The Ore Extension of Group-Cograded Hopf Coquasigroups

Abstract

The aim of this paper is the Ore extension of group-cograded Hopf coquasigroups. This paper first shows a categorical interpretation and some examples of group-cograded Hopf coquasigroups, and then provides a necessary and sufficient condition for the Ore extension of group-cograded Hopf coquasigroups to be group-cograded Hopf coquasigroups. Finally, a certain isomorphism between Ore extensions are considered.

1. Introduction

A version of the non-commutative polynomial ring, introduced by Ore in [1], has become one of the most basic and useful constructions in ring theory. Such a polynomial ring is now referred to as the Ore extension. From the perspective of quantum groups [2] and Hopf algebras [3,4], the Ore extension is very important for constructing examples of Hopf algebras that are neither commutative nor cocommutative. These extensions are also called skew polynomial rings.

In recent years, the Ore extension has been widely applied to other branches of Hopf algebra. Many new examples (usually finite dimensional) with special properties are constructed through Ore extension, such as pointed Hopf algebras, co-Frobenius Hopf algebras, Artin-Schelter regular algebraS [5] and quasitriangular Hopf algebras.

In [6], Aleksandr N. Panov introduced the Hopf–Ore extensions and gave the necessary and sufficient conditions for the Ore extension of a Hopf algebra to be a Hopf algebra. Zhao Lihui and Lu Diming [7] generalized the theory of Hopf–Ore extensions on Hopf algebras to multiplier Hopf algebras. Li Chao and Li Jinqi [8] introduced the concept of Ambikew Hopf $\pi$-coalgebras, which can be obtained from Hopf $\pi$-coalgebras through twice Ore extensions. Jiao Zhengming [9] further generalized the Hopf-Ore extensions theory to Hopf coquasigroups. Afterwards, Wang Dingguo and Lu Daowei [10] extended the Ore extension of Hopf algebras to the Hopf group coalgebras.

Therefore, there is a natural question: Does the Ore extension still hold for group-cograded Hopf coquasigroups? This is the motivation of our paper. For this question, we provide a positive answer in this paper. The first matter we have to resolve is how to define the extension of group-cograded Hopf coquasigroups. This paper is organized as follows.

In Section 2, we recall some concepts that will be used in the following section, such as Hopf coquasigroups, Ore extensions, and the Turaev category.

In Section 3, we provide the definition of group-cograded Hopf coquasigroups and provide a categorical interpretation and some interesting examples.

In Section 4, we introduce the concept of the Ore extension for group-cograded Hopf coquasigroups and provide an equivalent condition that characterizes them as still being group-cograded Hopf coquasigroups. Finally, an isomorphism theorem of group-cograded Hopf coquasigroups is presented.

2. Preliminaries

In this section, we review some basic definitions that need to be used in the following, such as Hopf coquasigroups, the Ore extension of algebras, the Ore extension of Hopf coquasigroups, and the Turaev category. Throughout this article, all spaces we considered are over a fixed field k.

2.1. Hopf Coquasigroups and Ore Extensions

First, let us recall the definition of the Ore extension of an algebra from [2]. Let A be an algebra, and let $\tau$ be an algebra endomorphism of A. A linear endomorphism $\delta$ of A is called an $\tau$-derivation of A if for any $a,b\in A$,

$$\begin{array}{c}\hfill \delta \left(ab\right)=\delta \left(a\right)b+\tau \left(a\right)\delta \left(b\right).\end{array}$$

This condition implies $\delta \left(1\right)=0$.

The Ore extension $R=A[y;\tau ,\delta ]$ of A is an algebra generated by the algebra A and the variable y with the relation

$$\begin{array}{c}\hfill ya=\tau \left(a\right)y+\delta \left(a\right),\end{array}$$

(1)

for all $a\in A$.

Recall from [11] that a Hopf coquasigroup H is a unital associative algebra equipped with counital $ϵ:H\to k$ and algebra homomorphisms $\mathsf{\Delta }:H\to H\otimes H$ and linear map $S:H\to H$ such that

$$\begin{array}{ccc}& & (m\otimes id)(S\otimes id\otimes id)(id\otimes \mathsf{\Delta })\mathsf{\Delta }=1\otimes id=(m\otimes id)(id\otimes S\otimes id)(id\otimes \mathsf{\Delta })\mathsf{\Delta },\hfill \end{array}$$

(2)

$$\begin{array}{ccc}& & (id\otimes m)(id\otimes id\otimes S)(\mathsf{\Delta }\otimes id)\mathsf{\Delta }=id\otimes 1=(id\otimes m)(id\otimes S\otimes id)(\mathsf{\Delta }\otimes id)\mathsf{\Delta }.\hfill \end{array}$$

(3)

In this paper, we use Sweedler notation. The conditions (2) and (3) are expressed as

$$\begin{array}{ccc}& & S\left(h_{\left(1\right)}\right)h_{\left(2\right)\left(1\right)}\otimes h_{\left(2\right)\left(2\right)}=1\otimes h=h_{\left(1\right)}S\left(h_{\left(2\right)\left(1\right)}\right)\otimes h_{\left(2\right)\left(2\right)},\hfill \\ & & h_{\left(1\right)\left(1\right)}\otimes S\left(h_{\left(1\right)\left(2\right)}\right)h_{\left(2\right)}=h\otimes 1=h_{\left(1\right)\left(1\right)}\otimes h_{\left(1\right)\left(2\right)}S\left(h_{\left(2\right)}\right),\hfill \end{array}$$

for all $h\in H$.

Let H be a Hopf coquasigroup. $R=H[y;\tau ,\delta ]$ is called the Hopf coquasigroup-Ore extension in [9] if R is a Hopf coquasigroup with sub-Hopf coquasigroup H, and there are $r_1,r_2\in H$ such that $\mathsf{\Delta }\left(y\right)=y\otimes r_1+r_2\otimes y$.

2.2. The Turaev Category

The Turaev category as an special symmetric monoidal category was introduced by Caenepeel in [12]. Let K be a commutative ring. A Turaev K-module is a couple $\underline{M}=(X,M)$, where X is a set, and $M={\left(M_x\right)}_{x\in X}$ is a family of K-modules indexed by X. A morphism between two T-modules $(X,M)$ and $(Y,N)$ is a couple $\underline{\phi }=(f,\phi )$, where $f:Y\to X$ is a function, and $\phi ={({\phi }_y:M_{f_{\left(y\right)}}\to N_y)}_{y\in Y}$ is a family of linear maps indexed by Y. The composition of $\underline{\phi }:\underline{M}\to \underline{N}$ and $\underline{\phi }:\underline{N}\to \underline{P}=(Z,P)$ is defined as follows:

$$\begin{array}{ccc}& & \underline{\psi }\circ \underline{\phi }=(f\circ g,{({\psi }_z\circ {\phi }_{g\left(z\right)})}_{z\in Z}).\hfill \end{array}$$

The category of Turaev K-modules is called the Turaev category and denoted by ${\mathcal{T}}_K$. When K is the field k, then ${\mathcal{T}}_K={\mathcal{T}}_k$.

3. Group-Cograded Hopf Coquasigroups

Let G be a group with the unit 1. First, we introduce group-cograded Hopf coquasigroups.

Definition 1. 

$H=({⨁}_{p\in G}{H}_p,m,\mu ,\mathsf{\Delta },ϵ,S)$ is called a group-cograded Hopf coquasigroup over field k if the following conditions hold:

(1) 

Each $H_p$ is an unital associative k-algebra with multiplication $m_p$ and unit ${\mu }_p$.

$H_p{H}_q=0$ whenever $p,q\in G$ and $p\ne q$, and ${\mu }_p\left(1_k\right)=1_p$.

(2) 

Comultiplication Δ is a family of algebra homomorphisms ${\{{\mathsf{\Delta }}_{p,q}:H_{pq}\to H_p\otimes H_q\}}_{p,q\in G}$, and counit $ϵ:H_1\to k$ is an algeba homomorphism in the sense that, for $p\in G,$

$$\begin{array}{ccc}\hfill (i{d}_{H_p}\otimes ϵ){\mathsf{\Delta }}_{p,1}& =& (ϵ\otimes i{d}_{H_p}){\mathsf{\Delta }}_{1,p}=i{d}_{H_p},\hfill \\ \hfill ϵ\left(1_1\right)& =& 1_k.\hfill \end{array}$$

(3) 

Antipode S is an algebra anti-homomorphism with $S={\{S_p:H_p\to H_{p^{-1}}\}}_{p\in G}$, and for any $p,q\in G$,

$$\begin{array}{ccc}& & (m_q\otimes i{d}_{H_p})(S_{q^{-1}}\otimes i{d}_{H_q}\otimes i{d}_{H_p})(i{d}_{H_{q^{-1}}}\otimes {\mathsf{\Delta }}_{q,p}){\mathsf{\Delta }}_{q^{-1},qp}={\mu }_q\otimes i{d}_{H_p}\hfill \\ & =& (m_q\otimes i{d}_{H_p})(i{d}_{H_q}\otimes S_{q^{-1}}\otimes i{d}_{H_p})(i{d}_{H_q}\otimes {\mathsf{\Delta }}_{q^{-1},p}){\mathsf{\Delta }}_{q,q^{-1}p},\hfill \\ & & (i{d}_{H_p}\otimes m_q)(i{d}_{H_p}\otimes i{d}_{H_q}\otimes S_{q^{-1}})({\mathsf{\Delta }}_{p,q}\otimes i{d}_{H_{q^{-1}}}){\mathsf{\Delta }}_{pq,q^{-1}}=i{d}_{H_p}\otimes {\mu }_q\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& =& (i{d}_{H_p}\otimes m_q)(i{d}_{H_p}\otimes S_{q^{-1}}\otimes i{d}_{H_q})({\mathsf{\Delta }}_{p,q^{-1}}\otimes i{d}_{H_q}){\mathsf{\Delta }}_{p{q}^{-1},q}.\hfill \end{array}$$

(5)

Remark 1. 

