Generalized Hopf–Ore Extensions of Hopf Group-Coalgebras

Abstract

In this paper, we introduce the concept of a generalized Hopf–Ore extension of a Hopf group-coalgebra and give the necessary and sufficient conditions for the Ore extension of a Hopf group-coalgebra to be a Hopf group-coalgebra. Moreover, an isomorphism theorem on generalized Hopf group-coalgebra Ore extensions is given and specific cases in a simple type called special Hopf group-coalgebra Ore extensions are also considered. Our results are generalizations of Hopf–Ore extensions on Hopf algebras and also are very useful to construct new Hopf group-coalgebras.

1. Introduction

Ore’s extensions, also known as skew polynomial rings, are a well-studied particular example of ring extensions. Their systematic study began with Ore’s “Theory of Non-Commutative Polynomials” in 1933 [1]. Ore extension plays an important role in the construction of noncommutative rings and quantum groups. In the framework of the classification of finite-dimensional Hopf algebras, the interest in Ore’s extensions has been revived in the late 1990s by Beattie, Dăscălescu, Grünenfelder, Năstăsescu, and Nenciu, who took advantage of this construction to provide many new examples with special properties, such as pointed Hopf algebras, co-Frobenius Hopf algebras, quasitriangular Hopf algebras, and so on (cf. [2,3,4,5,6]). In 2002, following this intense activity, Panov raised the question when an Ore extension of a Hopf algebra is again a Hopf algebra, and Panov [7] introduced the concept of Hopf–Ore extensions and gave its classifications. In 2015, Brown, O’Hagan, Zhang, and Zhuang [8] defined a more general Hopf–Ore extension, gave its classification, and studied the properties of iterated Hopf–Ore extensions. Recently, there have been many studies on Hopf–Ore extensions, References [9,10,11,12,13], and these show that Hopf–Ore extensions play a very important role in the study of infinite-dimensional noetherian Hopf algebras of low Gelfand–Kirillov dimension (cf. [14,15,16,17]).

Hopf group-coalgebras were introduced by V. G. Turaev in [18,19] when he studied the Hennings invariant of main $\pi$-bundle over the three-dimensional manifold. Hopf group-coalgebras as generalizations of the usual coalgebras and Hopf algebras were widely applied in the homotopy quantum field and vector bundles, such as constructing Hennings-like and Kuperberg-like invariants. In [20,21], Virelizier started an algebraic study of this topic, proved that a Hopf group-coalgebra is essentially a Hopf algebra in a Turaev module category, and the concept of quasitriangular Hopf group-coalgebra was introduced. Subsequently, Hopf group-coalgebras have attracted the attention of algebraists, such as [22,23,24,25,26,27]. In 2014, Wang and Lu [28,29] studied the theory of Hopf–Ore extensions on Hopf group-coalgebras. Inspired by the above works, we generalize the theory of generalized Hopf–Ore extensions in [8,12] to Hopf group-coalgebras and give a necessary and sufficient condition for Ore extensions of Hopf group-coalgebras to become Hopf group-coalgebras. Moreover, we study isomorphism between generalized Hopf group-coalgebra Ore extensions.

The paper is organized as follows. In Section 2, we recall the definitions and some basic results which will be used later. In Section 3, we first define the generalized Hopf group-coalgebras Ore extensions. In Theorem 1, we give a sufficient and necessary condition for an Ore extension of a Hopf group-coalgebra to become a Hopf group-coalgebra. We obtain necessary and sufficient conditions for the isomorphics between generalized Hopf group-coalgebra Ore extensions in Proposition 3. In Section 4, we consider these problems on the special Hopf group-coalgebra Ore extension, which is a simple explicit form of generalized Hopf group-coalgebra Ore extensions, and obtain necessary and sufficient conditions in Theorem 2. The isomorphic conditions are given in Proposition 5. Then, we obtain some corollaries and give some examples.

2. Preliminaries

Throughout this paper, let $\pi$ be a discrete group (with neutral element 1) and $\mathbb{k}$ be a base field. All algebras are supposed to be over $\mathbb{k}$. Let ${\tau }_{U,V}:U\otimes V\to V\otimes U$, $u\otimes v\mapsto v\otimes u$, denotes the flip on $\mathbb{k}$-space U and V. If A is an augmented algebra, denote the augmentation ideal of A by $A^{+}$ and write $A^{\times }$ for the unit group of A. We first review some definitions and terminology from [1,7,8,30], etc.

Let A be a $\mathbb{k}$-algebra, $\sigma$ be an endomorphism of A and $\delta \in En{d}_{\mathbb{k}}\left(A\right)$. $\delta$ is called a (left) σ-derivation of A, if $\delta$ satisfies the $\sigma$-Leibniz rule, i.e., $\delta \left(ab\right)=\sigma \left(a\right)\delta \left(b\right)+\delta \left(a\right)b$ for all $a,b\in A$. The Ore extension $B=A[z;\sigma ,\delta ]$ of the $\mathbb{k}$-algebra A is generated by the variable z and the algebra A with the relation

$$za=\sigma \left(a\right)z+\delta \left(a\right),$$

(1)

for all $a\in A$.

In 2003, Panov [7] introduced the concept of Hopf–Ore extensions. Let A and $B=A[z;\sigma ,\delta ]$ be two Hopf algebras. The Hopf algebra B is called the Hopf–Ore extension of A if $\Delta \left(z\right)=z\otimes r_1+r_2\otimes z$ for some $r_1,r_2\in A$ and A is a Hopf subalgebra of B. Recently, Brown, O’Hagan, Zhang, and Zhuang studied the more general Hopf–Ore extensions in [8] and You, Wang, and Chen also discussed a generalized case in [12].

Now, we review some definitions and lemmas about Hopf group-coalgebras.

Definition 1

([20]). A group-coalgebra (or $\pi$-coalgebra) is a family $C={\left\{C_{\alpha }\right\}}_{\alpha \in \pi }$ of $\mathbb{k}$-spaces endowed with a family $\mathbb{k}$-linear maps $\Delta ={\{{\Delta }_{\alpha ,\beta }:C_{\alpha \beta }\to C_{\alpha }\otimes C_{\beta }\}}_{\alpha ,\beta \in \pi }$ (the comultiplication) and a $\mathbb{k}$-linear map $\epsilon :C_1\to \mathbb{k}$ (the counit) such that

(i)

Δ is a coassociative in the sense that, for all $\alpha ,\beta ,\gamma \in \pi$,

$$({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{C_{\gamma }}){\Delta }_{\alpha \beta ,\gamma }=({\mathrm{Id}}_{C_{\alpha }}\otimes {\Delta }_{\beta ,\gamma }){\Delta }_{\alpha ,\beta \gamma };$$

(ii)

$({\mathrm{Id}}_{C_{\alpha }}\otimes \epsilon ){\Delta }_{\alpha ,1}={\mathrm{Id}}_{C_{\alpha }}=(\epsilon \otimes {\mathrm{Id}}_{C_{\alpha }}){\Delta }_{1,\alpha }$ for all $\alpha \in \pi$.

Sweedler’s Notation. We extend the Sweedler notation as follows: for all $\alpha ,\beta \in \pi$ and $c\in C_{\alpha \beta }$, we write ${\Delta }_{\alpha ,\beta }\left(c\right)={\displaystyle \sum _{\left(c\right)}}{c}_{(1,\alpha )}\otimes c_{(2,\beta )}\in C_{\alpha }\otimes C_{\beta }$. For convenience, we usually omit the summation and write ${\Delta }_{\alpha ,\beta }\left(c\right)=c_{(1,\alpha )}\otimes c_{(2,\beta )}$. The coassociativity axiom gives that, $c_{(1,\alpha \beta )(1,\alpha )}\otimes c_{(1,\alpha \beta )(2,\beta )}\otimes c_{(2,\gamma )}=c_{(1,\alpha )}\otimes c_{(2,\beta \gamma )(1,\beta )}\otimes c_{(2,\beta \gamma )(2,\gamma )}$, for all $\alpha ,\beta ,\gamma \in \pi$ and $c\in C_{\alpha \beta \gamma }$. This element in $C_{\alpha }\otimes C_{\beta }\otimes C_{\gamma }$ is written as $c_{(1,\alpha )}\otimes c_{(2,\beta )}\otimes c_{(3,\gamma )}$.

Definition 2

([20]). A Hopf group-coalgebra (or Hopf $\pi$-coalgebra) is a group-coalgebra $H=({\left\{H_{\alpha }\right\}}_{\alpha \in \pi },\Delta ,\epsilon )$ endowed with a family $S=\{S_{\alpha }:H_{\alpha }\to H_{{\alpha }^{-1}}\}$ of $\mathbb{k}$-linear maps (the antipode) such that

(i)

Each $H_{\alpha }$ is an algebra with multiplication $m_{\alpha }$ and $1_{\alpha }$;

(ii)

$\epsilon :H_1\to \mathbb{k}$ and ${\Delta }_{\alpha ,\beta }:H_{\alpha \beta }\to H_{\alpha }\otimes H_{\beta }$ (for all $\alpha ,\beta \in \pi$) are algebra homomorphisms;

(iii)

$\forall \alpha \in \pi ,m_{\alpha }(S_{{\alpha }^{-1}}\otimes {\mathrm{Id}}_{H_{\alpha }}){\Delta }_{{\alpha }^{-1},\alpha }=\epsilon 1_{\alpha }=m_{\alpha }({\mathrm{Id}}_{H_{\alpha }}\otimes S_{{\alpha }^{-1}}){\Delta }_{\alpha ,{\alpha }^{-1}}:H_1\to H_{\alpha }$.

