# Fingerprint Based mmWave Positioning System Aided by Reconfigurable
Intelligent Surface

Tuo Wu, Cunhua Pan, Yijin Pan, Hong Ren, Maged Elkashlan, and Cheng-Xiang
Wang, _Fellow IEEE_ (Corresponding author: Cunhua Pan). T. Wu and M.Elkashlan
are with the School of Electronic Engineering and Computer Science at Queen
Mary University of London, London E1 4NS, U.K. (Email:{tuo.wu,
maged.elkashlan}@qmul.ac.uk). C. Pan, Y. Pan and H. Ren are with the National
Mobile Communications Research Laboratory, Southeast University, Nanjing
210096, China. C.-X. Wang is with the National Mobile Communications Research
Laboratory, Southeast University, Nanjing 210096, China, and also with the
Purple Mountain Laboratories, Nanjing, 211111, China. (Email: {cpan, panyj,
hren, chxwang}@seu.edu.cn).

###### Abstract

Reconfigurable intelligent surface (RIS) is a promising technique for
millimeter wave (mmWave) positioning systems. In this paper, we consider
multiple mobile users (MUs) positioning problem in the multiple-input
multiple-output (MIMO) time-division duplex (TDD) mmWave systems aided by the
RIS. We derive the expression for the space-time channel response vector
(STCRV) as a novel type of fingerprint. The STCRV consists of the multipath
channel characteristics, e.g., time delay and angle of arrival (AOA), which is
related to the position of the MU. By using the STCRV as input, we propose a
novel residual convolution network regression (RCNR) learning algorithm to
output the estimated three-dimensional (3D) position of the MU. Specifically,
the RCNR learninng algorithm includes a data processing block to process the
input STCRV, a normal convolution block to extract the features of STCRV, four
residual convolution blocks to further extract the features and protect the
integrity of the features, and a regression block to estimate the 3D position.
Extensive simulation results are also presented to demonstrate that the
proposed RCNR learning algorithm outperforms the traditional convolution
neural network (CNN).

###### Index Terms:

Reconfigurable intelligent surface (RIS), intelligent reflecting surface,
positioning, radio localization.

##  I Introduction

Localization-related industries demand high levels of localization accuracy
[1], e.g., mobile user sensing [2]. It is worth pointing out that prevalent
global positioning system (GPS) localization accuracy, even in ideal
conditions, is approximately 5 meters, which falls short of meeting the
stringent requirements of location-sensitive applications. Hence, wireless
positioning systems in the millimeter wave (mmWave) band were advocated by
some researchers as a way to improve the positioning performance [3]. However,
due to the sensitivity of mmWave signals to blockages, high-precision
positioning is difficult to maintain [4].

Reconfigurable intelligent surface (RIS) is an emerging technique for mmWave
positioning systems with several advantages [5, 6, 7, 8, 9, 10]. First, the
RIS can reconstruct a new line-of-sight (LoS) communication link if the direct
link was blocked by obstacles [7]. Second, the RIS provides reliable and high-
precision estimation with low energy consumption [9]. Finally, the RIS saves
hardware costs when deploying a wireless positioning reference compared with
the access point (AP) [8]. Thus, wireless positioning algorithms aided by the
RIS is a promising enabler for sixth-generation (6G) wireless systems [10].

Currently, wireless positioning algorithms aided by the RIS have been studied
by some researchers, including two-step positioning algorithms and fingerprint
based algorithms [11, 12]. For two-step positioning algorithms, the channel
parameters, e.g., time delay and angle of arrival (AOA), are estimated at the
first step. At the second step, the channel parameters can be used to derive
the three-dimensional (3D) position of the mobile user (MU) by using the
geometry relationship. For instance, a near-field joint channel estimation and
localization algorithm was proposed in [11]. However, the two-step positioning
algorithms depend on the channel parameters estimation, which require line-of-
sight (LoS) measurement. These algorithms may not be suitable for indoor
localization as the LoS links may be blocked by obstacles. Without estimating
the channel parameters, fingerprint based positioning algorithms directly
predict the position by using the fingerprint (e.g., received signal strength
information (RSSI)). For example, [12] regarded RSSI as a type of fingerprint
to predict the MUs aided by the RIS. However, RSSI-based fingerprint
localization algorithms can be unstable due to the fast fading fluctuation.
Recently, some researchers proposed the channel state information (CSI) as the
fingerprint, due to its potential to enhance the positioning accuracy compared
with RSSI [13].

Against the above background, the main contributions of this paper are
summarized as follows:

  * 1)

For the fingerprint based mmWave positioning system, we propose a new type of
fingerprint, space-time channel response vector (STCRV), which consists of
multipath channel characteristics. The proposed STCRV fingerprint is closely
related to the position of the MU.

  * 2)

Utilizing the STCRV as the wireless positioning fingerprint, we propose a
novel residual convolution network regression (RCNR) learning algorithm to
estimate the 3D positions of the MUs. Specifically, STCRV is processed by a
data processing block at first. Then, a normal convolution block is then used
to extract the features of the output from the data processing block.
Consequently, four residual convolution blocks are utilized to further extract
the features and protect the integrity. Finally, the 3D position is estimated
through a regression block.

  * 3)

Simulation results are provided to evaluate the performance of the proposed
RCNR learning algorithm. The proposed algorithm outperforms the CNN in terms
of root mean square error (RMSE).

##  II System Model and Problem Formulation

We consider an RIS-aided 3D massive multiple-input multiple-output (MIMO)
time-division duplex (TDD) mmWave positioning system, where the MUs send pilot
signals to the AP to locate the positions of the MUs aided by an RIS. In
addition, we assume that the direct channels between the AP and the MUs are
blocked by some obstacles, such as thick walls..

Figure 1: RIS-aided positioning system model.

As we can see from Fig. 1, the AP is placed on the left side of the wall with
the center located at ${\bf p}=[x_{p},y_{p},z_{p}]^{T}$. Moreover, the AP is
assumed to be equipped with a uniform planar array (UPA) with
$M_{x,z}=M_{x}\times M_{z}$ antennas, where $M_{x}$ and $M_{z}$ denote the
numbers of antennas along the x-axis and the z-axis, respectively.
Additionally, the RIS is assumed to be placed at the front side of the wall
with the center located at ${\bf s}=[x_{s},y_{s},z_{s}]^{T}$. The UPA-based
RIS has $N_{y,z}=N_{y}\times N_{z}$ reflecting elements along the y-axis and
the z-axis, respectively. Furthermore, there are $N_{u}$ MUs , each of which
is equipped with a single antenna. The MUs are located at ${\bf
u}_{i}=[x_{i},y_{i},z_{i}]^{T}$, $i=1,2,\cdots,N_{u}$.

It is assumed that the number of propagation paths between the $i$th MU and
the RIS is $N_{p}$, the AOA of the $p$th path from the $i$th MU to the RIS can
be decomposed into the elevation angle $0\leq\theta_{p,i}\leq\pi$ in the
vertical direction, and the azimuth angle $0\leq\phi_{p,i}\leq\pi$ in the
horizontal direction. As a result, the array response vector at the RIS can be
expressed as [14]

$\displaystyle{\bf a}_{R_{a}}(\theta_{p,i},\phi_{p,i})={\bf
a}_{R_{a}}^{(e)}(\theta_{p,i})\otimes{\bf
a}_{R_{a}}^{(a)}(\theta_{p,i},\phi_{p,i}),$ (1)

where $\otimes$ denotes the the Kronecker product. Moreover, we have

$\displaystyle{\bf a}_{R_{a}}^{(e)}(\theta_{p,i})=[1,e^{\frac{-j2\pi
d_{r}\cos\theta_{p,i}}{\lambda_{c}}},...,e^{\frac{-j2\pi(N_{y}-1)d_{r}\cos\theta_{p,i}}{\lambda_{c}}}]^{T},$
(2)

and

$\displaystyle{\bf a}_{R_{a}}^{(a)}(\theta_{p,i},\phi_{p,i})=$
$\displaystyle[1,e^{\frac{-j2\pi
d_{r}\sin\theta_{p,i}\cos\phi_{p,i}}{\lambda_{c}}},...,$ $\displaystyle
e^{\frac{-j2\pi(N_{z}-1)d_{r}\sin\theta_{p,i}\cos\phi_{p,i}}{\lambda_{c}}}]^{T},$
(3)

where $d_{r}$ and $\lambda_{c}$ denote the distance between the adjacent
elements of the RIS and the carrier wavelength, respectively. Then, the
channel from the $i$th MU to the RIS, denoted as ${\bf g}_{i}$, can be modeled
as

$\displaystyle{\bf g}_{i}=\sum^{N_{p}}_{p=1}\alpha_{p,i}{\bf
a}_{R_{a}}(\theta_{p,i},\phi_{p,i}),$ (4)

where $\alpha_{p,i}$ denotes the complex channel gain of the $p$th path.

Figure 2: The structure of the residual convolution network regression (RCNR)
learning algorithm.

Similarly, it is assumed that the number of propagation paths between the AP
and the RIS is $N_{j}$. The angle of departure (AOD) of the $j$th path from
the RIS to the AP can be decomposed into the elevation angle
$0\leq\theta_{j}\leq\pi$ in the vertical direction and the azimuth angle
$0\leq\phi_{j}\leq\pi$ in the horizontal direction. Hence, the array response
vector ${\bf a}_{R_{d}}(\theta_{j},\phi_{j})$ can be written as

$\displaystyle{\bf a}_{R_{d}}(\theta_{j},\phi_{j})={\bf
a}_{R_{d}}^{(e)}(\theta_{j})\otimes{\bf
a}_{R_{d}}^{(a)}(\theta_{j},\phi_{j}),$ (5)

with

$\displaystyle{\bf a}_{R_{d}}^{(e)}(\theta_{j})=[1,e^{\frac{-j2\pi
d_{r}\cos\theta_{j}}{\lambda_{c}}},...,e^{\frac{-j2\pi(N_{y}-1)d_{r}\cos\theta_{j}}{\lambda_{c}}}]^{T},$
(6)

and

$\displaystyle{\bf a}_{R_{d}}^{(a)}(\theta_{j},\phi_{j})=$
$\displaystyle[1,e^{\frac{-j2\pi
d_{r}\sin\theta_{j}\cos\phi_{j}}{\lambda_{c}}},...,$ $\displaystyle
e^{\frac{-j2\pi(N_{z}-1)d_{r}\sin\theta_{j}\cos\phi_{j}}{\lambda_{c}}}]^{T}.$
(7)

For the RIS-AP link, the AOA of the $j$th path can be decomposed into the
elevation angle $0\leq\psi_{j}\leq\pi$ in the vertical direction and the
azimuth angle $0\leq\omega_{j}\leq\pi$ in the horizontal direction. Therefore,
the array response vector ${\bf a}_{B}(\psi_{j},\omega_{j})$ can be written as

$\displaystyle{\bf a}_{B}(\psi_{j},\omega_{j})={\bf
a}_{B}^{(e)}(\psi_{j})\otimes{\bf a}_{B}^{(a)}(\psi_{j},\omega_{j}),$ (8)

with

$\displaystyle{\bf a}_{B}^{(e)}(\psi_{j})=[1,e^{\frac{-j2\pi
d_{b}\cos\psi_{j}}{\lambda_{c}}},...,e^{\frac{-j2\pi(M_{x}-1)d_{b}\cos\psi_{j}}{\lambda_{c}}}]^{T},$
(9)

and

$\displaystyle{\bf a}_{B}^{(a)}(\psi_{j},\omega_{j})=$
$\displaystyle[1,e^{\frac{-j2\pi
d_{b}\sin\psi_{j}\cos\omega_{j}}{\lambda_{c}}},...,$ $\displaystyle
e^{\frac{-j2\pi(M_{z}-1)d_{b}\sin\psi_{j}\cos\omega_{j}}{\lambda_{c}}}]^{T},$
(10)

where $d_{b}$ denotes the distance of the antennas of the AP.

