1e-8
Warmup steps | 5000
Effective batch size | 1024
Table 4: Parameters used for LM training. Figure 11: Loss dynamic during LM
training.


# Zhukovsky-Volterra top and quantisation ideals

A. Mikhailov1, T. Skrypnyk1,2
1 School of Mathematics, University of Leeds, LS2 9JT, UK ,
2 Bogolyubov Institute for Theoretical Physics, Metrologichna st.14-b, 03143,
Kiev, Ukraine

###### Abstract

In this letter, we revisit the quantisation problem for a fundamental model of
classical mechanics—the Zhukovsky-Volterra top. We have discovered a four-
parametric pencil of compatible Poisson brackets, comprising two quadratic and
two linear Poisson brackets. Using the quantisation ideal method, we have
identified two distinct quantisations of the Zhukovsky-Volterra top. The first
type corresponds to the universal enveloping algebras of $so(3)$, leading to
Lie-Poisson brackets in the classical limit. The second type can be regarded
as a quantisation of the four-parametric inhomogeneous quadratic Poisson
pencil. We discuss the relationships between the quantisations obtained in our
paper, Sklyanin’s quantisation of the Euler top, and Levin-Olshanetsky-Zotov’s
quantisation of the Zhukovsky-Volterra top.

Keywords: quantum top, quantisation ideal, quadratic Poisson brackets,
Sklyanin algebra.

##  1 Introduction

The classical and quantum tops are fundamental models in physics. The
anisotropic Zhukovsky-Volterra [1, 2] and Euler tops [3] stand out as the
simplest yet non-trivial examples. In the classical case, they describe the
motion of a free rigid body in the presence or absence of an external field.
In the quantum domain, they characterise the dynamics of an isolated spinning
particle, an atom or a nucleus subjected to a constant external field, and
contribute to the description of phase transitions in atomic nuclei in the
Lipkin-Meshkov-Glick model (see [4] and references therein). Their
significance in classical and quantum mechanics motivates us to revisit the
problem of quantisation using a novel approach.

The classical Zhukovsky-Volterra system [1, 2] is a dynamical system in three-
dimensional phase space with coordinates $S_{\alpha},\
\alpha\in\overline{1,3}$ (representing rotational momenta or classical spin),
described by the following system of three ordinary differential equations:

$\frac{dS_{\alpha}}{dt}=2(j_{\beta}-j_{\gamma})S_{\beta}S_{\gamma}+2(k_{\beta}S_{\gamma}-k_{\gamma}S_{\beta}),$
(1)

where $\alpha,\beta,\gamma$ represent a cyclic permutation of the indices
$1,2,3$, $j_{\alpha}$ are parameters of anisotropy (reciprocals of the
components of the inertia tensor), and $k_{\alpha}$ are the constant
components of the external field.

It is well-known that the Zhukovsky-Volterra (and Euler) top admits a pencil
of compatible Lie–Poisson brackets. By extending the phase space with a new
coordinate $S_{0}$, which is a constant of motion of the dynamical system, one
can also construct quadratic Poisson brackets on the resulting four-
dimensional space. For the Euler top, the later coincides with the famous
Sklyanin algebra [10]. In the case of the Zhukovsky-Volterra top, it
represents a known modification of the Sklyanin algebra [6].

In this paper, we demonstrate that Zhukovsky-Volterra and Euler tops admit a
second inhomogeneous quadratic Poisson bracket. Furthermore, we establish that
all four Poisson brackets are mutually compatible in the sense of Magri [9],
and define a family of Poisson brackets:

$\begin{array}[]{l}\\{S_{\alpha},S_{\beta}\\}_{a,b,c,d}=2\left((c+d\,j_{\alpha})S_{0}S_{\gamma}+(a+b\,j_{\gamma})S_{\gamma}+d\,k_{\gamma}S_{0}+b\,k_{\gamma}\right),\\\
\\{S_{0},S_{\alpha}\\}_{a,b,c,d}=2e\,(j_{\beta}-j_{\gamma})S_{\beta}S_{\gamma}+2e\,(k_{\beta}S_{\gamma}-k_{\gamma}S_{\beta}),\end{array}$
(2)

where $a$, $b$, $c$, $d$ and $e$ are arbitrary parameters of the family (if
$e\neq 0$, then by a rescaling of the bracket one can set $e=1$ without a loss
of generality). The existence of the second quadratic structure in these
models, along with the compatibility of all four Poisson brackets, seems to be
a novel result to the best of our knowledge. If $e\neq 0$, the variable
$S_{0}$ plays the role of the Hamiltonian for the entire family of the
brackets.

We address the quantisation problem through a novel approach based on the
concept of quantisation ideal, initially introduced in [12] and further
developed in [13]-[16]. This approach is tailored for dynamical systems
defined on free associative algebras. As a preliminary step, we lift the
Zhukovsky-Volterra system to the free algebra
$\mathcal{A}=\mathbb{C}\langle\hat{S}_{0},\hat{S}_{1},\hat{S}_{2},\hat{S}_{3}\rangle$
using the same Lax representation as applied to the matrix-valued system [17].
The resulting system defines a derivation
$\frac{d}{dt}\,:\,\mathcal{A}\rightarrow\mathcal{A}$ of the algebra.
Subsequently, we seek a quantisation ideal $\mathcal{J}\subset\mathcal{A}$,
which is an ideal satisfying two conditions:

  1. (i)

the ideal $\mathcal{J}$ is $\frac{d}{dt}$–stable:
$\frac{d}{dt}\mathcal{J}\subset\mathcal{J}$;

  2. (ii)

the quotient algebra $\mathcal{A}/\mathcal{J}$ admits a basis of normally
ordered monomials.

The quotient algebra $\mathcal{A}/\mathcal{J}$ is then said to be a quantum
algebra for the system. The first condition implies that $\frac{d}{dt}$
decends to a derivation of the quantum algebra, defining the quantum
Zhukovsky-Volterra system with commutation relations determined by the
generators of the quantisation ideal.

As a candidate for a quantisation ideal, we consider the ideal $\mathcal{I}$
genrated by the set of polynomials:

$\begin{array}[]{ll}\mathcal{I}=\langle\hat{S}_{\alpha}\hat{S}_{\beta}-\hat{S}_{\beta}\hat{S}_{\alpha}+A_{\gamma}(\hat{S}_{0}\hat{S}_{\gamma}+\hat{S}_{\gamma}\hat{S}_{0})+K_{\gamma}\hat{S}_{0}+M_{\gamma}\hat{S}_{\gamma}+N_{\gamma},\\\
\hat{S}_{\gamma}\hat{S}_{0}-\hat{S}_{0}\hat{S}_{\gamma}+B_{\gamma}(\hat{S}_{\alpha}\hat{S}_{\beta}+\hat{S}_{\beta}\hat{S}_{\alpha})+({l}_{\alpha}\hat{S}_{\beta}-{l}_{\beta}\hat{S}_{\alpha})\,|\,\alpha,\beta,\gamma\in\overline{1,3}\rangle.\end{array}$

These polynomials generalise the commuation relations in the Sklyanin algebra
[10] and the algebra obtained in [6]. The ideal $\mathcal{I}$ is parametried
by the set of 18 constants
$A_{\alpha},B_{\alpha},K_{\alpha},M_{\alpha},N_{\alpha}$ and $l_{\alpha},\
\alpha\in\overline{1,3}$. The fulfillment of conditions (i) and (ii) leads to
a system of algebraic equations on these parameters, which we have solved in
the paper to derive the most general quantisation ideal of the form
$\mathcal{I}$.

In the simplest case $A_{\gamma}=B_{\gamma}=0$ condition (ii) is satisfied due
to the Poincare-Birkchoff-Witt Theorem. Ideals satisfying condition (i) almost
immideatly give rise to a quantum algebra isomorphic to the universal
enveloping algebra given by (2) with $c=d=e=0$, and with the central element
$\hat{S}_{0}$.

