Original Article PAM, DL, SOTA, DIP, SIRS, PSNR, GAN, DDPM, ILVR, MCG, DPS,
SSIM, LPIPS Bogazici University Physics Department, Istanbul, Turkey
<EMAIL_ADDRESS>
# Speeding up Photoacoustic Imaging using Diffusion Models

Irem Loc  Bogazici University Physics Department, Istanbul, Turkey  Mehmet
Burcin Unlu  Faculty of Engineering, Ozyegin University, Istanbul, Turkey
Faculty of Aviation and Aeronautical Sciences, Ozyegin University, Istanbul,
Turkey

###### Abstract

Background: Photoacoustic Microscopy (PAM) integrates optical and acoustic
imaging, offering enhanced penetration depth for detecting optical-absorbing
components in tissues. Nonetheless, challenges arise in scanning large areas
with high spatial resolution. With speed limitations imposed by laser pulse
repetition rates, the potential role of computational methods is highlighted
in accelerating PAM imaging.

Purpose: We are proposing a novel and highly adaptable DiffPam algorithm that
utilizes diffusion models for speeding up the photoacoustic imaging process.

Method: We leveraged a diffusion model trained exclusively on natural images,
comparing its performance with an in-domain trained U-Net model using a
dataset focused on PAM images of mice brain microvasculature.

Results: Our findings indicate that DiffPam achieves comparable performance to
a dedicated U-Net model, without the need for a large dataset or training a
deep learning model. The study also introduces the efficacy of shortened
diffusion processes for reducing computing time without compromising accuracy.

Conclusion: This study underscores the significance of DiffPam as a practical
algorithm for reconstructing undersampled PAM images, particularly for
researchers with limited AI expertise and computational resources.

###### keywords:

Diffusion Models, Photoacoustic Imaging, Image Reconstruction

##  1 Introduction

Photoacoustic Microscopy (PAM) imaging, also called optoacoustic imaging, is a
rapidly emerging field of non-invasive medical imaging[1]. The principle is
based on detecting the ultrasonic waves generated by pressurized tissues by
laser beams. As the name suggests, it combines the advantages of both optical
and acoustic (ultrasonic) imaging techniques. The penetration depth is much
higher than purely optical imaging modalities while it can detect the optical
absorbing parts of the tissue such as hemoglobin, lipids, water, and other
light-absorbing chromophores[2].

Similar to other imaging techniques, there is a trade between scanned area,
scanning time, and spatial resolution. Scanning large areas with high
resolution increases the time complexity of the whole process. In the
literature, advanced scanning techniques are developed for high-speed PAM,
however, the main factor that restricts the scanning speed is the laser pulse
repetition rate[3]. Unless advanced lasers are used, the best option for
speeding up PAM imaging is reducing the scanning area and reconstructing the
image via computational methods.

For the past couple of years, deep learning (DL) has been used as the state-
of-the-art (SOTA) computational method for medical image reconstruction,
enhancement, denoising, and super-resolution. Many recent studies have
illustrated that DL methods can offer solutions for problems in PAM imaging
that are otherwise unsolved.

Zhao et al. [4], used a multi-task residual dense network to speed up optical
resolution PAM with low pulse laser energy. They addressed the issues of image
denoising, super-resolution, and vascular enhancement simultaneously through
multi-supervised learning. Sharma et al. [5] employed supervised fully dense
U-Net to enhance out-of-focus acoustic resolution PAM images. DiSpirito III et
al. [6], approached the reconstruction of undersampled PAM images, i. e. an
inpainting problem. They demonstrated that using supervised fully dense U-Net
they were able to reconstruct artificially downsampled PAM images. All
techniques up to now require a large set of PAM images to train the neural
network. On the contrary, Vu et al.[7] use the deep image prior (DIP) [8]
technique which uses an untrained neural network as an image prior. They
reconstructed the undersampled PAM images, similar to DiSpirito III et al.
[6]. Based on our knowledge, this study is the only one which does not require
supervised training.

In this study, we are proposing a novel method that utilizes a pre-trained
diffusion model to reconstruct the degraded PAM image. This approach leverages
the effectiveness of neural networks pre-trained on natural image data while
eliminating the need for an extensive corpus of PAM images and mitigating the
cumbersome training processes. Notably, our methodology demonstrates a degree
of adaptability, offering the freedom to select the specific degradation
operation, whether it be super-resolution, denoising, or inpainting.
Furthermore, our proposed framework can integrate and benefit from the
existing enhancement techniques.

##  2 Background

###  2.1 Inverse Problems in Image Processing

Reconstructing the degraded PAM image can be considered as a subset of inverse
problems in image processing, which are by nature ill-posed. We can select the
degradation operation as lowering the resolution, masking or noise adding,
corresponding super-resolution, inpainting, or denoising, respectively. In
this study, we approached the problem as a modified version of single image
super-resolution (SIRS) and inpainting, separately. The term super-resolution
refers to techniques to enhance the resolution of an image. Similarly, image
inpainting is a set of techniques that restore missing pixels in an image. In
the literature, there are various approaches for solving SIRS and inpainting,
the main ones are interpolation and DL-based methods. In recent years, deep
learning-based approaches have proven to be superior to interpolation methods.
Early studies on super-resolution/inpainting utilizing deep learning employed
PSNR-based methods. Today, many SISR/inpainting techniques take advantage of
deep generative models. The SOTA generative models that are used for super-
resolution/inpainting are encoder-decoder networks, generative adversarial
networks (GANs), flow-based models, and denoising diffusion probabilistic
models (DDPMs).

Deep generative models can be used for both conditional and unconditional
image generation. Unconditional diffusion models learn the underlying
distribution of natural images to generate an image from pure noise. On the
other hand, conditional generation requires a condition that can be in the
form of text, class label, or image. For super-resolution or inpainting
applications, degraded images can be used as the condition.

Conditional generation can be achieved by using the condition directly in the
training process or leveraging the power of pre-trained unconditional
generative models. Both approaches demonstrated remarkable performance in
inverse problems in image processing [9]. The former requires a large corpus
of high-quality images of interest, the specific design of the model, and the
training process. In this study, we used the latter approach, since it is more
robust and flexible.

###  2.2 Denoising Diffusion Probabilistic Models

DDPMs were first introduced by Sohl-Dickstein [10] and then popularized by Ho
et al [11], as a parametrized Markov chain with variational inference. Markov
process defines a stochastic process where the future only depends on the
present. Dhariwal and Nichol [12] showed that DDPMs can achieve superior image
quality and beat GANs. The diffusion process in DDPMs is inspired by the
thermodynamic process of diffusion. DDPM has forward and backward diffusion
processes. The forward process is defined as adding Gaussian noise to an image
and the backward process is trying to obtain the original image by using the
noisy one (Figure 1).

Figure 1: Forward $q(x_{t}|x_{t-1})$ and backward $p_{\theta}(x_{t-1}|x_{t})$
diffusion processes. $X_{0}$ is the original image, where $X_{T}$ is the
Gaussian noise when $T\rightarrow\infty$.

For the image generation, the backward process is started from pure Gaussian
noise at $t=T$ and repeated until we obtain $X_{0}$ at $t=0$.

###  2.3 Conditional Generation from Unconditional Diffusion Model

The sequential nature of the diffusion models enables researchers to interfere
with the generation process. There are several studies based on the principle
of altering the generation process to utilize an unconditional model to
generate conditional images. In general, a measured image is defined as a
noisy version of the real image transformed by a degradation operation.

$y=Ax+n$ (1)

Degradation operation ($A$) is selected to mimic the relation of measured
value ($y$) and real value ($x$) and ($n$) represents the measurement noise.
The degradation depends on the inverse problem. For super-resolution, the
operation will be downsampling with Gaussian blurring. For inpainting, the
transformation is the multiplication of a mask operator.

Choi et al.[13] defined a process called Iterative Latent Variable Refinement
(ILVR). Proposed algorithm is to generate $x^{\prime}_{t-1}$ from $x_{t}$
using unconditional generation, then update predicted $x^{\prime}_{t-1}$ using
the condition:

$x_{t-1}=x^{\prime}_{t-1}+A^{T}(y_{t-1})-AA^{T}(x^{\prime}_{t-1})$ (2)

where $A^{T}$ is the reverse operation to degradation, e.g. upsampling for
super-resolution. This process can be deemed as a projection operation onto
the desired image manifold. The measurement is considered noiseless ($n=0$) in
this setting.

Chung et al. (a)[14] proposed an improvement to projection-based approaches
named Manifold Constraint Gradient (MCG) correction. They devised a series of
constraints to ensure that the gradient of the measurement remains on the
manifold.

Projection-based approaches may perform well in some cases, but they have
downsides. Since the predicted image is projected onto the measurement, noise
in the measurement is amplified at each iteration. Chung et al. (b)[15]
developed an alternative approach that uses gradient-based correction to the
predicted image, named Diffusion Posterior Sampling (DPS). In this way, the
noise in the measurement is handled during the gradient process, unlike
projection-based methods.

$x_{t-1}=x^{\prime}_{t-1}-\eta\nabla_{x_{t-1}}||y-A(x^{\prime}_{0})||^{2}_{2}$
(3)

$\eta$ is the scaling factor for DPS, which determines the strength of the
measurement in the final image. Chung et al. (b)[15] found that $\eta$ should
be between 0.1 and 1, experimentally. Smaller $\eta$ values lead to
hallucination in the generated images, while larger values may produce
artifacts.