By using the array response vector ${\bf a}_{R_{d}}(\theta_{j},\phi_{j})$ in
(5) and ${\bf a}_{B}(\psi_{j},\omega_{j})$ in (8), the channel matrix of the
AP with the RIS can be formulated as

$\displaystyle{\bf H}=\sum_{j=1}^{N_{j}}\beta_{j}{\bf
a}_{R_{d}}(\theta_{j},\phi_{j}){\bf a}^{H}_{B}(\psi_{j},\omega_{j}),$ (11)

where $\beta_{j}$ denotes the channel gain of the $j$th path.

Denote ${\bf\Psi}_{t}\in\mathbb{C}^{N_{y,z}\times N_{y,z}}$ as the phase shift
matrix of the RIS in time slot $t$. It is assumed that the MUs transmit pilot
sequences of length $\tau$ via the RIS to the AP. During the uplink
transmission of the $i$th MU, in time slot $t$, $1\leq t\leq\tau$, the
received signal from the $i$th MU at the AP can be written as

$\displaystyle{\bf y}_{i}(t)={\bf H}{\bm{\Psi}}_{t}{\bf
g}_{i}\sqrt{p}s_{i}(t)+{\bf n}_{i}(t),$ (12)

where $s_{i}(t)$ denotes the pilot signal from the $i$th MU, ${\bf
n}_{i}(t)\in\mathbb{C}^{M_{x,z}\times 1}$ $\thicksim$
$\mathcal{CN}(0,\delta^{2}{\bf I})$ represents additive white Gaussian noise
(AWGN) with power $\delta^{2}$ at the AP. Here, $p$ denotes the transmit power
of the $i$th MU. According to the expression of the received signal from the
$i$th MU, we can define the space-time channel response vector (STCRV) of the
$i$th MU as

$\displaystyle{\bf h}_{i}={\bf H}{\bm{\Psi}}_{t}{\bf g}_{i},$ (13)

which consists of the channel parameters (e.g., channel gain and AOA/AOD). The
STCRVs are unique for different positions, hence the STCRVs can be regarded as
a new type of CSI fingerprint of the MU. According to the definition of
directly positioning algorithms [13], the STCRVs can be used to estimate the
positions of the MUs. Therefore, by denoting the estimated position of the
$i$th MU as $\hat{\bf u}_{i}=[\hat{x}_{i},\hat{y}_{i},\hat{z}_{i}]^{T}$, we
have

$\displaystyle\hat{\bf u}_{i}=f({\bf h}_{i}),$ (14)

where $f(\cdot)$ denotes the complex non-linear function between the STCRV and
the estimated position of the $i$th MU. Hence, the regression problem of
estimating the positions of the MUs can be formulated as

$\displaystyle\min_{\hat{\bf u}_{i}\atop
i=1,\cdots,N_{u}}\quad\frac{1}{N_{u}}\sum_{i=1}^{N_{u}}(\hat{\bf u}_{i}-{\bf
u}_{i})^{2}.$ (15)

##  III Residual Convolution Network Regression Learning Algorithm

According to the regression problem in (15), the closed-form expression of the
3D position of the MU is not available by using traditional optimization
methods. Hence, a regression learning algorithm to predict the positions of
MUs is developed in this section. Deep learning based method, popular in image
recognition of computer science, can be applied to the STCRV fingerprint
positioning since the STCRV can be seen as an image. Therefore, we propose a
novel residual convolution network regression (RCNR) learning algorithm to
represent the function $f(\cdot)$ in (14) and solve the regression problem
(15).

###  III-A Regression-oriented Positioning

In computer science, convolution neural network (CNN) is often used for image
classification with the last layer being activated by a softmax function [15].
As an alternative to the classification function, CNN can also be viewed as a
regression function if the softmax layer is replaced by a fully connected
layer containing an activation function. As an advanced network of CNN,
residual convolution network (RCN) can also be used as regression function by
substituting a fully connected layer to the last softmax layer.

(a) Data processing (DP) block.

(b) Regression block.

Figure 3: Two blocks used for the proposed algorithm.

###  III-B The structure of the RCNR learning algorithm

Fig. 2 shows the details of the proposed algorithm. As shown in Fig. 2, the
proposed algorithm consists of a data processing block, a normal convolution
block, four residual convolution blocks, and a regression block. The
descriptions of these blocks will be introduced as follows.

####  III-B1 Data Processing Block

The data processing (DP) block is designed to process the input STCRV. As we
can see from Fig. 3-(a), the DP block includes two layers: a data decomposing
(DD) layer, and a data reshape (DR) layer.

First, since the STCRV is a complex vector, we design the DD layer to
decompose ${\bf h}_{i}$ into two parts, the real value vector ${\bf
h}^{(r)}_{i}$ and the imaginary valure vector ${\bf h}^{(i)}_{i}$, which can
be expressed as

$\displaystyle{\bf h}_{i}={\bf h}^{(r)}_{i}+j{\bf h}^{(i)}_{i}.$ (16)

Then, according to the expression of the STCRV in (13), the STCRV consists of
the features of the horizontal angle domain and the vertical angle domain of
the AP. To further extract these features, we design the DR layer to reshape
the two vectors ${\bf h}^{(r)}_{i}\in\mathbb{C}^{M_{x,z}\times 1}$ and ${\bf
h}^{(i)}_{i}\in\mathbb{C}^{M_{x,z}\times 1}$ as two space-time channel
response matrices (STCRM), which are denoted as ${\bf
H}^{(r)}_{i}\in\mathbb{C}^{M_{x}\times M_{z}}$ and ${\bf
H}^{(i)}_{i}\in\mathbb{C}^{M_{x}\times M_{z}}$, respectively. To be specific,
we reshape these two vectors by arranging the elements of vector into $M_{x}$
rows and $M_{z}$ columns.

####  III-B2 Normal Convolution Block

(a) Normal convolution (NC) block.

(b) Residual convolution (RC) block.

Figure 4: Two kinds of convolution block used for the proposed algorithm.

The normal convolution (NC) block is designed to extract the features of the
STCRM. As we can see from Fig. 4-(a), the NC block consists of three layers: a
convolution (Con) layer, a batch normalization (BN) layer, and a max pooling
(MP) layer.

Due to the large number of antennas at the AP, the STCRM has a large
dimension. Therefore, if the STCRM is input directly into a deep neural
network (DNN) consisting of fully connected layers, a large number of weight
parameters of the DNN should be trained. To improve the efficiency of the
training neural network, the Con layer is proposed [15]. The Con layer has
multiple filters sliding over it for a given input STCRM so that the features
of STCRM can be extracted [15]. As a result, the number of weight parameters
to be trained can be reduced.

Moreover, when training DNN, the input for each layer changes as the weight
parameters of the previous layers change, leading to a reduction in the
convergence rate. Hence, the BN layer is designed to normalize the input data
so that the convergence rate can be improved.

Furthermore, the MP layer is designed to reduce the complexity of further
layers, which is similar to reducing the resolution in the field of computer
science.

####  III-B3 Residual Convolution Block

In the field of computer science, increasing the number of the Con layers will
allow the neural network to extract more features [15]. However, according to
the experiment in [16], when the neural network reaches a certain depth, the
problem of gradient explosion and gradient disappearance will appear, which
leads to a worse optimization effect and lower accuracy of the proposed neural
network. Hence, to improve the estimation accuracy, the residual convolution
network is proposed to enable the deeper neural network to train well and
obtain a better optimization effect [16].

Inspired by the residual convolution network of computer science, the residual
convolution (RC) block is designed to further extract the features of STCRV
and protect the integrity of features in our proposed RCNR learning algorithm.
As shown in Fig. 4-(b). The RC block includes three Con layers and two BN
layers.

As we can see from Fig. 4-(b), the RC block starts with two Con layers. Each
Con layer is followed by a BN layer and a ReLU activation function. Then we
skip these 2 convolutional operations through the cross-layer datapath and add
the input directly before the final ReLU activation function. As a result, the
integrity of the features is protected and the degradation of the neural
network can be solved.

####  III-B4 Regression Block

The regression block is designed to output the estimated positions of the MUs.
As shown in Fig. 3-(b), the regression block includes an average pooling (AP)
layer and a fully connected (FC) layer.

The AP layer is used to reshape the output of the RC blocks for the final FC
layer by taking the average of each feature from the RC block [15]. Adding the
AP layer between the RC block and the FC layer avoids the large number of
weight parameters introduced by the FC layer. As a result, the AP layer
reduces overfitting while improving the convergence rate.

The FC layer is used to combine the features from the NC blocks and output the
estimated positions of the MUs. The FC layer multiplies the input by a
parametric weight matrix and then adds a bias vector. By using ${\bm{\Omega}}$
to denote the output before the FC layer, the estimated position after the FC
layer can be written as

$\displaystyle\hat{\bf u}_{i}={\bf W}\textrm{vec}\\{{\bm{\Omega}}\\}+{\bf b},$
(17)

where ${\bf W}$ and ${\bf b}$ are the parametric weight matrix and bias vector
respectively that can be learned together with the training of the RCNR
learning algorithm.

##  IV Simulation Results

In this section, simulation results are provided to evaluate the performance
of the proposed RCNR algorithm. The software Wireless Insite [17] is used to
simulate the mmWave positioning system aided by the RIS. For the AP, the
number of antennas is set to $M_{x,z}=M_{x}\times M_{z}=255\times 255$.
Besides, the center of the AP is located at ${\bf p}=(-10,-5,2.5$ m$)$, and
the distance between the antennas $d_{b}$ is set to 0.2 m. For the RIS, the
number of the elements is set to $N_{y,z}=N_{y}\times N_{z}=255\times 255$.
Moreover, the center of the RIS is located at ${\bf s}=(-5.10,-1.43,2$ m$)$
and the distance of the elements $d_{r}$ is set to $0.2$ m. For the MUs, we
assume that the MUs are uniformly distributed in the grid of $9.6$ m in length
and $5.8$ m in width. Furthermore, we have set up three grids in total, and
the heights of the grids are $1.4$ m, $1.5$ m, and $1.6$ m, respectively. In
addition, the distance of the MUs is $0.2$ m and the transmit power of the MUs
is $10$ dBm. The Ray tracing propagation model is selected to simulate the
communication system from the MUs via the RIS to the AP. The phase shifts of
the elements at the RIS are set to a unity matrix.

Figure 5: Train Loss of the RCNR learning algorithm. Figure 6: Test Loss of
the RCNR learning algorithm.

In Fig. 5 and Fig. 6, the convergence behavior of the proposed RCNR algorithm
is demonstrated. Both Fig. 5 and Fig. 6 have one curve, named ‘train Loss’ and
‘test Loss’, respectively. These curves denote the train loss and the test
loss of the proposed RCNR learning algorithm. The loss can be obtained by

$\displaystyle Loss=\frac{1}{N}\sum_{({\bf x},{\bf y})\in D}({\bf y}-\hat{\bf
y})^{2},$ (18)

where $D$ denotes the dataset, $({\bf x},{\bf y})$ denotes the sample of the
dataset, ${\bf x}$ represents the input STCRV and ${\bf y}$ denotes the 3D
coordinate of the MU. $\hat{\bf y}$ denotes the prediction of the 3D
coordinate of the MU. $N$ denotes the number of samples of the dataset.

It can be observed from Fig. 5 and Fig. 6 that both train loss and test loss
decrease when the number of epochs increases, which means that the proposed
RCNR algorithm is learning to estimate the positions of the MUs.

Figure 7: Test RMSE of the RCNR learning algorithm.