(1) 

In the following, we use the Sweelder notation for the comultiplication: For any $p,q\in G$ and $h_{pq}\in H_{pq}$,

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,q}\left(h_{pq}\right)=h_{(1,p)}\otimes h_{(2,q)}.\end{array}$$

The conditions (4) and (5) are then expressed as

$$\begin{array}{ccc}& & S_{q^{-1}}\left(h_{(1,q^{-1})}\right)h_{(21,q)}\otimes h_{(22,p)}=1_q\otimes h_p=h_{(1,q)}{S}_{q^{-1}}\left(h_{(21,q^{-1})}\right)\otimes h_{(22,p)},\hfill \\ & & h_{(11,p)}\otimes h_{(12,q)}{S}_{q^{-1}}\left(h_{(2,q^{-1})}\right)=h_p\otimes 1_q=h_{(11,p)}\otimes S_{q^{-1}}\left(h_{(12,q^{-1})}\right)h_{(2,q)}.\hfill \end{array}$$

(2) 

If the comultiplication Δ of group-cograded Hopf coquasigroup H is coassociative, then H is actually a Hopf group-coalgebra introduced in [13,14].

(3) 

If $H={⨁}_{p\in Q}{H}_p$ is a group-cograded Hopf coquasigroup with each component $H_p$ is finite dimensional, then $H^{*}={⨁}_{p\in Q}{H}_p^{*}$ is a group-graded Hopf quasigroup introduced in [15], with Q being a group.

As shown in [12,16], Hopf group coalgbras introduced in [17] (or group-graded Hopf quasigroups) are Hopf algebras (or Hopf quasigroups) in the Turaev category ${\mathcal{T}}_k$. We give a categorical interpretation for group-cograded Hopf coquasigroups as follows.

Proposition 1. 

If $H={⨁}_{p\in G}{H}_p$ is a group-cograded Hopf coquasigroup, then $(G,H)$ is a Hopf coquasigroup in the Turaev category ${\mathcal{T}}_k$.

Proof of Proposition 1. 

As H is a group-cograded Hopf coquasigroup and G is a group, then we can provide $\underline{H}=(G,H)$ a counital coalgebra structure $(\underline{H},\underline{\mathsf{\Delta }},\underline{ϵ})$ by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \underline{H}& \stackrel{\underline{ϵ}}{⟶}\underline{k}\hfill \\ \hfill G& \stackrel{i}{⟵}(\ast )\hfill \\ \hfill H_1=H_{i\left(e\right)}& \stackrel{ϵ}{⟶}k,\hfill \end{array}\mathrm{and}\begin{array}{cc}\hfill \underline{H}& \stackrel{\underline{\mathsf{\Delta }}}{⟶}\underline{H}\otimes \underline{H}\hfill \\ \hfill G& \stackrel{\eta }{⟵}G\times G\hfill \\ \hfill H_{pq}=H_{\eta (p,q)}& \stackrel{{\mathsf{\Delta }}_{p,q}}{⟶}{H}_p\otimes H_q.\hfill \end{array}\end{array}$$

We have

$$\begin{array}{c}\hfill \begin{array}{ccccc}\hfill \underline{H}& \stackrel{\underline{\mathsf{\Delta }}}{⟶}\hfill & \hfill \underline{H}\otimes \underline{H}& \stackrel{\underline{ϵ}\otimes \underline{id}}{⟶}\hfill & \hfill \underline{H}\\ \hfill G& \stackrel{\eta }{⟵}\hfill & \hfill G\times G& \stackrel{(i,G)}{⟵}\hfill & \hfill G\\ \hfill H_p& \stackrel{{\mathsf{\Delta }}_{e,p}}{⟶}\hfill & \hfill H_1\otimes H_p& \stackrel{ϵ\otimes id}{⟶}\hfill & \hfill H_p,\end{array}\mathrm{and}\begin{array}{ccccc}\hfill \underline{H}& \stackrel{\underline{\mathsf{\Delta }}}{⟶}\hfill & \hfill \underline{H}\otimes \underline{H}& \stackrel{\underline{id}\otimes \underline{ϵ}}{⟶}\hfill & \hfill \underline{H}\\ \hfill G& \stackrel{\eta }{⟵}\hfill & \hfill G\times G& \stackrel{(G,\mathrm{i})}{⟵}\hfill & \hfill G\\ \hfill H_p& \stackrel{{\mathsf{\Delta }}_{p,\mathrm{e}}}{⟶}\hfill & \hfill H_p\otimes H_1& \stackrel{id\otimes ϵ}{⟶}\hfill & \hfill H_p.\end{array}\end{array}$$

We can also provide $(G,H)$ a coalgebra structure $(\underline{H},\underline{\mathsf{\Delta }},\underline{ϵ})$ by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \underline{k}& \stackrel{\underline{\mu }}{⟶}\underline{H}\hfill \\ \hfill (\ast )& \stackrel{e}{⟵}G\hfill \\ \hfill k& \stackrel{{\mu }_p}{⟶}{H}_p,\hfill \end{array}\mathrm{and}\begin{array}{cc}\hfill \underline{H}\otimes \underline{H}& \stackrel{\underline{\mathrm{m}}}{⟶}\underline{H}\hfill \\ \hfill G\times G& \stackrel{\delta }{⟵}G\hfill \\ \hfill H_p\otimes H_p& \stackrel{{\mu }_p}{⟶}{H}_p,\hfill \end{array}\end{array}$$

such that $(\underline{\mathsf{\Delta }},\underline{ϵ})$ are algebra maps.

Let $s:G\to G,s\left(g\right)=g^{-1}$. We can then consider a map $\underline{S}=(s,S)$ in the Turaev category as the antipode of $\underline{H}$, where S is the antipode of group-cograded Hopf coquasigroup H. Next, we will confirm that $\underline{S}$ satisfies the conditions (2) and (3) of a Hopf quasigroup.

$$\begin{array}{c}\hfill \begin{array}{ccccccccc}\hfill \underline{H}& \stackrel{\underline{\mathsf{\Delta }}}{⟶}\hfill & \hfill \underline{H}\otimes \underline{H}& \stackrel{\underline{id}\otimes \underline{\mathsf{\Delta }}}{⟶}\hfill & \hfill \underline{H}\otimes \underline{H}\otimes \underline{H}& \stackrel{\underline{S}\otimes \underline{id}\otimes \underline{id}}{⟶}\hfill & \hfill \underline{H}\otimes \underline{H}\otimes \underline{H}& \stackrel{\underline{m}\otimes \underline{id}}{⟶}\hfill & \hfill \underline{H}\otimes \underline{H}\\ \hfill G& \stackrel{\eta }{⟵}\hfill & \hfill G\times G& \stackrel{(G,\eta )}{⟵}\hfill & \hfill G\times G\times G& \stackrel{(s,G,G)}{⟵}\hfill & \hfill G\times G\times G& \stackrel{(\delta ,G)}{⟵}\hfill & \hfill G\times G\\ \hfill H_p& \stackrel{{\mathsf{\Delta }}_{q,q^{-1}p}}{⟶}\hfill & \hfill D& \stackrel{id\otimes {\mathsf{\Delta }}_{q^{-1},p}}{⟶}\hfill & \hfill E& \stackrel{S_q\otimes id\otimes id}{⟶}\hfill & \hfill F& \stackrel{m_{q^{-1}}\otimes id}{⟶}\hfill & \hfill H_{q^{-1}}\otimes H_p\end{array}\end{array}$$

where D is $H_q\otimes H_{q^{-1}p}$, E is $H_q\otimes H_{q^{-1}}\otimes H_p$, F is $H_{q^{-1}}\otimes H_{q^{-1}}\otimes H_p$, and

$$\begin{array}{ccc}\hfill \underline{k}\otimes \underline{H}& \stackrel{\underline{\mu }\otimes \underline{id}}{⟶}& \underline{H}\otimes \underline{H}\hfill \\ \hfill (*)\times G& \stackrel{i\otimes G}{⟵}& G\times G\hfill \\ \hfill k\otimes H_p& \stackrel{{\mu }_p\otimes id}{⟶}& H_p\otimes H_p.\hfill \end{array}$$

Since H is a group-cograded Hopf coquasigroup, we have $(m\otimes id)(S\otimes id\otimes id)(id\otimes \mathsf{\Delta })\mathsf{\Delta }=1\otimes id$. Thus, the left-hand side of Equation (2) holds, and the right-hand side is similar.