###  2.4 Come-Closer-Diffuse-Faster

Although the pre-trained diffusion models are robust and domain-adaptable,
they are computationally expensive. Producing even a single image is a time-
consuming process. Chung et al. (c)[16] claim that for the inverse problems,
starting the generation from pure Gaussian noise is redundant. Using an
estimate of initial image $x_{0}$, the reverse diffusion process can start at
a timestep $t<<T$. They referred to this process as Come-Closer-Diffuse-
Faster. An initial estimate can be as simple as an interpolation, or as fine-
detailed as the output of a fully-trained neural network. As the initial
estimate improves, the diffusion process can be shortened. This allows us to
make use of existing deep learning models and go beyond their performance.

##  3 Method

###  3.1 Dataset

For all experiments, we have employed OpenAI’s unconditional DDPM model [17]
trained with ImageNet 256x256 dataset[18]. Since we are using a pre-trained
diffusion model, training and validation of the diffusion models are beyond
the scope of this study.

For testing the proposed method in this study, the Duke PAM [19] dataset is
used. Duke PAM is an open-source database acquired by an optical resolution
PAM system at a wavelength of 532 nm in the Photoacoustic Imaging Lab at Duke
University. This PAM system has a lateral resolution of 5 $\mu$m and an axial
resolution of 15 $\mu$m [6]. This dataset consists largely of in vivo images
of mouse brain microvasculature, from which 18 images are randomly selected as
the test set.

Given that the selected diffusion model is trained with 256x256 images,
central cropping is applied to make input image dimensions 256x256. Then,
images are normalized before being passed to the diffusion model. These
normalized images are considered to be the ground truth images ($X_{ref}$).

At this step, our study is divided into three paths to generate input images
($y$):

  1. 1.

For super-resolution, $1/4$ downsampling operation is performed and followed
by low-pass filtering to prevent anti-aliasing using the Resizer Python
package developed by Shocher et al.[20]. Therefore, the resulting image has
6.25% of the pixels of the original image.

  2. 2.

For inpainting with a uniform undersampling, every four rows out of five is
replaced with a zero, leaving 20% of the original pixels.

  3. 3.

For inpainting with a random undersampling, 80% of the pixels are replaced
with zero, leaving 20% of the original pixels.

###  3.2 Neural Network as an Approximate Solution

The previous works mainly focused on reconstructing uniformly undersampled PAM
images[6, 7]. They approached it as an inpainting problem and used U-Net
shaped neural networks for a solution. In this study, we demonstrated that
existing solutions can be benefitted and improved using our algorithm. To this
end, we trained a U-Net model, which was subsequently subjected to comparative
evaluation and involved in our proposed algorithmic framework.

For the training, we used the 337 training samples in the Duke PAM dataset.
For regularization, the data augmentation processes, namely, random crops,
horizontal and vertical flips with 30% probability, random rotations up to 20
degrees, and random Gaussian blur are applied to each image in the training
set. Then images are normalized and used as the ground truth. Input images are
constructed by applying a uniform undersampling to the ground truth image,
which replaces every four rows out of five with a zero, leaving 20% of the
original pixels. Then, empty rows are filled with bilinear interpolation.
Figure 2 illustrates an example image with uniform masking and the resulting
bilinear interpolation.

Figure 2: An example mice brain microvasculature image from Duke PAM dataset.
a) Original (ground truth) image, b) uniformly undersampled (every four rows
out of five is replaced with a zero) and c) missing pixels are filled via
bilinear interpolation.

The network was optimized using Adam optimizer[21] of Pytorch optimizer
package and $l_{1}$ loss function. We used the PyTorch framework version 1.11.
The hyperparameter tuning is achieved by splitting 20% of the training data as
the validation set.

###  3.3 The Reconstruction Algorithm

The steps of the algorithm are the following:

  * •

The operation ($A$) transforms the real value ($X$) into the measured value
($y$). For super-resolution, the operation will be downsampling. For
inpainting, the transformation is the multiplication of a mask operator.

$y=AX_{ref}$

  * •

Optional: An approximate solution to the inverse problem is required. The
better we approximate the real value, the fewer diffusion steps would be
needed. The solution depends on the transform operator $A$. But in general,
interpolation operations or a simple trained neural network as an approximator
can be utilized. Depending on the approximate solution, the starting point
($N<T$) for the reverse diffusion process can be settled.

  * •

After having $X_{N}$, an unconditional diffusion model is employed to generate
$X_{N-1}$. Then, using DPS conditioning the image generation process is
iterated in the direction of interest.

Table 1 summarizes the experiments conducted in this study, regarding the
selection of the measurement operations ($A$), approximate solutions and the
starting point for the diffusion processes ($N$).

Inverse Problem | Measurement Operation | Approximate Solution | Starting Point (N) |
---|---|---|---|---
Inpainting (uniform) | Uniform Undersampling | - | T |
Inpainting (uniform) | Uniform Undersampling | Bilinear Interpolation | T/2 |
Inpainting (uniform) | Uniform Undersampling | Pre-trained U-Net | T/5 |
Inpainting (Random) | Random Undersampling | - | T |
Super Resolution | Downsampling + Gaussian Blurring | - | T |
Table 1: The experimens conducted in this study, regarding the selection of
the measurement operations ($A$), approximate solutions and the starting point
for the diffusion processes ($N$).

The diffusion model that we used in this study is trained with $T=1000$
diffusion steps. Depending on the quality of the approximate solution, we have
selected $N=500$ for bilinear and bicubic interpolation and $N=200$ for pre-
trained U-Net outputs.

Image production time is highly correlated with the sample step size.
Selecting $N=T/k$ reduces the computing time by a factor of $k$.

###  3.4 Evaluation

The quality of reconstructed images is evaluated using three key metrics:
PSNR, structural similarity index (SSIM), and learned perceptual image patch
similarity (LPIPS). PSNR is the logarithm of the ratio between the peak signal
value and the mean squared error (MSE) between the reference and the
reconstructed images. The PSNR goes to infinity when MSE is equal to zero. On
the other hand, SSIM is designed to simulate any image distortion using a
combination of three factors: loss of correlation, luminance distortion, and
contrast distortion[22]. The SSIM values range between zero and one. Zero SSIM
means no correlation between the images, and one refers to identical images.
While PSNR and SSIM are widely acknowledged, they may occasionally fall short
in aligning with human perceptual judgments. In response to this, LPIPS, as
developed by Richard Zhang et al.[23], leverages deep neural networks to offer
a more refined measure of image similarity that is closer to human perception.
Contrary to PSNR and SSIM, the lower LPIPS scores mean better image quality.
In this study, the Scikit-image framework is employed to calculate PSNR and
SSIM metrics, whereas the lpips Python package with AlexNet is used to compute
LPIPS scores.

##  4 Results

###  4.1 Inpainting

For inpainting, 18 images are randomly selected from the test set, and
subjected to uniform and random masking leaving 20% effective pixels. Then,
images are reconstructed using pre-trained U-Net and diffusion model
conditioned on undersampled image (DiffPam). Four DiffPam experiments are
conducted for inpainting, as mentioned in the Methods section.

Producing a single 256x256 image from the $T=1000$ sample steps took 18 min on
average on AWS Sagemaker Studio Lab accelerated computing. Starting from the
halfway, image reconstruction time can be reduced to 9 min. Even further,
starting from the U-Net model output, the time can be reduced up to 3.6 min.

Three out of four inpainting experiments were conducted using uniform masking,
with differing approximate solutions. Figure 3 demonstrates the ground truth
(fully-sampled) PAM image, bilinear interpolation, and the results of
reconstruction from the undersampled version. For simplicity, only the
diffusion model started with bilinear input (B-DiffPam) is illustrated in the
figure.

Figure 3: An example PAM image from our test set. a) Original (ground truth)
image, b) bilinear interpolation applied to the uniformly undersampled image
c) reconstructed image by pre-trained U-Net d) reconstructed image by DiffPam
using (b) as the input.

The result statistics in Figure 4 illustrate that, in all metrics, there is no
statistical significance between the domain-specifically trained U-Net and
DiffPam performances. In addition, DiffPam slightly improved the median SSIM,
when applied to U-Net outputs (U-DiffPam).

Figure 4: The uniform undersampling experimental results in terms of a) PSNR,
b) SSIM and c) LPIPS metrics. DiffPam refers to the reconstruction of
uniformly undersampled image using the diffusion model from pure noise.
B-DiffPam refers to the diffusion process starting from the halfway with
bilinear as input, lastly U-DiffPam refers to the diffusion process using the
output of U-Net model.

As an alternative to uniform masking, we offer scanning randomly distributed
pixels with an equal number of scanned pixels. We have reconstructed the
selected test images with random masking using $T=1000$ sample steps in the
same diffusion model and compared the results with uniform masking. Table 2
demonstrates the PSNR, SSIM, and LPIPS metrics of undersampled and
reconstructed images with two masking methods. As seen in the table, there are
no significant differences between the metrics of the undersampled images.
Nonetheless, when we reconstructed images, the results of random masking
outperformed the ones of uniform masking in all metrics.

|  PSNR |  SSIM |  LPIPS
---|---|---|---
Undersampled (uniform) |  7.699±2.962 |  0.096±0.062 |  0.854±0.078
Undersampled (random) |  7.760±2.973 |  0.102±0.063 |  1.223±0.103
Reconstructed (uniform) |  23.250±3.261 |  0.758±0.079 |  0.160±0.056
Reconstructed (random) |  28.395±3.714 |  0.881±0.047 |  0.094±0.021
Table 2: The metric comparison between inputs and outputs of different masking
strategies. Figure 5: Visual comparison of random and uniform inpainting
DiffPam results. a) Original (ground truth) image b) undersampled image with
random masking c) reconstructed image from the random mask d) undersampled
image with uniform masking e) reconstructed image from the uniform mask. The
second row illustrates the focus of the upper box, whereas the third row
demonstrates the focus of the lower box. e1) and e2) contain artifacts in the
microvasculature, where c1) and c2) resembles the ground truth.