Lemma 1

([20]). Let $H=({\left\{H_{\alpha }\right\}}_{\alpha \in \pi },\Delta ,\epsilon ,S)$ be a Hopf π-coalgebra. Then

(i)

$\forall \alpha ,\beta \in \pi ,{\Delta }_{{\beta }^{-1},{\alpha }^{-1}}{S}_{\alpha \beta }={\tau }_{H_{{\alpha }^{-1}},H_{{\beta }^{-1}}}(S_{\alpha }\otimes S_{\beta }){\Delta }_{\alpha ,\beta }$;

(ii)

$\epsilon S_1=\epsilon$;

(iii)

$\forall \alpha \in \pi ,a,b\in H_{\alpha },S_{\alpha }\left(ab\right)=S_{\alpha }\left(b\right)S_{\alpha }\left(a\right)$;

(iv)

$\forall \alpha \in \pi ,S_{\alpha }\left(1_{\alpha }\right)=1_{{\alpha }^{-1}}$.

Definition 3

([20]). A $\pi$-grouplike element of a Hopf group-coalgebra H is a family $g={\left\{g_{\alpha }\right\}}_{\alpha \in \pi }\in {\prod }_{\alpha \in \pi }{H}_{\alpha }$ such that ${\Delta }_{\alpha ,\beta }\left(g_{\alpha \beta }\right)=g_{\alpha }\otimes g_{\beta }$ for all $\alpha ,\beta \in \pi$ and $\epsilon \left(g_1\right)=1_{\mathbb{k}}$ (or equivalently $g_1\ne 0$).

Note that $g_1$ is then a (usual) grouplike element of the Hopf algebra $H_1$. The set $G\left(H\right)$ of $\pi$-grouplike elements of H is a group (with respect to the multiplication and unit of the product monoid ${\prod }_{\alpha \in \pi }{H}_{\alpha }$) and if $g={\left\{g_{\alpha }\right\}}_{\alpha \in \pi }\in G\left(H\right)$, then $g^{-1}={\left\{S_{{\alpha }^{-1}}\left(g_{a^{-1}}\right)\right\}}_{\alpha \in \pi }$.

Definition 4.

For $g,h\in G\left(H\right)$, an element $a={\left\{a_{\alpha }\right\}}_{\alpha \in \pi }\in {\prod }_{\alpha \in \pi }{H}_{\alpha }$ is a$(g,h)$-$\pi$-primitive elementif ${\Delta }_{\alpha ,\beta }\left(a_{\alpha \beta }\right)=a_{\alpha }\otimes g_{\beta }+h_{\alpha }\otimes a_{\beta }$ for all $\alpha ,\beta \in \pi$. Let $P_{g,h}\left(H\right)$ denote the set of all $(g,h)$-π-primitive elements of H. When $g=h=1$, a $(g,h)$-π-primitive element is simply called a π-primitive element of H, and $P_{1,1}\left(H\right)$ is simply written as $P\left(H\right)$.

Proposition 1.

Let $H=({\left\{H_{\alpha }\right\}}_{\alpha \in \pi },\Delta ,\epsilon ,S)$ be a Hopf group-coalgebra and $g,h\in G\left(H\right)$, $a\in P_{g,h}\left(H\right)$.

(i)

$\epsilon \left(g_1\right)=\epsilon \left(h_1\right)=1$ and $\epsilon \left(a_1\right)=0$,

(ii)

$S_{\alpha }\left(g_{\alpha }\right)=g_{{\alpha }^{-1}}^{-1}$, $S_{\alpha }\left(h_{\alpha }\right)=h_{{\alpha }^{-1}}^{-1}$ and $S_{\alpha }\left(a_{\alpha }\right)=-h_{{\alpha }^{-1}}^{-1}{a}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}$.

Proof. 

The proof is straightforward.    □

3. Generalized Hopf Group-Coalgebra Ore Extension

In this section, we generalize the theory of generalized Hopf–Ore extension on Hopf algebras to Hopf group-coalgebras. We first define generalized Hopf–Ore extension on Hopf group-coalgebras, and then study necessary and sufficient conditions for a generalized Hopf–Ore extension to be a Hopf group-coalgebra.

Definition 5.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra. A generalized Hopf group-coalgebra Ore extensionof A is a family $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ such that

(i)

H is a Hopf group-coalgebra with Hopf group-subcoalgebra A;

(ii)

For all $\alpha \in \pi$, $H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]$ be Ore extension of $A_{\alpha }$;

(iii)

For all $\alpha ,\beta \in \pi$, there are $h_{\alpha }\in A_{\alpha }$, $g_{\beta }\in A_{\beta }$ and $v_{\alpha ,\beta },w_{\alpha ,\beta }\in A_{\alpha }\otimes A_{\beta }$ such that

$${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)=z_{\alpha }\otimes g_{\beta }+h_{\alpha }\otimes z_{\beta }+v_{\alpha ,\beta }(z_{\alpha }\otimes z_{\beta })+w_{\alpha ,\beta }.$$

(2)

When $\pi =\left\{1\right\}$, the generalized Hopf group-coalgebra Ore extension is just the Hopf Ore extension in [8]. Note that one recovers the definition of Hopf group-coalgebra Ore extension in [29] when $x_{\alpha }\otimes y_{\beta }=0$ for all $\alpha ,\beta \in \pi$. In the next section, we will study the case where $w_{\alpha ,\beta }$ has only one summation term.

Lemma 2.

Let $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra with Hopf group-subcoalgebra A. If $A_{\alpha }$ is a domain for all $\alpha \in \pi$, i.e., $A_{\alpha }$ is a nonzero ring in which $ab=0$ implies $a=0$ or $b=0$, then

$${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)=s_{\alpha ,\beta }(z_{\alpha }\otimes 1_{\beta })+t_{\alpha ,\beta }(1_{\alpha }\otimes z_{\beta })+v_{\alpha ,\beta }(z_{\alpha }\otimes z_{\beta })+w_{\alpha ,\beta },$$

(3)

where $s_{\alpha ,\beta },t_{\alpha ,\beta },v_{\alpha ,\beta },w_{\alpha ,\beta }\in A_{\alpha }\otimes A_{\beta }$.

Proof. 

Since $H_{\alpha }$ is an Ore extension of $A_{\alpha }$, for all $\alpha \in \pi$, $H_{\alpha }$ is a free left $A_{\alpha }$-module with a basis ${\left\{y_{\alpha }^i\right\}}_{i\ge 0}$. Thus, we can write

$${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)=\sum _{i,j\ge 0}{w}_{\alpha ,\beta }^{(i,j)}{z}_{\alpha }^i\otimes z_{\beta }^j,$$

(4)

where each $w_{\alpha ,\beta }^{(i,j)}\in A_{\alpha }\otimes A_{\beta }$, $i,j=0,1,2,3,\cdots$.

Set $j_0=max\left\{j\right|w_{\alpha ,\beta }^{(i,j)}\ne 0,i,j\ge 0\}$, and $i_0=max\left\{i\right|w_{\alpha ,\beta }^{(i,j_0)}\ne 0,i\ge 0\}$. For all $\alpha \in \pi$ and $a_{\alpha }\in A_{\alpha }$, define $deg{a}_{\alpha }=0$ and $deg{z}_{\alpha }=1$, and extend it to define the lexicographic order for the elements of $H\otimes H$ and $H\otimes H\otimes H$. This is called degree. Thence, the maximal degree component of Equation (4) is $w_{\alpha ,\beta }^{(i_0,j_0)}{z}_{\alpha }^{i_0}\otimes z_{\beta }^{j_0}$.

For all $\alpha ,\beta ,\gamma \in \pi$, applying $({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }}){\Delta }_{\alpha \beta ,\gamma }$ to $z_{\alpha \beta \gamma }$, we obtain

$$\begin{array}{cc}\hfill ({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }}){\Delta }_{\alpha \beta ,\gamma }\left(z_{\alpha \beta \gamma }\right)& =({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})(\sum _{i,j\ge 0}{w}_{\alpha \beta ,\gamma }^{(i,j)}{z}_{\alpha \beta }^i\otimes z_{\gamma }^j)\hfill \\ \hfill & =\sum _{i,j\ge 0}({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})\left(w_{\alpha \beta ,\gamma }^{(i,j)}\right)\left({(\sum _{s,t\ge 0}{w}_{\alpha ,\beta }^{(s,t)}{z}_{\alpha }^s\otimes z_{\beta }^t)}^i\otimes z_{\gamma }^j\right),\hfill \end{array}$$

(5)

with maximal degree component $\theta z_{\alpha }^{i_0^2}\otimes z_{\beta }^{i_0{j}_0}\otimes z_{\gamma }^{j_0}$, where

$$\theta =({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})\left(w_{\alpha \beta ,\gamma }^{(i_0,j_0)}\right)(w_{\alpha ,\beta }^{(i_0,j_0)}({\sigma }_{\alpha }^{i_0}\otimes {\sigma }_{\beta }^{j_0})\left(w_{\alpha ,\beta }^{(i_0,j_0)}\right)\cdots {({\sigma }_{\alpha }^{i_0}\otimes {\sigma }_{\beta }^{j_0})}^{i_0-1}\left(w_{\alpha ,\beta }^{(i_0,j_0)}\right)\otimes 1_{\gamma }).$$

Similarly, applying $({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma }){\Delta }_{\alpha ,\beta \gamma }$ to $z_{\alpha \beta \gamma }$, we have

$$\begin{array}{cc}\hfill ({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma }){\Delta }_{\alpha ,\beta \gamma }\left(z_{\alpha \beta \gamma }\right)& =({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })(\sum _{i,j\ge 0}{w}_{\alpha ,\beta \gamma }^{(i,j)}{z}_{\alpha }^i\otimes z_{\beta \gamma }^j)\hfill \\ \hfill & =\sum _{i,j\ge 0}({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })\left(w_{\alpha ,\beta \gamma }^{(i,j)}\right)\left(z_{\alpha }^i\otimes {(\sum _{s,t\ge 0}{w}_{\beta ,\gamma }^{(s,t)}{z}_{\beta }^s\otimes z_{\gamma }^t)}^j\right),\hfill \end{array}$$