The proof of Equation (3) for $\underline{H}$ is similar to the first one. □

In the end of this section, we provide some examples of group-cograded Hopf coquasigroups.

Example 1. 

Let H be a Hopf coquasigroup. Let $H^G={⨁}_{p\in G}{H}_p$, and let G be the homomorphism group of H, where, for each $p\in G$, the algebra $H_p$ is a copy of H. Fix an identification isomorphism of algebras $i_p:H\to H_p$. For $p,q\in G$, we define a comultiplication ${\mathsf{\Delta }}_{p,q}:H_{pq}\to H_p\otimes H_q$ by

$$\begin{array}{ccc}& & {\mathsf{\Delta }}_{p,q}\left(i_{pq}\left(h\right)\right)=i_p\left(h_{\left(1\right)}\right)\otimes i_q\left(h_{\left(2\right)}\right),\hfill \end{array}$$

where $h\in H$. The counit $ϵ:H_1\to k$ is defined by ${ϵ}_1\left(i_1\left(h\right)\right)=ϵ\left(h\right)\in k$ for $h\in H$. For $p\in G$, the antipode $S_p:H_p\to H_{p^{-1}}$ is given by

$$\begin{array}{ccc}& & S_p\left(i_p\left(h\right)\right)=i_{p^{-1}}\left(S\left(h\right)\right),\hfill \end{array}$$

where $h\in H$. It is easy to confirm that $H^G$ is a group-cograded Hopf coquasigroup.

Example 2. 

Let $H={⨁}_{p\in Q}{H}_p$ be a Q-graded Hopf quasigroup introduced in [15], where each component $H_p$ is finitely dimensional and Q is a group. Thus, $H^{*}={⨁}_{p\in Q}{H}_p^{*}$ is a group-cograded Hopf coquasigroup.

Example 3. 

Let $H^G$ be a group-cograded Hopf coquasigroup introduced in Example 1. Set ${\tilde{H}}^G$ as the same family of algebra ${(H_p=H)}_{p\in G}$ with the same counit, the comultiplication ${\tilde{\mathsf{\Delta }}}_{p,q}:H_{pq}\to H_p\otimes H_q$, and the antipode ${\tilde{S}}_p:H_p\to H_{p^{-1}}$ defined by

$$\begin{array}{ccc}& & {\tilde{\mathsf{\Delta }}}_{p,q}\left(i_{pq}\left(h\right)\right)=i_p\left(q\left(h_{\left(1\right)}\right)\right)\otimes i_q\left(h_{\left(2\right)}\right),\hfill \\ & & {\tilde{S}}_p\left(i_p\left(h\right)\right)=i_{p^{-1}}\left(p\left(S\left(h\right)\right)\right)=i_{p^{-1}}\left(S\left(p\left(h\right)\right)\right),\hfill \end{array}$$

where $h\in H$. ${\tilde{H}}^G$ then becomes a group-cograded Hopf coquasigroup following from the definition.

4. The Ore Extension

The aim of this section is to prove a criterion for an Ore extension of a group-cograded Hopf coquasigroup to be a group-cograded Hopf coquasigroup with a coproduct satisfying the relation (6) below. First, we introduce the definition of an Ore extension of a group-cograded Hopf coquasigroup.

Definition 2. 

Let $H={⨁}_{p\in G}{H}_p$ be a group-cograded Hopf coquasigroup. The family $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ of k-spaces is called a group-cograded Hopf coquasigroup-Ore extension if $R={⨁}_{p\in G}{R}_p$ is also a group-cograded Hopf coquasigroup, where, for any $p\in G$, $H_p[y_p;{\tau }_p,{\delta }_p]$ is the Ore extension of $H_p$, and there are $r_p^1,r_p^2\in H_p$ such that

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,q}\left(y_{pq}\right)=y_p\otimes r_q^2+r_p^1\otimes y_q.\end{array}$$

Note that $R_1=H_1[y_1;{\tau }_1,{\delta }_1]\}$ is the Hopf coquasigroup-Ore extension in the sense of [9]. By Definition 1, for $y_q\in H_q$, we have

$$\begin{array}{ccc}& & (m_q\otimes i{d}_{H_p})(S_{q^{-1}}\otimes i{d}_{H_q}\otimes i{d}_{H_p})(i{d}_{H_{q^{-1}}}\otimes {\mathsf{\Delta }}_{q,p}){\mathsf{\Delta }}_{q^{-1},qp}\left(y_p\right)\hfill \\ & =& 1_q\otimes y_p=(m_q\otimes i{d}_{H_p})(i{d}_{H_q}\otimes S_{q^{-1}}\otimes i{d}_{H_p})(i{d}_{H_q}\otimes {\mathsf{\Delta }}_{q^{-1},p}){\mathsf{\Delta }}_{q,q^{-1}p}\left(y_p\right),\hfill \\ & & (i{d}_{H_p}\otimes m_q)(i{d}_{H_p}\otimes i{d}_{H_q}\otimes S_{q^{-1}})({\mathsf{\Delta }}_{p,q}\otimes i{d}_{H_{q^{-1}}}){\mathsf{\Delta }}_{pq,q^{-1}}\left(y_p\right)\hfill \\ & =& y_p\otimes 1_q=(i{d}_{H_p}\otimes m_q)(i{d}_{H_p}\otimes S_{q^{-1}}\otimes i{d}_{H_q})({\mathsf{\Delta }}_{p,q^{-1}}\otimes i{d}_{H_q}){\mathsf{\Delta }}_{p{q}^{-1},q}\left(y_p\right).\hfill \end{array}$$

That is,

$$\begin{array}{ccc}& & S_{q^{-1}}\left(y_{q^{-1}}\right)r_{(1,q)}^2\otimes r_{(2,p)}^2+S_{q^{-1}}\left(r_{q^{-1}}^1\right)y_q\otimes r_p^2+S_{q^{-1}}\left(r_{q^{-1}}^1\right)r_q^1\otimes y_p\hfill \\ & =& 1_q\otimes y_p=y_q{S}_{q^{-1}}\left(r_{(1,q^{-1})}^2\right)\otimes r_{(2,p)}^2+r_q^1{S}_{q^{-1}}\left(y_{q^{-1}}\right)\otimes r_p^2+r_q^1{S}_{q^{-1}}\left(r_{q^{-1}}^1\right)\otimes y_p,\hfill \\ & & y_p\otimes r_q^2{S}_{q^{-1}}\left(r_{q^{-1}}^2\right)+r_p^1\otimes y_q{S}_{q^{-1}}\left(r_{q^{-1}}^2\right)+r_{(1,p)}^1\otimes r_{(2,q)}^1{S}_{q^{-1}}\left(y_{q^{-1}}\right)\hfill \\ & =& y_p\otimes 1_q=y_p\otimes S_{q^{-1}}\left(r_{q^{-1}}^2\right)r_q^2+r_p^1\otimes S_{q^{-1}}\left(y_{q^{-1}}\right)r_q^2+r_{(1,p)}^1\otimes S_{q^{-1}}\left(r_{(2,q^{-1})}^1\right)y_q.\hfill \end{array}$$

Therefore, we have

$$\begin{array}{ccc}& & S_{q^{-1}}\left(r_{q^{-1}}^1\right)r_q^1=r_q^1{S}_{q^{-1}}\left(r_{q^{-1}}^1\right)=1_q,r_q^2{S}_{q^{-1}}\left(r_{q^{-1}}^2\right)=S_{q^{-1}}\left(r_{q^{-1}}^2\right)r_q^2=1_q,\hfill \\ & & S_{q^{-1}}\left(y_{q^{-1}}\right)r_{(1,q)}^2\otimes r_{(2,p)}^2+S_{q^{-1}}\left(r_{q^{-1}}^1\right)y_q\otimes r_p^2\hfill \\ & =& y_q{S}_{q^{-1}}\left(r_{(1,q^{-1})}^2\right)\otimes r_{(2,p)}^2+r_q^1{S}_{q^{-1}}\left(y_{q^{-1}}\right)\otimes r_p^2=0,\hfill \\ & & r_p^1\otimes y_q{S}_{q^{-1}}\left(r_{q^{-1}}^2\right)+r_{(1,p)}^1\otimes r_{(2,q)}^1{S}_{q^{-1}}\left(y_{q^{-1}}\right)\hfill \\ & =& r_p^1\otimes S_{q^{-1}}\left(y_{q^{-1}}\right)r_q^2+r_{(1,p)}^1\otimes S_{q^{-1}}\left(r_{(2,q^{-1})}^1\right)y_q=0.\hfill \end{array}$$

If $r^1,r^2$ satisfy the following conditions

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left(r_{pq}^1\right)& =& r_p^1\otimes r_q^1,\hfill \\ \hfill {\mathsf{\Delta }}_{p,q}\left(r_{pq}^2\right)& =& r_p^2\otimes r_q^2,\hfill \end{array}$$

then we have

$$\begin{array}{ccc}\hfill {\left(r_q^i\right)}^{-1}& =& S_{q^{-1}}\left(r_{q^{-1}}^i\right),i=1,2,\hfill \\ \hfill (S_{q^{-1}}\left(y_{q^{-1}}\right)r_q^2+S_{q^{-1}}\left(r_{q^{-1}}^1\right)y_q)\otimes r_p^2& =& (y_q{S}_{q^{-1}}\left(r_{q^{-1}}^2\right)+r_q^1{S}_{q^{-1}}\left(y_{q^{-1}}\right))\otimes r_p^2=0,\hfill \\ \hfill r_p^1\otimes (y_q{S}_{q^{-1}}\left(r_{q^{-1}}^2\right)+r_q^1{S}_{q^{-1}}\left(y_{q^{-1}}\right))& =& r_p^1\otimes (S_{q^{-1}}\left(y_{q^{-1}}\right)r_q^2+S_{q^{-1}}\left(r_{q^{-1}}^1\right)y_q)=0.\hfill \end{array}$$