Upon examining the reconstructed images, we noted the presence of certain
artifacts in microvasculature exclusively in uniformly masked images, a
phenomenon absent in their randomly masked counterparts (Figure 5). Although
we are aware that acquiring signals from random locations in PAM is a more
challenging task than orienting row by row, the benefits of achieving it may
surpass its hardship.

###  4.2 Super Resolution

If acquiring signals in random positions is deemed an impossible task, we
suggest increasing the spatial solution. The corresponding operation might be
a downsampling with Gaussian blurring in the image processing domain. We
experimented with this setting using six images in the test set. We were able
to improve the PSNR by 18.2% and SSIM by 31.7%, and lower the LPIPS by 54.0%
on average. Figure 6 illustrates the ground truth, downsampled and
reconstructed images, and pixel value change across the red line.

Figure 6: The example result of super-resolution from 4x downsampled image. a)
Original (ground truth) image, b) 4x downsampled image, c) reconstructed image
by DiffPam, d) pixel values across the red line in the images. DiffPam is able
to reconstruct the downsampled images using prior knowledge of natural images.

##  5 Discussion

In this study, we demonstrated that the proposed algorithm DiffPam, adeptly
reconstructs undersampled PAM images without the need of a large dataset or
training a deep learning model. The diffusion model which is solely trained
with the ImageNet database of natural images, was able to perform on par with
the U-Net model specifically trained with mice brain microvasculature image
database. Furthermore, for users with an existing trained model, our algorithm
provides a means to enhance results without interfering with the original
model.

As the days go by, the advancing artificial intelligence technology constantly
demands more computing power; however, accessing high computing power remains
a challenge among scientists, especially with limited resources. We value the
removal of these obstacles in the progress of science. Hence, we demonstrated
that the diffusion process can be shortened to reduce the amount of computing
and processing time, with limited or zero accuracy loss.

Since the speed of the scanning axes of PAM imaging may differ greatly,
skipping every few lines in the slower axis is a more convenient way of
speeding up the process of acquiring images. uniform masking of PAM images,
which is commonly used in the literature, is a way of representing this
process. Nonetheless, removing rows or columns periodically from an image,
results in disappearing components in the frequency (Fourier) domain (Figure
7). Thus, some valuable information is lost in the process, making the
lossless recovery impossible. The information loss is the primary cause of the
artifacts in reconstructed images, which are illustrated in the section 4.
These artifacts may cause misinterpretation of medical data, which can be
hazardous. Hence, we highlight that adapting random scanning of pixels
generates more accurate and reliable results than uniform sampling. An
alternative approach can be employing larger spatial resolution, and then
using a super-resolution approach to reconstruct finer details. In this study,
we aimed to demonstrate the limitations of all approaches and leave the
decision to the researchers.

Figure 7: Visual comparison of a) original, b) randomly masked, and c)
uniformly masked microvasculature photoacoustic images and their Fourier
components (d-f). The Fourier components of the uniformly masked image (c)
present unmatched coefficients compared to the original image (d).

The primary result of our study is the potential utilization of the proposed
algorithm by researchers with limited expertise or computational resources in
artificial intelligence. The diffusion model employed in our investigation is
openly accessible and can be substituted with any other proficient diffusion
model. Our source code is accessible on a public GitHub
repository111https://github.com/iremzog/diffpam. For scientists facing
constraints in accessing accelerated computing, the option of fast-tracked,
truncated diffusion steps presents a viable solution for reducing computation
time.

Our study is limited to a single image dataset of narrow scope, namely mice
brain microvasculature. Experiments are made with a relatively small number of
images, due to limited access to accelerated computing resources. Nonetheless,
we hope that our study will shed light on further research in this field.

##  6 Conclusion

In conclusion, our study introduces the DiffPam algorithm, showcasing its
efficiency in reconstructing undersampled PAM images without the need for
extensive datasets or deep learning model training. The diffusion model,
exclusively trained on the natural image database, demonstrated comparable
performance to an in-domain trained neural network. We also address the
escalating demand for computing power in advancing AI technology, emphasizing
the significance of overcoming obstacles in accessing high computational
resources for scientific progress. We propose the reduction of computing time
through shortened diffusion processes without compromising accuracy.
Additionally, our exploration of random or uniform sampling techniques in PAM
imaging underscores the superiority of random sampling in preserving valuable
information. We acknowledge the limitations of our study, confined to a
specific image dataset, and a relatively small sample size due to resource
constraints. However, we anticipate that our findings will inspire further
research in this domain, offering researchers with limited resources a
valuable algorithmic tool for PAM image reconstruction. Our algorithm’s
accessibility is underscored by the availability of source code on a public
GitHub repository, providing a practical way for researchers to implement and
extend our work.

## conflict of interest

The authors declare no conflict of interest.

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#  Formal integration of complete Rota-Baxter Lie algebras

Maxim Goncharov  Sobolev Institute of Mathematics, Acad. Koptyug ave. 4,
Novosibirsk, Russia<EMAIL_ADDRESS>,  Pavel Kolesnikov  Sobolev
Institute of Mathematics, Acad. Koptyug ave. 4, Novosibirsk, Russia
<EMAIL_ADDRESS>,  Yunhe Sheng  Department of Mathematics, Jilin
University, Changchun 130012, Jilin, China<EMAIL_ADDRESS>and  Rong Tang
Department of Mathematics, Jilin University, Changchun 130012, Jilin, China
<EMAIL_ADDRESS>
###### Abstract.

In this paper, first we revisit the formal integration of Lie algebras, which
give rise to braces in some special cases. Then we establish the formal
integration theory for complete Rota-Baxter Lie algebras, that is, we show
that there is a Rota-Baxter group with the underlying group structure given by
the Baker-Campbell-Hausdorff formula, associated to any complete Rota-Baxter
Lie algebra.

###### Key words and phrases:

Rota-Baxter Lie algebra, Rota-Baxter group, Rota-Baxter Hopf algebra, brace

††footnotetext: 2020 Mathematics Subject Classification. 22E60, 16T25, 17B38,
16T05,

##  1\. Introduction

The notion of Rota-Baxter operators on associative algebras was introduced by
G. Baxter [7] and they are applied in the Connes-Kreimer’s algebraic approach
to renormalization of quantum field theory [8], noncommutative symmetric
functions and noncommutative Bohnenblust-Spitzer identities [10], splitting of
operads [3], double Lie algebras [12] and etc. See the book [14] for more
details. In the Lie algebra context, Rota-Baxter operators of weight 0 lead to
the classical Yang-Baxter equation and integrable systems [2, 17], Rota-Baxter
operators of weight 1 are in one-to-one correspondence with solutions of the
modified Yang-Baxter equation [22] and give rise to factorizations of Lie
algebras. Moreover, from a more algebraic approach, a Rota-Baxter operator
naturally gives rise to a pre-Lie algebra or a post-Lie algebra, and play
important roles in mathematical physics [1, 5].

To study the global symmetry of Rota-Baxter Lie algebras, the notion of Rota-
Baxter operators on groups was introduced in [15], and further studied in [6].
One can obtain Rota-Baxter operators of weight 1 on Lie algebras from that on
Lie groups by differentiation. Then in [16], it was shown that every Rota-
Baxter Lie algebra of weight 1 can be integrated to a local Rota-Baxter Lie
group. It remains to be an open problem whether every Rota-Baxter Lie algebra
of weight 1 can be integrated to a Rota-Baxter Lie group. On the other hand,
the notion of Rota-Baxter operators on cocommutative Hopf algebras was given
in [13] such that many classical results still hold in the Hopf algebra level.
Then the notions of post-groups and post-Hopf algebras were introduced in [4]
and [18], which are the split algebraic structures underlying Rota-Baxter
operators on groups and Hopf algebras respectively. In particular, a
commutative post-group is called a pre-group, which is equivalent to a brace.
The notion of a brace was introduced by Rump in [21] to understand the
structure of set-theoretical solutions of the Yang-Baxter equation, which was
proposed by Drinfel’d [9].

In this paper, we use the formal integration approach to study integration of
Rota-Baxter Lie algebras. For this purpose, we first revisit formal
integration of Lie algebras, and characterize group-like elements in the
completion of the universal enveloping algebra of a filtered Lie algebra using
the exponential map explicitly. For a complete Lie algebra
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$, the formal integration
$(\mathfrak{g},*)$ is a group, where the group structure $*$ is given by the
Baker-Campbell-Hausdorff formula. As applications, we show that formal
integration of certain nilpotent Lie algebras give rise to braces naturally,
which lead to possible applications in the Yang-Baxter equation and pre-Lie
algebras [21, 23]. Then we show that the universal enveloping algebra of a
filtered Rota-Baxter Lie algebra is a filtered Rota-Baxter Hopf algebra.
Finally, by completing the filtered Rota-Baxter universal enveloping algebra,
we obtain a Rota-Baxter group. We summarize it in the following diagram:

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In particular, for a complete Rota-Baxter Lie algebra
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$, we obtain a Rota-Baxter
group $(\mathfrak{g},*,\mathfrak{R})$, which serves as the formal integration.