(6)

with maximal degree component ${\theta }^{\prime }{z}_{\alpha }^{i_0}\otimes z_{\beta }^{i_0{j}_0}\otimes z_{\gamma }^{j_0^2}$, where

$${\theta }^{\prime }=({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })\left(w_{\alpha ,\beta \gamma }^{(i_0,j_0)}\right)(1_{\alpha }\otimes w_{\beta ,\gamma }^{(i_0,j_0)}({\sigma }_{\beta }^{i_0}\otimes {\sigma }_{\gamma }^{j_0})\left(w_{\beta ,\gamma }^{(i_0,j_0)}\right)\cdots {({\sigma }_{\beta }^{i_0}\otimes {\sigma }_{\gamma }^{j_0})}^{j_0-1}\left(w_{\beta ,\gamma }^{(i_0,j_0)}\right)).$$

Since $A_{\alpha }\otimes A_{\beta }$ is a domain for all $\alpha ,\beta \in \pi$, $\theta \ne 0$ and ${\theta }^{\prime }\ne 0$. Since Equation (5) equals Equation (6) by coassociativity, we obtain $j_0=0$ or 1. Similarly, the maximal value of i such that $w_{\alpha ,\beta }^{(i,j)}\ne 0$ for some j is $i=0$ or 1. Therefore, Equation (3) holds for some $s_{\alpha ,\beta },t_{\alpha ,\beta },v_{\alpha ,\beta }$, and $w_{\alpha ,\beta }$ in $A_{\alpha }\otimes A_{\beta }$.    □

In the above lemma, we raise the question whether the conclusion (3) can be replaced by Equation (2)? The answer is positive when $A_{\alpha }$ is still noetherian for all $\alpha \in \pi$ (cf. [9]). Therefore, we might as well continue with the notation of Definition 5.

Lemma 3.

Let $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ be a generalized Hopf group-coalgebra Ore extension of Hopf group-subcoalgebra $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$. If $A_{\alpha }$ is a domain for all $\alpha \in \pi$, then

(i)

For all $\alpha ,\beta \in \pi$, if $\exists {\gamma }_0\in \pi$ such that $v_{{\gamma }_0,\alpha \beta }=0$ or $w_{\alpha ,\beta }=0$, then g is a π-grouplike element; if $\exists {\gamma }_0\in \pi$ such that $v_{\alpha \beta ,{\gamma }_0}=0$ or $w_{\alpha ,\beta }=0$, then h is a π-grouplike element;

(ii)

$v_{\alpha ,\beta }\in \mathbb{k}$ for all $\alpha ,\beta \in \pi$;

(iii)

For all $\alpha ,\beta ,\gamma \in \pi$, we have

$$h_{\alpha }\otimes w_{\beta ,\gamma }+({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })\left(w_{\alpha ,\beta \gamma }\right)=w_{\alpha ,\beta }\otimes g_{\gamma }+({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})\left(w_{\alpha \beta ,\gamma }\right).$$

Proof. 

(i) For all $\alpha ,\beta ,\gamma \in \pi$, applying $({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }}){\Delta }_{\alpha \beta ,\gamma }$ to $z_{\alpha \beta \gamma }$ and using (2), we obtain

$$\begin{array}{c}{\displaystyle ({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }}){\Delta }_{\alpha \beta ,\gamma }\left(z_{\alpha \beta \gamma }\right)}\hfill \\ =({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})(h_{\alpha \beta }\otimes z_{\gamma }+z_{\alpha \beta }\otimes g_{\gamma }+v_{\alpha \beta ,\gamma }(z_{\alpha \beta }\otimes z_{\gamma })+w_{\alpha \beta ,\gamma })\hfill \\ ={\Delta }_{\alpha ,\beta }\left(h_{\alpha \beta }\right)\otimes z_{\gamma }+(h_{\alpha }\otimes z_{\beta }+z_{\alpha }\otimes g_{\gamma }+v_{\alpha ,\beta }(z_{\alpha }\otimes z_{\beta })+w_{\alpha ,\beta })\otimes g_{\gamma }\hfill \\ +({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})\left(v_{\alpha \beta ,\gamma }\right)((h_{\alpha }\otimes z_{\beta }+z_{\alpha }\otimes g_{\beta }+v_{\alpha ,\beta }(z_{\alpha }\otimes z_{\beta })+w_{\alpha ,\beta })\otimes z_{\gamma })\hfill \\ +({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})\left(w_{\alpha \beta ,\gamma }\right).\hfill \end{array}$$

(7)

Similarly, applying $({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma }){\Delta }_{\alpha ,\beta \gamma }$ to $z_{\alpha \beta \gamma }$, we have

$$\begin{array}{c}{\displaystyle ({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma }){\Delta }_{\alpha ,\beta \gamma }\left(z_{\alpha \beta \gamma }\right)}\hfill \\ =({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })(h_{\alpha }\otimes z_{\beta \gamma }+z_{\alpha }\otimes g_{\beta \gamma }+v_{\alpha ,\beta \gamma }(z_{\alpha }\otimes z_{\beta \gamma })+w_{\alpha ,\beta \gamma })\hfill \\ =h_{\alpha }\otimes (h_{\beta }\otimes z_{\gamma }+z_{\beta }\otimes g_{\gamma }+v_{\beta ,\gamma }(z_{\beta }\otimes z_{\gamma })+w_{\beta ,\gamma })+z_{\alpha }\otimes {\Delta }_{\beta ,\gamma }\left(g_{\beta ,\gamma }\right)\hfill \\ +({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })\left(v_{\alpha ,\beta \gamma }\right)(z_{\alpha }\otimes (h_{\beta }\otimes z_{\gamma }+z_{\beta }\otimes g_{\gamma }+v_{\beta ,\gamma }(z_{\beta }\otimes z_{\gamma })+w_{\beta ,\gamma }))\hfill \\ +({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })\left(w_{\alpha ,\beta \gamma }\right).\hfill \end{array}$$

(8)

Equating coefficients in $A_{\alpha }\otimes A_{\beta }\otimes A_{\gamma }$ of $1_{\alpha }\otimes 1_{\beta }\otimes z_{\gamma }$ in (7) and (8) yields

$${\Delta }_{\alpha ,\beta }\left(h_{\alpha ,\beta }\right)\otimes 1_{\gamma }+({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})\left(v_{\alpha \beta ,\gamma }\right)(w_{\alpha ,\beta }\otimes 1_{\gamma })=h_{\alpha }\otimes h_{\beta }\otimes 1_{\gamma }.$$

(9)

Similarly, consider coefficients of $z_{\alpha }\otimes 1_{\beta }\otimes 1_{\gamma }$ in Equations (7) and (8) to obtain

$$1_{\alpha }\otimes g_{\beta }\otimes g_{\gamma }=1_{\alpha }\otimes {\Delta }_{\beta ,\gamma }\left(g_{\beta \gamma }\right)+({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })\left(v_{\alpha ,\beta \gamma }\right)(1_{\alpha }\otimes w_{\beta ,\gamma }).$$

(10)

Suppose that there is ${\gamma }_0\in \pi$ such that $v_{\alpha \beta ,{\gamma }_0}=0$ or $w_{\alpha ,\beta }=0$. Then, Equations (9) and (10) yield ${\Delta }_{\alpha ,\beta }\left(h_{\alpha \beta }\right)=h_{\alpha }\otimes h_{\beta }$; if there is ${\gamma }_0\in \pi$ such that $v_{{\gamma }_0,\alpha \beta }=0$ or $w_{\alpha ,\beta }=0$, then Equations (9) and (10) infer ${\Delta }_{\alpha ,\beta }\left(g_{\alpha \beta }\right)=g_{\alpha }\otimes g_{\beta }$. Hence, part (i) holds.

(ii) We will prove in two steps. We first show $v_{\alpha ,1},v_{1,\alpha }\in \mathbb{k}$ for all $\alpha \in \pi$, and then show $v_{\alpha ,\beta }\in \mathbb{k}$ for all $\alpha ,\beta \in \pi$.