We treat the generic case $A_{\gamma}\neq 0,B_{\gamma}\neq 0$ separately.
Firstly, we identify and solve equations for the coefficients of the ideal to
fulfill condition (ii) for quadratic and cubic monomials (Theorem 3.1).
Condition (i) imposes firther constraints on the coefficients of the ideal
(Propositions 3.2 and 3.3). Finally we propose a reparametrisation of the
coefficients that resolve all the constraints and is convenient for a
classical limit.

Ultimately, we obtain a general quantisation ideal that leads to commutation
relations:

$\begin{array}[]{ll}\phantom{.}&[\hat{S}_{\alpha},\hat{S}_{\beta}]=-\frac{h}{1+h^{2}j_{\gamma}\bigl{(}C+(j_{\beta}+j_{\alpha})D\bigr{)}}\Bigl{(}(C+Dj_{\gamma})(\hat{S}_{0}\hat{S}_{\gamma}+\hat{S}_{\gamma}\hat{S}_{0})+2(A+Bj_{\gamma})\hat{S}_{\gamma}+2Dk_{\gamma}\hat{S}_{0}+2Bk_{\gamma}\Bigr{)},\\\
&[\hat{S}_{\gamma},\hat{S}_{0}]=-h\Bigl{(}{(j_{\alpha}-j_{\beta})}(\hat{S}_{\alpha}\hat{S}_{\beta}+\hat{S}_{\beta}\hat{S}_{\alpha})+2({k}_{\alpha}\hat{S}_{\beta}-{k}_{\beta}\hat{S}_{\alpha})\Bigr{)}.\end{array}$
(3)

Here $h=-i\hbar$ can be regarded as a quantisation (Planck) constant.

The five parameters $h,\,A,\,B,\,C,\,D$ defines the quantum algebra. The
classical limit corresponds to $h=0$, in which the algebra becomes
commutative. We may assume that the parameters
$A=A(h),\,B=B(h),\,C=C(h),\,D=D(h)$ are analytic functions of the variable $h$
at $h=0$. They represent a trajectory to the classical boundary in the space
of parameters. Evaluating the standard classical limit of the commuation
relations (3) results in the Poisson brackets family (2), where
$a=A(0),\,b=B(0),\,c=C(0),\,d=D(0),\,e=1$.

The quantum Sklyanin algebra [10] corresponds to the case of the Euler top
($k_{\alpha}=0$), with homogeneous quadratic commutation relations having
$A=B=0,\,e=1$ and certain choices of functions $C(h)$ and $D(h)$ (see Section
3.6.1). In the quantisation of the Zhukovsky-Volterra by Levin-Olshnetsky-
Zotov [6], the commutation relations do not have the form (3), and they do not
satisfy condition (i) for the lifted equation on $\mathcal{A}$, but satisfy
condition (ii). Their quantum system is a deformation of the classical one,
i.e. they deform both the commutative algebra and the equation of motion.

The structure of the present paper is as follows: in Section 2, we explore the
classical Zhukovsky-Volterra (and Euler) top, its integrals and four
compatible Poisson structures. In Section 3, we apply the quantization ideal
method and identify a five-dimensional variety of quantizations. Finally, in
Section 4, we provide a brief summary and discuss the open problems.

##  2 Classical Zhukovsky-Volterra and Euler tops

###  2.1 Equation of motion and its first integrals

Motion of a rigid body in a constant external field, known as the classical
Zhukovsky-Volterra top, can be characterised by the vector of angular momenta
$(S_{1},S_{2},S_{3})\in\mathbb{R}^{3}$, whose components satisfy the following
system of ordinary differential equations

$\frac{dS_{\alpha}}{dt}=2(j_{\beta}-j_{\gamma})S_{\beta}S_{\gamma}+2(k_{\beta}S_{\gamma}-k_{\gamma}S_{\beta}).$
(4)

Here $(k_{1},k_{2},k_{3})$ denotes a vector of a constant external field and
$j_{1},j_{2},j_{3}$ are reciprocals of inertia momentum of the diagonal
inertia tensor. In system (4) and thereafter we assume that indices
$(\alpha,\beta,\gamma)$ represent a cyclic permutation of the table $(1,2,3)$.

The classical Zhukovsky-Volterra top (4) admits two first integrals

$C=\frac{1}{2}\sum\limits_{\alpha=1}^{3}S^{2}_{\alpha},\qquad
H=\frac{1}{2}\sum\limits_{\alpha=1}^{3}j_{\alpha}S^{2}_{\alpha}+\sum\limits_{\alpha=1}^{3}k_{\alpha}S_{\alpha}.$
(5)

When the external field vanishes ($k_{\alpha}=0$), system (4) reduces to the
classical Euler top.

###  2.2 Linear Poisson pencil

It is well known that system (4) is Hamiltonian

$\frac{dS_{\alpha}}{dt}=\\{S_{\alpha},H\\}_{1}.$

with respect to the standard linear Poisson structure on $so^{*}(3)$:

$\\{S_{\alpha},S_{\beta}\\}_{1}=2S_{\gamma},$ (6)

and the Hamiltonian $H$. It is also Hamiltonian

$\frac{dS_{\alpha}}{dt}=\\{C,S_{\alpha}\\}^{\prime}_{1}.$

with respect to the linear inhomogeneous Poisson structure

$\\{S_{\alpha},S_{\beta}\\}^{\prime}_{1}=2(j_{\gamma}S_{\gamma}+k_{\gamma}),$
(7)

and the Hamiltonian $C$.

The functions $C$ and $H$ are Casimir functions of the brackets $\\{\ ,\
\\}_{1}$ and $\\{\ ,\ \\}^{\prime}_{1}$, respectively, i.e.,

$\\{C,S_{\alpha}\\}_{1}=\\{H,S_{\alpha}\\}^{\prime}_{1}=0,\quad\alpha\in\overline{1,3}.$

The brackets $\\{\ ,\ \\}_{1}$ and $\\{\ ,\ \\}^{\prime}_{1}$ are compatible
in the sense of Magri [9], meaning that a linear combination of the brackets

$\\{S_{\alpha},S_{\beta}\\}_{a,b}=2(a+j_{\gamma}b)S_{\gamma}+2k_{\gamma}b,$
(8)

with arbitrary constant parameters $a$ and $b$ is a Poisson bracket. The
Casimir function of this bracket is the following function

$C_{a,b}=aC+bH.$

###  2.3 Extension of the phase space and quadratic Poisson structures

Let us now consider quadratic Poisson structure for the Zhukovsky-Volterra
top. For this purpose we need to extend the phase space of the model with a
new variable $S_{0}$, which is a constant of motion ($\frac{dS_{0}}{dt}=0$)
and a central element of two linear brackets:

$\\{S_{0},S_{\alpha}\\}_{1}=0,\quad\\{S_{0},S_{\alpha}\\}^{\prime}_{1}=0,\qquad\alpha\in\overline{1,3}.$

In the extended phase space we are looking for an inhomogeneous quadratic
Poisson structure of the form:

$\\{S_{\alpha},S_{\beta}\\}=2a_{\gamma}S_{0}S_{\gamma}+2K_{\gamma}S_{0},$ (9a)
$\\{S_{0},S_{\alpha}\\}=e\bigl{(}2(j_{\beta}-j_{\gamma})S_{\beta}S_{\gamma}+2(k_{\beta}S_{\gamma}-k_{\gamma}S_{\beta})\bigr{)},$
(9b)

where $a_{\alpha}$, $K_{\gamma}$ are some constants.

###### Proposition 2.1

(i) The brackets (9) satisfy the Jacobi identity iff

$\ a_{\alpha}=c+j_{\alpha}d,\quad
K_{\alpha}=k_{\alpha}d,\quad\alpha\in\overline{1,3}.$ (10)

(ii) The Poisson brackets given by (9) with the structure constants defined by
(10) are compatible with the pencil of linear-constant Poisson brackets $\\{\
,\ \\}_{a,b}$ defined by (8) for any values of $c$, $d$, $a$, $b$, $e$.