The paper is organized as follows. In Section 2, we recall the notions of
filtered vector spaces, filtered Lie algebras, filtered Hopf algebras and
their relations. In Section 3, we revisit formal integration of Lie algebras,
and establish the connection with braces. In Section 4, first we show that the
universal enveloping algebra of a filtered Rota-Baxter Lie algebra is a
filtered Rota-Baxter Hopf algebra. Then using the trick of completion, we
associate to every filtered Rota-Baxter Lie algebra a Rota-Baxter group.

Throughout the paper, ${\mathbf{k}}$ is an arbitrary field of characteristic
zero.

##  2\. Preliminaries: filtered vector spaces, Lie algebras, and Hopf
algebras.

In this section, we recall the notions of filtered vector spaces, filtered Lie
algebras, filtered Hopf algebras and the relations between filtered Lie
algebras and filtered Hopf algebras.

######  Definition 2.1.

[11] A filtered vector space is a pair $(V,\mathcal{F}_{\bullet}V)$, where $V$
is a vector space and $\mathcal{F}_{\bullet}V$ is a descending filtration of
the vector space $V$ such that

$V=\mathcal{F}_{0}V\supset\mathcal{F}_{1}V\supset\cdots\supset\mathcal{F}_{n}V\supset\cdots.$

Let $(V,\mathcal{F}_{\bullet}V)$ and $(W,\mathcal{F}_{\bullet}W)$ be filtered
vector spaces. A homomorphism
$f:(V,\mathcal{F}_{\bullet}V)\to(W,\mathcal{F}_{\bullet}W)$ is a linear map
$f:V\to W$ such that $f(\mathcal{F}_{n}V)\subset\mathcal{F}_{n}W,~{}n\geq 0.$

Denote by $\mathsf{fVect}$ the category of filtered vector spaces. For
filtered vector spaces $(V,\mathcal{F}_{\bullet}V)$ and
$(W,\mathcal{F}_{\bullet}W)$, there is a filtration on $V\otimes W$ given by

(1) $\displaystyle\mathcal{F}_{n}(V\otimes
W)\coloneqq\sum_{i+j=n}\mathcal{F}_{i}V\otimes\mathcal{F}_{j}W\subset V\otimes
W.$

Thus, $(V\otimes W,\mathcal{F}_{\bullet}(V\otimes W))$ is a filtered vector
space, called the tensor product of filtered vector spaces
$(V,\mathcal{F}_{\bullet}V)$ and $(W,\mathcal{F}_{\bullet}W)$. For the ground
field ${\mathbf{k}}$, we define
$\mathcal{F}_{0}{\mathbf{k}}={\mathbf{k}},\mathcal{F}_{n}{\mathbf{k}}=\\{0\\},~{}n\geq
1$. It follows that $(\mathsf{fVect},\otimes,{\mathbf{k}})$ is a
${\mathbf{k}}$-linear symmetric monoidal category.

######  Proposition 2.2.

Let $V$ be a vector space equipped with two filtrations
$\mathcal{F}_{\bullet}V$ and $\mathcal{F}^{\prime}_{\bullet}V$. Define the
intersection of filtrations $(\mathcal{F}\cap\mathcal{F}^{\prime})_{\bullet}V$
as

$(\mathcal{F}\cap\mathcal{F}^{\prime})_{n}V=\mathcal{F}_{n}V\cap\mathcal{F}^{\prime}_{n}V.$

Then $(V,(\mathcal{F}\cap\mathcal{F}^{\prime})_{\bullet}V)$ is also a filtered
vector space.

######  Definition 2.3.

[19] A filtered Lie algebra is a pair
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$, where $\mathfrak{g}$ is a
Lie algebra and $\mathcal{F}_{\bullet}\mathfrak{g}$ is a descending filtration
of the Lie algebra $\mathfrak{g}$ such that
$\mathfrak{g}=\mathcal{F}_{1}\mathfrak{g}\supset\mathcal{F}_{2}\mathfrak{g}\supset\ldots\supset\mathcal{F}_{n}\mathfrak{g}\supset\ldots$
and

(2)
$[\mathcal{F}_{n}\mathfrak{g},\mathcal{F}_{m}\mathfrak{g}]\subset\mathcal{F}_{m+n}\mathfrak{g}$

for all $m,n\geq 1$.

Obviously, if $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ is a filtered
Lie algebra, then for any $n\geq 1$, $\mathcal{F}_{n}\mathfrak{g}$ is an ideal
in $\mathfrak{g}$.

Now we give two natural examples of filtration structures on a Lie algebra.

######  Example 2.4.

For an arbitrary Lie algebra $\mathfrak{g}$, one can define a trivial
filtration by

$\mathcal{F}_{n}\mathfrak{g}=\mathfrak{g},\quad n\geq 1.$

######  Example 2.5.

By the standard filtration on a Lie algebra $\mathfrak{g}$, we will mean the
following filtration $\mathcal{F}_{\bullet}\mathfrak{g}$ on $\mathfrak{g}$

$\mathcal{F}_{n}\mathfrak{g}=\mathfrak{g}^{n},\quad n\geq 1,$

where $\mathfrak{g}^{n}=[\mathfrak{g},\mathfrak{g}^{n-1}]$ for $n>1$ and
$\mathfrak{g}^{1}=\mathfrak{g}$.

######  Proposition 2.6.

Let ($\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ and
$(\mathfrak{g},\mathcal{F}^{\prime}_{\bullet}\mathfrak{g})$ be two structures
of filtered Lie algebra on the same Lie algebra $\mathfrak{g}$. Then
$(\mathfrak{g},(\mathcal{F}\cap\mathcal{F}^{\prime})_{\bullet}\mathfrak{g})$
is also a filtered Lie algebra.

Note that (2) means that the product
$[\cdot,\cdot]:\mathfrak{g}\otimes\mathfrak{g}\longrightarrow\mathfrak{g}$ is
a homomorphism of filtered vector spaces.

For a Hopf algebra $H=(H,\mu,\Delta,\eta,\epsilon,S)$, we will use the
following notations:

  * •

$\mu:H\otimes H\mapsto H$ is a multiplication,

  * •

$\Delta:H\mapsto H\otimes H$ is a comultiplication,

  * •

$\eta:{\mathbf{k}}\mapsto H$ is a unit,

  * •

$\epsilon:H\mapsto{\mathbf{k}}$ is a counit,

  * •

$S:H\mapsto H$ is the antipode.

Given a coalgebra $(A,\Delta)$, we will use the Sweedler’s notation: for any
$a\in A$

$\Delta(a)=\sum a_{(1)}\otimes a_{(2)}.$

Given a Hopf algebra $H$, by $P(H)$ we will denote the set of primitive
elements of $H$

$P(H)=\\{x\in H|\ \Delta(x)=x\otimes 1+1\otimes x\\}.$

######  Definition 2.7.

A filtered Hopf algebra is a Hopf algebraic object in the category
$(\mathsf{fVect},\otimes,{\mathbf{k}})$. More precisely, it is a Hopf algebra
$(H,\mu,\Delta,\eta,\epsilon,S)$ endowed with a filtration
$\mathcal{F}_{\bullet}H$ such that:

  * (i)

all maps $(\mu,\Delta,\eta,\epsilon,S)$ are homomorphisms of the corresponding
filtered vector spaces.

  * (ii)

$\ker(\epsilon)=\mathcal{F}_{1}H$.

Note that the second condition $\ker(\epsilon)=\mathcal{F}_{1}H$ is equivalent
to the condition that $H/\mathcal{F}_{1}H={\mathbf{k}}$.

The following conclusion is straightforward.

######  Proposition 2.8.

[11, 19] Let $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ be a filtered
Lie algebra. Then the universal enveloping algebra $U(\mathfrak{g})$ is a
filtered Hopf algebra with the filtration
$\mathcal{F}_{\bullet}U(\mathfrak{g})$ defined as

(3) $\displaystyle\mathcal{F}_{0}U(\mathfrak{g})=U(\mathfrak{g}),$ (4)
$\displaystyle\mathcal{F}_{n}U(\mathfrak{g})=span\\{x_{1}\ldots x_{k}|\ k\geq
1,\sum_{i=1}^{k}n_{i}\geq n,\ x_{i}\in\mathcal{F}_{n_{i}}\mathfrak{g}\
\\},\quad\ n\geq 1.$

The filtered Hopf algebra
$(U(\mathfrak{g}),\mathcal{F}_{\bullet}U(\mathfrak{g}))$ is called the
filtered universal enveloping algebra of the filtered Lie algebra
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$.

Consider the universal enveloping algebra of the filtered Lie algebras given
in Example 2.4 and Example 2.5, we have the following examples.

######  Example 2.9.

The filtered universal enveloping algebra
$(U(\mathfrak{g}),\mathcal{F}_{\bullet}U(\mathfrak{g}))$ of the trivial
filtered Lie algebra $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ given
in Example 2.4 is given by

$\displaystyle\mathcal{F}_{0}U(\mathfrak{g})$ $\displaystyle=$ $\displaystyle
U(\mathfrak{g}),$ $\displaystyle\mathcal{F}_{n}U(\mathfrak{g})$
$\displaystyle=$ $\displaystyle\bar{U}(\mathfrak{g})=\ker(\epsilon),\quad\
n\geq 1.$

######  Example 2.10.

The filtered universal enveloping algebra
$(U(\mathfrak{g}),\mathcal{F}_{\bullet}U(\mathfrak{g}))$ of the standard
filtered Lie algebra $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ given
in Example 2.5 is given by

$\displaystyle\mathcal{F}_{0}U(\mathfrak{g})$ $\displaystyle=$ $\displaystyle
U(\mathfrak{g}),$ $\displaystyle\mathcal{F}_{n}U(\mathfrak{g})$
$\displaystyle=$ $\displaystyle span\\{x_{1}\ldots x_{k}|\ k\geq n,\
x_{i}\in\mathfrak{g}\\},\quad n\geq 1.$

We call this filtration the standard filtration on $U(\mathfrak{g})$.