Replacing the generating elements $y_p$ by $y_p^{\prime }=y_p{\left(r_p^2\right)}^{-1}$ and $r_p^1{\left(r_p^2\right)}^{-1}$ by $r_p$, we see that

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left(y_{pq}^{\prime }\right)& =& {\mathsf{\Delta }}_{p,q}\left(y_{pq}{\left(r_{pq}^2\right)}^{-1}\right)\hfill \\ & =& {\mathsf{\Delta }}_{p,q}\left(y_{pq}\right){\mathsf{\Delta }}_{p,q}\left({\left(r_{pq}^2\right)}^{-1}\right)\hfill \\ & =& (y_p\otimes r_q^2+r_p^1\otimes y_q)({\left(r_p^2\right)}^{-1}\otimes {\left(r_q^2\right)}^{-1})\hfill \\ & =& y_p{\left(r_p^2\right)}^{-1}\otimes r_q^2{\left(r_q^2\right)}^{-1}+r_p^1{\left(r_p^2\right)}^{-1}\otimes y_q{\left(r_q^2\right)}^{-1}\hfill \\ & =& y_p^{\prime }\otimes 1_q+r_p\otimes y_q^{\prime }.\hfill \end{array}$$

Therefore, we always assume in what follows that the elements $y_p$ in the group-cograded Hopf coquasigroup-Ore extension satisfy the relations

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,q}\left(y_{pq}\right)=y_p\otimes 1_q+r_p\otimes y_q,\end{array}$$

(6)

for some elements $r_p\in H_p$. As usual, $A{d}_{r_p}\left(h\right)=r_{p}h{S}_{p^{-1}}\left(r_{p^{-1}}\right)=r_{p}h{\left(r_p\right)}^{-1}$.

Lemma 1. 

Let $H={⨁}_{p\in G}{H}_p$ be a group-cograded Hopf coquasigroup. If $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ is the group-cograded Hopf coquasigroup-Ore extension of H, then

$$\begin{array}{c}\hfill S_{p^{-1}}\left(y_{p^{-1}}\right)=-{\left(r_p\right)}^{-1}{y}_p,\end{array}$$

(7)

where ${\left(r_p\right)}^{-1}=S_{p^{-1}}\left(r_{p^{-1}}\right)$.

Proof of Lemma 1. 

By (6), we have

$$\begin{array}{c}\hfill m_p(S_{p^{-1}}\otimes i{d}_{H_p}){\mathsf{\Delta }}_{p^{-1},p}\left(y_1\right)=ϵ\left(y_1\right)1_p=0.\end{array}$$

Thus,

$$\begin{array}{c}\hfill S_{p^{-1}}\left(y_{p^{-1}}\right)+S_{p^{-1}}\left(r_{p^{-1}}\right)y_p=0.\end{array}$$

Therefore, $S_{p^{-1}}\left(y_{p^{-1}}\right)=-S_{p^{-1}}\left(r_{p^{-1}}\right)y_p=-{\left(r_p\right)}^{-1}{y}_p$. □

Following the above results, we obtain the main theorem of this paper.

Theorem 1. 

Let $H={⨁}_{p\in G}{H}_p$ be a group-cograded Hopf coquasigroup. The group-cograded Hopf coquasigroup $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ is then the group-cograded Hopf coquasigroup-Ore extension if and only if

(1) 

there is a character $\chi :H_1\to k$ such that for any $p\in G$, $h_p\in H_p$

$$\begin{array}{c}\hfill {\tau }_p\left(h_p\right)=\chi \left(h_{(1,1)}\right)h_{(2,p)};\end{array}$$

(8)

(2) 

the following relations hold:

$$\begin{array}{ccc}\hfill \chi \left(h_{(1,1)}\right)h_{(21,p)}\otimes h_{(22,q)}& =& A{d}_{r_p}\left(h_{(1,p)}\right)\chi \left(h_{(21,1)}\right)\otimes h_{(22,q)}\hfill \\ & =& \chi \left(h_{(11,1)}\right)h_{(12,p)}\otimes h_{(2,q)};\hfill \end{array}$$

(9)

(3) 

the ${\tau }_p$-derivation ${\delta }_p$ satisfies the relation

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(h_{p,q}\right)\right)={\delta }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)}+r_p{h}_{(1,p)}\otimes {\delta }_q\left(h_{(2,q)}\right).\end{array}$$

(10)

Proof of Theorem 1. 

The proof is divided into three parts. At Step 1, we show that the comultiplication $\mathsf{\Delta }={⨁}_{p\in G}{\mathsf{\Delta }}_p$ can be extended to $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ by (6) if and only if the relations (8)–(10) hold. At Step 2, we prove that $R_1$ admits an extension of the counit from $H_1$ (in fact this has been proved in [9] ). At Step 3, we show that R has antipode S extending the antipode ${S|}_H$ by (7).

1.

Comultiplication. Assume that the comultiplication ${\mathsf{\Delta }|}_H$ can be extended to $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ (6). The homomorphism Δ then preserves the relation

$$\begin{array}{c}\hfill y_p{h}_p={\tau }_p\left(h_p\right)y_p+{\delta }_p\left(h_p\right),\end{array}$$

(11)

for any $p\in G$ and $h_p\in H_p$, i.e.,

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,q}\left(y_{pq}\right){\mathsf{\Delta }}_{p,q}\left(h_{pq}\right)={\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right){\mathsf{\Delta }}_{p,q}\left(y_{pq}\right)+{\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(h_{pq}\right)\right),\end{array}$$

(12)

for any $h_{pq}\in H_{pq}$. We have

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left(y_{pq}\right){\mathsf{\Delta }}_{p,q}\left(h_{pq}\right)& =& (y_p\otimes 1_q+r_p\otimes y_q)(h_{(1,p)}\otimes h_{(2,q)})\hfill \\ & =& y_p{h}_{(1,p)}\otimes h_{(2,q)}+r_p{h}_{(1,p)}\otimes y_q{h}_{(2,q)}\hfill \\ & =& {\tau }_p\left(h_{(1,p)}\right)y_p\otimes h_{(2,q)}+{\delta }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)}\hfill \\ & & +r_p{h}_{(1,p)}\otimes {\tau }_q\left(h_{(2,q)}\right)y_q+r_p{h}_{(1,p)}\otimes {\delta }_q\left(h_{(2,q)}\right)\hfill \\ & =& ({\tau }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)})(y_p\otimes 1_q)\hfill \\ & & +(r_p{h}_{(1,p)}{r}_p^{-1}\otimes {\tau }_q\left(h_{(2,q)}\right))(r_p\otimes y_q)\hfill \\ & & +{\delta }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)}+r_p{h}_{(1,p)}\otimes {\delta }_q\left(h_{(2,q)}\right),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right){\mathsf{\Delta }}_{p,q}\left(y_{pq}\right)+{\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(h_{pq}\right)\right)\hfill \\ \hfill =& & {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)(y_p\otimes 1_q+r_p\otimes y_q)+{\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(h_{pq}\right)\right)\hfill \\ \hfill =& & {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)(y_p\otimes 1_q)+{\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)(r_p\otimes y_q)+{\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(h_{pq}\right)\right).\hfill \end{array}$$

It is clear that Δ preserves (11) if and only if the following relations hold:

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)& =& {\tau }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)& =& A{d}_{r_p}\left(h_{(1,p)}\right)\otimes {\tau }_q\left(h_{(2,q)}\right),\hfill \\ \hfill {\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(h_{pq}\right)\right)& =& {\delta }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)}+r_p{h}_{(1,p)}\otimes {\delta }_q\left(h_{(2,q)}\right),\hfill \end{array}$$

(14)

for any $h\in H_{pq}$. The last equation coincides with (10).