Step 1. For all $\alpha \in \pi$, we calculate the following equation with (2):

$$({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,1}){\Delta }_{\alpha ,1}\left(z_{\alpha }\right)=({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1}){\Delta }_{\alpha ,1}\left(z_{\alpha }\right).$$

(11)

We have

$$\begin{array}{c}{\displaystyle ({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,1}){\Delta }_{\alpha ,1}\left(z_{\alpha }\right)}\hfill \\ =({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,1})(h_{\alpha }\otimes z_1+z_{\alpha }\otimes g_1+v_{\alpha ,1}(z_{\alpha }\otimes z_1)+w_{\alpha ,1})\hfill \\ =h_{\alpha }\otimes {\Delta }_{1,1}\left(z_1\right)+z_{\alpha }\otimes {\Delta }_{1,1}\left(g_1\right)\hfill \\ +({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,1})\left(v_{\alpha ,1}\right)(z_{\alpha }\otimes {\Delta }_{1,1}\left(z_1\right))+({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,1})\left(w_{\alpha ,1}\right)\hfill \\ =h_{\alpha }\otimes (h_1\otimes z_1+z_1\otimes g_1+v_{1,1}(z_1\otimes z_1)+w_{1,1})+z_{\alpha }\otimes {\Delta }_{1,1}\left(g_1\right)\hfill \\ +({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,1})\left(v_{\alpha ,1}\right)(z_{\alpha }\otimes (h_1\otimes z_1+z_1\otimes g_1+v_{1,1}(z_1\otimes z_1))\hfill \\ +({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,1})\left(w_{\alpha ,1}\right)\hfill \end{array}$$

(12)

and

$$\begin{array}{c}{\displaystyle ({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1}){\Delta }_{\alpha ,1}\left(z_{\alpha }\right)}\hfill \\ =({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1})(h_{\alpha }\otimes z_1+z_{\alpha }\otimes g_1+v_{\alpha ,1}(z_{\alpha }\otimes z_1)+w_{\alpha ,1})\hfill \\ ={\Delta }_{\alpha ,1}\left(h_{\alpha }\right)\otimes z_1+{\Delta }_{\alpha ,1}\left(z_{\alpha }\right)\otimes g_1\hfill \\ +({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1})\left(v_{\alpha ,1}\right)({\Delta }_{\alpha ,1}\left(z_{\alpha }\right)\otimes z_1)+({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1})\left(w_{\alpha ,1}\right)\hfill \\ ={\Delta }_{\alpha ,1}\left(h_{\alpha }\right)\otimes z_1+(h_{\alpha }\otimes z_1+z_{\alpha }\otimes g_1+v_{\alpha ,1}(z_{\alpha }\otimes z_1)+w_{\alpha ,1})\otimes g_1\hfill \\ +({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1})\left(v_{\alpha ,1}\right)((h_{\alpha }\otimes z_1+z_{\alpha }\otimes g_1+v_{\alpha ,1}(z_{\alpha }\otimes z_1)+w_{\alpha ,1})\otimes z_1)\hfill \\ +({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1})\left(w_{\alpha ,1}\right).\hfill \end{array}$$

(13)

Thus, comparing the coefficients of $1_{\alpha }\otimes z_1\otimes z_1$ in Equations (12) and (13) obtains that

$$h_{\alpha }\otimes v_{1,1}=({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1})\left(v_{\alpha ,1}\right)(h_{\alpha }\otimes 1_1\otimes 1_1),$$

Case 1: $w_{\alpha ,1}=0$. Since $h_{\alpha }\ne 0$ by part (i) and Lemma 2 in [8] (Section 2.2) $v_{1,1}\in \mathbb{k}$. Thus, we have $({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1})\left(v_{\alpha ,1}\right)=1_{\alpha }\otimes v_{1,1}\in \mathbb{k}$. Suppose that $v_{\alpha ,1}={\sum }_i{c}_{\alpha }^{\left(i\right)}\otimes d_1^{\left(i\right)}\in A_{\alpha }\otimes A_1$, with $\left\{d_1^{\left(i\right)}\right\}$ linearly independent. For all integer i, ${\Delta }_{\alpha ,1}\left(c_{\alpha }^{\left(i\right)}\right)\otimes d_1^{\left(i\right)}\in \mathbb{k}$, and hence $d_1^{\left(i\right)}\in \mathbb{k}$ and ${\Delta }_{\alpha ,1}\left(c_{\alpha }^{\left(i\right)}\right)\in \mathbb{k}$. Thus, $c_{\alpha }^{\left(i\right)}=m_{\alpha }({\mathrm{Id}}_{H_{\alpha }}\otimes \epsilon ){\Delta }_{\alpha ,1}\left(c_{\alpha }^{\left(i\right)}\right)\in \mathbb{k}$. Therefore, $v_{\alpha ,1}\in \mathbb{k}$.

Case 2: $w_{\alpha ,1}\ne 0$. Comparing the coefficients of $z_{\alpha }\otimes 1_1\otimes 1_1$ in Equations (12) and (13), we obtain $1_{\alpha }\otimes g_1\otimes g_1=({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,1})\left(v_{\alpha ,1}\right)(1_{\alpha }\otimes w_{1,1})+1_{\alpha }\otimes {\Delta }_{1,1}\left(g_1\right)$. As a result, $v_{\alpha ,1}=1_{\alpha }\otimes v_1$ for some $v_1\in A_1$. Then, compare the coefficients of $1_{\alpha }\otimes 1_1\otimes z_1$ in the same equations to obtain $h_{\alpha }\otimes h_1\otimes 1_1={\Delta }_{\alpha ,1}\left(h_{\alpha }\right)\otimes 1_1+({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_1})\left(v_{\alpha ,1}\right)(w_{\alpha ,1}\otimes 1_1)$. Thus, $v_{\alpha ,1}=v_{\alpha }\otimes 1_1$ for some $v_{\alpha }\in A_{\alpha }$. It follows that $v_{\alpha ,1}=v(1_{\alpha }\otimes 1_1)$ for some $v\in \mathbb{k}$, i.e., $v_{\alpha ,1}\in \mathbb{k}$.

Considering $({\mathrm{Id}}_{H_1}\otimes {\Delta }_{1,\alpha }){\Delta }_{1,\alpha }\left(z_{\alpha }\right)=({\Delta }_{1,1}\otimes {\mathrm{Id}}_{H_{\alpha }}){\Delta }_{1,\alpha }\left(z_{\alpha }\right)$ analogously, we can obtain $v_{1,\alpha }\in \mathbb{k}$ for all $\alpha \in \pi$.

Step 2. Applying $({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }}){\Delta }_{\alpha ,\beta }$ and $({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,\beta }){\Delta }_{\alpha ,\beta }$ to $z_{\alpha \beta }$, respectively, we have

$$\begin{array}{c}{\displaystyle ({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }}){\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)}\hfill \\ =({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }})(h_{\alpha }\otimes z_{\beta }+z_{\alpha }\otimes g_{\beta }+v_{\alpha ,\beta }(z_{\alpha }\otimes z_{\beta })+w_{\alpha ,\beta })\hfill \\ ={\Delta }_{\alpha ,1}\left(h_{\alpha }\right)\otimes z_{\beta }+{\Delta }_{\alpha ,1}\left(z_{\alpha }\right)\otimes g_{\beta }\hfill \\ +({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }})\left(v_{\alpha ,\beta }\right)({\Delta }_{\alpha ,1}\left(z_{\alpha }\right)\otimes z_{\beta })+({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }})\left(w_{\alpha ,\beta }\right)\hfill \\ ={\Delta }_{\alpha ,1}\left(h_{\alpha }\right)\otimes z_{\beta }+(h_{\alpha }\otimes z_1+z_{\alpha }\otimes g_1+v_{\alpha ,1}(z_{\alpha }\otimes z_1)+w_{\alpha ,1})\otimes g_{\beta }\hfill \\ +({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }})\left(v_{\alpha ,\beta }\right)\left((h_{\alpha }\otimes z_1+z_{\alpha }\otimes g_1+v_{\alpha ,1}(z_{\alpha }\otimes z_1)+w_{\alpha ,1})\otimes z_{\beta }\right)\hfill \\ +({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }})\left(w_{\alpha ,\beta }\right)\hfill \end{array}$$

(14)

and

$$\begin{array}{c}{\displaystyle ({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,\beta }){\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)}\hfill \\ =({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,\beta })(h_{\alpha }\otimes z_{\beta }+z_{\alpha }\otimes g_{\beta }+v_{\alpha ,\beta }(z_{\alpha }\otimes z_{\beta })+w_{\alpha ,\beta })\hfill \\ =h_{\alpha }\otimes {\Delta }_{1,\beta }\left(z_{\beta }\right)+z_{\alpha }\otimes {\Delta }_{1,\beta }\left(g_{\beta }\right)\hfill \\ +({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,\beta })\left(v_{\alpha ,\beta }\right)(z_{\alpha }\otimes {\Delta }_{1,\beta }\left(z_{\beta }\right))+({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,\beta })\left(w_{\alpha ,\beta }\right)\hfill \\ =h_{\alpha }\otimes (h_1\otimes z_{\beta }+z_1\otimes g_{\beta }+v_{1,\beta }(z_1\otimes z_{\beta })+w_{1,\beta })+z_{\alpha }\otimes {\Delta }_{1,\beta }\left(g_{\beta }\right)\hfill \\ +({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,\beta })\left(v_{\alpha ,\beta }\right)\left(z_{\alpha }\otimes (h_1\otimes z_{\beta }+z_1\otimes g_{\beta }+v_{1,\beta }(z_1\otimes z_{\beta })+w_{1,\beta })\right)\hfill \\ +({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,\beta })\left(w_{\alpha ,\beta }\right).\hfill \end{array}$$

(15)

Case 1: $w_{\alpha ,1}=0$. Comparing the coefficients of $1_{\alpha }\otimes z_1\otimes z_{\beta }$ in Equations (14) and (15), we obtain $({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }})\left(v_{\alpha ,\beta }\right)(h_{\alpha }\otimes 1_1\otimes 1_{\beta })=h_{\alpha }\otimes v_{1,\beta }$. Since $h_{\alpha }\ne 0$ by (i) and $v_{1,\beta }\in \mathbb{k}$ by Step 1, $v_{\alpha ,\beta }=v_{\alpha }\otimes 1_{\beta }$ for some $v_{\alpha }\in A_{\alpha }$. A similar discussion of $z_{\alpha }\otimes z_1\otimes 1_{\beta }$ in the same equations yields $v_{\alpha ,\beta }=1_{\alpha }\otimes v_{\beta }$ for some $v_{\beta }\in A_{\beta }$. It follows that $v_{\alpha ,\beta }=v(1_{\alpha }\otimes 1_{\beta })$ for some $v\in \mathbb{k}$, i.e., $v_{\alpha ,\beta }\in \mathbb{k}$.