######  Proposition 2.11.

Let $(H,\mu,\Delta,\eta,\epsilon,S,\mathcal{F}_{\bullet}H)$ be a filtered Hopf
algebra. Then the set $P(H)$ of primitive elements is a filtered Lie algebra
with the induced filtration

$\mathcal{F}_{n}P(H)=\mathcal{F}_{n}H\cap P(H),\quad n\geq 1.$

###### Proof.

Since $P(H)\subset\ker(\epsilon)=\mathcal{F}_{1}H$, we have that

$\mathcal{F}_{1}P(H)=\mathcal{F}_{1}H\cap P(H)=\ker(\epsilon)\cap P(H)=P(H).$

For any $n\geq 1$, we have

$\mathcal{F}_{n}P(H)=\mathcal{F}_{n}H\cap P(H)\supset\mathcal{F}_{n+1}H\cap
P(H)=\mathcal{F}_{n+1}P(H).$

Also, for any $i,j\geq 1$,

$[\mathcal{F}_{i}P(H),\mathcal{F}_{j}P(H)]\subset[\mathcal{F}_{i}H,\mathcal{F}_{j}H]\subset\mathcal{F}_{i+j}H.$

Since $[P(H),P(H)]\subset P(H)$, we conclude that
$[\mathcal{F}_{i}P(H),\mathcal{F}_{j}P(H)]\subset\mathcal{F}_{i+j}P(H)$. The
proof is finished. ∎

######  Remark 2.12.

The induced filtration on $\mathfrak{g}$ from the standard filtration
$\mathcal{F}_{\bullet}U(\mathfrak{g})$ on the universal enveloping algebra
$U(\mathfrak{g})$ of a Lie algebra $\mathfrak{g}$ coincides with the standard
filtration on $\mathfrak{g}$.

##  3\. Formal integration of complete Lie algebras and braces

In this section, we revisit formal integrations of complete Lie algebras, in
which the completion of a filtered Hopf algebra is the main ingredient. In
particular, we give the concrete description of group-like elements in the
completion of the universal enveloping algebra of a filtered Lie algebra.
Moreover, we establish the connection between formal integration of complete
Lie algebras and braces.

######  Definition 3.1.

[11] A complete vector space is a filtered vector space
$(V,\mathcal{F}_{\bullet}V)$ such that the natural homomorphism

(5) $\displaystyle\Phi_{V}:V\to\underleftarrow{\lim}V/\mathcal{F}_{n}V$

is a linear isomorphism of vector spaces. Denote by $\mathsf{cVect}$ the
category of complete vector spaces.

Let $(V,\mathcal{F}_{\bullet}V)$ be a filtered vector space. Then the vector
space $\widehat{V}=\underleftarrow{\lim}V/\mathcal{F}_{n}(V)$ is a complete
vector space with the filtration given by

(6)
$\displaystyle\mathcal{F}_{n}(\widehat{V})=\ker\pi_{n},\,\,\pi_{n}:\underleftarrow{\lim}V/\mathcal{F}_{n}(V)\,\rightarrow\,V/\mathcal{F}_{n}(V).$

We call the complete vector space
$\big{(}\widehat{V},\mathcal{F}_{\bullet}\widehat{V}\big{)}$ the completion of
the filtered vector space $(V,\mathcal{F}_{\bullet}V)$. Let
$(V,\mathcal{F}_{\bullet}V)$ and $(W,\mathcal{F}_{\bullet}W)$ be complete
vector spaces. The completion of the filtered vector space $(V\otimes
W,\mathcal{F}_{\bullet}(V\otimes W))$ is called the complete tensor product of
$(V,\mathcal{F}_{\bullet}V)$ and $(W,\mathcal{F}_{\bullet}W)$. We denote the
complete tensor product of $(V,\mathcal{F}_{\bullet}V)$ and
$(W,\mathcal{F}_{\bullet}W)$ by
$(V\hat{\otimes}W,\mathcal{F}_{\bullet}(V\hat{\otimes}W))$.

######  Proposition 3.2.

[11, Proposition 7.3.11] Let $(V,\mathcal{F}_{\bullet}V)$ and
$(W,\mathcal{F}_{\bullet}W)$ be filtered vector spaces. Then the complete
vector space $\widehat{V{\otimes}W}$ is isomorphic to the complete vector
space $\widehat{V}\hat{\otimes}\widehat{W}$.

Moreover, $(\mathsf{cVect},\hat{\otimes},{\mathbf{k}})$ is a
${\mathbf{k}}$-linear symmetric monoidal category. By Proposition 3.2, one can
see that the completion functor $\widehat{(-)}$ is a symmetric monoidal
functor from the category $\mathsf{fVect}$ to the category $\mathsf{cVect}$.

######  Definition 3.3.

A filtered Lie algebra $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ is
called complete if there is an isomorphism of vector spaces
$\mathfrak{g}\cong\underleftarrow{\lim}~{}\mathfrak{g}/\mathcal{F}_{n+1}\mathfrak{g}$.

######  Definition 3.4.

A Lie algebra $\mathfrak{g}$ is called nilpotent of nilindex $n\in\mathbb{N}$,
if $\mathfrak{g}^{n}=0$.

######  Example 3.5.

Let $\mathfrak{g}$ be a nilpotent Lie algebra of nilindex $n$. Then the
standard filtered Lie algebra
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ is a complete Lie algebra.
The isomorphism between $\mathfrak{g}$ and
$\varprojlim\mathfrak{g}/\mathcal{F}_{n+1}\mathfrak{g}$ is given by

(7) $x\mapsto(\pi_{1}(x),\pi_{2}(x),\ldots,\pi_{n-1}(x),x,x,\ldots,x,\ldots),$

where
$\pi_{i}:\mathfrak{g}\rightarrow\mathfrak{g}/\mathcal{F}_{i+1}\mathfrak{g}$ is
the natural Lie algebra homomorphism.

Let $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ be a filtered Lie
algebra and
$(U(\mathfrak{g}),\Delta,\epsilon,S,\mathcal{F}_{\bullet}U(\mathfrak{g}))$ be
the filtered universal enveloping algebra of
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$. Let
$\pi_{n}:U(\mathfrak{g})\rightarrow
U(\mathfrak{g})/\mathcal{F}_{n+1}U(\mathfrak{g})$ ($n\geq-1$) be the natural
homomorphism. Let
$p_{n}:U(\mathfrak{g})/\mathcal{F}_{n+1}U(\mathfrak{g})\rightarrow
U(\mathfrak{g})/\mathcal{F}_{n}U(\mathfrak{g})$ ($n\geq 1$) be the map defined
by $p_{n}\circ\pi_{n}=\pi_{n-1}$, that is

$p_{n}(x+\mathcal{F}_{n+1}U(\mathfrak{g}))=x+\mathcal{F}_{n}U(\mathfrak{g}).$

Then $(U(\mathfrak{g})/\mathcal{F}_{n}U(\mathfrak{g}),p_{n})$ is an inverse
system of algebras.

Let
$\widehat{U}(\mathfrak{g})=\varprojlim(U(\mathfrak{g})/\mathcal{F}_{n}(U(\mathfrak{g}))$
be the completion of the algebra $U(\mathfrak{g})$. As an algebra,
$\widehat{U}(\mathfrak{g})$ is a subalgebra in $\prod\limits_{n\geq
0}U(\mathfrak{g})/\mathcal{F}_{n+1}U(\mathfrak{g})$, the direct product of
algebras $\left(U(\mathfrak{g})/\mathcal{F}_{n+1}U(\mathfrak{g})\right)_{n\geq
0}$ with component-wise operations. As a vector space,
$\widehat{U}(\mathfrak{g})$ consists of elements

$(x_{0},\ldots,x_{n},\ldots)\in\prod\limits_{n\geq
1}U(\mathfrak{g})/\mathcal{F}_{n+1}U(\mathfrak{g})$

satisfying $p_{n}(x_{n})=x_{n-1}$. Define a filtration on
$\widehat{U}(\mathfrak{g})$ as

(8) $\displaystyle\mathcal{F}_{n}(\widehat{U}(\mathfrak{g}))=\ker\pi_{n-1},\
n\geq 0.$

Consider the tensor product
$\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g})$ as a filtered
vector space with the filtration defined by (1). Let
$P_{n}:\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g})/\mathcal{F}_{n+1}(\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g}))\rightarrow\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g})/\mathcal{F}_{n}(\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g}))$
be the natural homomorphism. Then
$(\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g}),\mathcal{F}_{\bullet}(\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g})),P_{n})$
is an inverse system of vector spaces. Define the complete tensor product
$\widehat{\otimes}$ as:

$\widehat{U}(\mathfrak{g})\widehat{\otimes}\widehat{U}(\mathfrak{g})=\varprojlim(\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g})/\mathcal{F}_{n+1}(\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g}))).$

For any $n\geq 0$, define a map

$\Delta_{n}:U(\mathfrak{g})/\mathcal{F}_{n+1}U(\mathfrak{g})\rightarrow
U(\mathfrak{g})\otimes
U(\mathfrak{g})/\mathcal{F}_{n+1}\left(U(\mathfrak{g})\otimes
U(\mathfrak{g})\right)$

as

$\Delta_{n}\circ\pi_{n}=(\pi_{n}\otimes\pi_{n})\circ\Delta.$

In other words, for any $h\in U(\mathfrak{g})$:
$\Delta_{n}(\pi_{n}(h))=\pi_{n}(h_{(1)})\otimes\pi_{n}(h_{(2)})$. It is
straightforward to check that $\Delta_{n-1}\circ p_{n}=P_{n}\circ\Delta_{n}$.
In this case, we can define a map
$\widehat{\Delta}:\hat{U}(\mathfrak{g})\rightarrow\widehat{U}(\mathfrak{g})\hat{\otimes}\widehat{U}(\mathfrak{g})$:
for any $(x_{0},\ldots,x_{n},\ldots)\in\widehat{U}(\mathfrak{g})$ we put

$\widehat{\Delta}((x_{0},\ldots,x_{n},\ldots))=(\Delta_{0}(x_{0}),\ldots,\Delta_{n}(x_{n}),\ldots)\in\widehat{U}(\mathfrak{g})\hat{\otimes}\widehat{U}(\mathfrak{g}).$

Similarly, we can define completions of the other Hopf algebra operations
defined on $U(\mathfrak{g})$ and consider $\widehat{U}(\mathfrak{g})$ as a
complete Hopf algebra.