Furthermore, in order for the equation (9b) to coincide exactly with the
equation of motion we will further impose the normalization condition $e=1$.
This can be always achieved if $e\neq 0$. The the case $e=0$ can be
effectively reduced to the previously considered linear case. Thus we obtained
the Poisson brackets:

$\\{S_{\alpha},S_{\beta}\\}_{a,b,c,d}=2\bigl{(}(c+j_{\alpha}d)S_{0}S_{\gamma}+(a+j_{\gamma}b)S_{\gamma}+k_{\gamma}dS_{0}+k_{\gamma}b\bigr{)},$
(11a)
$\\{S_{0},S_{\alpha}\\}_{a,b,c,d}=2(j_{\beta}-j_{\gamma})S_{\beta}S_{\gamma}+2(k_{\beta}S_{\gamma}-k_{\gamma}S_{\beta}),$
(11b)

which depend on four arbitrary parameters $a,b,c,d$.

The above Proposition can be proven by solving equations on the coefficients
$a_{\gamma},K_{\gamma}$ that are obtained from the Jacobi identity. We derived
the brackets (11) as a classical limit of the general commutation relations
(44) outlined in Proposition 3.5. Consequently, the fulfillment of the Jacobi
identity is guaranteed for any choice of $a,b,c,d$.

Remark 1. Note that (11) represents a linear combination of two quadratic
brackets if we set $c+d=1$.

Remark 2. Observe that in the case $da-bc=0,\ c\neq 0,\ d\neq 0$, the linear
term in the equation (11a) can be obtained simply by shift of the element
$S_{0}$: $S_{0}\rightarrow S_{0}+\nu$, where $\nu=ac^{-1}=bd^{-1}$.

Applying the method of indeterminate coefficients, we have identified two
Casimir functions of the brackets (11):

$C_{1}=\sum\limits_{\alpha=1}^{3}(c+j_{\alpha}d)S_{\alpha}^{2}+2d\sum\limits_{\alpha=1}^{3}k_{\alpha}S_{\alpha}-2(da-
bc)S_{0},$ (12a)
$C_{2}=\sum\limits_{\alpha=1}^{3}S_{\alpha}^{2}+dS^{2}_{0}+2bS_{0}.$ (12b)

##  3 Quantum Zhukovsky-Volterra top and quantisation ideals

In order to apply the quantisation ideal approach we lift equations of the
classical commutative Zhukovsky–Volterra top to a free associative algebra
$\mathcal{A}=\mathbb{C}\langle\hat{S}_{0},\ldots,\hat{S}_{3}\rangle$ using a
Lax representation, which is similar to the commutative case.

In the simplest case of Lie type ideals the existence of a basis of normally
ordered monomials $\mathcal{B}=\langle
S_{0}^{n_{0}}S_{1}^{n_{1}}S_{2}^{n_{2}}S_{3}^{n_{3}}\,|\,n_{k}\in\mathbb{N}\rangle$
in the quotient algebra $\mathcal{A}/\mathcal{I}$, or a PBW basis, is
following from the Poincaré–Birkhoff–Witt Theorem. The stability condition (i)
leads to a quantisation that in the classical limit results in the pencil of
compatible linear Poisson brackets (8).

For a quadratic ideal $\mathcal{I}\subset\mathcal{A}$, we examine the
conditions emerging from the requirement of existence of a PBW basis in the
quotient algebra $\mathcal{A}/\mathcal{I}$ (condition (ii)) in order to find
equations on the parameters of the ideal. We focus on the subspaces spanned by
monomials
$\mathcal{B}_{N}=\langle\hat{S}_{0}^{n_{0}}\hat{S}_{1}^{n_{1}}\hat{S}_{2}^{n_{2}}\hat{S}_{3}^{n_{3}}\,|\,n_{0}+n_{1}+n_{2}+n_{3}\leqslant
N,\ n_{k}\in\mathbb{N}\rangle$ with $N=2$ and $N=3$.

Subsequently, we consider the stability condition (i). The quantisation
obtained in this way depends on five parameters, and in the commutative
classical limit results in a four parametric family of compatible Poisson
brackets (11). Finally we compare the results obtained with Sklyanin’s
quantisation of the Euler top and Levin–Olshanetsky–Zykov quantisation of the
Zhukovsky–Volterra top.

###  3.1 Equation of motion on free associative algebra

In the classical commutative case the Lax representation for the Zhukovsky-
Volterra model was discovered in [5] and studied in [6], [7]. The Lax pair
with matrix valued entries $S_{\alpha}$ was discussed in [17]. Here, we employ
the same Lax pair, replacing matrices $S_{\alpha}$ by elements
$\hat{S}_{\alpha}$ from the free algebra $\mathcal{A}$:

$\hat{L}=i\sum\limits_{\alpha=1}^{3}(u_{\alpha}\hat{S}_{\alpha}+\frac{k_{\alpha}}{u_{\alpha}})\sigma_{\alpha},\quad\hat{M}=i\sum\limits_{\alpha=1}^{3}u_{\beta}u_{\gamma}\hat{S}_{\alpha}\sigma_{\alpha},$
(13)

where $u_{\alpha}$, $\alpha\in\overline{1,3}$ are coordinates of a point on
the elliptic spectral curve

$u^{2}_{1}-u^{2}_{2}=j_{1}-j_{2},\qquad u^{2}_{2}-u^{2}_{3}=j_{2}-j_{3},$

and $\sigma_{\alpha}$ are standard Pauli matrices. The Lax equation

$\frac{d\hat{L}}{dt}=[\hat{L},\hat{M}],$ (14)

leads to the dynamical system

$\frac{d\hat{S}_{\alpha}}{dt}=(j_{\beta}-j_{\gamma})(\hat{S}_{\beta}\hat{S}_{\gamma}+\hat{S}_{\gamma}\hat{S}_{\beta})+2({k}_{\beta}\hat{S}_{\gamma}-{k}_{\gamma}\hat{S}_{\beta}),\qquad\alpha\in\overline{1,3},$
(15)

that together with the equations

$\frac{d\hat{S}_{0}}{dt}=0,$ (16)

represent a lift of the of the classical commutative Zhukovsky-Volterra top
(4) to the free algebra $\mathcal{A}$.

###  3.2 Quantisation ideals of Lie type

Let us at first consider a Lie type ideal $\mathcal{J}$, generated by
polynomials:

$\mathcal{J}=\langle
f_{\gamma}=\hat{S}_{\alpha}\hat{S}_{\beta}-\hat{S}_{\beta}\hat{S}_{\alpha}-a_{\gamma}\hat{S}_{\gamma}-b_{\gamma},\
g_{\gamma}=\hat{S}_{0}\hat{S}_{\gamma}-\hat{S}_{\gamma}\hat{S}_{0}\,|\,\alpha,\beta,\gamma\in\overline{1,3}\rangle,$
(17)

where $a_{\gamma},b_{\gamma}$ are six arbitrary constants and $\hat{S}_{0}$ is
a central element of the algebra $\mathcal{A}$. It follws from the Poincare-
Birkchoff-Witt theorem that the quotient algebra $\mathcal{A}/\mathcal{J}$
admits a basis of normally ordered monomials $\mathcal{B}=\langle
S_{0}^{n_{0}}S_{1}^{n_{1}}S_{2}^{n_{2}}S_{3}^{n_{3}}\,|\,n_{k}\in\mathbb{N}\rangle$,
which is refered as a PBW basis.

The ideal (17) is a quantisation ideal for system (15), (16) if $\mathcal{J}$
it is stable with respect to the dynamics.

###### Proposition 3.1

The ideal $\mathcal{J}$ is $\frac{d}{dt}$–stable, i.e.
$\frac{d\mathcal{J}}{dt}\subset\mathcal{J}$ if and only if

$a_{\gamma}=2(a+j_{\gamma}b),\quad b_{\gamma}=2k_{\gamma}b,$ (18)

where $a,b$ are arbitrary constants.