To evaluate the performance of the proposed RCNR algorithm, we present the
RMSE comparison of the RCNR algorithm with 3 RC blocks, the RCNR algorithm
with 4 RC blocks, the CNN algorithm with 3 NC blocks, and the CNN algorithm
with 4 NC blocks, denoted as ‘CNN, 3 NC blocks’, ‘CNN, 4 NC blocks’, ‘RCNR, 3
RC blocks’, and ‘RCNR, 4 RC blocks’. As shown in Fig. 7, with the increase in
the number of convolution blocks, the RMSE becomes lower, which means that the
proposed algorithm with more RC blocks can achieve higher accuracy.
Furthermore, the proposed RCNR algorithm outperforms the CNN algorithm. With
more convolution blocks, our proposed RCNR algorithm performs better than CNN.

##  V Conclusion

In this paper, we studied the MU positioning problem in MIMO TDD mmWave
systems aided by the RIS. We derived the expression for STCRV at the AP as a
new type of fingerprint. In addition, by using the STCRV fingerprint as input,
we proposed a novel RCNR algorithm to predict the 3D position of the MU.
Extensive simulation results were presented to demonstrate the superiority of
the proposed RCNR algorithm over the CNN algorithm.

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# The structure of connected (graded) Hopf algebras revisited

C.-C. Li, G.-S. Zhou  Li
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
E-mail<EMAIL_ADDRESS>

Zhou
School of Mathematics and Statistics, Ningbo University, Ningbo 315211, China
E-mail<EMAIL_ADDRESS>
###### Abstract.

Let $H$ be a connected graded Hopf algebra over a field of characteristic zero
and $K$ an arbitrary graded Hopf subalgebra of $H$. We show that there is a
family of homogeneous elements of $H$ and a total order on the index set that
satisfy several desirable conditions, which reveal some interesting
connections between $H$ and $K$. As one of its consequences, we see that $H$
is a graded iterated Hopf Ore extension of $K$ of derivation type provided
that $H$ is of finite Gelfand-Kirillov dimension. The main tool of this work
is Lyndon words, along the idea developed by Lu, Shen and the second-named
author in [24].

###### Key words and phrases:

connected Hopf algebra, Hopf Ore extension, Lyndon word, Gelfand-Kirillov
dimension

###### 2010 Mathematics Subject Classification:

16T05, 68R15, 16P90, 16S15, 16W50

## Introduction

Throughout the paper, we work over a fixed field denoted by $k$ which is of
characteristic zero. All vector spaces, algebras, coalgebras and unadorned
tensors are over $k$.

A _connected Hopf algebra_ is a Hopf algebra with coradical of dimension one.
Though the condition of being connected is quite restrictive, it has rich
examples from diverse fields of mathematics. Two well-known classes of
examples are the universal enveloping algebras of Lie algebras and the
coordinate rings of unipotent algebraic groups, which are cocommutative and
commutative respectively. Another celebrated class of examples appear in
combinatorics, including the Hopf algebras of symmmectric functions, of quasi-
symmetric functions and of permutations [10, 1, 2]. These Hopf algebras tend
to have bases naturally parametrized by combinatorial objects (such as
partitions, compositions, permutations, etc.), and they are always of infinite
Gelfand-Kirillov dimension (GK dimension, for short). Recently, connected Hopf
algebras of GK dimensions up to $4$ are completely classified by Wang, Zhang
and Zhuang [27, 22], when the base field is algebraically closed. It is a part
of the longstanding project of classifying Hopf algebras that satisfy some
finiteness conditions. Some new interesting connected Hopf algebras occur in
the classification results.

The motivation of this paper is to understand the structure of connected Hopf
algebras. We do it by considering the subclass of _connected graded Hopf
algebras_ , which are Hopf algebras $H$ equipped with a grading
$H=\bigoplus_{n\geq 0}H_{n}$ such that $H$ is simultaneously a graded algebra
and a graded coalgebra, the antipode preserves the given grading and moreover
$H_{0}$ is one dimensional. It is shown in [24, Theorem A] that every
connected graded Hopf algebra has _PBW generators_ which satisfy some
desirable conditions. By PBW generators of an algebra $A$ we mean a family of
elements of $A$ indexed by a totally ordered set such that the nondecreasing
product of these indexed elments form a linear basis of $A$. In this paper, we
prove a relative version of [24, Theorem A] as follows.

######  Theorem A.

(Theorem 3.1) Let $H$ be a connected graded Hopf algebra and $K$ a graded Hopf
subalgebra. Then there is an indexed family $(z_{\gamma})_{\gamma\in\Gamma}$
of homogeneous elements of $H$ and a total order $\leq$ on $\Gamma$ such that

  1. (1)

$\Delta(z_{\gamma})\in 1\otimes z_{\gamma}+z_{\gamma}\otimes
1+(H_{+}^{<\gamma}\otimes H_{+}^{<\gamma})_{\deg(z_{\gamma})}$ for every index
$\gamma\in\Gamma$, where $H^{<\gamma}$ is the subalgebra of $H$ generated by
$K$ and $\\{~{}z_{\delta}~{}|~{}\delta\in\Gamma,~{}\delta<\gamma~{}\\}$;

  2. (2)

$[z_{\gamma},H^{<\gamma}]\subseteq H^{<\gamma}$ for every index
$\gamma\in\Gamma$;

  3. (3)

$\\{~{}z_{\gamma_{1}}\cdots z_{\gamma_{n}}~{}|~{}n\geq
0,~{}\gamma_{1},\cdots,\gamma_{n}\in\Gamma,~{}\gamma_{1}\leq\cdots\leq\gamma_{n}~{}\\}$
is a homogeneous basis of $H$ as a left graded $K$-module as well as a right
graded $K$-module;

  4. (4)

$\operatorname{\rm GKdim}H=\operatorname{\rm GKdim}K+\\#(\Gamma)$, where
$\\#(\Gamma)$ is the number of elements of $\Gamma$.

The above theorem has some interesting consequences. For example, it follows
that every commutative connected (graded) Hopf algebra is a (graded)
polynomial algebra over any of its (graded) Hopf subalgebras in some family of
(graded) variables, see Corollary 3.2 and Corollary 3.5. As special cases,
some classical results are reconfirmed: (1) every commutative affine connected
Hopf algebra is a polynomial algebra [4, Theorem 0.1]; (2) the shuffle
algebras are graded polynomial algebras [19, Theorem 3.1.1]; (3) the Hopf
algebra of quasi-symmetric functions is a graded polynomial algebra over the
Hopf algebra of symmetric fuctions [15, Corollary 2.2]. We expect more
applications of Theorem A for the study of group schemes and combinatorial
Hopf algebras.

Another interesting consequence of Theorem A concerns Hopf Ore extensions.
This line of research begun in [18] and has been made great progress recently
[5, 11, 6, 24]. Recall that an _$n$ -step iterated Hopf Ore extension_ (_n-
step IHOE_ , for short) of a Hopf algebra $R$ is a Hopf algebra $H$ which
contains $R$ as a Hopf subalgebra and there is a chain of Hopf subalgebras
$R=H^{0}\subset H^{1}\subset\cdots\subset H^{n}=H$ with each of the extensions
$H^{i}\subset H^{i+1}$ being an Ore extension as algebras (Definition 3.6).
According to [24, Theorem B], connected graded Hopf algebras of GK dimension
$d$ are all graded $d$-step IHOEs’ of the base field. In this paper, we
generalize [24, Theorem B] as follows.

######  Theorem B.

(Theorem 3.7) Let $H$ be a connected graded Hopf algerba of GK dimension $d$
and $K$ a graded Hopf subalgebra of GK dimension $e$. Then $H$ is a graded
$(d-e)$-step IHOE of $K$ of derivation type.

Note that connected Hopf algebras of finite GK dimension are not necessarilly
IHOEs’ of the base field. However, the known counterexamples are all the
universal enveloping algebras of certain Lie algebras [5, Example 3.1 (iv)].
By the classification results obtained in [27, 22], connected Hopf algebras of
GK dimensions up to $4$ are all IHOEs’ of the universal enveloping algebra of
its primitive space. In view of the above theorem, it is natural to ask the
following question.

######  Question C.

Is every connnected Hopf algebra of finite GK dimension can be obtained as an
IHOE of the universal enveloping algebra of its primitive space?

Now let us sketch the proof of Theorem A. First we choose a set $X$ of
homogeneous generators for $H$ such that $X^{\prime}:=X\cap K$ generates $K$,
and then fix a well order on $X$ satisfying certain conditions related to
$X^{\prime}$. Denote by $I$ (resp. $J$) the ideal of defining relations on $X$
(resp. $X^{\prime}$) for $H$ (resp. $K$). In addition, let $\mathcal{N}_{I}$
(resp. $\mathcal{N}_{J}$) be the set of $I$-irreducible (resp.
$J$-irreducible) Lyndon words on $X$ (resp. $X^{\prime}$). By the
combinatorial properties of Lyndon words and the standard bracketing on words,
one may construct from $X$ a new family of homogeneous generators
$(z_{\gamma})_{\gamma\in\mathcal{N}_{I}}$ for $H$. The coalgebra structure of
$H$ assures that this new family of generators satisfy some desirable
conditions (Proposition 2.4). Note that several key parts of this observation
was already obtained in [24, Proposition 2.4], which is not easy to prove and
has its own idea trace back to [12, 21]. The decisive new observation for this
work is the equality $\mathcal{N}_{J}=\mathcal{N}_{I}\cap\langle
X^{\prime}\rangle$, due to the feature of Lyndon words (Lemma 2.7). Combine
the above-mentioned two observations as well as some others, we conclude that
the family $(z_{\gamma})_{\gamma\in\Gamma}$, where
$\Gamma:=\mathcal{N}_{I}\backslash\mathcal{N}_{J}$, together with the
restriction of the lexicographic order on $\Gamma$ satisfy the requirements.

The paper is organized as follows. In section 1, we recall the definition and
basic facts of Lyndon words. In section 2, we study the structure of connected
graded algebras that admit pseudo-comultiplications (Definition 2.1) by taking
advantage of the excellent combinatorial properties of Lyndon words. This
section is the technical part of the paper and as a matter of fact, Theorem A
is a direct consequence of Theorem 2.2. The final section is devoted to study
the structure of connected (graded) Hopf algebras. In particular, the above
two main theorems are proved there.

##  1\. Preliminaries

In this section, we introduce some basic notations that will be used in the
sequel, recall the definition of Lyndon words and present some of their well-
known properties for reader’s convenience. The materials of Lyndon words are
mostly extracted or modified from [7, 12, 14, 24].

Throughout this section, $X$ stands for a well-ordered alphabet. If
$X=\\{x_{1},\cdots,x_{\theta}\\}$ for some positive integer $\theta$, then $X$
is tacitly equipped with the natural order $x_{1}<\cdots<x_{\theta}$. We
denote by $\langle X\rangle$ the set of all words on $X$. The empty word is
denoted by $1$. The _length_ of a word $u$ is denoted by $|u|$.

Let $u,\,v$ be two words on $X$. We call $v$ a factor of $u$ if there exist
words $w_{1},w_{2}$ on $X$ such that $w_{1}vw_{2}=u$. If $w_{1}$ (resp.
$w_{2}$) can be chosen to be the empty word then $v$ is called a _prefix_
(resp. _suffix_) of $u$. A factor (resp. prefix, resp. suffix) $v$ of $u$ is
called proper if $v\neq u$. The _lexicographic order_ on $\langle X\rangle$,
denoted by $\leq_{\rm lex}$, is defined as follows. For words $u,v\in\langle
X\rangle$,

$\displaystyle u<_{\rm
lex}v\,\,\Longleftrightarrow\,\,\left\\{\begin{array}[]{llll}v\text{ is a
proper prefix of }u,\quad\text{or}&&\\\ u=rxs,\,v=ryt,\,\text{ with }x,y\in
X;\,x<y;\,r,s,t\in\langle X\rangle.\end{array}\right.$

Clearly, it is a total order compatible with the concatenation of words from
left but not from right. For example if $x,y\in X$ with $x>y$, one has
$x>_{\rm lex}x^{2}$ but $xy<_{\rm lex}x^{2}y$.