We now show that (13) and (14) imply (8) and (9). Let $\chi \left(h_1\right):=ϵ\left({\tau }_1\left(h_1\right)\right)$. It is clear that $\chi \left(h_1\right)\in k$. One can regard χ as a mapping $\chi :H_1\to k$. If, for any $p\in G$, τ is an endomorphism, it follows that

$$\begin{array}{ccc}\hfill \chi \left(h_1{g}_1\right)& =& ϵ\left({\tau }_1\left(h_1{g}_1\right)\right)=ϵ\left({\tau }_1\left(h_1\right)\right)ϵ\left({\tau }_1\left(g_1\right)\right)=\chi \left(h_1\right)\chi \left(g_1\right),\hfill \\ \hfill \chi (h_1+g_1)& =& ϵ\left({\tau }_1(h_1+g_1)\right)=ϵ\left({\tau }_1\left(h_1\right)\right)+ϵ\left({\tau }_1\left(g_1\right)\right)=\chi \left(h_1\right)+\chi \left(g_1\right).\hfill \end{array}$$

For ${\tau }_p\left(h_p\right)\in H_p$ by (13), we have

$$\begin{array}{c}\hfill (i{d}_{H_1}\otimes {\mathsf{\Delta }}_{1,p}){\mathsf{\Delta }}_{1,p}\left({\tau }_p\left(h_p\right)\right)={\tau }_1\left(h_{(1,1)}\right)\otimes h_{(21,1)}\otimes h_{(22,p)}.\end{array}$$

Therefore, by (4), we have

$$\begin{array}{c}\hfill S_1\left({\tau }_1\left(h_{(1,1)}\right)\right)\left(h_{(21,1)}\right)\otimes h_{(22,p)}=1_1\otimes {\tau }_p\left(h_p\right)={\tau }_1\left(h_{(1,1)}\right)S_1\left(\left(h_{(21,1)}\right)\right)\otimes h_{(22,p)}.\end{array}$$

(15)

Using the formula above, one can recover τ from χ:

$$\begin{array}{ccc}\hfill \chi \left(h_{(1,1)}\right)h_{(2,p)}& =& ϵ\left({\tau }_1\left(h_{(1,1)}\right)\right)h_{(2,p)}\hfill \\ & =& ϵ\left({\tau }_1\left(h_{(1,1)}\right)\right)ϵ\left(h_{(21,1)}\right)h_{(22,p)}\hfill \\ & =& ϵ\left({\tau }_1\left(h_{(1,1)}\right)\right)ϵ\left(S_1\left(h_{(21,1)}\right)\right)h_{(22,p)}\hfill \\ & =& ϵ\left({\tau }_1\left(h_{(1,1)}\right)S_1\left(h_{(21,1)}\right)\right)h_{(22,p)}\hfill \\ & \stackrel{\left(15\right)}{=}& ϵ\left(1_1\right){\tau }_p\left(h_p\right)\hfill \\ & =& {\tau }_p\left(h_p\right).\hfill \end{array}$$

This proves (8). Substituting ${\tau }_p\left(h_p\right)$ into (13), we obtain

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)& =& {\tau }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)}\hfill \\ & \stackrel{\left(8\right)}{=}& \chi \left(h_{11,1}\right)h_{(12,p)}\otimes h_{(2,q)},\hfill \end{array}$$

and substituting ${\tau }_p\left(h_p\right)$ into (14), we obtain

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)& \stackrel{\left(8\right)}{=}& {\mathsf{\Delta }}_{p,q}\left(\chi \left(h_{(1,1)}\right)h_{(2,pq)}\right)\hfill \\ & =& \chi \left(h_{(1,1)}\right)h_{(21,p)}\otimes h_{(22,q)}\hfill \\ & \stackrel{\left(14\right)}{=}& A{d}_{r_p}\left(h_{(1,p)}\right)\otimes {\tau }_q\left(h_{(2,q)}\right)\hfill \\ & \stackrel{\left(8\right)}{=}& A{d}_{r_p}\left(h_{(1,p)}\right)\otimes \chi \left(h_{(21,1)}\right)h_{(22,q)}.\hfill \end{array}$$

Hence,

$$\begin{array}{c}\hfill \chi \left(h_{(1,1)}\right)h_{(21,p)}\otimes h_{(22.q)}=A{d}_{r_p}\left(h_{(1,p)}\right)\chi \left(h_{(21,1)}\right)\otimes h_{(22,q)}=\chi \left(h_{(11,1)}\right)h_{(12,p)}\otimes h_{(2,q)}.\end{array}$$

This proves (9). We have proved that the conditions (8)–(10) are necessary conditions of the comultiplication.

If the conditions (8)–(10) hold, then

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)& \stackrel{\left(8\right)}{=}& {\mathsf{\Delta }}_{p,q}\left(\chi \left(h_{(1,1)}\right)h_{(2,pq)}\right)\hfill \\ & =& \chi \left(h_{(1,1)}\right)h_{(21,p)}\otimes h_{(22,q)}\hfill \\ & \stackrel{\left(9\right)}{=}& \chi \left(h_{(11,1)}\right)h_{(12,p)}\otimes h_{(2,q)}\hfill \\ & \stackrel{\left(8\right)}{=}& {\tau }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)},\hfill \\ \hfill {\mathsf{\Delta }}_{p,q}\left({\tau }_{pq}\left(h_{pq}\right)\right)& \stackrel{\left(8\right)}{=}& {\mathsf{\Delta }}_{p,q}\left(\chi \left(h_{(1,1)}\right)h_{(2,pq)}\right)\hfill \\ & =& \chi \left(h_{(1,1)}\right)h_{(21,p)}\otimes h_{(22,q)}\hfill \\ & \stackrel{\left(9\right)}{=}& A{d}_{r_p}\left(h_{(1,p)}\right)\chi \left(h_{(21,1)}\right)\otimes h_{(22,q)}\hfill \\ & \stackrel{\left(8\right)}{=}& A{d}_{r_p}\left(h_{(1,p)}\right)\otimes {\tau }_q\left(h_{(2,q)}\right).\hfill \end{array}$$

This proves that the relations (13) and (14) hold and that the comultiplication ${\mathsf{\Delta }|}_H$ can be extended to a homomorphism $\mathsf{\Delta }:R\to R\otimes R$, and Δ is not required to be coassociative.

2.

$Counit.$ For this section, from [9], we know that, as $R_1$ admits a comultiplication, there is a counit extending ${ϵ|}_{H_1}$ and satisfying $ϵ\left(y_1\right)=0$. It follows that ϵ admits an extension to R if and only if

$$\begin{array}{c}\hfill ϵ\left({\delta }_1\left(h_1\right)\right)=0,\end{array}$$

for any $h_1\in H_1$.

3.

$Antipode.$ Let R be as in Step 1. Recall that $S={⨁}_{p\in G}{S}_p:H_p\to H_{p^{-1}}$, with $S_p$ being an antiautomorphism. If R admits an antipode S that can be extended from H to R by means of (7), then S satisfies (4) and (5) and preserves (11). This means that, for any $h_p\in H_p$,

$$\begin{array}{ccc}& & (m_q\otimes i{d}_{H_p})(S_{q^{-1}}\otimes i{d}_{H_q}\otimes i{d}_{H_p})(i{d}_{H_{q^{-1}}}\otimes {\mathsf{\Delta }}_{q,p}){\mathsf{\Delta }}_{q^{-1},qp}\left(y_p{h}_p\right)=1_q\otimes y_p{h}_p\hfill \\ & & =(m_q\otimes i{d}_{H_p})(i{d}_{H_q}\otimes S_{q^{-1}}\otimes i{d}_{H_p})(i{d}_{H_q}\otimes {\mathsf{\Delta }}_{q^{-1},p}){\mathsf{\Delta }}_{q,q^{-1}p}\left(y_p{h}_p\right),\hfill \\ & & (i{d}_{H_p}\otimes m_q)(i{d}_{H_p}\otimes i{d}_{H_q}\otimes S_{q^{-1}})({\mathsf{\Delta }}_{p,q}\otimes i{d}_{H_{q^{-1}}}){\mathsf{\Delta }}_{pq,q^{-1}}\left(y_{p}h\right)=y_p{h}_p\otimes 1_q\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& & =(i{d}_{H_p}\otimes m_q)(i{d}_{H_p}\otimes S_{q^{-1}}\otimes i{d}_{H_q})({\mathsf{\Delta }}_{p,q^{-1}}\otimes i{d}_{H_q}){\mathsf{\Delta }}_{p{q}^{-1},q}\left(y_p{h}_p\right),\hfill \end{array}$$

(17)

and

$$\begin{array}{c}\hfill S_p\left(h_p\right)S_p\left(y_p\right)=S_p\left(y_p\right)S_p\left({\tau }_p\left(h_p\right)\right)+S_p\left({\delta }_p\left(h_p\right)\right).\end{array}$$

(18)