Sketch of the proof: The stability conditions can be obtained from the
requirement that the time derivatives of the ideal generators belong to the
ideal. The conditions of stability of the generators $g_{\gamma}$ do not
impose any constraints on the constants $a_{\gamma},b_{\gamma}$. It follows
from the conditions $\frac{df_{\gamma}}{dt}\subset\mathcal{J},\
\gamma\in\overline{1,3}$ that

$(j_{1}-j_{2})a_{3}+(j_{2}-j_{3})a_{1}+(j_{3}-j_{1})a_{2}=0,\qquad
b_{\gamma}(j_{\alpha}-j_{\beta})=k_{\gamma}(a_{\alpha}-a_{\beta}),\quad\alpha,\beta,\gamma\in\overline{1,3}.$

The above implies (18). $\Box$

The statement of the Proposition means that the quantisation ideal of the Lie
type effectively depend on two parameters

$\mathcal{J}_{(a,b)}=\langle\hat{S}_{\alpha}\hat{S}_{\beta}-\hat{S}_{\beta}\hat{S}_{\alpha}-2(a+j_{\gamma}b)\hat{S}_{\gamma}-2k_{\gamma}b,\
\hat{S}_{0}\hat{S}_{\gamma}-\hat{S}_{\gamma}\hat{S}_{0}\,|\,\alpha,\beta,\gamma\in\overline{1,3}\rangle,$

that in the classical limit reduces to the Poisson pencil (8). The center of
the quantum algebra $\mathcal{A}/\mathcal{J}_{(a,b)}$ is generated by
$\hat{S}_{0}$ and $a\hat{C}+b\hat{H}$, where

$\hat{C}=\frac{1}{2}\sum\limits_{\alpha=1}^{3}\hat{S}^{2}_{\alpha},\qquad\hat{H}=\frac{1}{2}\sum\limits_{\alpha=1}^{3}j_{\alpha}\hat{S}^{2}_{\alpha}+\sum\limits_{\alpha=1}^{3}k_{\alpha}\hat{S}_{\alpha}.$
(19)

The specification $a=i\hbar$ and $b=0$ leads to the standard commutation
relations

$[\hat{S}_{\alpha},\hat{S}_{\beta}]=2i\hbar\hat{S}_{\gamma},\qquad[\hat{S}_{0},\hat{S}_{\alpha}]=0$

for $so(3)$ quantum systems. On the algebra
$\mathcal{A}/\mathcal{J}_{(i\hbar,0)}$ the quantum Zhukovsky-Volterra system
(15), (16) can be presented in the Heisenberg from

$i\hbar\frac{d\hat{S}_{\alpha}}{dt}=[\hat{S}_{\alpha},\hat{H}].$ (20)

The center of the quantum algebra $\mathcal{A}/\mathcal{J}_{(i\hbar,0)}$ is
generated by the elements $\hat{S}_{0}$ and $\hat{C}$. The classical limit in
this case results in the Poisson brackets (6), Hamiltonian $H$ and Casimir
element $C$ (5).

The second choice of specification $a=0$ and $b=i\hbar$ leads to commutation
relations

$[\hat{S}_{\alpha},\hat{S}_{\beta}]^{\prime}=2i\hbar(j_{\gamma}\hat{S}_{\gamma}+k_{\gamma}),\qquad[\hat{S}_{0},\hat{S}_{\alpha}]^{\prime}=0$

on the algebra $\mathcal{A}/\mathcal{J}_{(0,i\hbar)}$. Here we use “prime” in
$[\cdot,\cdot]^{\prime}$ to emphasize that the multiplication rules the
algebras $\mathcal{A}/\mathcal{J}_{(i\hbar,0)}$ and
$\mathcal{A}/\mathcal{J}_{(0,i\hbar)}$ are different.

In the algebra $\mathcal{A}/\mathcal{J}_{(0,i\hbar)}$ the center is generated
by the elements $\hat{S}_{0},\hat{H}$ (19), and the element $\hat{C}$ becomes
the Hamiltonian for the quantum Zhukovsky-Volterra system

$i\hbar\frac{d\hat{S}_{\alpha}}{dt}=[\hat{C},\hat{S}_{\alpha}]^{\prime}\,.$
(21)

The classical limit in the case of algebra
$\mathcal{A}/\mathcal{J}_{(0,i\hbar)}$ results in the Poisson brackets (7),
Hamiltonian $C$ and Casimir element $H$ (5).

###  3.3 Quadratic ideals: the PBW condition

We start with consideration of a quite general ideal
$\mathcal{I}\subset\mathcal{A}$ genrated by quadratic polynomials:

$\begin{array}[]{ll}\mathcal{I}=\langle
F_{\gamma}=\hat{S}_{\alpha}\hat{S}_{\beta}-\hat{S}_{\beta}\hat{S}_{\alpha}+A_{\gamma}(\hat{S}_{0}\hat{S}_{\gamma}+\hat{S}_{\gamma}\hat{S}_{0})+K_{\gamma}\hat{S}_{0}+M_{\gamma}\hat{S}_{\gamma}+N_{\gamma},\\\
G_{\gamma}=\hat{S}_{\gamma}\hat{S}_{0}-\hat{S}_{0}\hat{S}_{\gamma}+B_{\gamma}(\hat{S}_{\alpha}\hat{S}_{\beta}+\hat{S}_{\beta}\hat{S}_{\alpha})+({l}_{\alpha}\hat{S}_{\beta}-{l}_{\beta}\hat{S}_{\alpha})\,|\,\alpha,\beta,\gamma\in\overline{1,3}\rangle.\end{array}$
(22)

These polynomials generalise the commutation relations in the Sklyanin algebra
[10] and the algebra obtained in [6].

The first problem is to find conditions on 18 parameters
$A_{\alpha},B_{\alpha},K_{\alpha},M_{\alpha},N_{\alpha}$ and $l_{\alpha},\
\alpha\in\overline{1,3}$ which guarantee the existence of the normally ordered
monomial basis in the subspaces of quadratic and cubic polynomials in the
quotient algebra $\mathcal{A}/\mathcal{I}$.

###### Theorem 3.1

  1. 1.

Quadratic polynomials in variables $\hat{S}_{\alpha},\ \alpha\in\\{0,1,2,3\\}$
admit normal ordering, modulo the ideal $\mathcal{I}$, i.e. a unique
representation in the monomial basis $\mathcal{B}_{2}$, iff

$A_{1}B_{1}\neq-1,\quad A_{2}B_{2}\neq 1,\quad A_{3}B_{3}\neq-1.$ (23)

  2. 2.

Cubic polynomials admit normal ordering, modulo the ideal $\mathcal{I}$, if
conditions (23) satisfied and

$\displaystyle\sum\limits_{\gamma=1}^{3}A_{\gamma}B_{\gamma}+\prod\limits_{\gamma=1}^{3}A_{\gamma}B_{\gamma}=0,$
(25) $\displaystyle B_{1}+B_{2}+B_{3}=0,$ $\displaystyle K_{\alpha}$
$\displaystyle=$
$\displaystyle\frac{(A_{\beta}-A_{\gamma}+B_{\alpha}A_{\beta}A_{\gamma})}{B_{\alpha}(1+A_{\beta}B_{\beta}A_{\gamma}B_{\gamma})}l_{\alpha},$
(26) $\displaystyle M_{\alpha}$ $\displaystyle=$ $\displaystyle 2\nu
A_{\alpha}-\mu\bigl{(}\frac{3+A_{\beta}B_{\beta}-A_{\gamma}B_{\gamma}+A_{\beta}B_{\beta}A_{\gamma}B_{\gamma}}{1+A_{\beta}B_{\beta}A_{\gamma}B_{\gamma}}\bigr{)},$
(27) $\displaystyle N_{\alpha}$ $\displaystyle=$
$\displaystyle{\nu}K_{\alpha},\qquad\alpha\in\overline{1,3},$ (28)
$\displaystyle B_{\alpha}\neq 0,\quad A_{\alpha}^{2}B_{\alpha}^{2}\neq
1,\qquad\alpha\in\overline{1,3},$ (29)
$\displaystyle\left(1+A_{1}B_{1}-A_{2}B_{2}-A_{1}A_{2}B_{2}B_{1}\right)^{2}+16A_{1}A_{2}B_{1}B_{2}\neq
0,$ (30)

where $\mu$, $\nu$ are arbitrary parameters, and indices $\alpha$, $\beta$,
$\gamma$ are cyclic permutation of the set $1,2,3$.