Case 2: $w_{\alpha ,1}\ne 0$. Comparing the coefficients of $1_{\alpha }\otimes 1_1\otimes z_{\beta }$ in Equations (14) and (15), we have ${\Delta }_{\alpha ,1}\left(h_{\alpha }\right)\otimes 1_{\beta }+({\Delta }_{\alpha ,1}\otimes {\mathrm{Id}}_{H_{\beta }})\left(v_{\alpha ,\beta }\right)(w_{\alpha ,1}\otimes 1_{\beta })=h_{\alpha }\otimes h_1\otimes 1_{\beta }.$ Compare the right-most tensor in the above equation to obtain $v_{\alpha ,\beta }=v_{\alpha }\otimes 1_{\beta }$ for some $v_{\alpha }\in A_{\alpha }$. Comparing the coefficients of $z_{\alpha }\otimes 1_1\otimes 1_{\beta }$ in the same equations, we obtain $1_{\alpha }\otimes g_1\otimes g_{\beta }=1_{\alpha }\otimes {\Delta }_{1,\beta }\left(g_{\beta }\right)+({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{1,\beta })\left(v_{\alpha ,\beta }\right)(1_{\alpha }\otimes w_{1,\beta }).$ A similar consideration on the left-most tensor yields $v_{\alpha ,\beta }=1_{\alpha }\otimes v_{\beta }$ for some $v_{\beta }\in A_{\beta }$. It follows that $v_{\alpha ,\beta }=v(1_{\alpha }\otimes 1_{\beta })$ for some $v\in \mathbb{k}$, i.e., $v_{\alpha ,\beta }\in \mathbb{k}$.

In short, we always have $v_{\alpha ,\beta }\in \mathbb{k}$ for all $\alpha ,\beta \in \pi$.

Finally, part (iii) follows by comparing the coefficients of $1_{\alpha }\otimes 1_1\otimes 1_{\beta }$ in Equations (7) and (8).    □

Lemma 4.

Let $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ be a generalized Hopf group-coalgebra Ore extension of Hopf group-coalgebra $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ and write $w_{\alpha ,\beta }={\sum }_i{w}_{\alpha }^{\left(i\right)}\otimes w_{\beta }^{\left(i\right)}\in A_{\alpha }\otimes A_{\beta }$. Then,

(i)

If $A_{\alpha }$ is a domain for all $\alpha \in \pi$, $S_{\alpha }\left(z_{\alpha }\right)=p_{{\alpha }^{-1}}{z}_{{\alpha }^{-1}}+q_{{\alpha }^{-1}}$ where $p_{{\alpha }^{-1}}\in A_{{\alpha }^{-1}}^{\times }$ and $q_{{\alpha }^{-1}}\in A_{{\alpha }^{-1}}$.

For the remainder of the Lemma, assume that the antipode has the form given in (i).

(ii)

$v_{\alpha ,{\alpha }^{-1}}=0$ for all $\alpha \in \pi$.

(iii)

$g,h$ are π-grouplike elements.

(iv)

$p_{{\alpha }^{-1}}=-h_{{\alpha }^{-1}}^{-1}{\sigma }_{{\alpha }^{-1}}\left(g_{{\alpha }^{-1}}^{-1}\right)$ and $q_{{\alpha }^{-1}}=h_{{\alpha }^{-1}}^{-1}(\epsilon \left(z_1\right)-{\delta }_{{\alpha }^{-1}}\left(g_{{\alpha }^{-1}}^{-1}\right)-\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right))$ for all $\alpha \in \pi$.

Proof. 

(i) Since $H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]$ is an Ore extension of $A_{\alpha }$ for all $\alpha \in \pi$,

$$z_{\alpha }{a}_{\alpha }={\sigma }_{\alpha }\left(a_{\alpha }\right)z_{\alpha }+{\delta }_{\alpha }\left(a_{\alpha }\right),\forall a_{\alpha }\in A_{\alpha }.$$

Applying $S_{\alpha }$ to the above equation shows that

$$S_{\alpha }\left(a_{\alpha }\right)S_{\alpha }\left(z_{\alpha }\right)=S_{\alpha }\left(z_{\alpha }\right)S_{\alpha }\left({\sigma }_{\alpha }\left(a_{\alpha }\right)\right)+S_{\alpha }{\delta }_{\alpha }\left(a_{\alpha }\right).$$

Replace $a_{\alpha }$ by ${\sigma }_{\alpha }^{-1}{S}_{\alpha }^{-1}\left(a_{{\alpha }^{-1}}\right)$ to obtain

$$S_{\alpha }\left(z_{\alpha }\right)a_{{\alpha }^{-1}}=S_{\alpha }{\sigma }_{\alpha }^{-1}{S}_{\alpha }^{-1}\left(a_{{\alpha }^{-1}}\right)S_{\alpha }\left(z_{\alpha }\right)-S_{\alpha }{\delta }_{\alpha }{\sigma }_{\alpha }^{-1}{S}_{\alpha }^{-1}\left(a_{{\alpha }^{-1}}\right).$$

Thus, $H_{{\alpha }^{-1}}$ can be seen as an Ore extension of $A_{{\alpha }^{-1}}$ generated by $A_{{\alpha }^{-1}}$ and $S_{\alpha }\left(z_{\alpha }\right)$. This means that $H_{{\alpha }^{-1}}=A_{{\alpha }^{-1}}[S_{\alpha }\left(z_{\alpha }\right);S_{\alpha }{\sigma }_{\alpha }^{-1}{S}_{\alpha }^{-1},-S_{\alpha }{\delta }_{\alpha }{\sigma }_{\alpha }^{-1}{S}_{\alpha }^{-1}]$. In fact, $H_{{\alpha }^{-1}}$ is also an Ore extension $A_{{\alpha }^{-1}}[z_{{\alpha }^{-1}};{\sigma }_{{\alpha }^{-1}},{\delta }_{{\alpha }^{-1}}]$. Since $A_{{\alpha }^{-1}}$ is a domain, part (i) follows by considering the expression for $z_{{\alpha }^{-1}}$ as a polynomial of $S_{\alpha }\left(z_{\alpha }\right)$ with coefficients in $A_{{\alpha }^{-1}}$.

(ii) By the property of the antipode, we obtain

$$\begin{array}{cc}\hfill \epsilon \left(z_1\right)& =m_{{\alpha }^{-1}}({\mathrm{Id}}_{{\alpha }^{-1}}\otimes S_{\alpha }){\Delta }_{{\alpha }^{-1},\alpha }\left(z_{{\alpha }^{-1}\alpha }\right)\hfill \\ \hfill & =m_{{\alpha }^{-1}}({\mathrm{Id}}_{{\alpha }^{-1}}\otimes S_{\alpha })(h_{{\alpha }^{-1}}\otimes z_{\alpha }+z_{{\alpha }^{-1}}\otimes g_{\alpha }+v_{{\alpha }^{-1},\alpha }(z_{{\alpha }^{-1}}\otimes z_{\alpha })+w_{{\alpha }^{-1},\alpha })\hfill \\ \hfill & =h_{{\alpha }^{-1}}{p}_{{\alpha }^{-1}}{z}_{{\alpha }^{-1}}+h_{{\alpha }^{-1}}{q}_{{\alpha }^{-1}}+z_{{\alpha }^{-1}}{S}_{\alpha }\left(g_{\alpha }\right)\hfill \\ \hfill & +v_{{\alpha }^{-1},\alpha }{z}_{{\alpha }^{-1}}{p}_{{\alpha }^{-1}}{z}_{{\alpha }^{-1}}+v_{{\alpha }^{-1},\alpha }{z}_{{\alpha }^{-1}}{q}_{{\alpha }^{-1}}+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)\hfill \\ \hfill & =h_{{\alpha }^{-1}}{p}_{{\alpha }^{-1}}{z}_{{\alpha }^{-1}}+h_{{\alpha }^{-1}}{q}_{{\alpha }^{-1}}+z_{{\alpha }^{-1}}{S}_{\alpha }\left(g_{\alpha }\right)+v_{{\alpha }^{-1},\alpha }{\sigma }_{{\alpha }^{-1}}\left(p_{{\alpha }^{-1}}\right)z_{{\alpha }^{-1}}^2\hfill \\ \hfill & +v_{{\alpha }^{-1},\alpha }{\delta }_{{\alpha }^{-1}}\left(p_{{\alpha }^{-1}}\right)z_{{\alpha }^{-1}}+v_{{\alpha }^{-1},\alpha }{z}_{{\alpha }^{-1}}{q}_{{\alpha }^{-1}}+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right),\hfill \end{array}$$

(16)

where we use Lemma 3(ii) at the third equality. The only term of degree 2 in $z_{{\alpha }^{-1}}$ on the right-hand side of the above equation is $v_{{\alpha }^{-1},\alpha }{\sigma }_{{\alpha }^{-1}}\left(p_{{\alpha }^{-1}}\right)z_{{\alpha }^{-1}}^2.$ Since this is zero and $p_{{\alpha }^{-1}}\in A_{{\alpha }^{-1}}^{\times }$, $v_{{\alpha }^{-1},\alpha }=0$.

(iii) For all $\alpha ,\beta \in \pi$, set ${\gamma }_0={\left(\alpha \beta \right)}^{-1}$. By part (ii), $v_{\alpha \beta ,{\gamma }_0}=0=v_{{\gamma }_0,\alpha \beta }$. Then, part (iii) follows from Lemma 3(i).