######  Lemma 1.1.

For any integer $n\geq 1$, the retriction of $\leq_{\operatorname{\rm lex}}$
on the set of all words on $X$ of degree less than or equal to $n$ is a well
order.

###### Proof.

It suffices to see every nonincreasing sequence $u_{1}\geq_{\operatorname{\rm
lex}}u_{2}\geq_{\operatorname{\rm lex}}\cdots$ of words of degree $\leq n$
stablizes eventually. We prove it by induction on $n$. If $u_{r}\neq 1$ for
some $r$ then $u_{s}\neq 1$ for $s\geq r$, so we may assume first that all
$u_{r}$ are nonempty. Since $\leq_{\operatorname{\rm lex}}$ coincides with
$\leq$ on $X$, the result is true for $n=1$. For $n\geq 2$, consider the
decompositions $u_{r}=v_{r}w_{r}$ for $r\geq 1$, where $v_{r}\in X$. Clearly,

$v_{1}\geq v_{2}\geq v_{3}\geq\cdots$

by the definition of $\leq_{\operatorname{\rm lex}}$. Since $\leq$ is a well
order on $X$, there exists $p\geq 1$ such that $v_{p}=v_{p+1}=\cdots.$ So we
get a nonincreasing sequence $w_{p}\geq_{\operatorname{\rm
lex}}w_{p+1}\geq_{\operatorname{\rm lex}}\cdots$ of words of degree $\leq
n-1$. By induction, there exists $q\geq p$ such that $w_{q}=w_{q+1}=\cdots$,
hence $u_{q}=u_{q+1}=\cdots$. ∎

######  Definition 1.2.

A word $u\in\langle X\rangle$ is called Lyndon if $u$ is nonempty and $u>_{\rm
lex}wv$ for every factorization $u=vw$ with $v,w$ nonempty. The set of all
Lyndon words on $X$ is denoted by $\mathbb{L}=\mathbb{L}(X)$.

For a word $u$ of length $\geq 2$, define $u_{R}\in\langle X\rangle$ to be the
lexicographically largest proper suffix of $u$ and define $u_{L}\in\langle
X\rangle$ by the decomposition $u=u_{L}u_{R}$. The pair of words

$\operatorname{\rm Sh}(u):=(u_{L},u_{R})$

is called the Shirshov factorization of $u$. As an example, $\operatorname{\rm
Sh}(x_{2}^{2}x_{1}x_{2}x_{1})=(x_{2}^{2}x_{1},x_{2}x_{1})$.

Lyndon words share many excellent combinatorial properties, we collect below
some of them for reader’s interests. Due to the different favor of the
definition of the lexicographic order in literatures, the statements that
presented may be adjusted accordingly from the given references.

######  Proposition 1.3.

  1. (L1)

([14, Proposition 5.1.2], [12, Lemma 2]) A nonempty word $u$ is Lyndon if and
only if it is lexicographically greater than any one of its proper nonempty
suffices.

  2. (L2)

([14, Proposition 5.1.3] $\&$ [7, Lemma 4.3 (3)]) Let
$u=w_{1}w_{2},\,v=w_{2}w_{3}$ be Lyndon words. If $u>_{\rm lex}v$ (which holds
in priori when $w_{2}\neq 1$), then $w_{1}w_{2}w_{3}$ is a Lyndon word.

  3. (L3)

([14, Proposition 5.1.3]) Let $u$ be word of length $\geq 2$. Then $u$ is a
Lyndon word if and only if $u_{L}$ and $u_{R}$ are both Lyndon words and
$u_{L}>_{\rm lex}u_{R}$.

  4. (L4)

([14, Proposition 5.1.4]) Let $u,v$ be Lyndon words such that $u>_{\rm lex}v$.
Then $\operatorname{\rm Sh}(uv)=(u,v)$ if and only if either $u$ is a letter
or $u$ is of length $\geq 2$ with $u_{R}\leq_{\rm lex}v$.

  5. (L5)

([14, Theorem 5.1.5]) Every nonempty word $u$ can be written uniquely as a
(lexicographically) nondecreasing product $u=u_{1}u_{2}\cdots u_{r}$ of Lyndon
words.

  6. (L6)

([7, Lemma 4.5]) Let $u=u_{1}u_{2}\cdots u_{r}$ be a nondecreasing product of
Lyndon words. If $v$ is a Lyndon factor of $u$ then $v$ is a factor of some
$u_{i}$. $\Box$

Let $k\langle X\rangle$ be the free algebra on $X$, which has all words on $X$
as a linear basis. Elements of $k\langle X\rangle$ are frequently called
(noncommutative) polynomials on $X$ (with coefficients in $k$). In the sequel,
let us assume that the free algebra $k\langle X\rangle$ is connected graded
with each letter homogeneous of positive degree. The degree of a homogeneous
polynomial $f$ is denoted by $\deg(f)$.

We need some elementary notions from (noncommutative) Groebner basis theory.
The _graded lex order_ on $\langle X\rangle$, denoted by
$\leq_{\operatorname{\rm glex}}$, is defined as follows. For words $u,\,v$,

$\displaystyle u<_{\rm\operatorname{\rm
glex}}v\,\,\Longleftrightarrow\,\,\left\\{\begin{array}[]{llll}\deg(u)<\deg(v),\quad\text{or}&&\\\
\deg(u)=\deg(v)\,\text{ and }u<_{\operatorname{\rm lex}}v.\end{array}\right.$

Clearly, it is an _admissible order_ on $\langle X\rangle$, i.e. a well order
on $\langle X\rangle$ that compatible with the concatenation of words from
both sides. The _leading word_ of a nonzero polynomial $f$, denoted by
$\operatorname{\rm LW}(f)$, is the largest word that occurs in $f$ (with
respect to $\leq_{\operatorname{\rm glex}}$). The coeffcient of
$\operatorname{\rm LW}(f)$ in $f$ is called the _leading coefficient_ of $f$.
For any set $F\subseteq k\langle X\rangle$, we write $\operatorname{\rm
LW}(F):=\\{~{}\operatorname{\rm LW}(f)~{}|~{}f\in F,\;f\neq 0~{}\\}.$

Let $I$ be an ideal of $k\langle X\rangle$. A word on $X$ is called _$I$
-reducible_ if it is the leading word of some polynomials in $I$. A word that
is not $I$-reducible is called _$I$ -irreducible_. An _obstruction_ of $I$
(with respect to $\leq_{\operatorname{\rm glex}}$) is an $I$-recucible word
whose proper factors are all $I$-irreducible. We denote by $\mathcal{N}_{I}$
the set of all $I$-irreducible Lyndon words, by $\mathcal{O}_{I}$ the set of
all obstructions of $I$, and by $\mathcal{B}_{I}$ the set of all
lexicographically nondecreasing product of Lyndon words in $\mathcal{N}_{I}$.

######  Lemma 1.4.

Let $I$ be an ideal of $k\langle X\rangle$. The set of all $I$-irreducible
words is included in $\mathcal{B}_{I}$, and these two sets are equal if and
only if $\mathcal{O}_{I}$ consists of Lyndon words.

###### Proof.

It is an easy consequence of Proposition 1.3 (L5, L6) and the fact that every
$I$-reducible word has a factor in $\mathcal{O}_{I}$. ∎

##  2\. Pseudo-comultiplications on connected graded algebras

The primary interest of this work is the structure of connected (graded) Hopf
algebras. The techniques that employed or developed in this paper are aimed to
deal with it. In our process, it turns out that the coassociativity of the
comulplication plays no role in some key observations. In this section, we
study the structure of connected graded algebras $A$ which admit algebra
homomorphisms $A\to A\otimes A$ that share some features of the
comultiplication of connected graded Hopf algerbas. One of the main tools is
the Lyndon words reviewed in the previous section.

Recall that a _connected graded algebra_ is an algebra $A$ on which a
decomposition $A=\bigoplus_{n\geq 0}A_{n}$ is specified such that
$A_{0}=k\,1_{A}$ and $A_{i}\cdot A_{j}\subseteq A_{i+j}$ for $i,j\geq 0$. We
write $A_{+}:=\bigoplus_{n\geq 1}A_{n}$ in the sequel.

######  Definition 2.1.

Let $A=\bigoplus_{n\geq 0}A_{n}$ be a connected graded algebra. A _pseudo-
comultiplication_ on $A$ is a homomorphism of graded algebras $\Delta:A\to
A\otimes A$ such that

$\Delta(a)\in 1\otimes a+a\otimes 1+(A_{+}\otimes A_{+})_{\deg(a)}$

for every homogeneous element $a$ of positive degree. Clearly, to check the
above condition it suffices to show it for some family of homogeneous
generators of $A$.

The whole section is devoted to prove the following result.

######  Theorem 2.2.

Let $A$ be a connected graded algebra and $B\subseteq A$ a graded subalgebra.
Let $\Delta:A\to A\otimes A$ be a pseudo-comultiplication on $A$ such that
$\Delta(B)\subseteq B\otimes B$. Then there is an indexed family
$(z_{\gamma})_{\gamma\in\Gamma}$ of homogeneous elements of $A$ of positive
degree and a total order $\leq$ on $\Gamma$ such that

  1. (1)

$\Delta(z_{\gamma})\in 1\otimes z_{\gamma}+z_{\gamma}\otimes
1+(A_{+}^{<\gamma}\otimes A_{+}^{<\gamma})_{\deg(z_{\gamma})}$ for every index
$\gamma\in\Gamma$, where $A^{<\gamma}$ is the subalgebra of $A$ generated by
$B$ and $\\{~{}z_{\delta}~{}|~{}\delta\in\Gamma,~{}\delta<\gamma~{}\\}$;

  2. (2)

$[z_{\gamma},A^{<\gamma}]\subseteq A^{<\gamma}$ for every index
$\gamma\in\Gamma$;

  3. (3)

$\\{~{}z_{\gamma_{1}}\cdots z_{\gamma_{n}}~{}|~{}n\geq
0,~{}\gamma_{1},\cdots,\gamma_{n}\in\Gamma,~{}\gamma_{1}\leq\cdots\leq\gamma_{n}~{}\\}$
is a homogeneous basis of $A$ as a left graded $B$-module as well as a right
graded $B$-module;

  4. (4)

$\operatorname{\rm GKdim}A=\operatorname{\rm GKdim}B+\\#(\Gamma)$, where
$\\#(\Gamma)$ is the number of elements of $\Gamma$.

The proof of this theorem will be addressed after some preparatory results.

######  Remark 2.3.

The totally ordered set $(\Gamma,\leq)$ appeared in Theorem 2.2 may have no
smallest element. Let $A$ be the universal enveloping algebra of the Lie
algebra with basis $\\{e_{n}\\}_{n\geq 1}$ and Lie bracket
$[e_{i},e_{j}]=(j-i)e_{i+j}$. Then $A$ is a connected graded algebra with
$e_{n}$ of degree $n$. Take $B$ to be the trivial subalgebra $k\cdot 1_{A}$.
Then the family $(e_{n})_{n\in\Gamma}$ with $\Gamma=\\{1,2,3,\cdots\\}$ and
the total order $\leq$ on $\Gamma$ given by $1>2>3>\cdots$ satisfy the
requirements. Clearly, in this case, $(\Gamma,\leq)$ has no smallest element.