Sketch of the Proof. (1.) We regard $F_{\gamma}=0,\
G_{\gamma}=0,\gamma\in\\{1,2,3\\}$ as a system of six linear equations with
respect to the quadratic monomials $\hat{S}_{i}\hat{S}_{j},\ i>j$ which are
not normally ordered. This system admits a unique solution if and only if the
conditions (23) are satisfied. Its solution enables us to span any polynomial
of degree less or equal to two in the basis $\mathcal{B}_{2}$ of the normally
ordered monomials, modulo the ideal $\mathcal{J}$.

(2) The set of possible $64$ cubic monomials contains $20$ normally ordered
monomials. The rest $44$ unordered monomials can be expressed in the basis
$\mathcal{B}_{3}$ solving the system of $48$ polynomial equations

$\hat{S}_{\beta}F_{\alpha}=0,\quad
F_{\alpha}\hat{S}_{\beta}=0,\quad\hat{S}_{\beta}G_{\alpha}=0,\quad
G_{\alpha}\hat{S}_{\beta}=0,\quad\alpha\in\\{1,2,3\\},\
\beta\in\\{0,1,2,3\\}.$

The solution of the above system enables one to represent any cubic polynomial
in $\mathcal{A}$ in the basis $\mathcal{B}_{3}$ uniquely, modulo the ideal
$\mathcal{I}$. The resolvability conditions for this overdetermined system of
linear equations leads to (25)-(30). $\Box$

Remark 3. The quantum ideals and PBW conditions for the quadratic structures
of the quantum Euler top are obtained by putting $l_{\alpha}=0$,
$K_{\alpha}=0$, $N_{\alpha}=0$, $\alpha\in\overline{1,3}$ in the formulae
above.

###  3.4 Stability of the quadratic ideal

####  3.4.1 Dynamical stability of the ideal and projective parametrisation

We will denote $\mathcal{J}$ the ideal $\mathcal{I}$ (22), whose parameters
satisfy conditions (23)-(30) (Theorem 3.1). The stability of the ideal with
respect to the dynamics (15), (16) impose further constraints.

###### Proposition 3.2

The ideal $\mathcal{J}$ is $\frac{d}{dt}$–stable iff

$B_{\alpha}=h(j_{\beta}-j_{\gamma}),\quad l_{\alpha}=2hk_{\alpha},$ (31)

where $h$ is an arbitrary constant.

There is a convenient parametrisation of parameters $A_{\alpha}$, satisfying
(25) with $B_{\alpha}$ satisfying $(\ref{B2})$.

###### Proposition 3.3

The coefficients $A_{\alpha}$ satisfying condition (25) with the constants
$B_{\alpha}$ defined by (31) can be parametrized as follows:

$A_{\alpha}=\frac{1}{hJ_{\alpha}}\frac{J_{\beta}-J_{\gamma}}{j_{\beta}-j_{\gamma}},\qquad\alpha\in\overline{1,3},$
(32)

where $J_{\alpha}$ satisfy the following inequalities:

$J_{\alpha}\neq 0,\quad J_{\alpha}\neq
J_{\beta}+J_{\gamma},\qquad\alpha\in\overline{1,3},$ (33)
$\left(J_{1}+J_{2}-J_{3}\right){}^{4}+16J_{1}J_{2}\left(J_{1}-J_{3}\right)\left(J_{2}-J_{3}\right)\neq
0$ (34)

and are arbitrary otherwise.

The statement of Proposition 3.3 can be checked by a direct substitution of
(31), (32) in (25). Conditions (33), (34) represent inequalities (29),(30),
where $A_{\alpha}$ is given by (32).

Remark 4. Observe that there exists another then (32) parametrization of
$A_{\alpha}$, namely:

$A_{\alpha}=-\frac{1}{h\tilde{J}_{\alpha}}\frac{\tilde{J}_{\beta}-\tilde{J}_{\gamma}}{j_{\beta}-j_{\gamma}},\qquad\alpha\in\overline{1,3},$
(35)

The parametrizations (32), (35) are equivalent. The equivalence is achieved by
an invertible map [18]:

$\tilde{J}_{\alpha}=J_{\alpha}(J_{\alpha}-J_{\beta}-J_{\gamma}).$

The structure constants $K_{\delta}$, as it follows from the formula (26) and
the above form of $A_{\alpha}$, $B_{\alpha}$, $l_{\alpha}$, are:

$K_{\delta}=-\frac{2}{h}\frac{k_{\delta}}{J_{\delta}}\frac{\sum\limits_{\alpha=1}^{3}j_{\alpha}(J_{\beta}-J_{\gamma})}{\prod\limits_{\alpha=1}^{3}(j_{\beta}-j_{\gamma})},\qquad\delta\in\overline{1,3}.$
(36)

The structure constants $M_{\alpha}$, as it follows from the formula (27) are
the following:

$M_{\alpha}=\frac{2\nu}{hJ_{\alpha}}\frac{J_{\beta}-J_{\gamma}}{j_{\beta}-j_{\gamma}}+\mu\frac{(J_{1}+J_{2}+J_{3})}{J_{\alpha}},\qquad\alpha\in\overline{1,3}.$
(37)

In the result the generators of the ideal $\mathcal{J}$ acquire the following
explicit form:

$\hat{S}_{\alpha}\hat{S}_{\beta}-\hat{S}_{\beta}\hat{S}_{\alpha}+\frac{1}{hJ_{\gamma}}\frac{J_{\alpha}-J_{\beta}}{j_{\alpha}-j_{\beta}}\bigl{(}(\hat{S}_{0}+\nu)\hat{S}_{\gamma}+\hat{S}_{\gamma}(\hat{S}_{0}+\nu)\bigr{)}+K_{\gamma}(\hat{S}_{0}+\nu)+\frac{\mu\sum\limits_{\alpha=1}^{3}J_{\alpha}}{J_{\gamma}}\hat{S}_{\gamma},$
(38a)
$\hat{S}_{\gamma}\hat{S}_{0}-\hat{S}_{0}\hat{S}_{\gamma}+h\bigl{(}(j_{\alpha}-j_{\beta})(\hat{S}_{\alpha}\hat{S}_{\beta}+\hat{S}_{\beta}\hat{S}_{\alpha})+2({k}_{\alpha}\hat{S}_{\beta}-{k}_{\beta}\hat{S}_{\alpha})\bigr{)},\qquad\gamma\in\overline{1,3},$
(38b)

where the indices $\alpha,\beta,\gamma$ constitute the cyclic permutations of
the indices $1,2,3$ and the ideal parameters $(J_{1}:J_{2}:J_{3},n,m,h)$
belong to the space $\mathbb{C}P^{2}\times\mathbb{C}^{3}$.

####  3.4.2 The Casimir elements

Let us now describe the Casimir elements of the algebra (22) with the
structure constants (31)-(36).