(iv) We first calculate the antipodes of $h_{\alpha }$ and $g_{\alpha }$. Applying ${\mathrm{Id}}_{H_1}\otimes \epsilon$ to ${\Delta }_{1,1}\left(h_1\right)=h_1\otimes h_1$, we obtain $h_1=h_1\epsilon \left(h_1\right).$ It follows that $\epsilon \left(h_1\right)=1$. Then, applying ${\mathrm{Id}}_{H_{{\alpha }^{-1}}}\otimes S_{\alpha }$ to ${\Delta }_{{\alpha }^{-1},\alpha }\left(h_1\right)=h_{{\alpha }^{-1}}\otimes h_{\alpha }$, we obtain $h_{{\alpha }^{-1}}{S}_{\alpha }\left(h_{\alpha }\right)=\epsilon \left(h_1\right)=1.$ Thus, $S_{\alpha }\left(h_{\alpha }\right)=h_{{\alpha }^{-1}}^{-1}$. We similarly have $S_{\alpha }\left(g_{\alpha }\right)=g_{{\alpha }^{-1}}^{-1}$.

Plugging $S_{\alpha }\left(g_{\alpha }\right)=g_{{\alpha }^{-1}}^{-1}$ into Equation (16) and using (ii), we have

$$\begin{array}{cc}\hfill \epsilon \left(z_1\right)& =h_{{\alpha }^{-1}}{p}_{{\alpha }^{-1}}{z}_{{\alpha }^{-1}}+h_{{\alpha }^{-1}}{q}_{{\alpha }^{-1}}+z_{{\alpha }^{-1}}{S}_{\alpha }\left(g_{\alpha }\right)+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)\hfill \\ \hfill & =h_{{\alpha }^{-1}}{p}_{{\alpha }^{-1}}{z}_{{\alpha }^{-1}}+z_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}+h_{{\alpha }^{-1}}{q}_{{\alpha }^{-1}}+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)\hfill \\ \hfill & =h_{{\alpha }^{-1}}{p}_{{\alpha }^{-1}}{z}_{{\alpha }^{-1}}+{\sigma }_{{\alpha }^{-1}}\left(g_{{\alpha }^{-1}}^{-1}\right)z_{{\alpha }^{-1}}+{\delta }_{{\alpha }^{-1}}\left(g_{{\alpha }^{-1}}^{-1}\right)+h_{{\alpha }^{-1}}{q}_{{\alpha }^{-1}}+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right).\hfill \end{array}$$

Considering the coefficients of $z_{{\alpha }^{-1}}$ and the constant term, respectively, part (iv) holds.    □

The following theorem gives a sufficient and necessary condition for an Ore extension of a Hopf group-coalgebra to have a generalized Hopf group-coalgebra Ore extension structure, which generalizes the analogous theorems in [7,8,29]. First of all, the polynomial variable of a skew polynomial extension is far from uniquely determined. For example, if $H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]$ is a skew polynomial algebra and ${\lambda }_{\alpha }\in \mathbb{k}$, then ${\delta }_{\alpha ,{\lambda }_{\alpha }}:={\delta }_{\alpha }+{\lambda }_{\alpha }({\mathrm{Id}}_{A_{\alpha }}-{\sigma }_{\alpha })$ is another ${\sigma }_{\alpha }$-derivation of A, as is easily checked, and $H_{\alpha }=A_{\alpha }[z_{\alpha }+{\lambda }_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha ,{\lambda }_{\alpha }}]$. Therefore, we may assume without loss of generality when studying a generalized Hopf group-coalgebra Ore extension $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ such that $\epsilon \left(z_1\right)=0$. Furthermore, for any $g_{\alpha }\in A_{\alpha }^{\times }$, replacing $z_{\alpha }$ by $g_{\alpha }^{-1}{z}_{\alpha }$, and writing ${Ad}_{g_{\alpha }^{-1}}$ to denote conjugation by $g_{\alpha }^{-1}$, i.e., ${Ad}_{g_{\alpha }^{-1}}\left(a\right)=g_{\alpha }^{-1}a{S}_{{\alpha }^{-1}}\left(g_{{\alpha }^{-1}}^{-1}\right)\stackrel{\mathrm{if}g\in G\left(A\right)}{=}{g}_{\alpha }^{-1}a{g}_{\alpha }$, one easily checks that $H={\{H_{\alpha }=A_{\alpha }[g_{\alpha }^{-1}{z}_{\alpha };{Ad}_{g_{\alpha }^{-1}}{\sigma }_{\alpha },g_{\alpha }^{-1}{\delta }_{\alpha }]\}}_{\alpha \in \pi }$.

Theorem 1.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra.

(i)

Let $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ be a generalized Hopf group-coalgebra Ore extension of A. For all $\alpha \in \pi$, suppose that

$$S_{\alpha }\left(z_{\alpha }\right)=p_{{\alpha }^{-1}}{z}_{{\alpha }^{-1}}+q_{{\alpha }^{-1}},$$

(17)

where $p_{{\alpha }^{-1}}\in A_{{\alpha }^{-1}}^{\times }$ and $q_{{\alpha }^{-1}}\in A_{{\alpha }^{-1}}$. Write $w_{\alpha ,\beta }=\sum w_{\alpha }\otimes w_{\beta }\in A_{\alpha }\otimes A_{\beta }$, with $\left\{w_{\alpha }\right\}$ and $\left\{w_{\beta }\right\}$ chosen to be k-linearly independent subsets of $A_{\alpha }$ and $A_{\beta }$. Then, the following hold.

(a)

For all $\alpha ,\beta \in \pi$, $v_{\alpha ,{\alpha }^{-1}}=0$ and $g,h\in G\left(A\right)$.

(b)

For all $\alpha ,\beta \in \pi$, after a change of the variables $\left\{z_{\alpha }\right\}$ and corresponding adjustments to ${\sigma }_{\alpha }$, ${\delta }_{\alpha }$ and $w_{\alpha ,\beta }$, we may have

$$\epsilon \left(z_1\right)=0$$

(18)

and

$${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)=z_{\alpha }\otimes 1_{\beta }+h_{\alpha }\otimes z_{\beta }+w_{\alpha ,\beta }.$$

(19)

For the remainder of (i), we assume that item (b) holds.

(c)

For all $\alpha \in \pi$,

$$S_{\alpha }\left(z_{\alpha }\right)=-h_{{\alpha }^{-1}}^{-1}(z_{{\alpha }^{-1}}+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)).$$

(20)

(d)

There is a character $\chi :A_1\to \mathbb{k}$ such that

$${\sigma }_{\alpha }\left(a\right)=\chi \left(a_{(1,1)}\right)a_{(2,\alpha )}={Ad}_{h_{\alpha }}\left(a_{(1,\alpha )}\right)\chi \left(a_{(2,1)}\right)$$

(21)

for all $a\in A_{\alpha }$, $\alpha \in \pi$.

(e)

The σ-derivation δ satisfies the relation

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right)& ={\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,\beta )}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{\beta }\left(a_{(2.\beta )}\right)\hfill \\ \hfill & +w_{\alpha ,\beta }{\Delta }_{\alpha ,\beta }\left(a\right)-{\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha ,\beta }\left(a\right)\right)w_{\alpha ,\beta },\hfill \end{array}$$

(22)

for all $a\in A_{\alpha \beta }$, $\alpha ,\beta \in \pi$.

(f)

The element $w_{\alpha ,1}\in A_{\alpha }\otimes A_1^{+}$ and $w_{1,\alpha }\in A_1^{+}\otimes A_{\alpha }$ for all $\alpha \in \pi$, and satisfies the equations

$$\sum {}_i{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)w_{\alpha }^{\left(i\right)}=h_{\alpha }^{-1}\sum {}_i{w}_{\alpha }^{\left(i\right)}{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right),$$

(23)

$$h_{\alpha }\otimes w_{\beta ,\gamma }+({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma })\left(w_{\alpha ,\beta \gamma }\right)=w_{\alpha ,\beta }\otimes 1_{\gamma }+({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }})\left(w_{\alpha \beta ,\gamma }\right),$$

(24)

for all $\alpha ,\beta ,\gamma \in \pi$.

(ii)

Suppose for all $\alpha ,\beta \in \pi$ given $h\in G\left(A\right)$, $w_{\alpha ,\beta }\in A_{\alpha }\otimes A_{\beta }$, a $\mathbb{k}$-algebra automorphism ${\sigma }_{\alpha }$ of $A_{\alpha }$ and a ${\sigma }_{\alpha }$-derivation ${\delta }_{\alpha }$ of $A_{\alpha }$ such that these data satisfy (d), (e) and (f) in (i). Then, the skew polynomial algebra $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ admits a structure of Hopf group-coalgebra with A as a Hopf group-subcoalgebra, and with $z_{\alpha }$ satisfying (17), (a), (b) and (c) in (i). As a consequence, H is a generalized Hopf group-coalgebra Ore extension of A.

Proof. 

(i) Part (a) happens to be Lemma 4(ii) and (iii), and part (b) follows from the discussion before the theorem. Given (b), part (c) is exactly Lemma 4(iv).