Let us proceed to show Theorem 2.2. We continue to use the notations and
conventions employed in Section 1. So $X$ stands for a well-ordered alphabet
and the free algebra $k\langle X\rangle$ is connected graded with letters
homogeneous of positive degrees. In addition to the Lyndon words, another
brick to construct the required family of generators in Theorem 2.2 is the so-
called _standard bracketing_ on $\langle X\rangle$, which is the map
$[-]:\langle X\rangle\to k\langle X\rangle$ defined as follows. First set
$[1]=1$ and $[x]:=x$ for $x\in X$; and then for words $u$ of length $\geq 2$,
inductively set

$\displaystyle[u]=\left\\{\begin{array}[]{ll}~{}[[u_{L}],[u_{R}]],&u\text{ is
Lyndon},\\\ ~{}[u_{L}][u_{R}],&u\text{ is not Lyndon}.\end{array}\right.$

Here, $[-,-]:k\langle X\rangle\times k\langle X\rangle\to k\langle X\rangle$
stands for the commutator operation on $k\langle X\rangle$ defined by
$[f,g]=fg-gf.$ Note that $[u_{1}u_{2}\cdots
u_{n}]=[u_{1}][u_{2}]\cdots[u_{n}]$ for Lyndon words $u_{1}\leq_{\rm
lex}u_{2}\leq_{\rm lex}\cdots\leq_{\rm lex}u_{n}$.

For a word $w$, we denote by $k\langle X|I\rangle^{<w}$ (resp. $k\langle
X|I\rangle^{\leq w}$) the subalgebra of $k\langle X\rangle$ generated by

$\\{~{}[u]~{}|~{}u\in\mathcal{N}_{I},~{}u<_{\operatorname{\rm
lex}}w~{}\\}\quad\Big{(}\text{ resp.
}\\{~{}[u]~{}|~{}u\in\mathcal{N}_{I},~{}u\leq_{\operatorname{\rm
lex}}w~{}\\}~{}\Big{)}.$

Note that they are all graded subalgebras of $k\langle X\rangle$.

######  Proposition 2.4.

Let $I$ be a homogeneous proper ideal of $k\langle X\rangle$ and $A:=k\langle
X\rangle/I$. Assume that there exists a pseudo-comultiplication $\Delta$ on
$k\langle X\rangle$ such that $\Delta(I)\subseteq k\langle X\rangle\otimes
I+I\otimes k\langle X\rangle$.

  1. (1)

$\mathcal{B}_{I}$ equals to the set of $I$-irreduicble words, and
$\mathcal{O}_{I}$ consists of Lyndon words.

  2. (2)

$[v]\in k\langle X|I\rangle^{<v}_{\deg(v)}+I$ for every $I$-reducible Lyndon
words $v$.

  3. (3)

$\Delta([v])\in 1\otimes[v]+[v]\otimes 1+\big{(}(k\langle
X|I\rangle_{+}^{<v}+I)\otimes(k\langle
X|I\rangle_{+}^{<v}+I)\big{)}_{\deg(v)}$ for every $v\in\mathcal{N}_{I}$.

  4. (4)

$[u][v]-[v][u]\in k\langle X|I\rangle_{\deg(uv)}^{\leq uv}+I$ for every pair
$u,v\in\mathcal{N}_{I}$ with $u>_{\operatorname{\rm lex}}v$.

  5. (5)

$\\{~{}[w]+I\;|\;w\in\mathcal{B}_{I}~{}\\}$ is a homogeneous basis of $A$.

  6. (6)

$\operatorname{\rm GKdim}A=\\#(\mathcal{N}_{I})$, where $\\#(\mathcal{N}_{I})$
is the number of elements of $\mathcal{N}_{I}$.

###### Proof.

Parts (2), (4) and (5) are obtained in [24, Proposition 2.4], whose proof is a
bit of complicate. By Part (5) and the fact that $\operatorname{\rm
LW}([u])=u$ for any word $u$ on $X$, it is not hard to see that
$\\{~{}w+I\;|\;w\in\mathcal{B}_{I}~{}\\}$ is a basis of $A$. By Lemma 1.4,
Part (1) follows. By [24, Proposition 2.3], one has

$\Delta([v])\in 1\otimes[v]+[v]\otimes 1+(k\langle X\rangle_{+}^{<v}\otimes
k\langle X\rangle_{+}^{<v})_{\deg(v)},$

where $k\langle X\rangle^{<v}$ is the subalgebra of $k\langle X\rangle$
generated by $\\{\,[u]\;|\;u<_{\operatorname{\rm
lex}}v,\,u\in\mathbb{L}~{}\\}.$ Note that

$k\langle X|I\rangle_{+}^{<v}+I=k\langle X\rangle_{+}^{<v}+I$

by Part (2), one then obtains Part (3). Finally we show Part (6). For any
finite subset $\Xi$ of $\mathcal{N}_{I}$, let $A_{\Xi}$ be the subalgebra of
$A$ generated by $\\{[v]+I~{}|~{}v\in\Xi~{}\\}$. For any integer $n$, let
$d_{\Xi}(n)$ be the number of nondecreasing sequences $(v_{1},\cdots,v_{p})$
in $\Xi$ such that $\deg(v_{1}\cdots v_{p})=n$. Then

$\sum_{n\geq 0}\dim(A_{\Xi})_{n}~{}t^{n}\geq\sum_{n\geq
0}d_{\Xi}(n)~{}t^{n}=\prod_{v\in\Xi}\big{(}1-t^{\deg(v)}\big{)}^{-1}$

by Part (5) and a simple combinatorial argument, where the inequality means
that the difference of the series has no negative coefficients. Consequently,

($*$) $\operatorname{\rm
GKdim}(A_{\Xi})=\limsup_{n\to\infty}\log_{n}\sum_{i\leq
n}\dim(A_{\Xi})_{i}\geq\limsup_{n\to\infty}\log_{n}\sum_{i\leq
n}d_{\Xi}(i)=\\#(\Xi).$

Here, the first equality is by [13, Lemma 6.1 (b)] and the second equality is
by [3, Proposition 2.21]. It follows that if $\mathcal{N}_{I}$ is infinite
then $\operatorname{\rm GKdim}(A)\geq\operatorname{\rm
GKdim}(A_{\Xi})\geq\\#(\Xi)$ for any finite subset $\Xi$ of $\mathcal{N}_{I}$
and hence $\operatorname{\rm GKdim}(A)=\infty$. If $\mathcal{N}_{I}$ is finite
then $A=A_{\mathcal{N}_{I}}$ and the inequality in ($*$) becomes an equality
for $\Xi=\mathcal{N}_{I}$ by Part (5), whence $\operatorname{\rm
GKdim}(A)=\operatorname{\rm GKdim}(A_{\mathcal{N}_{I}})=\\#(\mathcal{N}_{I})$.
∎

######  Remark 2.5.

Connected graded algebras of the form $A=k\langle X\rangle/I$ with
$\mathcal{O}_{I}$ consists of Lyndon words are studied in [8, 23, 26], where
some important homological and algebraic invariants of $A$ are described in
terms of $\mathcal{N}_{I}$. One interesting source of such algerbas are Artin-
Schelter regular algebras of global dimension $\leq 5$ that satisfy some extra
conditions [25, Theorem 8.1]. Proposition 2.4 provides us with another natural
source of such algebras, including all connected graded Hopf algebras.

######  Lemma 2.6.

Let $I$ be a homogeneous ideal of $k\langle X\rangle$ and $A:=k\langle
X\rangle/I$. Assume that there exists a pseudo-comultiplication $\Delta$ on
$k\langle X\rangle$ such that $\Delta(I)\subseteq k\langle X\rangle\otimes
I+I\otimes k\langle X\rangle$. Then for every sequence $(w_{1},\ldots,w_{m})$
of Lyndon words in $\mathcal{N}_{I}$, the coset $[w_{1}]\cdots[w_{m}]+I$ in
$A$ is a linear combination of cosets of the form $[v_{1}]\cdots[v_{p}]+I$
with $v_{1},\ldots,v_{p}\in\mathcal{N}_{I}$ satisfying that

$v_{1}\leq_{\operatorname{\rm lex}}\cdots\leq_{\operatorname{\rm
lex}}v_{p}\leq_{\operatorname{\rm lex}}\max\\{w_{1},\ldots
w_{m}\\}\quad\text{and}\quad\deg(v_{1}\cdots v_{p})=\deg(w_{1}\cdots w_{m}),$

where $\max\\{w_{1},\ldots w_{m}\\}$ is the lexicographically maximal word
among $w_{1},\cdots,w_{m}$.

###### Proof.

Let $N_{I}^{(\infty)}$ be the set of finite sequences in $\mathcal{N}_{I}$,
including the empty sequence. Let $\leq$ be the partial order on
$\mathcal{N}_{I}^{(\infty)}$ defined as follows. For $a=(a_{1},\cdots,a_{m})$
and $b=(b_{1},\cdots,b_{n})$ in $\mathcal{N}_{I}^{(\infty)}$, $a<b$ if and
only if $\sum_{i=1}^{m}\deg(a_{i})=\sum_{j=1}^{n}\deg(b_{j})$ but there exists
an integer $p\leq\min\\{m,n\\}$ such that $a_{i}=b_{i}$ for $i<p$ and
$a_{p}<_{\rm lex}b_{p}$. Clearly, $\leq$ is compatible with the left and right
concatenation of finite sequences. In addition, $\leq$ satisfies the
descending chain condition by Lemma 1.1. Now we proceed to show the result by
induction on $(w_{1},\cdots,w_{m})$ with respect to $\leq$. If
$(w_{1},\cdots,w_{m})$ is nondecreasing, there is nothing to prove. Otherwise,
assume $w_{i}>_{\rm lex}w_{i+1}$ for some $i$. By Proposition 2.4 (4),

$\displaystyle[w_{1}]\cdots[w_{m}]$ $\displaystyle\in$
$\displaystyle[w_{1}]\cdots[w_{i-1}][w_{i+1}][w_{i}][w_{i+2}]\cdots[w_{m}]$
$\displaystyle+\sum_{(c_{1},\cdots,c_{s})}\lambda_{(c_{1},\cdots,c_{s})}^{w_{i},w_{i+1}}~{}[w_{1}]\cdots[w_{i-1}][c_{1}]\cdots[c_{s}][w_{i+2}]\cdots[w_{m}]+I,$

where $(c_{1},\cdots,c_{s})$ runs over finite sequences in $\mathcal{N}_{I}$
such that

$\deg(c_{1}\cdots c_{s})=\deg(w_{i}w_{i+1})\quad\text{ and
}\quad\max\\{c_{1},\cdots,c_{s}\\}\leq_{\operatorname{\rm
lex}}w_{i}w_{i+1}<_{\operatorname{\rm lex}}\max\\{w_{1},\cdots,w_{m}\\}.$

It follows that
$\max\\{w_{1},\cdots,w_{i-1},c_{1},\cdots,c_{s},w_{i+2},\cdots,w_{m}\\}\leq_{\operatorname{\rm
lex}}\max\\{w_{1},\cdots,w_{m}\\}$ and

$\displaystyle(w_{1},\cdots,w_{i-1},c_{1},\cdots,c_{s},w_{i+2},\cdots,w_{m})<(w_{1},\cdots,w_{m}).$

Also we have
$(w_{1},\cdots,w_{i-1},w_{i+1},w_{i},w_{i+2},\cdots,w_{m})<(w_{1},\cdots,w_{m})$.
By the induction hypothesis, $[w_{1}]\cdots[w_{m}]+I$ has the required
decomposition. ∎

The following result is a crucial observation for the proof of Theorem 2.2.
First let us recall more terminologies from standard Groebner basis theory. A
set $F\subseteq k\langle X\rangle$ is called _reduced_ (with respect to
$\leq_{\operatorname{\rm glex}}$) if $0\not\in F$ and for every $f\in F$, the
leading coefficient of $f$ is $1$ and all words occur in $f$ has no factor in
$\operatorname{\rm LW}(F\backslash\\{f\\})$. For an ideal $I$ of $k\langle
X\rangle$, a _Groebner basis_ of $I$ (with respect to $\leq_{\operatorname{\rm
glex}}$) is a subset $\mathcal{G}\subseteq I$ such that every $I$-reducible
word has a factor in $\operatorname{\rm LW}(\mathcal{G})$. It is well-known
that every ideal admits a unique reduced Groebner basis [20, Proposition 14].
Note that all these definitons and facts as mentioned are workable with
respect to any admissible orders on $\langle X\rangle$.