###### Proposition 3.4

The following elements:

$\hat{C_{1}}=-\frac{\prod\limits_{\alpha=1}^{3}(J_{\beta}-J_{\gamma})}{\prod\limits_{\alpha=1}^{3}(j_{\beta}-j_{\gamma})}\frac{(\hat{S}_{0}+\nu)^{2}}{h^{4}\prod\limits_{\delta=1}^{3}J_{\delta}}+\sum\limits_{\alpha=1}^{3}\frac{J_{\beta}-J_{\gamma}}{j_{\beta}-j_{\gamma}}\frac{\hat{S}^{2}_{\alpha}}{h^{2}J_{\alpha}}+\sum\limits_{\alpha=1}^{3}\frac{K_{\alpha}}{h}\hat{S}_{\alpha}-\mu\frac{(\sum\limits_{\alpha=1}^{3}J_{\alpha})(\sum\limits_{\alpha=1}^{3}j_{\alpha}J_{\alpha}(J_{\beta}-J_{\gamma}))}{h^{3}\prod\limits_{\delta=1}^{3}J_{\delta}\prod\limits_{\alpha=1}^{3}(j_{\beta}-j_{\gamma})}\hat{S}_{0},$
(39)

and

$\hat{C_{2}}=-\sum\limits_{\alpha=1}^{3}\frac{1}{h^{2}}\frac{J_{\beta}-J_{\gamma}}{j_{\beta}-j_{\gamma}}(J_{\alpha}-J_{\beta}-J_{\gamma})\hat{S}^{2}_{\alpha}+\\\
2\sum\limits_{\alpha=1}^{3}\frac{k_{\alpha}}{h^{2}}\frac{j_{\alpha}(J_{\beta}-J_{\gamma})(J_{\alpha}-J_{\beta}-J_{\gamma})-j_{\beta}(J_{\gamma}-J_{\alpha})J_{\gamma}-j_{\gamma}(J_{\alpha}-J_{\beta})J_{\beta}}{(j_{1}-j_{2})(j_{3}-j_{1})(j_{2}-j_{3})}\hat{S}_{\alpha}-\frac{\mu(\sum\limits_{\alpha=1}^{3}J_{\alpha})(\sum\limits_{\alpha=1}^{3}j_{\alpha}(J_{\beta}-J_{\gamma}))}{h^{3}\prod\limits_{\alpha=1}^{3}(j_{\beta}-j_{\gamma})}\hat{S}_{0}$
(40)

are central elements of the algebra of the algebra (22) with the structure
constants (31)-(36).

Idea of the Proof. The Casimir elements are found using the method of the
indeterminate coefficients in the assumption of the linear-quadratic form of
the Casimirs.

Remark 5. From (38b) it follows that Heisenberg equation of motion with
respect to $\hat{S}_{0}$:

$i\hbar\frac{d\hat{S}_{\gamma}}{dt}=[\hat{S}_{\gamma},\hat{S}_{0}],\quad\gamma\in\overline{1,3}$

coincides — on the quotient algebra — with the dynamical equations (15) if and
only if $h=-i\hbar$.

###  3.5 The variety of quantum algebras and the classical limit

####  3.5.1 Affine re-parametrisation of the ideal

In this subsection we give another parametrisation of the ideal that yields a
quantum analogue of the Poisson pencil structure. It is based on the
observation that a simultaneous re-scaling of the parameters
$J_{\alpha}\to\hat{J}_{\alpha}=QJ_{\alpha},\ Q\neq 0$ does not affect the
ideal generated by the polynomials (38).

###### Lemma 3.1

Let
$\sum\limits_{\alpha=1}^{3}J_{\alpha}j_{\beta}j_{\gamma}(j_{\beta}-j_{\gamma})\neq
0$. Then up to projective equivalence $\hat{J}_{\alpha}=QJ_{\alpha}$ the
structure constants $\hat{J}_{\alpha}$ can be parametrised as follows:

$\hat{J}_{\alpha}=1+h^{2}j_{\alpha}\Bigl{(}C+(j_{\beta}+j_{\gamma})D\Bigr{)},\qquad\alpha\in\overline{1,3}.$
(41)

where $C$ and $D$ are arbitrary complex parameters.

Proof. The sysetem of three equations on variables $C,D$ and $Q$

$QJ_{\alpha}=1+h^{2}j_{\alpha}\Bigl{(}C+(j_{\beta}+j_{\gamma})D\Bigr{)},\qquad\alpha\in\overline{1,3}$

admits a unique solution

$C=\sum\limits_{\alpha=1}^{3}\frac{{J}_{\alpha}j_{\alpha}(j_{\beta}-j_{\gamma})}{h^{2}\sum\limits_{\alpha=1}^{3}J_{\alpha}j_{\beta}j_{\gamma}(j_{\beta}-j_{\gamma})},\quad
D=-\frac{\sum\limits_{\alpha=1}^{3}{J}_{\alpha}(j_{\beta}-j_{\gamma})}{h^{2}\sum\limits_{\alpha=1}^{3}J_{\alpha}j_{\beta}j_{\gamma}(j_{\beta}-j_{\gamma})},$
(42)

and

$Q=\frac{\prod\limits_{\alpha=1}^{3}(j_{\beta}-j_{\gamma})}{\sum\limits_{\alpha=1}^{3}J_{\alpha}j_{\beta}j_{\gamma}(j_{\beta}-j_{\gamma})}.$

$\Box$

Remark 6. In terms of the paramters $C,D$ the inequalities (33), (34) take the
form

$\begin{array}[]{l}\left(h^{2}\left(C\left(j_{1}+j_{2}-j_{3}\right)+2Dj_{1}j_{2}\right)+1\right){}^{4}+\\\
16h^{4}\left(j_{1}-j_{3}\right)\left(j_{2}-j_{3}\right)\left(C+Dj_{1}\right)\left(C+Dj_{2}\right)\left(h^{2}j_{2}\left(C+D\left(j_{1}+j_{3}\right)\right)+1\right)\left(h^{2}j_{1}\left(C+D\left(j_{2}+j_{3}\right)\right)+1\right)\neq
0,\\\ \\\
1+h^{2}\left(C\left(j_{\alpha}+j_{\beta}-j_{\gamma}\right)+2Dj_{\alpha}j_{\beta}\right)\neq
0,\quad 1+h^{2}j_{\alpha}\left(C+D\left(j_{\beta}+j_{\gamma}\right)\right)\neq
0,\quad\alpha\in\overline{1,3}.\end{array}$ (43)

The following Proposition is the main result of the present article:

###### Proposition 3.5

A quantisation ideal of the form (22) for the Zhukovsky–Volterra top (15),
(16) leads to a quadratic quantum algebra with the commutation relations:

$[\hat{S}_{\alpha},\hat{S}_{\beta}]=-\frac{2h}{1+h^{2}j_{\gamma}\bigl{(}C+(j_{\beta}+j_{\alpha})D\bigr{)}}\Bigl{(}(C+Dj_{\gamma})\frac{(\hat{S}_{0}\hat{S}_{\gamma}+\hat{S}_{\gamma}\hat{S}_{0})}{2}+(A+Bj_{\gamma})\hat{S}_{\gamma}+Dk_{\gamma}\hat{S}_{0}+Bk_{\gamma}\Bigr{)},$
(44a)
$[\hat{S}_{\gamma},\hat{S}_{0}]=-h\Bigl{(}{(j_{\alpha}-j_{\beta})}(\hat{S}_{\alpha}\hat{S}_{\beta}+\hat{S}_{\beta}\hat{S}_{\alpha})+2({k}_{\alpha}\hat{S}_{\beta}-{k}_{\beta}\hat{S}_{\alpha})\Bigr{)},\qquad\gamma\in\overline{1,3},$
(44b)

where parameters $h,\,A,\,B,\,C,\,D$ satisfy the inequalities (43) and
arbitrary otherwise.