(d)&(e) The proofs of these are similar to the proof of [29] (Theorem 3.5). In short, to obtain the first equality in (21), we first write down the conditions on $\Delta$ which ensure that the relations $z_{\alpha }a={\sigma }_{\alpha }\left(a\right)z_{\alpha }+{\delta }_{\alpha }\left(a\right)$, i.e.,

$${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right){\Delta }_{\alpha ,\beta }\left(a\right)={\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha ,\beta }\left(a\right)\right){\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)+{\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha ,\beta }\left(a\right)\right),$$

(25)

for all $a\in A_{\alpha ,\beta }$, $\alpha ,\beta \in \pi$. We have

$$\begin{array}{c}{\displaystyle {\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right){\Delta }_{\alpha ,\beta }\left(a\right)}\hfill \\ =(h_{\alpha }{a}_{(1,\alpha )}{h}_{\alpha }^{-1}\otimes {\sigma }_{\beta }\left(a_{(2,\beta )}\right))(h_{\alpha }\otimes z_{\beta })+({\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,\beta )})(z_{\alpha }\otimes 1_{\beta })\hfill \\ +{\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,\beta )}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{\beta }\left(a_{(2,\beta )}\right)+w_{\alpha ,\beta }{\Delta }_{\alpha ,\beta }\left(a\right)\hfill \end{array}$$

(26)

and

$$\begin{array}{c}{\displaystyle {\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha ,\beta }\left(a\right)\right){\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)+{\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha ,\beta }\left(a\right)\right)}\hfill \\ ={\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)(h_{\alpha }\otimes z_{\beta })+{\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)(z_{\alpha }\otimes 1_{\beta })\hfill \\ +{\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)w_{\alpha ,\beta }+{\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right).\hfill \end{array}$$

(27)

It is clear that (25) holds if and only if the following hold:

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)& ={\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,\beta )},\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)& =h_{\alpha }{a}_{(1,\alpha )}{h}_{\alpha }^{-1}\otimes {\sigma }_{\beta }\left(a_{(2,\beta )}\right),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right)& ={\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,\beta )}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{\beta }\left(a_{(2,\beta )}\right)\hfill \\ \hfill & +w_{\alpha ,\beta }{\Delta }_{\alpha ,\beta }\left(a\right)-{\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)w_{\alpha ,\beta },\hfill \end{array}$$

for all $\alpha ,\beta \in \pi$ and $a\in A_{\alpha ,\beta }$. The last equation above is exactly (22). Define a family of maps ${\{{\chi }_{\alpha }:A_1\to A_{\alpha }\}}_{\alpha \in \pi }$ by ${\chi }_{\alpha }\left(a\right)={\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)S_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right),a\in A_1$ for all $a\in A_1$. The proofs of the following equations are similar to [29] (Theorem 3.5):

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,1}\left({\chi }_{\alpha }\left(a\right)\right)={\chi }_{\alpha }\left(a\right)\otimes 1_1,& {\Delta }_{1,\alpha }\left({\chi }_{\alpha }\left(a\right)\right)={\chi }_1\left(a\right)\otimes 1_{\alpha },\hfill \\ \hfill {\chi }_{\alpha }\left(a\right)=\epsilon \left({\sigma }_1\left(a\right)\right)1_{\alpha },& {\chi }_{\alpha }\left(a\right)=\epsilon \left({\chi }_1\left(a\right)\right)1_{\alpha }.\hfill \end{array}$$

By [7] (Theorem 1.3), ${\chi }_1\left(a\right)\in \mathbb{k}$ for all $a\in A_1$. It follows that one can regard ${\chi }_{\alpha }$ as a composition of the mapping ${\chi }_1:A_1\to \mathbb{k}$ and injection $\mathbb{k}\to A_{\alpha }$, and hence ${\chi }_1$ is a character of $A_1$ as is easily checked. Therefore, one can recover ${\sigma }_{\alpha }$ from ${\chi }_{\alpha }$, and by Equations (28) and (29), one can show (d).

(f) The equation (24) is the very Lemma 4(iii). For all $\alpha \in \pi$, applying $({\mathrm{Id}}_{H_{\alpha }}\otimes \epsilon ){\Delta }_{\alpha ,1}={\mathrm{Id}}_{H_{\alpha }}=(\epsilon \otimes {\mathrm{Id}}_{H_{\alpha }}){\Delta }_{1,\alpha }$ to $z_{\alpha }$, we have $({\mathrm{Id}}_{H_{\alpha }}\otimes \epsilon )\left(w_{\alpha ,1}\right)=0=(\epsilon \otimes {\mathrm{Id}}_{H_{\alpha }})\left(w_{1,\alpha }\right)$. It follows that $w_{\alpha ,1}\in A_{\alpha }\otimes A_1^{+}$ and $w_{1,\alpha }\in A_1^{+}\otimes A_{\alpha }$ for all $\alpha \in \pi$ by the linear independence of $\left\{w_{\alpha }\right\}$ and $\left\{w_{\beta }\right\}$. In order to show Equation (23), we calculate $m_{\alpha }(S_{{\alpha }^{-1}}\otimes {\mathrm{Id}}_{H_{\alpha }}){\Delta }_{{\alpha }^{-1},\alpha }\left(z_1\right)=\epsilon \left(z_1\right)$ by (b) and (c). We obtain

$$\begin{array}{cc}\hfill 0& =\epsilon \left(z_1\right)\hfill \\ \hfill & =m_{\alpha }(S_{{\alpha }^{-1}}\otimes {\mathrm{Id}}_{H_{\alpha }})(h_{{\alpha }^{-1}}\otimes z_{\alpha }+z_{{\alpha }^{-1}}\otimes 1_{\alpha }+w_{{\alpha }^{-1},\alpha })\hfill \\ \hfill & =S_{{\alpha }^{-1}}\left(h_{{\alpha }^{-1}}\right)z_{\alpha }+S_{{\alpha }^{-1}}\left(z_{{\alpha }^{-1}}\right)+\sum {}_i{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)w_{\alpha }^{\left(i\right)}\hfill \\ \hfill & =h_{{\alpha }^{-1}}^{-1}{z}_{\alpha }-h_{{\alpha }^{-1}}^{-1}(z_{\alpha }+\sum {}_i{w}_{\alpha }^{\left(i\right)}{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right))+\sum {}_i{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)w_{\alpha }^{\left(i\right)}\hfill \\ \hfill & =\sum {}_i{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)w_{\alpha }^{\left(i\right)}-\sum {}_i{w}_{\alpha }^{\left(i\right)}{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right),\hfill \end{array}$$

It follows that Equation (23) holds.

(ii) From the given data, we can define comultiplication $\Delta ={\left\{{\Delta }_{\alpha }\right\}}_{\alpha \in \pi }$ on $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ satisfying Equations (18) and (19) in (b). Then, we prove H is a Hopf group-coalgebra with A as a Hopf group-subcoalgebra, the proof process is carried out in four steps.

Step 1. It is clear that $H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]$ is an algebra for all $\alpha \in \pi$.

Step 2. We show the comultiplication ${\Delta |}_A$ can be extended to H. This means that Equation (25) holds. We only need to proof Equations (28) and (29) by the proof (d) and (e) in part (i). These hold because as follows:

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)& ={\Delta }_{\alpha ,\beta }\left(\chi \left(a_{(1,1)}\right)a_{(2,\alpha \beta )}\right)\hfill \\ \hfill & =\chi \left(a_{(1,1)}\right){\Delta }_{\alpha ,\beta }\left(a_{(2,\alpha \beta )}\right)\hfill \\ \hfill & =\chi \left(a_{(1,1)}\right)a_{(2,\alpha )}\otimes a_{(3,\beta )}\hfill \\ \hfill & ={\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,\beta )},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)& ={\Delta }_{\alpha ,\beta }\left(h_{\alpha \beta }{a}_{(1,\alpha \beta )}{h}_{\alpha \beta }^{-1}\chi \left(a_{(2,1)}\right)\right)\hfill \\ \hfill & ={\Delta }_{\alpha ,\beta }\left(h_{\alpha \beta }\right){\Delta }_{\alpha ,\beta }\left(a_{(1,\alpha \beta )}\right){\Delta }_{\alpha ,\beta }\left(h_{\alpha \beta }^{-1}\right)\chi \left(a_{(2,1)}\right)\hfill \\ \hfill & =h_{\alpha }{a}_{(1,\alpha )}{h}_{\alpha }^{-1}\otimes h_{\beta }{a}_{(2,\beta )}{h}_{\beta }^{-1}\chi \left(a_{(3,1)}\right)\hfill \\ \hfill & =h_{\alpha }{a}_{(1,\alpha )}{h}_{\alpha }^{-1}\otimes {\sigma }_{\beta }\left(a_{(2,\beta )}\right).\hfill \end{array}$$

Step 3. To show that $\epsilon :H_1\to \mathbb{k}$ is an algebra homomorphism, we need to confirm that $\epsilon$ respects the relation (1) for $H_1$; that is to say, we should show that $\epsilon \left({\delta }_1\left(a\right)\right)=0$ for all $a\in A_1$. Since $(H_1,m_1,1_1,{\Delta }_{1,1},\epsilon ,S_1)$ is an Ore extension of a usual Hopf algebra, the proof is similar to [7,8].

Step 4. Define an antipode $S={\left\{S_{\alpha }\right\}}_{\alpha \in \pi }$ on H. H admits an antipode S which can be extended from A to H via Equation (20) if and only if S preserves $z_{\alpha }a={\sigma }_{\alpha }\left(a\right)z_{\alpha }+{\delta }_{\alpha }\left(a\right)$, i.e.,

$$S_{\alpha }\left(a\right)S_{\alpha }\left(z_{\alpha }\right)=S_{\alpha }\left(z_{\alpha }\right)S_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)$$

(30)

for all $a\in A_{\alpha }$, $\alpha \in \pi$.