We will now prove (16) and (17):

$$\begin{array}{ccc}& & (m_q\otimes i{d}_{H_p})(S_{q^{-1}}\otimes i{d}_{H_q}\otimes i{d}_{H_p})(i{d}_{H_{q^{-1}}}\otimes {\mathsf{\Delta }}_{q,p}){\mathsf{\Delta }}_{q^{-1},qp}\left(y_p{h}_p\right)\hfill \\ & \stackrel{\left(11\right)}{=}& (m_q\otimes i{d}_{H_p})(S_{q^{-1}}\otimes i{d}_{H_q}\otimes i{d}_{H_p})(i{d}_{H_{q^{-1}}}\otimes {\mathsf{\Delta }}_{q,p})\hfill \\ & & {\mathsf{\Delta }}_{q^{-1},qp}({\tau }_p\left(h_p\right)y_p+{\delta }_p\left(h_p\right))\hfill \\ & \stackrel{\left(6\right)\mathrm{and}\left(13\right)}{=}& (m_q\otimes i{d}_{H_p})(S_{q^{-1}}\otimes i{d}_{H_q}\otimes i{d}_{H_p})(i{d}_{H_{q^{-1}}}\otimes {\mathsf{\Delta }}_{q,p})\hfill \\ & & ({\tau }_{q^{-1}}\left(h_{(1,q^{-1})}\right)y_{q^{-1}}\otimes h_{(2,qp)}+{\tau }_{q^{-1}}\left(h_{(1,q^{-1})}\right)r_{q^{-1}}\otimes h_{(2,qp)}{y}_{qp}\hfill \\ & & +{\mathsf{\Delta }}_{q^{-1},qp}\left({\delta }_p\left(h_p\right)\right))\hfill \\ & =& (m_q\otimes i{d}_{H_p})(S_{q^{-1}}\otimes i{d}_{H_q}\otimes i{d}_{H_p})(i{d}_{H_{q^{-1}}}\otimes {\mathsf{\Delta }}_{q,p})\hfill \\ & & ({\tau }_{q^{-1}}\left(h_{(1,q^{-1})}\right)y_{q^{-1}}\otimes h_{(2,qp)}+{\tau }_{q^{-1}}\left(h_{(1,q^{-1})}\right)r_{q^{-1}}\otimes h_{(2,qp)}{y}_{qp})\hfill \\ & & +(m_q\otimes i{d}_{H_p})(S_{q^{-1}}\otimes i{d}_{H_q}\otimes i{d}_{H_p})(i{d}_{H_{q^{-1}}}\otimes {\mathsf{\Delta }}_{q,p}){\mathsf{\Delta }}_{q^{-1},qp}\left({\delta }_p\left(h_p\right)\right)\hfill \\ & \stackrel{\left(4\right)}{=}& S_{q^{-1}}\left(y_{q^{-1}}\right)S_{q^{-1}}\left({\tau }_{q^{-1}}\left(h_{(1,q^{-1})}\right)\right)h_{(21,q)}\otimes h_{(22,p)}\hfill \\ & & +S_{q^{-1}}\left(r_{q^{-1}}\right)S_{q^{-1}}\left({\tau }_{q^{-1}}\left(h_{(1,q^{-1})}\right)\right)h_{(21,q)}{y}_q\otimes h_{(22,p)}\hfill \\ & & +S_{q^{-1}}\left(r_{q^{-1}}\right)S_{q^{-1}}\left({\tau }_{q^{-1}}\left(h_{(1,q^{-1})}\right)\right)h_{(21,q)}{r}_q\otimes h_{(22,p)}{y}_p+1_q\otimes {\delta }_p\left(h_p\right)\hfill \\ & \stackrel{\left(15\right)}{=}& S_{q^{-1}}\left(y_{q^{-1}}\right)\otimes {\tau }_p\left(h\right)+S_{q^{-1}}\left(r_{q^{-1}}\right)y_q\otimes {\tau }_p\left(h\right)+S_{q^{-1}}\left(r_{q^{-1}}\right)r_q\otimes {\tau }_p\left(h\right)y_p\hfill \\ & & +1_q\otimes {\delta }_p\left(h_p\right)\hfill \\ & =& S_{q^{-1}}\left(y_{q^{-1}}\right)\otimes {\tau }_p\left(h_p\right)+S_{q^{-1}}\left(r_{q^{-1}}\right)y_q\otimes {\tau }_p\left(h_p\right)+1_q\otimes {\tau }_p\left(h\right)y_p+1_q\otimes {\delta }_p\left(h_p\right)\hfill \\ & =& (S_{q^{-1}}\left(y_{q^{-1}}\right)+S_{q^{-1}}\left(r_{q^{-1}}\right)y_q)\otimes {\tau }_p\left(h_p\right)+1_q\otimes ({\tau }_p\left(h_p\right)y_p+{\delta }_p\left(h_p\right))\hfill \\ & =& 1_q\otimes y_p{h}_p.\hfill \end{array}$$

This proves (16). Relation (17) can be proved similarly.

For (18), it follows from (7) that, for any $h\in H_p$,

$$\begin{array}{ccc}\hfill -S_p\left(h_p\right){\left(r_{p^{-1}}\right)}^{-1}{y}_{p^{-1}}& =& -{\left(r_{p^{-1}}\right)}^{-1}{y}_{p^{-1}}{S}_p\left({\tau }_p\left(h_p\right)\right)+S_p\left({\delta }_p\left(h_p\right)\right),\hfill \\ \hfill -S_p\left(h_p\right){\left(r_{p^{-1}}\right)}^{-1}{y}_{p^{-1}}& =& -{\left(r_{p^{-1}}\right)}^{-1}{\tau }_{p^{-1}}\left(S_p\left({\tau }_p\left(h_p\right)\right)\right)y_{p^{-1}}\hfill \\ & & -{\left(r_{p^{-1}}\right)}^{-1}{\delta }_{p^{-1}}\left(S_p\left({\tau }_p\left(h_p\right)\right)\right)+S_p\left({\delta }_p\left(h_p\right)\right).\hfill \end{array}$$

Condition (18) holds if and only if the following two conditions hold:

$$\begin{array}{ccc}\hfill S_p\left(h_p\right){\left(r_{p^{-1}}\right)}^{-1}& =& {\left(r_{p^{-1}}\right)}^{-1}{\tau }_{p^{-1}}\left(S_p\left({\tau }_p\left(h_p\right)\right)\right),\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill r_{p^{-1}}{S}_p\left({\delta }_p\left(h_p\right)\right)& =& {\delta }_{p^{-1}}\left(S_p\left({\tau }_p\left(h_p\right)\right)\right).\hfill \end{array}$$

(20)

We will now prove (19). We have

$$\begin{array}{ccc}& & {\tau }_{p^{-1}}\left(S_p\left({\tau }_p\left(h_p\right)\right)\right)\hfill \\ & \stackrel{\left(8\right)}{=}& {\tau }_{p^{-1}}\left(S_p\left(\chi \left(h_{(1,1)}\right)h_{(2,p)}\right)\right)\hfill \\ & =& \chi \left(h_{(1,1)}\right){\tau }_{p^{-1}}\left(S_p\left(h_{(2,p)}\right)\right)\hfill \\ & \stackrel{\left(8\right)}{=}& \chi \left(h_{(1,1)}\right)\chi \left(S_p{\left(h_{(2,p)}\right)}_{(1,1)}\right)S_p{\left(h_{(2,p)}\right)}_{(2,p^{-1})}\hfill \\ & =& \chi \left(h_{(1,1)}\right)\chi \left(S_1\left(h_{(22,1)}\right)\right)S_p\left(h_{(21,p)}\right)\hfill \\ & =& \chi \left(S_1\left(h_{(22,1)}\right)\right)S_p\left(\chi \left(h_{(1,1)}\right)h_{(21,p)}\right)\hfill \\ & \stackrel{\left(9\right)}{=}& \chi \left(S_1\left(h_{(22,1)}\right)\right)S_p\left(A{d}_{r_p}\left(h_{(1,p)}\right)\chi \left(h_{(21,1)}\right)\right)\hfill \\ & =& S_p\left(A{d}_{r_p}\left(h_{(1,p)}\right)\right)\chi \left(h_{(21,1)}{S}_1\left(h_{(22,1)}\right)\right)\hfill \\ & =& S_p\left(A{d}_{r_p}\left(h_p\right)\right)\hfill \\ & =& r_{p^{-1}}{S}_p\left(h_p\right){\left(r_{p^{-1}}\right)}^{-1}\hfill \\ & =& A{d}_{r_{p^{-1}}}\left(S_p\left(h_p\right)\right).\hfill \end{array}$$

Our next objective is to prove (20). It follows from (8) that

$$\begin{array}{c}\hfill S_p\left({\tau }_p\left(h_p\right)\right)=S_p\left(\chi \left(h_{(1,1)}\right)h_{(2,p)}\right).\end{array}$$

Therefore, (20) can be represented in an equivalent form as

$$\begin{array}{c}\hfill r_{p^{-1}}{S}_p\left({\delta }_p\left(h_p\right)\right)=\chi \left(h_{(1,1)}\right){\delta }_{p^{-1}}\left(S_p\left(h_{(2,p)}\right)\right).\end{array}$$

(21)

We denote $L_p=r_{p^{-1}}{S}_p\left({\delta }_p\left(h_p\right)\right)$ and $M_p=\chi \left(h_{(1,1)}\right){\delta }_{p^{-1}}\left(S_p\left(h_{(2,p)}\right)\right)$.