######  Lemma 2.7.

Let $I$ be a homogeneous proper ideal of $k\langle X\rangle$, $X^{\prime}$ a
subset of $X$ and $J=I\cap k\langle X^{\prime}\rangle$. Assume that
$x_{1}<x_{2}$ for every $x_{1}\in X^{\prime}$ and $x_{2}\in X\backslash
X^{\prime}$, and assume there is a pseudo-comultiplication $\Delta$ on
$k\langle X\rangle$ such that $\Delta(k\langle X^{\prime}\rangle)\subseteq
k\langle X^{\prime}\rangle\otimes k\langle X^{\prime}\rangle$ and
$\Delta(I)\subseteq k\langle X\rangle\otimes I+I\otimes k\langle X\rangle$.
Then

$\displaystyle\mathcal{N}_{J}=\mathcal{N}_{I}\cap\langle
X^{\prime}\rangle,\quad\mathcal{O}_{J}=\mathcal{O}_{I}\cap\langle
X^{\prime}\rangle\quad\text{and}\quad\mathcal{G}_{J}=\mathcal{G}_{I}\cap
k\langle X^{\prime}\rangle,$

where $\mathcal{G}_{I}$ (resp. $\mathcal{G}_{J}$) is the reduced Groebner
basis of $I$ (resp. $J$) with respect to $\leq_{\operatorname{\rm glex}}$.

###### Proof.

First we show $\mathcal{N}_{J}=\mathcal{N}_{I}\cap\langle X^{\prime}\rangle$.
Let $v$ be a Lyndon word on $X^{\prime}$. Clearly, if $v$ is $J$-reducible
then it is $I$-reducible. Conversely, assume $v$ is $I$-reducible. Since the
biggest letter in a Lyndon word must occur in the left-most place, Lyndon
words on $X$ that lexicographically smaller than $v$ are necessarilly Lyndon
words on $X^{\prime}$. It is then clear that $v$ is $J$-reducible by
Proposition 2.4 (2).

Next we show $\mathcal{O}_{J}=\mathcal{O}_{I}\cap\langle X^{\prime}\rangle$.
By Proposition 2.4 (1), $\mathcal{O}_{I}$ and $\mathcal{O}_{J}$ consist of
Lyndon words. Note that a word on $X$ (resp. $X^{\prime}$) is $I$-irreducible
(resp. $J$-irreducible) if and only if its Lyndon factors are all so. For any
Lyndon word $u$ on $X^{\prime}$,

$\displaystyle u\in\mathcal{O}_{J}$ $\displaystyle\Longleftrightarrow$
$\displaystyle u\text{ is $J$-reducible and its proper Lyndon factors are all
$J$-irreducible}$ $\displaystyle\Longleftrightarrow$ $\displaystyle u\text{ is
$I$-reducible and its proper Lyndon factors are all $I$-irreducible}$
$\displaystyle\Longleftrightarrow$ $\displaystyle u\in\mathcal{O}_{I}.$

Here, the first and the third equivalences are clear, and the second
equivalence is by $\mathcal{N}_{J}=\mathcal{N}_{I}\cap\langle
X^{\prime}\rangle$ shown as above. The desired equality follows directly.

Finally, we show $\mathcal{G}_{J}=\mathcal{G}_{I}\cap k\langle
X^{\prime}\rangle$. Since $\operatorname{\rm
LW}(\mathcal{G}_{I})=\mathcal{O}_{I}$ and $\operatorname{\rm
LW}(\mathcal{G}_{J})=\mathcal{O}_{J}$ by standard Groebner basis theory, they
both consist of Lyndon words and $\operatorname{\rm
LW}(\mathcal{G}_{J})=\operatorname{\rm LW}(\mathcal{G}_{I})\cap\langle
X^{\prime}\rangle$. For any word $u\in\operatorname{\rm LW}(\mathcal{G}_{J})$,
let $f_{u}$ (resp. $g_{u}$) be the unique element of $\mathcal{G}_{J}$ (resp.
$\mathcal{G}_{I}$) with leading word $u$. By the definition of
$\mathcal{G}_{J}$, all words occur in $f_{u}-u$ have no factor in
$\operatorname{\rm LW}(\mathcal{G}_{J})$, and hence they have no factor in
$\operatorname{\rm LW}(\mathcal{G}_{I})$. Thus $f_{u}-u$ is a linear
combination of $I$-irreducible words. Consequently, $f_{u}=g_{u}$ for any
$u\in\operatorname{\rm LW}(\mathcal{G}_{J})$ and the desired equality follows
directly. ∎

######  Remark 2.8.

For a proper ideal $I$ of $k\langle X\rangle$ and a subset $X^{\prime}\subset
X$, knowing a generating set $G$ for $I$, it is not necessarilly true that
$G^{\prime}=G\cap k\langle X^{\prime}\rangle$ is a generating set for $J=I\cap
k\langle X^{\prime}\rangle$. To make it happen, a well-known choice of $G$ is
an arbitray Groebner basis of $I$ with respect to an arbitrary admissible
order $\sigma$ on $\langle X\rangle$ that eliminates $X\backslash X^{\prime}$
in the sense of [20, Definition 17]. It is because that $G^{\prime}$ is a
Groebner basis of $J$ with respect to the restriction of $\sigma$ to $\langle
X^{\prime}\rangle$ (see [20, Proposition 19]). Nevertheless, Lemma 2.7
provides more choices of $G$ when $I$ satisfy specific conditions.

Now we are ready to prove Theorem 2.2.

###### Proof of Theorem 2.2.

By choosing a set of homogeneous generators of $A$ of positive degree which
contains a generating set of $B$, we may fix a set $X$ of graded variables, a
decomposition $X=X^{\prime}\cup X^{\prime\prime}$, and a homogeneous ideal $I$
of $k\langle X\rangle$ such that $A=k\langle X\rangle/I$ as graded algebras,
the cosets of variables in $X^{\prime}$ form a generating set of $B$ and the
cosets of variables in $X^{\prime\prime}$ lie in $A\backslash B$. Further,
equip $X$ with a well order $\leq$ as follows. First fix a well order
$\leq^{\prime}_{r}$ on $X^{\prime}_{r}:=\\{~{}x\in
X^{\prime}~{}|~{}\deg(x)=r~{}\\}$ and a well order $\leq^{\prime\prime}_{r}$
on $X^{\prime\prime}_{r}:=\\{~{}x\in X^{\prime\prime}~{}|~{}\deg(x)=r~{}\\}$
for each integer $r\geq 1$; then for $x_{1},x_{2}\in X$,

$\displaystyle x_{1}\leq
x_{2}\,\,\Longleftrightarrow\,\,\left\\{\begin{array}[]{llll}x_{1}\in
X^{\prime}\text{ and }x_{2}\in X^{\prime\prime},\quad\text{or}\\\
x_{1},x_{2}\in X^{\prime}\text{ and
}\left\\{\begin{array}[]{llll}\deg(x_{1})<\deg(x_{2}),\quad\text{or}&&\\\
\deg(x_{1})=\deg(x_{2})\,\text{ and
}x_{1}\leq^{\prime}_{\deg(x_{1})}x_{2},\end{array}\right.\text{or}\\\
x_{1},x_{2}\in X^{\prime\prime}\text{ and
}\left\\{\begin{array}[]{llll}\deg(x_{1})<\deg(x_{2}),\quad\text{or}&&\\\
\deg(x_{1})=\deg(x_{2})\,\text{ and
}x_{1}\leq^{\prime\prime}_{\deg(x_{1})}x_{2},\end{array}\right.\end{array}\right.$

In addition, one may readily lift $\Delta$ to a triangular comultiplication
$\tilde{\Delta}:k\langle X\rangle\to k\langle X\rangle\otimes k\langle
X\rangle$ such that

$\tilde{\Delta}(k\langle X^{\prime}\rangle)\subseteq k\langle
X^{\prime}\rangle\otimes k\langle X^{\prime}\rangle\quad\text{ and
}\quad\tilde{\Delta}(I)\subseteq k\langle X\rangle\otimes I+I\otimes k\langle
X\rangle.$

Clearly, the canonical projection from $k\langle X\rangle$ to $A$ induces a
graded algebra isomorphism

$k\langle X^{\prime}\rangle/J\cong B,\quad\text{ where }J=I\cap k\langle
X^{\prime}\rangle.$

Let $\Gamma:=\mathcal{N}_{I}\backslash\mathcal{N}_{J}$ and define $\leq$ to be
the restriction of $\leq_{\operatorname{\rm lex}}$ on $\Gamma$. For every
$\gamma\in\Gamma$, let

$z_{\gamma}:=[\gamma]+I.$

Since $\mathcal{N}_{J}=\mathcal{N}_{I}\cap\langle X^{\prime}\rangle$ by Lemma
2.7, the left-most letter of every Lyndon word in $\Gamma$ must belong to
$X^{\prime\prime}$. Therefore, every Lyndon word in $\mathcal{N}_{J}$ is
lexicographically smaller than that in $\Gamma$. Note that $B$ is generated by
$\\{~{}[v]+I~{}|~{}v\in\mathcal{N}_{J}~{}\\}$, Parts (1) and (2) follow by
Proposition 2.4 (3, 4). Compare the basis of $A$ and $B$ that constructed from
$\mathcal{N}_{I}$ and $\mathcal{N}_{J}$ respectively as in Proposition 2.4
(5), it is easy to see that $A$ is a free left $B$-module with a basis as the
given set in Part (3). Since $\operatorname{\rm GKdim}A=\\#(\mathcal{N}_{I})$
and $\operatorname{\rm GKdim}B=\\#(\mathcal{N}_{J})$ by Proposition 2.4 (6),
Part (4) follows directly.