Given
$x=(0,x_{1},\ldots,x_{n},\ldots)\in\mathcal{F}_{1}\widehat{U}(\mathfrak{g})$,
we can define the exponential map
$\exp:\mathcal{F}_{1}\widehat{U}(\mathfrak{g})\rightarrow\widehat{U}(\mathfrak{g})$
as

(9)
$\exp(x)=\left(1,1+x_{1},1+x_{2}+\frac{x_{2}^{2}}{2},\ldots,\sum\limits_{i=0}^{n}\frac{x_{n}^{i}}{i!},\ldots\right)=1+x+\frac{x^{2}}{2}+\ldots+\frac{x^{n}}{n!}+\ldots.$

######  Definition 3.6.

By $G$, we will denote the group of group-like elements of
$\widehat{U}(\mathfrak{g})$.

Since $\mathfrak{g}$ is naturally embedded into $U(\mathfrak{g})$ and
$\mathcal{F}_{n}\mathfrak{g}\subset\mathcal{F}_{n}U(\mathfrak{g})$, we may
consider the completion
$\widehat{\mathfrak{g}}=\varprojlim(\mathfrak{g}/\mathcal{F}_{n}\mathfrak{g})$
as a subspace in $\mathcal{F}_{1}\widehat{U}(\mathfrak{g})$.

######  Proposition 3.7.

$g\in G$ if and only if $g=\exp(x)$ for some $x\in\widehat{\mathfrak{g}}$.

###### Proof.

Let $g=\exp(x)$ for some $x\in\widehat{\mathfrak{g}}$. In order to prove that
$g\in G$, it is enough to note that for any $x\in\mathfrak{g}$,

$\Delta\left(1+x+\ldots+\frac{x^{n}}{n!}\right)\equiv\left(1+x+\ldots+\frac{x^{n}}{n!}\right)\otimes\left(1+x+\ldots+\frac{x^{n}}{n!}\right)\
\textrm{mod}\ \mathcal{F}_{n+1}(U(\mathfrak{g})\otimes U(\mathfrak{g})).$

Let $g\in G$, $g=(g_{0},g_{1},\ldots,g_{n},\ldots)$. Since $g_{0}\in
U(\mathfrak{g})/\mathcal{F}_{1}(U(\mathfrak{g}))\cong{\mathbf{k}}$, we may
assume that $g_{0}=\pi_{0}(\alpha)$ for some $\alpha\in{\mathbf{k}}$. Without
loss of generality, we will write $\alpha$ instead of $\pi_{n}(\alpha)$ for
all $n\geq 0$. Since $g\in G$, then $\widehat{\epsilon}(g)=1$ and
$g_{i}=1+t_{i}$, where $t_{i}\in\pi_{i}(\mathcal{F}_{1}U(\mathfrak{g}))$.

Let $g_{1}=1+x_{1}$, where $x_{1}=\pi_{1}(t_{1})$, $t_{1}\in\ker(\pi_{0})$. By
the definition of the filtration $\mathcal{F}_{n}U(\mathfrak{g})$ in
Proposition 2.8, we may assume that $t_{1}\in\mathfrak{g}$.

Consider $g_{2}$. We can present
$g_{2}=1+\pi_{2}(t_{1})+\frac{1}{2}\pi_{2}(t_{1})^{2}+y_{2}$, where
$y_{2}=\pi_{2}(z_{2})$ for some
$z_{2}\in\ker\pi_{1}=\mathcal{F}_{2}U(\mathfrak{g})$. Note that
$p_{2}(\frac{1}{2}\pi_{2}(t_{1}^{2})+y_{2})=0$. Then

$\displaystyle\Delta_{2}(g_{2})$ $\displaystyle=$ $\displaystyle 1\otimes
1+\pi_{2}(t_{1})\otimes 1+1\otimes\pi_{2}(t_{1})$
$\displaystyle+\frac{1}{2}(\pi_{2}(t_{1})^{2}\otimes
1+2\pi_{2}(t_{1})\otimes\pi_{2}(t_{1})+1\otimes\pi_{2}(t_{1})^{2})+\Delta_{2}(y_{2}).$

On the other hand, since $t_{1}\otimes z_{2}$ and $z_{2}\otimes z_{2}$ are in
$\mathcal{F}_{3}(\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g}))$,
we have

$\displaystyle g_{2}\otimes g_{2}$ $\displaystyle=$ $\displaystyle 1\otimes
1+t_{1}\otimes 1+1\otimes t_{1}$ $\displaystyle+\frac{1}{2}(t_{1}^{2}\otimes
1+1\otimes t_{1}^{2}+2t_{1}\otimes t_{1})+1\otimes z_{2}+z_{2}\otimes
1+\mathcal{F}_{3}(\widehat{U}(\mathfrak{g})\otimes\widehat{U}(\mathfrak{g})).$

Thus, $\Delta_{2}(y_{2})=1\otimes y_{2}+y_{2}\otimes 1$, and we may take
$z_{2}\in\mathcal{F}_{2}\mathfrak{g}$. Note that
$\pi_{2}((t_{1}+z_{2})^{2})=\pi_{2}(t_{1}^{2})$. Thus, we can take
$t_{2}=t_{1}+z_{2}\in\mathfrak{g}$ and get that
$g_{2}=1+x_{2}+\frac{1}{2}x_{2}^{2}$, where $x_{2}=\pi_{2}(t_{2})$ satisfies
$p_{2}(x_{2})=x_{1}$.

Now we can do the same trick with the other components of $\mathfrak{g}$ and
obtain that

$g=(1,1+x_{1},1+x_{2}+\frac{x_{2}^{2}}{2},\ldots,\sum\limits_{i=0}^{n}\frac{x_{n}^{i}}{i!},\ldots)=\exp(x),$

where $x=(0,x_{1},\ldots,x_{n},\ldots)\in\widehat{\mathfrak{g}}$. ∎

Now we have two bijections: $\exp:\widehat{\mathfrak{g}}\rightarrow G$ and
$\log=\exp^{-1}:G\rightarrow\widehat{\mathfrak{g}}$. Consequently, the Baker-
Campbell-Hausdorff formula gives rise to a group structure on the set
$\widehat{\mathfrak{g}},$ which is the main ingredient in the study of formal
integration of Rota-Baxter Lie algebras in the next section.

######  Proposition 3.8.

Let $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ be a filtered Lie
algebra and $\widehat{\mathfrak{g}}$ be its completion. Then there is a group
structure
$*:\widehat{\mathfrak{g}}\times\widehat{\mathfrak{g}}\,\rightarrow\,\widehat{\mathfrak{g}}$
on $\widehat{\mathfrak{g}}$ which is given by

(10) $\displaystyle
x*y\triangleq\mathsf{BCH}(x,y)=\log\big{(}\exp(x)\cdot\exp(y)\big{)}\ \
\text{for all }x,y\in\widehat{\mathfrak{g}}.$

Furthermore, both $\exp$ and $\log$ are group isomorphisms.

######  Remark 3.9.

If $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ is a complete Lie
algebra, the group $(\mathfrak{g},*)$ is viewed as its formal integration.

######  Example 3.10.

Let $\mathfrak{g}$ be the 3-dimensional Heisenberg Lie algebra with the basis
$x,y,z$ satisfying

$[x,y]=z,\quad[z,x]=[z,y]=0.$

We have that $\mathfrak{g}^{3}=0$ and the standard filtration
$\mathcal{F}_{1}\mathfrak{g}=\mathfrak{g}$,
$\mathcal{F}_{2}\mathfrak{g}=span\\{z\\}$, $\mathcal{F}_{3}\mathfrak{g}=0$. So
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ is a complete Lie algebra.
Since $\mathfrak{g}^{3}=0$, the group structure $*$ on $\mathfrak{g}$ is given
by

(11) $x*y=x+y+[x,y].$

At the end of this section, we give an application of formal integration of
complete Lie algebras, and establish the connection with braces.

######  Definition 3.11.

A brace is a triple $(B,+,*)$, where $(B,+)$ is an abelian group and $(B,*)$
is a group such that the following compatibility condition holds:

(12) $x*(y+z)+x=x*y+x*z,\quad\forall x,y,z\in B.$

######  Theorem 3.12.

Let $\mathfrak{g}$ be a nilpotent Lie algebra. If $\mathfrak{g}^{3}=0,$ then
$(\mathfrak{g},+,*)$ is a brace, where $+$ is natural abelian group structure
on the vector space $\mathfrak{g}$ and the group structure $*$ is given by
(10).