Remark 7. The parameters $\nu$, $\mu$ are related with the constants
$A,\,B,\,C,\,D,\,h$ as follows:

$\nu D=B,\quad 2\nu
hC+\mu\Bigl{(}3+h^{2}\bigl{(}2D\sum\limits_{\alpha=1}^{3}j_{\beta}j_{\gamma}+C\sum\limits_{\alpha=1}^{3}j_{\alpha}\bigr{)}\Bigr{)}=2hA.$
(45)

####  3.5.2 The classical limit

Let us now assume that the functions $A$, $B$, $C$, $D$ are analytical
functions of $h$:

$A=a+\mathcal{O}(h),\quad B=b+\mathcal{O}(h),\quad C=c+\mathcal{O}(h),\quad
D=d+\mathcal{O}(h).$ (46)

Under such the assumption the quantum algebra (44) is a quantum deformation of
the classical inhomogeneous quadratic Poisson algebra with the Poisson
brackets (11) labeled by four parameters $a$, $b$, $c$, $d$. Indeed, using
(44) and the expansions (46), it is easy to see that in the limit
$h\rightarrow 0$ the right-hand-side of (44) is exactly $\\{\ ,\
\\}_{a,b,c,d}$ multiplied by $-h$ ( i.e. by $i\hbar$). The inequalities (43)
are obviously satisfied in the neighbourhood of $h=0$.

In terms of parameters $A$, $B$, $C$, $D$, the central elements (47)-(48) of
the quantum algebra take the form:

$\hat{C_{1}}=-\frac{h^{2}\prod\limits_{\alpha=1}^{3}(C+Dj_{\alpha})(D\hat{S}^{2}_{0}+2B\hat{S}_{0})}{D\prod\limits_{\alpha=1}^{3}(1+h^{2}j_{\alpha}(C+(j_{\beta}+j_{\gamma})D))}+\sum\limits_{\alpha=1}^{3}\frac{(C+Dj_{\alpha})\hat{S}_{\alpha}^{2}+2Dk_{\alpha}\hat{S}_{\alpha}}{(1+h^{2}j_{\alpha}(C+(j_{\beta}+j_{\gamma})D))}-\\\
-2(AD-
CB)\frac{(D+h^{2}(C^{2}+CD\sum\limits_{\alpha=1}^{3}j_{\alpha}+D^{2}\sum\limits_{\alpha=1}^{3}j_{\beta}j_{\gamma}))}{D\prod\limits_{\alpha=1}^{3}(1+h^{2}j_{\alpha}(C+(j_{\beta}+j_{\gamma})D))}\hat{S}_{0},$
(47)
$\hat{C_{2}}=\sum\limits_{\alpha=1}^{3}(C+Dj_{\alpha})(1+h^{2}(C(j_{\beta}+j_{\gamma}-j_{\alpha})+Dj_{\beta}j_{\gamma}))\hat{S}_{\alpha}^{2}+2\sum\limits_{\alpha=1}^{3}(D+h^{2}(C^{2}+DCj_{\alpha}-D^{2}j_{\beta}j_{\gamma}))k_{\alpha}\hat{S}_{\alpha}\\\
-2(AD-CB)\hat{S}_{0}\,.$ (48)

The classical limit of the central elements $\hat{C_{1}}$ and $\hat{C_{2}}$
yield Casimir elements (12) of the quadratic Poisson bracket (11):

$C_{1}=\lim\limits_{\hbar\rightarrow
0}{\hat{C}_{1}}=\lim\limits_{\hbar\rightarrow
0}{\hat{C}_{2}}=\sum\limits_{\alpha=1}^{3}(c+dj_{\alpha})S_{\alpha}^{2}+2d\sum\limits_{\alpha=1}^{3}k_{\alpha}S_{\alpha}-2(ad-
bc)S_{0}$ (49) $C_{2}=\lim\limits_{\hbar\rightarrow
0}\frac{1}{h^{2}}\frac{D}{\prod\limits_{\alpha=1}^{3}(C+Dj_{\alpha})}\Bigl{(}{\hat{C}_{2}}-\frac{D\prod\limits_{\alpha=1}^{3}(1+h^{2}j_{\alpha}(C+(j_{\beta}+j_{\gamma})D))}{(D+h^{2}(C^{2}+CD\sum\limits_{\alpha=1}^{3}j_{\alpha}+D^{2}\sum\limits_{\alpha=1}^{3}j_{\beta}j_{\gamma}))}{\hat{C}_{1}}\Bigr{)}=dS^{2}_{0}+\sum\limits_{\alpha=1}^{3}S_{\alpha}^{2}+2bS_{0}.$
(50)

###  3.6 Comparison with the existing algebras.

####  3.6.1 Sklyanin algebra

The quantum Sklyanin algebra [10] corresponds to the case of the purely
quadratic structure of quantum anisotropic Euler’s top, i.e to the case
$\mu=\nu=0$ and $k_{\alpha}=0$, $\alpha\in\overline{1,3}$. The corresponding
ideal generators (38) are simplified to the following form:

$F_{\gamma}=\hat{S}_{\alpha}\hat{S}_{\beta}-\hat{S}_{\beta}\hat{S}_{\alpha}+\frac{1}{hJ_{\gamma}}\frac{J_{\alpha}-J_{\beta}}{j_{\alpha}-j_{\beta}}\bigl{(}\hat{S}_{0}\hat{S}_{\gamma}+\hat{S}_{\gamma}\hat{S}_{0}\bigr{)},\qquad\gamma\in\overline{1,3},$
(51a)
$G_{\gamma}=\hat{S}_{\gamma}\hat{S}_{0}-\hat{S}_{0}\hat{S}_{\gamma}+h(j_{\alpha}-j_{\beta})(\hat{S}_{\alpha}\hat{S}_{\beta}+\hat{S}_{\beta}\hat{S}_{\alpha}),$
(51b)

The algebra with commutation relations (51) is equivalent to the Sklyanin
algebra obtained from the quantum group considerations, where

$J_{\alpha}=\frac{1+2j_{\alpha}h^{2}+((j_{\beta}+j_{\gamma})j_{\alpha}-j_{\beta}j_{\gamma})h^{4}}{(1+h^{2}j_{\alpha})},\qquad\alpha\in\overline{1,3}.$
(52)

The above expression for $J_{\alpha}$ coincides with the one obtained by
Sklyanin after the re-parametrisation $h^{2}=1/\wp(i\hbar)$, where $\wp$ is a
Weierstrass elliptic function with
$g_{2}=-4(j_{1}j_{2}+j_{2}j_{3}+j_{3}j_{1}),\ g_{3}=-4j_{1}j_{2}j_{3}$,
assuming $j_{1}+j_{2}+j_{3}=0$.

It follows from Lemma 3.1 that corresponding functions $C=C(h)$ and $D=D(h)$
are :

$C=\frac{1-h^{4}\sum\limits_{\alpha=1}^{3}j_{\beta}j_{\gamma}-2h^{6}j_{1}j_{2}j_{3}}{1+h^{6}(\sum\limits_{\alpha=1}^{3}j^{3}_{\alpha}+j_{1}j_{2}j_{3})-h^{8}\sum\limits_{\alpha=1}^{3}j^{2}_{\beta}j^{2}_{\gamma}},$
$D=\frac{h^{2}(3+h^{4}\sum\limits_{\alpha=1}^{3}j_{\beta}j_{\gamma})}{1+h^{6}(\sum\limits_{\alpha=1}^{3}j^{3}_{\alpha}+j_{1}j_{2}j_{3})-h^{8}\sum\limits_{\alpha=1}^{3}j^{2}_{\beta}j^{2}_{\gamma}}.$

In this case $c=1$, $d=0$, i.e. in the classical limit we obtain the first
quadratic (Sklyanin) brackets.