Next we show $A$ is spanned by the given set in Part (3) as a right
$B$-module. For $m\geq 1$, let $\Gamma_{m}$ be the set of all nondecreasing
finite sequences $(\gamma_{1},\cdots,\gamma_{q})$ in $\Gamma$ with
$\sum_{i=1}^{q}\deg(z_{\gamma_{i}})\leq m$. Clearly,

$A_{m}=\bigoplus_{(\gamma_{1},\cdots,\gamma_{q})\in\Gamma_{m}}A_{m}^{(\gamma_{1},\cdots,\gamma_{q})},\text{
where
}A_{m}^{(\gamma_{1},\cdots,\gamma_{q})}:=B_{m-\sum_{i=1}^{q}\deg(z_{\gamma_{i}})}\cdot(z_{\gamma_{1}}\cdots
z_{\gamma_{q}}).$

Let $\leq_{m}$ be the total order on $\Gamma_{m}$ defined as follows. For
$\overline{\gamma}=(\gamma_{1},\cdots,\gamma_{s})$ and
$\overline{\delta}=(\delta_{1},\cdots,\delta_{t})$ in $\Gamma_{m}$,
$\overline{\gamma}<_{m}\overline{\delta}$ if and only if $\overline{\gamma}$
is a proper right part of $\overline{\delta}$ or there exists an integer
$l\leq\min\\{m,n\\}$ such that $\gamma_{m-i}=\delta_{n-i}$ for $i<l$ and
$\gamma_{m-l}<\delta_{n-l}$. By a similar argument as that in Lemma 1.1, it is
not hard to see $\leq_{m}$ is a well-order on $\Gamma_{m}$ with the empty
sequence $\emptyset$ as the smallest element. Note that

$A_{m}^{\emptyset}=B_{m}.$

For any sequence
$(\gamma_{1},\cdots,\gamma_{s},{\small\overbrace{\gamma,\cdots,\gamma}^{p})}\in\Gamma_{m}$,
where $\gamma_{s}<\gamma$ and $p\geq 1$, one has by Part (2) that

$\lambda\cdot(z_{\gamma_{1}}\cdots
z_{\gamma_{s}}z_{\gamma}^{p})\in(z_{\gamma_{1}}\cdots
z_{\gamma_{s}}z_{\gamma}^{p})\cdot\lambda+\sum_{i=0}^{p-1}A^{<\gamma}\cdot
z_{\gamma}^{i},\quad\quad\lambda\in B.$

Since $A^{<\gamma}$ is spanned by

(2.5) $\displaystyle\\{~{}z_{\gamma_{1}}\cdots z_{\gamma_{n}}~{}|~{}n\geq
0,~{}\gamma_{1},\cdots,\gamma_{n}\in\Gamma,~{}\gamma_{1}\leq\cdots\leq\gamma_{n}<\gamma~{}\\}$

as a graded left $B$-module by Lemma 2.6, it follows that

$A_{m}^{(\gamma_{1},\cdots,\gamma_{s},{\tiny\overbrace{\gamma,\cdots,\gamma}^{p})}}\subseteq(z_{\gamma_{1}}\cdots
z_{\gamma_{s}}z_{\gamma}^{p})\cdot
B+\sum\nolimits_{(\delta_{1},\cdots,\delta_{t},{\tiny\overbrace{\gamma,\cdots,\gamma}^{i})})}A_{m}^{(\delta_{1},\cdots,\delta_{t},{\tiny\overbrace{\gamma,\cdots,\gamma}^{i})}},$

where
$(\delta_{1},\cdots,\delta_{t},{\small\overbrace{\gamma,\cdots,\gamma}^{i})}$
runs over elements in $\Gamma_{m}$ with $i<p$ and $\delta_{t}<\gamma$. Note
that

$(\delta_{1},\cdots,\delta_{t},{\small\overbrace{\gamma,\cdots,\gamma}^{i})}\,<_{m}\,(\gamma_{1},\cdots,\gamma_{s},{\small\overbrace{\gamma,\cdots,\gamma}^{p})},$

so by induction on $\Gamma_{m}$ with respect to $\leq_{m}$,
$A_{m}^{(\gamma_{1},\cdots,\gamma_{q})}$ is contained in the span of the given
set in Part (3) as a right $B$-module for all
$(\gamma_{1},\cdots,\gamma_{q})\in\Gamma_{m}$, and therefore $A=\sum_{m\geq
0}A_{m}$ is too.

Finally, we show the given set in Part (3) is right linearly independent over
$B$. Assume not, then

$\sum_{p=0}^{m}\sum_{(\gamma_{1},\cdots,\gamma_{s})}(z_{\gamma_{1}}\cdots
z_{\gamma_{s}}z_{\gamma}^{p})\cdot\lambda_{(\gamma_{1},\cdots,\gamma_{s})}^{(p)}=0,\quad\quad\lambda_{(\gamma_{1},\cdots,\gamma_{s})}^{(p)}\in
B,$

for some $\gamma\in\Gamma$ and $m\geq 1$, where
$(\gamma_{1},\cdots,\gamma_{s})$ runs over all nondecreasing finite sequences
in $\\{~{}\delta\in\Gamma~{}|~{}\delta<\gamma~{}\\}$ and
$\lambda_{(\gamma_{1},\cdots,\gamma_{s})}^{(m)}\neq 0$ for some index
$(\gamma_{1},\cdots,\gamma_{s})$. Note that

$(z_{\gamma_{1}}\cdots
z_{\gamma_{s}}z_{\gamma}^{p})\cdot\lambda_{(\gamma_{1},\cdots,\gamma_{s})}^{(p)}\in\lambda_{(\gamma_{1},\cdots,\gamma_{s})}^{(p)}\cdot(z_{\gamma_{1}}\cdots
z_{\gamma_{s}}z_{\gamma}^{p})+\sum_{i=0}^{p-1}A^{<\gamma}\cdot
z_{\gamma}^{i},$

so

$0\in\sum_{(\gamma_{1},\cdots,\gamma_{s})}\lambda_{(\gamma_{1},\cdots,\gamma_{s})}^{(m)}\cdot(z_{\gamma_{1}}\cdots
z_{\gamma_{s}}z_{\gamma}^{m})+\sum_{i=0}^{m-1}A^{<\gamma}\cdot
z_{\gamma}^{i}.$

Since the set (2.5) spans $A^{<\gamma}$ as a left $B$-module and the given set
in Part (3) is left $B$-linearly independent,
$\lambda_{(\gamma_{1},\cdots,\gamma_{s})}^{(m)}=0$ for all
$(\gamma_{1},\cdots,\gamma_{s})$, a contradiction. This completes the proof of
Part (3). ∎

##  3\. Applications to connected (graded) Hopf algebras

In this section we study the structure of connected (graded) Hopf algebras.

Let us begin by recalling some notations and definitions on Hopf algebras. For
a general Hopf algebra $H$, the usual notations $\Delta_{H}$,
$\varepsilon_{H}$ and $S_{H}$ are employed to denote the comultiplication,
counit and antipode of $H$ respectively. The _coradical_ of a Hopf algebra $H$
is defined to be the sum of all simple subcoalgebras of $H$. It is denoted by
$H_{(0)}$. Also, the _coradical filtration_ of $H$ ([16, Section 5.2]) is
denoted by $\\{H_{(n)}\\}_{n\geq 0}$. Note that the notations for coradical
and coradical filtration we used differ from that of [16]. For an element
$a\in H$, let $\rho_{H}(a)$ be the smallest number $n$ such that $a\in
H_{(n)}$. A Hopf algebra is called _connected_ if its coradical is one-
dimensional.

By a _graded Hopf algebra_ we mean a Hopf algebra $H$ equipped with a grading
$H=\bigoplus_{n\geq 0}H_{n}$ such that $H$ is both a graded algebra and a
graded coalgebra, and the antipode preserves the given grading. Such a graded
Hopf algebra is called _connected_ if $H_{0}=k$. It is easy to show that, when
$H$ is a connected graded Hopf algebra, one has
$\ker\varepsilon_{H}=H_{+}:=\bigoplus_{n\geq 1}H_{n}$ and

$\Delta_{H}(x)\in x\otimes 1+1\otimes x+(H_{+}\otimes H_{+})_{n},\quad x\in
H_{n}.$

So the comultiplication of a connected graded Hopf algebra is a pseudo-
comultiplication. Clearly connected graded Hopf algebras are connected Hopf
algebras. For a connected Hopf algebra $H$, the coradical filtration
$\\{H_{(n)}\\}_{n\geq 0}$ is a Hopf algebra filtration [16, p. 62], whence the
associated graded space $\operatorname{\rm gr}_{c}(H):=\bigoplus_{n\geq
0}H_{(n)}/H_{(n-1)}$ is a connected graded Hopf algebra in the natural way.

######  Theorem 3.1.

Let $H$ be a connected graded Hopf algebra and $K$ a graded Hopf subalgebra.
Then there is an indexed family $(z_{\gamma})_{\gamma\in\Gamma}$ of
homogeneous elements of $H$ and a total order $\leq$ on $\Gamma$ such that

  1. (1)

$\Delta(z_{\gamma})\in 1\otimes z_{\gamma}+z_{\gamma}\otimes
1+(H_{+}^{<\gamma}\otimes H_{+}^{<\gamma})_{\deg(z_{\gamma})}$ for every index
$\gamma\in\Gamma$, where $H^{<\gamma}$ is the subalgebra of $H$ generated by
$K$ and $\\{~{}z_{\delta}~{}|~{}\delta\in\Gamma,~{}\delta<\gamma~{}\\}$;

  2. (2)

$[z_{\gamma},H^{<\gamma}]\subseteq H^{<\gamma}$ for every index
$\gamma\in\Gamma$;

  3. (3)

$\\{~{}z_{\gamma_{1}}\cdots z_{\gamma_{n}}~{}|~{}n\geq
0,~{}\gamma_{1},\cdots,\gamma_{n}\in\Gamma,~{}\gamma_{1}\leq\cdots\leq\gamma_{n}~{}\\}$
is a homogeneous basis of $H$ as a left graded $K$-module as well as a right
graded $K$-module;

  4. (4)

$\operatorname{\rm GKdim}H=\operatorname{\rm GKdim}K+\\#(\Gamma)$, where
$\\#(\Gamma)$ is the number of elements of $\Gamma$.

###### Proof.

It is a direct consequence of Theorem 2.2. ∎

######  Corollary 3.2.

Let $H$ be a commutative connected graded Hopf algebra and $K$ a graded Hopf
subalgebra. Then $H$ is isomorphic as a graded algebra to the graded
polynomial algebra over $K$ in some family of graded variables.

###### Proof.

It is a direct connsequence of Theorem 3.1 (3). ∎

######  Corollary 3.3.

Let $H$ be a commutative Hopf algebra which is connected graded as an algebra.
Let $K$ be a Hopf subalgebra as well as a graded subalgebra. If
$(H_{+})^{2}\cap K=(K_{+})^{2}$ then $H$ is isomorphic as a graded algebra to
the graded polynomial algebra over $K$ in some family of graded variables.

###### Proof.

Let $\mathfrak{m}:=H_{+}=\sum_{n\geq 1}H_{n}$ and
$\mathfrak{n}:=K_{+}=\sum_{n\geq 1}K_{n}$. By the argument of [4, Lemma 2.1
(2)], to see the result we may assume $\mathfrak{m}=\ker(\varepsilon_{H})$, in
which case $\mathfrak{n}=\ker(\varepsilon_{K})$. By [9, Lemma 3.3],
$A=\bigoplus_{i\geq 0}\mathfrak{m}^{i}/\mathfrak{m}^{i+1}$ and
$B=\bigoplus_{i\geq 0}\mathfrak{n}^{i}/\mathfrak{n}^{i+1}$ are connected
graded Hopf algebras with comultiplication and counit defined in [9, Lemma
3.2]. Since $A$ (resp. $B$) is generated by
$A_{1}=\mathfrak{m}/\mathfrak{m}^{2}$ (resp.
$B_{1}=\mathfrak{n}/\mathfrak{n}^{2}$) and commutative, Theorem 3.1 (3) tells
us that $A$ (resp. $B$) is freely generated as a commutative algebra on any
choice of bases for $A_{1}=\mathfrak{m}/\mathfrak{m}^{2}$ (resp.
$B_{1}=\mathfrak{n}/\mathfrak{n}^{2}$). Assume $\mathfrak{m}^{2}\cap
K=\mathfrak{n}^{2}$. Then the canonical map $B_{1}\to A_{1}$ is injective, so
the canonical map $B\to A$ are injective homomorphism of graded Hopf algebras.
Therefore, we may consider $B$ as a graded Hopf subalgebra of $A$. Note that
$A$ is actually $\mathbb{Z}^{2}$-graded as an algebra with
$A_{(i,j)}=(\mathfrak{m}^{i}/\mathfrak{m}^{i+1})_{j}$, and $B$ a
$\mathbb{Z}^{2}$-graded subalgebra. Choose an indexed family of homogeneous
elements $\\{a_{\gamma}\\}_{\gamma\in\Gamma}$ of $\mathfrak{m}$ such that the
family of cosets $\\{~{}\overline{a_{\gamma}}~{}\\}_{\gamma\in\Gamma}$, where
$\overline{a_{\gamma}}:=a_{\gamma}+\mathfrak{m}^{2}$, forms a basis of a
complement of $B_{1}$ in $A_{1}$. Fix a total order $\leq$ on $\Gamma$. Then
the set