###### Proof.

We need to check that for all $x,y,z\in\mathfrak{g}$, the equality (12) holds.
We have

$\displaystyle x*(y+z)+x$ $\displaystyle=$ $\displaystyle
x+y+z+\frac{1}{2}[x,y+z]+x$ $\displaystyle=$
$\displaystyle(x+y+\frac{1}{2}[x,y])+(x+z+\frac{1}{2}[x,z])$ $\displaystyle=$
$\displaystyle x*y+x*z.$

The statement is proved. ∎

######  Example 3.13.

Let $\mathfrak{g}$ be the 3-dimensional Heisenberg Lie algebra given in
Example 3.10. Then $(\mathfrak{g},+,*)$ is a brace, where $+$ is the natural
abelian group structure on the vector space $\mathfrak{g}$ and the group
structure $*$ is given by (11).

##  4\. Formal integration of a complete Rota-Baxter Lie algebra

In this section first we show that the universal enveloping algebra of a
filtered Rota-Baxter Lie algebra is a filtered Rota-Baxter Hopf algebra. Then
by completing the Rota-Baxter operator in the filtered Rota-Baxter universal
enveloping algebra, we obtain a Rota-Baxter group. In particular, for a
complete Rota-Baxter Lie algebra
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$, we obtain a Rota-Baxter
group $(\mathfrak{g},*,\mathfrak{R})$, which serves as the formal integration.

######  Definition 4.1.

A Rota-Baxter Lie algebra is a pair $((\mathfrak{g},[\cdot,\cdot]),R)$, where
$(\mathfrak{g},[\cdot,\cdot])$ is a Lie algebra and
$R:\mathfrak{g}\longrightarrow\mathfrak{g}$ is a Rota-Baxter operator of
weight $1$, i.e.

(13) $[R(x),R(y)]=R\big{(}[R(x),y]+[x,R(y)]+[x,y]\big{)},\quad\forall
x,y\in\mathfrak{g}.$

######  Definition 4.2.

[15, Definition 2.1] A Rota-Baxter group is a pair $(G,\mathfrak{R})$, where
$G$ is a group and $\mathfrak{R}:G\rightarrow G$ is a map satisfying the
following equality:

(14)
$\mathfrak{R}(g)\mathfrak{R}(h)=\mathfrak{R}(g\mathfrak{R}(g)h\mathfrak{R}(g)^{-1}),\quad\forall
g,h\in G.$

######  Definition 4.3.

[13] A Rota-Baxter Hopf algebra is a pair $(H,\mathcal{R})$, where $H$ is a
cocommutative Hopf algebra and $\mathcal{R}:H\to H$ is a coalgebra
homomorphism satisfying the following equality:

(15)
$\displaystyle\mathcal{R}(h)\mathcal{R}(t)=\mathcal{R}\Big{(}h_{(1)}\mathcal{R}(h_{(2)})tS(\mathcal{R}(h_{(3)}))\Big{)},\,\,\forall
h,t\in H.$

Now we introduce the notions of filtered Rota-Baxter Lie algebras and filtered
Rota-Baxter Hopf algebras.

######  Definition 4.4.

A filtered Rota-Baxter Lie algebra is a triple
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$, where
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ is a filtered Lie algebra,
$R$ is a Rota-Baxter operator of weight 1 on $\mathfrak{g}$ and

$R(\mathcal{F}_{n}\mathfrak{g})\subset\mathcal{F}_{n}\mathfrak{g},\quad\forall
n\geq 1.$

A filtered Rota-Baxter Lie algebra
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$ is called complete if the
underlying filtered Lie algebra
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g})$ is complete.

######  Definition 4.5.

A filtered Rota-Baxter Hopf algebra is a triple
$(H,\mathcal{F}_{\bullet}H,\mathcal{R})$, where $(H,\mathcal{F}_{\bullet}H)$
is a filtered Hopf algebra and $\mathcal{R}$ is a Rota-Baxter operator on $H$
satisfying

$\mathcal{R}(\mathcal{F}_{n}H)\subset\mathcal{F}_{n}H,\quad\forall n\geq 0.$

######  Proposition 4.6.

Let $(H,\mathcal{F}_{\bullet}H,\mathcal{R})$ be a filtered Rota-Baxter Hopf
algebra. Then $(P(H),\mathcal{F}_{\bullet}H\cap P(H),\mathcal{R}|_{P(H)})$ is
a filtered Rota-Baxter Lie algebra.

###### Proof.

From [13] it follows that $\mathcal{R}(P(H))\subset P(H)$ and
$(P(H),\mathcal{R}|_{P(H)})$ is a Rota-Baxter Lie algebra. Since
$\mathcal{R}(\mathcal{F}_{n}H)\subset\mathcal{F}_{n}H$, we have

$\mathcal{R}(\mathcal{F}_{n}P(H))=\mathcal{R}(\mathcal{F}_{n}H\cap
P(H))\subset\mathcal{F}_{n}H\cap P(H)=\mathcal{F}_{n}P(H),$

as required. ∎

The connection between Rota-Baxter operators of weight 1 on a Lie algebra
$\mathfrak{g}$ and Rota-Baxter operators on the universal enveloping algebra
was obtained in [13]. More precisely, if $\mathfrak{g}$ is a Lie algebra and
$R:\mathfrak{g}\longrightarrow\mathfrak{g}$ is a Rota-Baxter operator of
weight 1, then a map $\mathcal{R}:U(\mathfrak{g})\longrightarrow
U(\mathfrak{g})$ defined as

(16) $\displaystyle\mathcal{R}(1)$ $\displaystyle=$ $\displaystyle 1,$ (17)
$\displaystyle\mathcal{R}(x)$ $\displaystyle=$ $\displaystyle R(x),\ \
x\in\mathfrak{g},$ (18) $\displaystyle\mathcal{R}(xh)$ $\displaystyle=$
$\displaystyle R(x)\mathcal{R}(h)-\mathcal{R}([R(x),h]),\ \ x\in\mathfrak{g},\
h\in U(\mathfrak{g})$

is well-defined and satisfy (15) (that is, $(U(\mathfrak{g}),\mathcal{R})$ is
a Rota-Baxter Hopf algebra).

Now we generalize this result to filtered setting.

######  Theorem 4.7.

Let $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$ be a filtered Rota-
Baxter Lie algebra and
$(U(\mathfrak{g}),\mathcal{F}_{\bullet}U(\mathfrak{g}))$ be the filtered
universal enveloping algebra given in Proposition 2.8. Then
$(U(\mathfrak{g}),\mathcal{F}_{\bullet}U(\mathfrak{g}),\mathcal{R})$ is a
filtered Rota-Baxter Hopf algebra.

###### Proof.

We need to prove that
$\mathcal{R}(\mathcal{F}_{n}U(\mathfrak{g}))\subset\mathcal{F}_{n}U(\mathfrak{g})$
for any $n\geq 1$.

Let $x_{1},\ldots,x_{m}\in\mathfrak{g}$ and $x_{1}\ldots
x_{m}\in\mathcal{F}_{n}U(\mathfrak{g})$. If $m=1$, then
$x_{1}\in\mathcal{F}_{n}\mathfrak{g}$. From (17) we have that

$\mathcal{R}(x_{1})=R(x_{1})\in\mathcal{F}_{n}\mathfrak{g}\subset\mathcal{F}_{n}U(\mathfrak{g}).$

We will use induction on $m$. Consider $x_{1}\ldots x_{m}$ with
$x_{i}\in\mathcal{F}_{k_{i}}\mathfrak{g}$ and $\sum\limits_{i}k_{i}\geq n$.
Then by (18)

$\displaystyle\mathcal{R}(x_{1}x_{2}\ldots x_{m})$ $\displaystyle=$
$\displaystyle R(x_{1})\mathcal{R}\big{(}x_{2}\ldots
x_{m})-\mathcal{R}([R(x_{1}),x_{2}\ldots x_{m}]\big{)}$ $\displaystyle=$
$\displaystyle R(x_{1})\mathcal{R}(x_{2}\ldots
x_{n})-\sum\limits_{i=2}^{m}\mathcal{R}\big{(}x_{2}\ldots
x_{i-1}[R(x_{1}),x_{i}]x_{i+1}\ldots x_{m}\big{)}.$

By the induction hypothesis and $R$ preserves the filtration on
$\mathfrak{g}$, we have

$R(x_{1})\mathcal{R}(x_{2}\ldots x_{m})\in\mathcal{F}_{n}U(\mathfrak{g}).$

Furthermore, it is obvious that for any $2\leq i\leq m$,

$\mathcal{R}(x_{2}\ldots x_{i-1}[R(x_{1}),x_{i}]x_{i+1}\ldots
x_{m})\in\mathcal{F}_{n}U(\mathfrak{g}).$

Thus, $\mathcal{R}(x_{1}\ldots x_{m})\in\mathcal{F}_{n}U(\mathfrak{g})$. ∎

######  Definition 4.8.

Let $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$ be a filtered Rota-
Baxter Lie algebra. Then the filtered Rota-Baxter Hopf algebra
$(U(\mathfrak{g}),\mathcal{F}_{\bullet}U(\mathfrak{g}),\mathcal{R})$ with the
filtration defined by (3)-(4) and the Rota-Baxter operator $\mathcal{R}$
defined by (16)-(18) is called the filtered Rota-Baxter universal enveloping
algebra.