####  3.6.2 Algebra of Levin-Olshanetsky-Zotov

Levin, Olshanetsky and Zotov proposed a quantisation of the Zhukovsky-Volterra
top ($k_{\alpha}\neq 0$) [6]. Basing on the reflection equation algebra they
found a quantum algebra defined by the following ideal:

$\mathcal{J}_{LOZ}=\langle\hat{S}_{\alpha}\hat{S}_{\beta}-\hat{S}_{\beta}\hat{S}_{\alpha}-i(\hat{S}_{0}\hat{S}_{\gamma}+\hat{S}_{\gamma}\hat{S}_{0}),\quad\hat{S}_{\gamma}\hat{S}_{0}-\hat{S}_{0}\hat{S}_{\gamma}+i\frac{(J_{\alpha}-J_{\beta})}{J_{\gamma}}(\hat{S}_{\alpha}\hat{S}_{\beta}+\hat{S}_{\beta}\hat{S}_{\alpha})+\frac{i}{J_{\gamma}}({l}_{\alpha}\hat{S}_{\beta}-{l}_{\beta}\hat{S}_{\alpha})\rangle.$
(53)

The re-scaling of the variables

$\hat{S}_{0}\rightarrow
ih{\hat{S}_{0}},\quad\hat{S}_{\alpha}\rightarrow\sqrt{J_{\alpha}J_{\beta}}\hat{S}_{\alpha},\quad{l}_{\alpha}\rightarrow
h\sqrt{J_{\alpha}J_{\beta}}{l}_{\alpha},\quad\alpha\in\overline{1,3}$

transforms the ideal (53) to the form

$\mathcal{J}=\langle\hat{S}_{\alpha}\hat{S}_{\beta}-\hat{S}_{\beta}\hat{S}_{\alpha}+\frac{h}{J_{\gamma}}(\hat{S}_{0}\hat{S}_{\gamma}+\hat{S}_{\gamma}\hat{S}_{0}),\quad\hat{S}_{\gamma}\hat{S}_{0}-\hat{S}_{0}\hat{S}_{\gamma}+\frac{(J_{\alpha}-J_{\beta})}{h}(\hat{S}_{\alpha}\hat{S}_{\beta}+\hat{S}_{\beta}\hat{S}_{\alpha})+({l}_{\alpha}\hat{S}_{\beta}-{l}_{\beta}\hat{S}_{\alpha})\rangle.$
(54)

The ideal $\mathcal{J}$ is a partial case of the ideal $\mathcal{I}$ (22),
corresponding to the following choice of the parameters

$A_{\alpha}=\frac{h}{J_{\alpha}},\
B_{\alpha}=\frac{{(J_{\beta}-J_{\gamma})}}{h},\ K_{\alpha}=0,\ M_{\alpha}=0,\
N_{\alpha}=0,\qquad\alpha\in\overline{1,3}.$ (55)

The parametrisation (55) satisfy the condition (25), (25) of our Theorem 3.1,
and therefore the quantum algebra of Levin, Olshanetsky and Zotov possess PBW
property up to monomials of the third order.

The Heisenberg equations with the Hamiltonian $\hat{S}_{0}$ in [6] are not
equivalent to the dynamical system (15) on the free algebra
$\mathcal{A}=\mathbb{C}\langle\hat{S}_{0},\ldots,\hat{S}_{3}\rangle$, in other
words, the coefficients $B_{\alpha}$ do not have the form (31), since the
ideal (54), (55) is not invariant with respect to the non-Abelian dynamics
(15). The quantisation presented in [6] can be regarded as a simultaneous
deformation of both the commutative algebra of functions on the phase space
and the constants $j_{\alpha}$ of the dynamical system (1). This deformation
depends on a single parameter $\hbar$

$J_{\alpha}=\frac{\sqrt{{(1+h^{2}j_{\beta}})(1+h^{2}j_{\gamma})}}{\sqrt{(1+h^{2}j_{\alpha})}},\qquad\alpha\in\overline{1,3}.$
(56)

where $h^{2}=1/\wp(i\hbar)$ with $g_{2}=-4(j_{1}j_{2}+j_{2}j_{3}+j_{3}j_{1}),\
g_{3}=-4j_{1}j_{2}j_{3}$, and $j_{1}+j_{2}+j_{3}=0$. In the classical limit
$\hbar\to 0$ we get

$J_{\alpha}=1+\hbar^{2}j_{\alpha}+\mathcal{O}(\hbar^{4}),\qquad\alpha\in\overline{1,3},$

the commutation relations yield the quadratic Poisson structure (11) with
$c=1$, $a=b=d=0$, and the limiting system coincides with (1).

##  4 Conclusion and Discussion

The results of this paper pose several interesting mathematical and physical
problems. The quantisation obtained depends on five parameters, one of which
can be identified with the Planck constant. This quantisation is a
generalisation of the commonly used deformation of the $so(3)$ standard
Poisson bracket and Sklyanin’s quadratic Poisson structure in the case of the
Euler top. In the classical limit, it results in a four-parametric family of
Poisson brackets, which lead to the same dynamical system as the Zhukovsky-
Volterra (and Euler) top, thereby yielding identical dynamics.

In the quantum case, the problem is more subtle. Although the equations of
motion formally coincide, the observables $\hat{S}_{\alpha}$ satisfy
commutation relations that essentially depend on a choice of the quantisation
parameters. We have reasons to believe that the resulting spectrum of the
Hamiltonian also depends on the choice of the parameters. In order to compare
our results with experimental data, it is necessary to develop a
representation theory for the obtained five-parametric algebra and solve the
spectral problem for the corresponding Hamiltonian.

##### Acknowledgements

The authors are grateful to E. Sklyanin, N. Reshetikhin, D. Gurevich, and A.
Zotov for their valuable discussions. The work of the second author was made
possible by an Isaac Newton Institute and London Mathematical Society
Solidarity grant, for which he expresses his gratitude. This article is
partially based on work from COST Action CaLISTA CA21109, supported by COST
(European Cooperation in Science and Technology), www.cost.eu.

## References

  * [1] Volterra V., Acta Math., 22, pp. 201-357. (1899).
  * [2] Zhukovsky N., in Collected works, 1, Moscow-Leningrad: Gostekhisdat, (1949) (in Russian).
  * [3] Landau L., Lifshitz E., Mechanics, 3rd ed., Pergamon Press, (1976).
  * [4] Skrypnyk T. SIGMA: Symmetry, Integrability and Geometry — Methods and Applications, 18, 074, (2022).
  * [5] Fedorov Yu. Regular and Chaotic Dynamics, 5, No 2, pp. 171-180 (2000).
  * [6] Levin A., Olshnetsky M., Zotov A., Communications in Mathematical Physics, 268, pp. 67-103 (2006).
  * [7] Skrypnyk T., Journal of Physics A, 50, 325206 (2017).
  * [8] Skrypnyk T. Nuclear Physics B, 988, 116102, (2023).
  * [9] Magri F. Journal of Mathematical Physics, 19, No 5, 1156-162 (1978).
  * [10] Sklyanin E., Funct. Anal. Appl., 16, No 4, pp. 263-270 (1982).
  * [11] Sklyanin E., Funct. Anal. Appl., 17, No 4, pp. 273-284 (1983).
  * [12] Mikhailov A. Russian Math. Surveys, 75, No 5, p. 978-980 (2020)
  * [13] Carpentier S., Mikhailov A., Wang J., Lett. Math. Phys., 112, No 5, p. 94 (2022).
  * [14] V.M. Buchstaber, A.V. Mikhailov,  KdV hierarchies and quantum Novikov’s equations.  Open Communications in Nonlinear Mathematical Physics, February 15, 2024, Special Issue in Memory of Decio Levi,  https://doi.org/10.46298/ocnmp.12684.
  * [15] Sylvain Carpentier, Alexander V. Mikhailov, and Jing Ping Wang.  Hamiltonians for the quantised Volterra hierarchy.  arXiv:2312.12077, 2023.
  * [16] A. V. Mikhailov and P. Vanhaecke.  Classical Poisson structures and deformations of noncommutative algebras.  In preparation, 2024.
  * [17] Levin A., Olshnetsky M., Zotov A., Journal of Physics A, 49, 395202, (2017).
  * [18] Zotov A. - private communications.
  * [19] Wigner E. Phys. Rev. 77, pp. 711-712, (1950).