From (10), we have

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,p^{-1}}\left({\delta }_1\left(h_1\right)\right)={\delta }_p\left(h_{(1,p)}\right)\otimes h_{(2,p^{-1})}+r_p{h}_{(1,p)}\otimes {\delta }_{p^{-1}}\left(h_{(2,p^{-1})}\right),\end{array}$$

and we apply $m_p(i{d}_{H_p}\otimes S_{p^{-1}})$ to the above equality. We thus obtain

$$\begin{array}{ccc}\hfill m_p(i{d}_{H_p}\otimes S_{p^{-1}})\left({\mathsf{\Delta }}_{p,p^{-1}}\left({\delta }_1\left(h_1\right)\right)\right)& =& m_p(i{d}_{H_p}\otimes S_{p^{-1}})({\delta }_p\left(h_{(1,p)}\right)\otimes h_{(2,p^{-1})}\hfill \\ & & +r_p{h}_{(1,p)}\otimes {\delta }_{p^{-1}}\left(h_{(2,p^{-1})}\right)),\hfill \\ \hfill 0=ϵ\left({\delta }_1\left(h_1\right)\right)1_p& =& {\delta }_p\left(h_{(1,p)}\right)S_{p^{-1}}\left(h_{(2,p^{-1})}\right)\hfill \\ & & +r_p{h}_{(1,p)}{S}_{p^{-1}}\left({\delta }_{p^{-1}}\left(h_{(2,p^{-1})}\right)\right).\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill -r_p^{-1}{\delta }_p\left(h_{(1,p)}\right)S_{p^{-1}}\left(h_{(2,p^{-1})}\right)=h_{(1,p)}{S}_{p^{-1}}\left({\delta }_{p^{-1}}\left(h_{(2,p^{-1})}\right)\right).\end{array}$$

(22)

Therefore, for any $h_{p^{-1}}\in H_{p^{-1}}$,

$$\begin{array}{ccc}\hfill L_{p^{-1}}& =& r_p{S}_{p^{-1}}\left({\delta }_{p^{-1}}\left(h_{p^{-1}}\right)\right)\hfill \\ & =& r_p{S}_{p^{-1}}\left(h_{(1,p^{-1})}\right)h_{(21,p)}{S}_{p^{-1}}\left({\delta }_{p^{-1}}\left(h_{(22,p^{-1})}\right)\right)\hfill \\ & \stackrel{\left(22\right)}{=}& -r_p{S}_{p^{-1}}\left(h_{(1,p^{-1})}\right){\left(r_p\right)}^{-1}{\delta }_p\left(h_{(21,p)}\right)S_{p^{-1}}\left(h_{(22,p^{-1})}\right)\hfill \\ & =& -A{d}_{r_p}\left(S_{p^{-1}}\left(h_{(1,p^{-1})}\right)\right){\delta }_p\left(h_{(21,p)}\right)S_{p^{-1}}\left(h_{(22,p^{-1})}\right).\hfill \end{array}$$

On the other hand, for any $h_1\in H_1$, we have $ϵ\left(h_1\right)1_p=h_{(1,p)}{S}_{p^{-1}}\left(h_{(2,p^{-1})}\right)$. The action by ${\delta }_p$ on both sides gives

$$\begin{array}{c}\hfill 0={\delta }_p\left(h_{(1,p)}\right)S_{p^{-1}}\left(h_{(2,p^{-1})}\right)+{\tau }_p\left(h_{(1,p)}\right){\delta }_p\left(S_{p^{-1}}\left(h_{(2,p^{-1})}\right)\right).\end{array}$$

(23)

We then have

$$\begin{array}{ccc}\hfill M_{p^{-1}}& =& \chi \left(h_{(1,1)}\right){\delta }_p\left(S_{p^{-1}}\left(h_{(2,p^{-1})}\right)\right)\hfill \\ & =& \chi \left(h_{(1,1)}\right){\tau }_p\left(S_{p^{-1}}\left(h_{(21,p^{-1})}\right)h_{(221,p)}\right){\delta }_p\left(S_{p^{-1}}\left(h_{(222,p^{-1})}\right)\right)\hfill \\ & =& \chi \left(h_{(1,1)}\right){\tau }_p\left(S_{p^{-1}}\left(h_{(21,p^{-1})}\right)\right){\tau }_p\left(h_{(221,p)}\right){\delta }_p\left(S_{p^{-1}}\left(h_{(222,p^{-1})}\right)\right)\hfill \\ & \stackrel{\left(23\right)}{=}& -\chi \left(h_{(1,1)}\right){\tau }_p\left(S_{p^{-1}}\left(h_{(21,p^{-1})}\right)\right){\delta }_p\left(h_{(221,p)}\right)S_{p^{-1}}\left(h_{(222,p^{-1})}\right)\hfill \\ & =& -{\tau }_p\left(S_{p^{-1}}\left(\chi \left(h_{(1,1)}\right)h_{(21,p^{-1})}\right)\right){\delta }_p\left(h_{(221,p)}\right)S_{p^{-1}}\left(h_{(222,p^{-1})}\right)\hfill \\ & =& -{\tau }_p\left(S_{p^{-1}}\left(\chi \left(h_{(11,1)}\right)h_{(12,p^{-1})}\right)\right){\delta }_p\left(h_{(21,p)}\right)S_{p^{-1}}\left(h_{(22,p^{-1})}\right)\hfill \\ & =& -\chi \left(h_{(11,1)}\right){\tau }_p\left(S_{p^{-1}}\left(h_{(12,p^{-1})}\right)\right){\delta }_p\left(h_{(21,p)}\right)S_{p^{-1}}\left(h_{(22,p^{-1})}\right)\hfill \\ & =& -\chi \left(h_{(11,1)}\right)A{d}_{r_p}\left(S_{p^{-1}}\left(h_{(122,p^{-1})}\right)\right)\chi \left(S_1\left(h_{(121,1)}\right)\right){\delta }_p\left(h_{(21,p)}\right)\hfill \\ & & S_{p^{-1}}\left(h_{(22,p^{-1})}\right)\hfill \\ & =& -\chi \left(h_{(11,1)}{S}_1\left(h_{(121,1)}\right)\right)A{d}_{r_p}\left(S_{p^{-1}}\left(h_{(122,p^{-1})}\right)\right){\delta }_p\left(h_{(21,p)}\right)S_{p^{-1}}\left(h_{(22,p^{-1})}\right)\hfill \\ & =& -A{d}_{r_p}\left(S_{p^{-1}}\left(h_{(1,p^{-1})}\right)\right){\delta }_p\left(h_{(21,p)}\right)S_{p^{-1}}\left(h_{(22,p^{-1})}\right).\hfill \end{array}$$

We therefore conclude that $L_p=M_p$. This proves both the relation (21) and the existence of the antipode.

□

Following this theorem, we can obtain some special results, e.g., Theorem 3.5 in [10] and Theorem 3.3 in [9].

Corollary 1. 

If group-cograded Hopf coquasigroup $H={⨁}_{p\in G}{H}_p$ satisfies coassociativity, then the group-cograded Hopf coquasigroup-Ore extension is the Hopf coquasigroup-Ore extension in the sense of [10]. The main conclusion (i.e., Theorem 1) in this article is Theorem 3.5 in [10].

Corollary 2. 

If the group G is trivial, i.e., $G=\left\{1\right\}$, then $R_1=H_1[y_1;{\tau }_1,{\delta }_1]\}$ is the Hopf coquasigroup-Ore extension in the sense of [9]. The main conclusion (i.e., Theorem 1) in this article is Theorem 3.3 in [9].

Example 4. 

Let Q be a Hopf coquasigroup with grouplike element $r^1,r^2$, and let $H={⨁}_{p\in G}{H}_p$ be a group-cograded Hopf coquasigroup with each branch $H_p=Q$ for any $p\in G$. The Ore extension of Q is $Q[y;\tau ,\delta ]$. Let $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ be a group-cograded Hopf coquasigroup with each branch $R_p=H_p[y_p;{\tau }_p,{\delta }_p]=Q[y;\tau ,\delta ]$. In the following, we show that R is a group-cograded Hopf coquasigroup-Ore extension.

1. 

As Q is a Hopf coquasigroup by Theorem 3.3(1) in [9], we have $\chi :Q\to k$. Since $H_1=H_p=Q$, for any $p\in G$, $h_p\in H_p$, we have

$$\begin{array}{c}\hfill {\tau }_p\left(h_p\right)=\chi \left(h_{(1,1)}\right)h_{(2,p)};\end{array}$$

2. 

Since each branch of H is equal to Q, and for $p,q\in G$, we have

$$\begin{array}{c}\hfill \chi \left(h_{(1,1)}\right)h_{(21,p)}\otimes h_{(22,q)}=\chi \left(h_{(1,1)}\right)h_{(21,1)}\otimes h_{(22,1)}.\end{array}$$

By Theorem 3.3(2) in [9], we have

$$\begin{array}{ccc}\hfill \chi \left(h_{(1,1)}\right)h_{(21,1)}\otimes h_{(22,1)}& =& A{d}_{r_1}\left(h_{(1,1)}\right)\chi \left(h_{(21,1)}\right)\otimes h_{(22,1)}\hfill \\ & =& \chi \left(h_{(11,1)}\right)h_{(12,1)}\otimes h_{(2,1)}.\hfill \end{array}$$

Therefore, we can obtain (9).

3. 

Similarly, for the ${\tau }_p$-derivation ${\delta }_p$, we can easily confirm that

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(h_{p,q}\right)\right)={\delta }_p\left(h_{(1,p)}\right)\otimes h_{(2,q)}+r_p{h}_{(1,p)}\otimes {\delta }_q\left(h_{(2,q)}\right)\end{array}$$

(24)

holds.

Following Theorem 1 in this paper, R is a group-cograded Hopf coquasigroup-Ore extension.

Proposition 2. 

If the group-cograded Hopf coquasigroup $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ is the group-cograded Hopf coquasigroup-Ore extension, then

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(r_{p,q}\right)r_{p,q}^{-1}\right)={\delta }_p\left(r_p\right)r_p^{-1}\otimes 1+r_p\otimes {\delta }_q\left(r_q\right)r_q^{-1}.\end{array}$$

Proof of Proposition 2. 