Let $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$ be a filtered Rota-
Baxter Lie algebra,
$(U(\mathfrak{g}),\mathcal{F}_{\bullet}U(\mathfrak{g}),\mathcal{R})$ be the
filtered Rota-Baxter universal enveloping algebra of
$(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$ given in Theorem 4.7, and
$\widehat{U}(\mathfrak{g})$ be the completion of the filtered universal
enveloping algebra $(U(\mathfrak{g}),\mathcal{F}_{\bullet}U(\mathfrak{g}))$.

Let
$\mathcal{R}_{n}:U(\mathfrak{g})/\mathcal{F}_{n+1}U(\mathfrak{g})\rightarrow
U(\mathfrak{g})/\mathcal{F}_{n+1}U(\mathfrak{g})$ be the induced map

$\mathcal{R}_{n}(x+\mathcal{F}_{n+1}U(\mathfrak{g}))=\mathcal{R}_{n}(\pi_{n}(x))=\pi_{n}(\mathcal{R}(x))=\mathcal{R}(x)+\mathcal{F}_{n+1}U(\mathfrak{g}).$

Obviously, $p_{n}\circ\mathcal{R}_{n}=\mathcal{R}_{n-1}\circ p_{n}$. Thus, we
can define a map

$\widehat{\mathcal{R}}=\varprojlim\mathcal{R}_{n}:\widehat{U}(\mathfrak{g})\rightarrow\widehat{U}(\mathfrak{g})$

as

$\widehat{\mathcal{R}}(x_{0},\ldots,x_{n},\ldots)=(\mathcal{R}_{0}(x_{0}),\ldots,\mathcal{R}_{n}(x_{n}),\ldots)\in\varprojlim(U(\mathfrak{g})/\mathcal{F}_{n+1}(U(\mathfrak{g})).$

Now we are ready to give the main result of the paper.

######  Theorem 4.9.

Let $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$ be a filtered Rota-
Baxter Lie algebra. Then the map
$\mathfrak{R}:\widehat{\mathfrak{g}}\rightarrow\widehat{\mathfrak{g}}$ defined
as

(19) $\mathfrak{R}(x)=\log(\widehat{\mathcal{R}}(\exp(x))),\quad\forall
x\in\widehat{\mathfrak{g}}$

is a Rota-Baxter operator on the group $(\widehat{\mathfrak{g}},*)$ given in
Proposition 3.8.

In particular, if $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$ is a
complete Rota-Baxter Lie algebra, then $(\mathfrak{g},*,\mathfrak{R})$ with
the map $\mathfrak{R}$ defined by (19) for all $x\in\mathfrak{g}$ is a Rota-
Baxter group, which is considered as the formal integration of the complete
Rota-Baxter Lie algebra $(\mathfrak{g},\mathcal{F}_{\bullet}\mathfrak{g},R)$.

To prove this theorem, we need the following results, which are interesting on
their own.

######  Proposition 4.10.

$\widehat{\mathcal{R}}$ is a Rota-Baxter operator on the complete Hopf algebra
$\widehat{U}(\mathfrak{g})$.

###### Proof.

Indeed, for any $n\geq 0$ and $x,y\in U(\mathfrak{g})$

$\mathcal{R}_{n}\left(\pi_{n}(x))\mathcal{R}_{n}(\pi_{n}(y))\right)=\pi_{n}((\mathcal{R}(x))\pi_{n}(\mathcal{R}(y))=\pi_{n}(\mathcal{R}(x)\mathcal{R}(y)).$

On the other hand,

$\displaystyle\mathcal{R}_{n}((\pi_{n}(x))_{(1)}\mathcal{R}_{n}(\pi_{n}(x))_{(2)}\pi_{n}(y)S(\mathcal{R}_{n}(\pi_{n}(x))_{(3)}))$
$\displaystyle=$
$\displaystyle\mathcal{R}_{n}(\pi_{n}(x_{(1)}\mathcal{R}(x_{(2)})yS(\mathcal{R}(x_{(3)}))$
$\displaystyle=$
$\displaystyle\pi_{n}(\mathcal{R}(x_{(1)}\mathcal{R}(x_{(2)})yS(\mathcal{R}(x_{(3)})))$
$\displaystyle=$ $\displaystyle\pi_{n}(\mathcal{R}(x)\mathcal{R}(y)).$

Thus, $\widehat{\mathcal{R}}$ satisfies (15). ∎

######  Proposition 4.11.

$\widehat{\mathcal{R}}(G)\subset G$, and the restriction
$\widehat{\mathcal{R}}|_{G}:G\rightarrow G$ is a Rota-Baxter operator on the
group $G$.

The proof of Theorem 4.9: By Proposition 4.10 and 4.11, $\mathfrak{R}$ is a
Rota-Baxter operator on the group $G$. Since $\log$ and $\exp$ are group
isomorphism between $G$ and $(\widehat{\mathfrak{g}},*)$, we have

$\displaystyle\mathfrak{R}(x)*\mathfrak{R}(y)$ $\displaystyle=$
$\displaystyle\log(\widehat{\mathcal{R}}(\exp(x)))*\log(\widehat{\mathcal{R}}(\exp(y)))$
$\displaystyle=$
$\displaystyle\log\Big{(}\widehat{\mathcal{R}}(\exp(x))\cdot\widehat{\mathcal{R}}(\exp(y))\Big{)}$
$\displaystyle=$
$\displaystyle\log\Big{(}\widehat{\mathcal{R}}\big{(}\exp(x)\cdot\widehat{\mathcal{R}}(\exp(x))\cdot\exp(y)\cdot\widehat{\mathcal{R}}(\exp(x))^{-1}\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\log\Big{(}\widehat{\mathcal{R}}\big{(}\exp(x)\cdot\exp(\mathfrak{R}(x))\cdot\exp(y)\cdot(\exp(\mathfrak{R}(x))^{-1})\big{)}\Big{)}$
$\displaystyle=$
$\displaystyle\log\Big{(}\widehat{\mathcal{R}}(\exp(x*\mathfrak{R}(x)*y*(\mathfrak{R}(x))^{-1}))\Big{)}$
$\displaystyle=$
$\displaystyle\mathfrak{R}(x*\mathfrak{R}(x)*y*(\mathfrak{R}(x))^{-1}),$

which implies that $\mathfrak{R}$ is a Rota-Baxter operator on the group
$(\widehat{\mathfrak{g}},*)$. The proof is completed.

We can find several components of
$\mathfrak{R}:\mathfrak{g}\rightarrow\mathfrak{g}$. Let
$x=(x_{1},x_{2}\ldots,x_{n},\ldots)\in\widehat{\mathfrak{g}}$. Then

$\displaystyle\mathfrak{R}(x)$ $\displaystyle=$
$\displaystyle\log(\widehat{\mathcal{R}}(\exp(x)))$ $\displaystyle=$
$\displaystyle\log((\widehat{\mathcal{R}}(1,1+x_{1},1+x_{2}+\frac{x_{2}^{2}}{2!},\ldots)$
$\displaystyle=$
$\displaystyle\log(\mathcal{R}(1),\mathcal{R}(1)+\mathcal{R}(x_{1}),\mathcal{R}(1)+\mathcal{R}(x_{2})+\mathcal{R}(\frac{x_{2}^{2}}{2})),\ldots$
$\displaystyle=$
$\displaystyle\log((1,1+R(x_{1}),1+R(x_{2})+\frac{1}{2}(R(x_{2})R(x_{2})-R([R(x_{2}),x_{2}]),\ldots).$

Thus,

(20)
$\mathfrak{R}((x_{1},\ldots,x_{n},\ldots))=(R(x_{1}),R(x_{2})-R([R(x_{2}),x_{2}]),\ldots)$

From (20) and (7) we have the following

######  Proposition 4.12.

Let $\mathfrak{g}$ be a nilpotent Lie algebra of nilindex $3$,
$\mathcal{F}_{\bullet}\mathfrak{g}$ be the standard filtration on
$\mathfrak{g}$, and $R:\mathfrak{g}\rightarrow\mathfrak{g}$ be a Rota-Baxter
operator preserving the filtration. Then the Rota-Baxter operator
$\mathfrak{R}$ on the group $(\mathfrak{g},*)$ is given by

$\mathfrak{R}(x)=R(x)-\frac{1}{2}R([R(x),x]).$

Note that Proposition 4.12 was independently obtained in [20], using a
different technique.

######  Example 4.13.

Consider the 3-dimensional Heisenberg Lie algebra given in Example 3.10.
Consider an operator $R$:

$R(x)=y,\ R(y)=x,\ R(z)=-z.$

It is straightforward to check that $R$ is a Rota-Baxter operator of weight 1
on $\mathfrak{g}$. Obviously, $R$ preserves the filtration, that is,
$(\mathfrak{g},R,\mathcal{F}_{\bullet}\mathfrak{g})$ is a complete Rota-Baxter
Lie algebra.

From Proposition 4.12, the map
$\mathfrak{R}:\mathfrak{g}\rightarrow\mathfrak{g}$ defined as

$\mathfrak{R}(\alpha x+\beta y+\gamma z)=\beta x+\alpha
y+\frac{1}{2}(-\alpha^{2}+\beta^{2}-2\gamma)z$

for all $\alpha,\beta,\gamma\in{\mathbf{k}}$, is a Rota-Baxter operator on the
group $(\mathfrak{g},*)$.

Acknowledgments. The first and the second authors were supported by RAS
Fundamental Research Program (FWNF-2022-0002), and the fourth author is
supported by NSFC (12371029).

Declaration of interests.  The authors have no conflicts of interest to
disclose.

Data availability.  Data sharing is not applicable to this article as no new
data were created or analyzed in this study.

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