$\\{~{}\overline{a_{\gamma_{1}}}\cdots\overline{a_{\gamma_{n}}}~{}|~{}n\geq
0,~{}\gamma_{1},\cdots,\gamma_{n}\in\Gamma,~{}\gamma_{1}\leq\cdots\leq\gamma_{n}~{}\\}$

is a homogeneous basis of $A$ as a $\mathbb{Z}^{2}$-graded $B$-module. By a
standard application of filtered-graded method and the observation that
$(\mathfrak{m}^{i})_{r}=0$ for integers $0\leq r<i$, it is not hard to
conclude that the set

$\\{~{}a_{\gamma_{1}}\cdots a_{\gamma_{n}}~{}|~{}n\geq
0,~{}\gamma_{1},\cdots,\gamma_{n}\in\Gamma,~{}\gamma_{1}\leq\cdots\leq\gamma_{n}~{}\\}$

forms a homogeneous basis of $H$ as a graded $K$-module. Since $H$ is
commutative, $H$ is a graded polynomial algebra over $K$ in some family of
graded variables. ∎

Let $H$ be a connected Hopf algebra and $K$ a Hopf subalgebra. By a _filter
basis_ of $H$ as a left (resp. right) $K$-module we mean a basis
$(a_{i})_{i\in I}$ of $H$ as a left (resp. right) $K$-module such that

$H_{(n)}=\sum_{i\in I}K_{(n-\rho_{H}(a_{i}))}\cdot a_{i}\quad(\text{resp.
}H_{(n)}=\sum_{i\in I}a_{i}\cdot K_{(n-\rho_{H}(a_{i}))}),\quad n\geq 0.$

######  Corollary 3.4.

Let $H$ be a connected Hopf algebra and $K$ a Hopf subalgebra. Then there is
an indexed family $(x_{\gamma})_{\gamma\in\Gamma}$ of elements of
$\ker(\varepsilon_{H})$ and a total order $\leq$ on $\Gamma$ such that
$\operatorname{\rm GKdim}H=\operatorname{\rm GKdim}K+\\#(\Gamma)$, where
$\\#(\Gamma)$ is the number of elements of $\Gamma$, and
$\\{~{}x_{\gamma_{1}}\cdots x_{\gamma_{n}}~{}|~{}n\geq
0,~{}\gamma_{1},\cdots,\gamma_{n}\in\Gamma,~{}\gamma_{1}\leq\cdots\leq\gamma_{n}~{}\\}$
is a filter basis of $H$ as a left $K$-module as well as a right $K$-module.

###### Proof.

By [16, Lemma 5.2.12], we may consider $\operatorname{\rm gr}_{c}(K)$ as a
Hopf subalgebra of $\operatorname{\rm gr}_{c}(H)$ in the natural way. Choose
an indexed family $\\{z_{\gamma}\\}_{\gamma\in\Gamma}$ of homogeneous elements
of $\operatorname{\rm gr}_{c}(H)$ of positive degrees and a total order $\leq$
on $\Gamma$ as in Theorem 3.1 for the pair $(\operatorname{\rm
gr}_{c}(H),\operatorname{\rm gr}_{c}(K))$. Then, for each index $\gamma$ with
$\deg(z_{\gamma})=n_{\gamma}$, pick an element $x_{\gamma}\in
H_{(n_{\gamma})}$ such that $x_{\gamma}+H_{(n_{\gamma}-1)}=z_{\gamma}$ in
$\operatorname{\rm gr}_{c}(H)$. Replacing $x_{\gamma}$ by
$x_{\gamma}-\varepsilon_{H}(x_{\gamma})$, one may assume
$x_{\gamma}\in\ker(\varepsilon_{H})$. Note that
$\rho_{H}(x_{\gamma})=n_{\gamma}$ and $\rho_{H}(x_{\gamma_{1}}\cdots
x_{\gamma_{n}})=\rho_{H}(x_{\gamma_{1}})+\cdots+\rho_{H}(x_{\gamma_{n}})$. By
[17, Lemma I.5.1 (3) ], the first two parts follow directly. In addition, by
[24, Proposition 3.4] and Theorem 3.1 (4), one has $\operatorname{\rm
GKdim}H=\operatorname{\rm GKdim}\operatorname{\rm gr}_{c}(H)=\operatorname{\rm
GKdim}\operatorname{\rm gr}_{c}(K)+\\#(\Gamma)=\operatorname{\rm
GKdim}K+\\#(\Gamma)$. ∎

######  Corollary 3.5.

Let $H$ be a commutative connected Hopf algebra and $K$ a Hopf subalgebra.
Then $H$ is isomorphic as an algebra to the polynomial algebra over $K$ in
some family of variables.

###### Proof.

It is a direct consequence of Corollary 3.4 (1). ∎

Now we turn to consider the sturcture of connected (graded) Hopf algebras of
finite GK dimension. First we recall the definition of Hopf Ore extensions and
iterated Hopf Ore extensions.

######  Definition 3.6.

Let $R$ be a Hopf algebra and $n\geq 2$.

  1. (1)

A Hopf algebra $H$ is called a _Hopf Ore extension_ (_HOE_ , for short) of $R$
if it contains $R$ as a Hopf subalgebra and is an Ore extension of $R$ as
algebras, that is $H=R[X;\sigma,\delta]$ for some algebra automorphism
$\sigma$ of $R$ and $\sigma$-derivation $\delta$ of $R$. We call the HOE $H$
is _of derivation type_ if $\sigma$ can be chosen to be the identity map of
$R$.

  2. (2)

A Hopf algebra $H$ is called an _( $n$-step) iterated Hopf Ore extension_ (_(
$n$-step) IHOE_, for short) of $R$ if it contains $R$ as a Hopf subalgebra and
there is a chain of Hopf subalgebras

$R=H^{0}\subset H^{1}\subset\cdots\subset H^{n}=H$

such that $H^{i}$ is an Ore extension of $H^{i-1}$ as algebras for
$i=1,\cdots,n$. We call the IHOE $H$ is _of derivation type_ if all sucessive
HOEs’ can be chosen to be of derivation type .

By convention, a $0$-step IHOE of $R$ is just $R$ itself; a $1$-step IHOE of
$R$ is just a HOE of $R$. One similarly define the concept of _graded HOE_ and
_graded $n$-step IHOE_ for graded Hopf algebras.

######  Theorem 3.7.

Let $H$ be a connected graded Hopf algerba of GK dimension $d$ and $K$ a
graded Hopf subalgebra of GK dimension $e$. Then $H$ is a graded $(d-e)$-step
IHOE of $K$ of derivation type.

###### Proof.

Let $l=d-e$. Fix an indexed family $\\{z_{\gamma}\\}_{\gamma\in\Gamma}$ of
homogeneous elements of $H$ of positive degrees and a total order $\leq$ on
$\Gamma$ as in Theorem 3.1. Since $\\#(\Gamma)=l$, we may assume
$\Gamma=\\{~{}1,\cdots,l~{}\\}$ equipped with the natural order. Let
$H^{0}=K$; let $H^{i}$ be the subalgebra of $H$ generated by $K$ and
$z_{1},\cdots,z_{i}$ for $i=1,\cdots,l$. Clearly, $H^{i}$ are all clearly Hopf
subalgebras of $H$ by Theorem 3.1 (1) and induction on $i$. Also, one may
readily conclude by Theorem 3.1 (2, 3) and induction on $i$ that the set

$\\{~{}1,~{}z_{i},~{}z_{i}^{2},\cdots~{}\\}$

is a homogeneous basis of $H^{i}$ as a graded left $H^{i-1}$-module for
$i=1,\cdots,d$. In addition, according to 3.1 (2), one may define a graded map
$\delta_{i}:H^{i-1}\to H^{i-1}$ of degree $\deg(z_{i})$, $i=1,\cdots,d$, by

$f\mapsto z_{i}\cdot f-f\cdot z_{i},\quad f\in H^{i-1}.$

It is easy to check that $\delta_{i}$ is a derivation of $H^{i-1}$ and
$H^{i}=H^{i-1}[z_{i};\delta_{i}].$ ∎

######  Corollary 3.8.

Let $H$ be a connected graded Hopf algerba of GK dimension $d$. Then $H$ is a
$(d-e)$-step graded IHOE of the universal enveloping algebra of $P(H)$ of
derivation type, where $P(H)$ is the primitive space of $H$ and $e$ is the
dimension of $P(H)$.

###### Proof.

Note that the universal enveloping algebra of $P(H)$ is canonically a graded
Hopf subalgebra of $H$ and of GK dimension $e$. The result then follows
directly by Theorem 3.7. ∎

######  Proposition 3.9.

Let $H$ be a connected Hopf algebra of GK dimension $d$ and $K$ a Hopf
subalgebra of GK dimension $d-1$. Let $r$ be the smallest number such that
$H_{(r)}\neq K_{(r)}$. Assume that $K$ is generated by $K_{(r-1)}$ as an
algebra. Then $H$ is a Hopf Ore extension of $K$.

###### Proof.

By Corollary 3.4, there’s an element $x\in H$ satisfying the following
conditions:

  1. (1)

$\\{1,x,x^{2},\cdots\\}$ forms a basis of $H$ as a left $K$-module;

  2. (2)

$\\{1,x,x^{2},\cdots\\}$ forms a basis of $H$ as a right $K$-module;

  3. (3)

$H_{(n)}=\sum_{i\geq 0}K_{(n-\rho_{H}(x)\cdot i)}\cdot x^{i}=\sum_{i\geq
0}x^{i}\cdot K_{(n-\rho_{H}(x)\cdot i)}$ for $n\geq 0$.

Then clearly, $\rho_{H}(x)=r$,

$\displaystyle x\cdot K_{(r-1)}\subseteq K_{(2r-1)}=K_{(r-1)}\cdot
x+K_{(2r-1)}\quad{and}\quad K_{(r-1)}\cdot x\subseteq K_{(2r-1)}=x\cdot
K_{(r-1)}+K_{(2r-1)}.$

Since $K$ is generated by $K_{(r-1)}$, it follows that

$\displaystyle x\cdot K\subseteq K\cdot x+K\quad{and}\quad K\cdot x\subseteq
x\cdot K+K.$

By (1) and $x\cdot K\subseteq K\cdot x+K$, one obtains that
$H=K[x;\sigma,\delta]$ for some endomorphism $\sigma$ of $K$ and
$\sigma$-derivation $\delta$ of $K$. It remains to show $\sigma$ is an
automorphism. If $\sigma(a)=0$ then $x\cdot a=\delta(a)$, and hence $a=0$ by
(2). So $\sigma$ is injective. Since $K\cdot x\subseteq x\cdot K+K$, for any
$a\in K$ one has

$a\cdot x=x\cdot a_{1}+a_{2}=\sigma(a_{1})\cdot x+\delta(a_{1})+a_{2}$

for some $a_{1},a_{2}\in K$. Then $a=\sigma(a_{1})$ by (1). So $\sigma$ is
surjective. ∎

Acknowledgments. The second-named author would like to thank Houyi Yu for
helpful discussions, particularly for calling his attention to [15]. G.-S.
Zhou is supported by the NSFC (Grant Nos. 11971141 & 11871186) and K. C. Wong
Magna Fund in Ningbo University.

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