Since ${\mathsf{\Delta }}_{p,q}\left(r_{pq}^1\right)=r_p^1\otimes r_q^1$ and $r_p=r_p^1{\left(r_p^2\right)}^{-1}$, it follows that

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left(r_{p,q}\right)& =& {\mathsf{\Delta }}_{p,q}\left(r_{pq}^1{\left(r_{pq}^2\right)}^{-1}\right)\hfill \\ & =& {\mathsf{\Delta }}_{p,q}\left(r_{pq}^1\right){\mathsf{\Delta }}_{p,q}\left({\left(r_{pq}^2\right)}^{-1}\right)\hfill \\ & =& (r_p^1\otimes r_q^1)({\left(r_p^2\right)}^{-1}\otimes {\left(r_q^2\right)}^{-1})\hfill \\ & =& r_p^1{\left(r_p^2\right)}^{-1}\otimes r_q^1{\left(r_q^2\right)}^{-1}\hfill \\ & =& r_p\otimes r_q.\hfill \end{array}$$

According to Formula (10), the following can be obtained:

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(r_{p,q}\right)r_{p,q}^{-1}\right)& =& {\mathsf{\Delta }}_{p,q}\left({\delta }_{pq}\left(r_{p,q}\right)\right){\mathsf{\Delta }}_{p,q}\left(r_{p,q}^{-1}\right)\hfill \\ & =& ({\delta }_p\left(r_p\right)\otimes r_q+r_p^2\otimes {\delta }_q\left(r_q\right))(r_p^{-1}\otimes r_q^{-1})\hfill \\ & =& {\delta }_p\left(r_p\right)r_p^{-1}\otimes 1_q+r_p\otimes {\delta }_q\left(r_q\right)r_q^{-1}.\hfill \end{array}$$

□

At the end of this section, let H and $H^{\prime }$ be group-cograded Hopf coquasigroups. Let $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ be the Ore extension of H with ${\mathsf{\Delta }}_{p,q}\left(y_{pq}\right)=y_p\otimes 1_q+r_p\otimes y_q$, and let $R^{\prime }={⨁}_{p\in G}{R}_p^{\prime }={⨁}_{p\in G}{H}_p^{\prime }[y_p^{\prime };{\tau }_p^{\prime },{\delta }_p^{\prime }]$ be the Ore extension of $H^{\prime }$ with ${\mathsf{\Delta }}_{p,q}\left(y_{pq}^{\prime }\right)=y_p^{\prime }\otimes 1_q+r_p^{\prime }\otimes y_q^{\prime }$. We will now discuss the isomorphism between Ore extensions of two group-cograded Hopf coquasigroups.

Theorem 2. 

Let R and $R^{\prime }$ be the group-cograded Hopf coquasigroup-Ore extension of H and $H^{\prime }$, respectively. $R^{\prime }$ is then isomorphic to R if there is an isomorphism $\phi :H\to H^{\prime }$ such that

$$\begin{array}{ccc}& & {\phi }_p\left(r_p\right)=r_p^{\prime },{\tau }_p^{\prime }\left({\phi }_p\left(h_p\right)\right)={\phi }_p\left({\tau }_p\left(h_p\right)\right),\hfill \\ & & {\delta }_p^{\prime }\left({\phi }_p\left(h_p\right)\right)={\phi }_p\left({\delta }_p\left(h_p\right)\right)+{\phi }_p\left({\tau }_p\left(h_p\right)\right)d_p-d_p{\phi }_p\left(h_p\right),\hfill \end{array}$$

where $d_p\in H_p^{\prime }$ such that ${\mathsf{\Delta }}_{p,q}\left(d_{pq}\right)=d_p\otimes 1_q+r_p^{\prime }\otimes d_q$.

Proof of Theorem 2. 

Let ${\overline{\phi }}_p\left(y_p\right)=y_p^{\prime }+d_p$. $\phi$ can then be extended from $\phi :H\to H^{\prime }$ to $\overline{\phi }:R\to R^{\prime }$. For all $h_p\in H_p$, we have

$$\begin{array}{ccc}\hfill {\overline{\phi }}_p\left(y_p{h}_p\right)& =& {\overline{\phi }}_p({\tau }_p\left(h_p\right)y_p+{\delta }_p\left(h_p\right))\hfill \\ & =& {\overline{\phi }}_p\left({\tau }_p\left(h_p\right)\right){\overline{\phi }}_p\left(y_p\right)+{\overline{\phi }}_p\left({\delta }_p\left(h_p\right)\right)\hfill \\ & =& {\overline{\phi }}_p\left({\tau }_p\left(h_p\right)\right)(y_p^{\prime }+d_p)+{\overline{\phi }}_p\left({\delta }_p\left(h_p\right)\right),\hfill \\ \hfill {\overline{\phi }}_p\left(y_p\right){\overline{\phi }}_p\left(h_p\right)& =& (y_p^{\prime }+d_p){\overline{\phi }}_p\left(h_p\right)\hfill \\ & =& y_p^{\prime }{\overline{\phi }}_p\left(h_p\right)+d_p{\overline{\phi }}_p\left(h_p\right)\hfill \\ & =& {\tau }_p^{\prime }\left({\overline{\phi }}_p\left(h_p\right)\right)y_p^{\prime }+{\delta }_p^{\prime }\left({\overline{\phi }}_p\left(h_p\right)\right)+d_p{\overline{\phi }}_p\left(h_p\right)\hfill \\ & =& {\tau }_p^{\prime }\left({\overline{\phi }}_p\left(h_p\right)\right)y_p^{\prime }+{\overline{\phi }}_p\left({\delta }_p\left(h_p\right)\right)+{\overline{\phi }}_p\left({\tau }_p\left(h_p\right)\right)d_p-d_p{\overline{\phi }}_p\left(h_p\right)+d_p{\overline{\phi }}_p\left(h_p\right)\hfill \\ & =& {\tau }_p^{\prime }\left({\overline{\phi }}_p\left(h_p\right)\right)(y_p^{\prime }+d_p)+{\overline{\phi }}_p\left({\delta }_p\left(h_p\right)\right)\hfill \\ & =& {\overline{\phi }}_p\left({\tau }_p\left(h_p\right)\right)(y_p^{\prime }+d_p)+{\overline{\phi }}_p\left({\delta }_p\left(h_p\right)\right).\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{c}\hfill {\overline{\phi }}_p\left(y_p{h}_p\right)={\overline{\phi }}_p\left(y_p\right){\overline{\phi }}_p\left(h_p\right).\end{array}$$

Since

$$\begin{array}{ccc}\hfill {\mathsf{\Delta }}_{p,q}\left({\overline{\phi }}_{pq}\left(y_{pq}\right)\right)& =& {\mathsf{\Delta }}_{p,q}(y_{pq}^{\prime }+d_{pq})\hfill \\ & =& {\mathsf{\Delta }}_{p,q}\left(y_{pq}^{\prime }\right)+{\mathsf{\Delta }}_{p,q}\left(d_{pq}\right)\hfill \\ & =& y_p^{\prime }\otimes 1_q+r_p^{\prime }\otimes y_q^{\prime }+d_p\otimes 1_q+r_p^{\prime }\otimes d_q\hfill \\ & =& (y_p^{\prime }+d_p)\otimes 1_q+r_p^{\prime }\otimes (y_q^{\prime }+d_q),\hfill \\ \hfill ({\overline{\phi }}_p\otimes {\overline{\phi }}_q){\mathsf{\Delta }}_{p,q}\left(y_{pq}\right)& =& ({\overline{\phi }}_p\otimes {\overline{\phi }}_q)(y_p\otimes 1_q+r_p\otimes y_q)\hfill \\ & =& {\overline{\phi }}_q\left(y_p\right)\otimes 1_q+{\overline{\phi }}_p\left(r_p\right)\otimes {\overline{\phi }}_q\left(y_p\right)\hfill \\ & =& (y_p^{\prime }+d_p)\otimes 1_q+r_p^{\prime }\otimes (y_q^{\prime }+d_q),\hfill \end{array}$$

it follows that

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_{p,q}\left({\overline{\phi }}_{pq}\left(y_{pq}\right)\right)=({\overline{\phi }}_p\otimes {\overline{\phi }}_q){\mathsf{\Delta }}_{p,q}\left(y_{pq}\right).\end{array}$$

We then have $R\cong R^{\prime }$. □

5. Conclusions

In this paper, a new algebraic structure is presented, which is crosslinked by Hopf group coalgebras and Hopf coquasigroups. We call it a group-cograded Hopf coquasigroup. Subsequently, we provided the Ore extension of a group-cograded Hopf coquasigroup. For a group-cograded Hopf coquasigroup $H={⨁}_{p\in G}{H}_p$, if $R={⨁}_{p\in G}{R}_p={⨁}_{p\in G}{H}_p[y_p;{\tau }_p,{\delta }_p]$ is a group-cograded Hopf coquasigroup-Ore extension, then $R={⨁}_{p\in G}{R}_p$ is also a group-cograded Hopf coquasigroup. A possible topic for further research is Ore extensions of Multiplier Hopf coquasigroups [18].