We substitute Equation (20) into Equation (30) and obtain

$$\begin{array}{c}{\displaystyle -S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}(z_{{\alpha }^{-1}}+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right))}\hfill \\ =-h_{{\alpha }^{-1}}^{-1}(z_{{\alpha }^{-1}}+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right))S_{\alpha }{\sigma }_{\alpha }\left(a\right)+S_{\alpha }{\delta }_{\alpha }\left(a\right).\hfill \end{array}$$

Thus, we have

$$\begin{array}{c}{\displaystyle S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}{z}_{{\alpha }^{-1}}+S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)}\hfill \\ =h_{{\alpha }^{-1}}^{-1}{z}_{{\alpha }^{-1}}{S}_{\alpha }{\sigma }_{\alpha }\left(a\right)-h_{{\alpha }^{-1}}^{-1}\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }{\sigma }_{\alpha }\left(a\right)+S_{\alpha }{\delta }_{\alpha }\left(a\right)\hfill \\ =h_{{\alpha }^{-1}}^{-1}{\sigma }_{{\alpha }^{-1}}\left(S_{\alpha }{\sigma }_{\alpha }\left(a\right)\right)z_{{\alpha }^{-1}}+h_{{\alpha }^{-1}}^{-1}{\delta }_{{\alpha }^{-1}}\left(S_{\alpha }{\sigma }_{\alpha }\left(a\right)\right)\hfill \\ +h_{{\alpha }^{-1}}^{-1}\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }{\sigma }_{\alpha }\left(a\right)-S_{\alpha }{\delta }_{\alpha }\left(a\right),\hfill \end{array}$$

Condition (30) holds if and only if the following two conditions hold:

$$S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}=h_{{\alpha }^{-1}}^{-1}{\sigma }_{{\alpha }^{-1}}\left(S_{\alpha }{\sigma }_{\alpha }\left(a\right)\right),$$

$$\begin{array}{c}{\displaystyle h_{{\alpha }^{-1}}{S}_{\alpha }{\delta }_{\alpha }\left(a\right)+h_{{\alpha }^{-1}}{S}_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)}\hfill \\ ={\delta }_{{\alpha }^{-1}}\left(S_{\alpha }{\sigma }_{\alpha }\left(a\right)\right)+\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right).\hfill \end{array}$$

(31)

The proof of the first equality above is similar to the proof in [29] (Theorem 3.5) and need not be repeated here. Now, we only show the second one. It follows from Equation (21) that we present Equation (31) in an equivalent form

$$\begin{array}{c}{\displaystyle h_{{\alpha }^{-1}}{S}_{\alpha }{\delta }_{\alpha }\left(a\right)+h_{{\alpha }^{-1}}{S}_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)}\hfill \\ =\chi \left(a_{(1,1)}\right){\delta }_{{\alpha }^{-1}}\left(S_{\alpha }{\sigma }_{\alpha }\left(a_{(2,\alpha )}\right)\right)+\chi \left(a_{(1,1)}\right)\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left({\sigma }_{\alpha }\left(a_{(2,\alpha )}\right)\right).\hfill \end{array}$$

We denote the left side of the above equation by $L_{{\alpha }^{-1}}$ and the right side by $R_{{\alpha }^{-1}}$. By the Equations (21) and (22), we have

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,{\alpha }^{-1}}\left({\delta }_1\left(a\right)\right)& ={\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,{\alpha }^{-1})}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)\hfill \\ \hfill & +w_{\alpha ,{\alpha }^{-1}}{\Delta }_{\alpha ,{\alpha }^{-1}}\left(a\right)-{\Delta }_{\alpha ,{\alpha }^{-1}}\left({\sigma }_1\left(a\right)\right)w_{\alpha ,{\alpha }^{-1}}\hfill \\ \hfill & ={\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,{\alpha }^{-1})}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)\hfill \\ \hfill & +\sum {}_i{w}_{\alpha }^{\left(i\right)}{a}_{(1,\alpha )}\otimes w_{{\alpha }^{-1}}^{\left(i\right)}{a}_{(2,{\alpha }^{-1})}-\sum {}_i{\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)w_{\alpha }^{\left(i\right)}\otimes a_{(2,{\alpha }^{-1})}{w}_{{\alpha }^{-1}}^{\left(i\right)}.\hfill \end{array}$$

We apply $m_{\alpha }({\mathrm{Id}}_{A_{\alpha }}\otimes S_{{\alpha }^{-1}})$ to the above equation and obtain

$$\begin{array}{c}{\displaystyle m_{\alpha }({\mathrm{Id}}_{A_{\alpha }}\otimes S_{{\alpha }^{-1}})\left({\Delta }_{\alpha ,{\alpha }^{-1}}\left({\delta }_1\left(a\right)\right)\right)}\hfill \\ =m_{\alpha }({\mathrm{Id}}_{A_{\alpha }}\otimes S_{{\alpha }^{-1}})({\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes a_{(2,{\alpha }^{-1})}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)\hfill \\ +\sum {}_i{w}_{\alpha }^{\left(i\right)}{a}_{(1,\alpha )}\otimes w_{{\alpha }^{-1}}^{\left(i\right)}{a}_{(2,{\alpha }^{-1})}-\sum {}_i{\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)w_{\alpha }^{\left(i\right)}\otimes a_{(2,{\alpha }^{-1})}{w}_{{\alpha }^{-1}}^{\left(i\right)}),\hfill \end{array}$$

and hence

$$\begin{array}{cc}\hfill 0& =\epsilon \left({\delta }_1\left(a\right)\right)1_{\alpha }\hfill \\ \hfill & ={\delta }_{\alpha }\left(a_{(1,\alpha )}\right)S_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)+h_{\alpha }{a}_{(1,\alpha )}{S}_{{\alpha }^{-1}}{\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)\hfill \\ \hfill & +\sum {}_i{w}_{\alpha }^{\left(i\right)}{a}_{(1,\alpha )}{S}_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)S_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)-\sum {}_i{\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)w_{\alpha }^{\left(i\right)}{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)S_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)\hfill \\ \hfill & ={\delta }_{\alpha }\left(a_{(1,\alpha )}\right)S_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)+h_{\alpha }{a}_{(1,\alpha )}{S}_{{\alpha }^{-1}}{\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)\hfill \\ \hfill & +\sum {}_i\epsilon \left(a\right)w_{\alpha }^{\left(i\right)}{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)-\sum {}_i{\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)w_{\alpha }^{\left(i\right)}{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)S_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right).\hfill \end{array}$$

It follows that

$$\begin{array}{c}{\displaystyle a_{(1,\alpha )}{S}_{{\alpha }^{-1}}{\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)+\sum {}_i{h}_{\alpha }^{-1}\epsilon \left(a\right)w_{\alpha }^{\left(i\right)}{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)}\hfill \\ =-h_{\alpha }^{-1}\left({\delta }_{\alpha }\left(a_{(1,\alpha )}\right)S_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)-\sum {}_i{\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)w_{\alpha }^{\left(i\right)}{S}_{{\alpha }^{-1}}\left(w_{{\alpha }^{-1}}^{\left(i\right)}\right)S_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)\right).\hfill \end{array}$$

We change the form of the above equation and apply it to the penultimate equality below

$$\begin{array}{cc}\hfill L_{{\alpha }^{-1}}& =h_{{\alpha }^{-1}}{S}_{\alpha }{\delta }_{\alpha }\left(a\right)+h_{{\alpha }^{-1}}{S}_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)\hfill \\ \hfill & =h_{{\alpha }^{-1}}\epsilon \left(a_{(1,1)}\right)S_{\alpha }{\delta }_{\alpha }\left(a_{(2,\alpha )}\right)+h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}\epsilon \left(a_{(2,1)}\right)\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)\hfill \\ \hfill & =h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)a_{(2,{\alpha }^{-1})}{S}_{\alpha }{\delta }_{\alpha }\left(a_{(3,\alpha )}\right)\hfill \\ \hfill & +h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}\epsilon \left(a_{(2,1)}\right)\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)\hfill \\ \hfill & =h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)\left(a_{(2,{\alpha }^{-1})}{S}_{\alpha }{\delta }_{\alpha }\left(a_{(3,\alpha )}\right)+h_{{\alpha }^{-1}}^{-1}\epsilon \left(a_{(2,1)}\right)\sum {}_i{w}_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)\right)\hfill \\ \hfill & =-h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}({\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)S_{\alpha }\left(a_{(3,\alpha )}\right)\hfill \\ \hfill & -\sum {}_i{\sigma }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)w_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left(a_{(3,\alpha )}\right))\hfill \\ \hfill & =-h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}{\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)S_{\alpha }\left(a_{(3,\alpha )}\right)\hfill \\ \hfill & +\sum {}_i{h}_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}{\sigma }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)w_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left(a_{(3,\alpha )}\right),\hfill \end{array}$$

Applying Equation (21) to the second item of the above equality, we obtain

$$\begin{array}{c}{\displaystyle \sum {}_i{h}_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}{\sigma }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)w_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left(a_{(3,\alpha )}\right)}\hfill \\ =\sum {}_i{h}_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)a_{(2,{\alpha }^{-1})}{h}_{{\alpha }^{-1}}^{-1}\chi \left(a_{(3,1)}\right)w_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left(a_{(4,\alpha )}\right)\hfill \\ =\sum {}_i{h}_{{\alpha }^{-1}}\epsilon \left(a_{(1,1)}\right)h_{{\alpha }^{-1}}^{-1}\chi \left(a_{(2,1)}\right)w_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left(a_{(3,\alpha )}\right)\hfill \\ =\sum {}_i\chi \left(a_{(1,1)}\right)w_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left(a_{(2,\alpha )}\right),\hfill \end{array}$$

Thence, we have

$$\begin{array}{cc}\hfill L_{{\alpha }^{-1}}=& -h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}{\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)S_{\alpha }\left(a_{(3,\alpha )}\right)\hfill \\ \hfill & +\sum {}_i\chi \left(a_{(1,1)}\right)w_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left(a_{(2,\alpha )}\right).\hfill \end{array}$$

Operation on the first item of Equation (20) is similar to [29] and we obtain

$$\chi \left(a_{(1,1)}\right){\delta }_{{\alpha }^{-1}}\left(S_{\alpha }{\sigma }_{\alpha }\left(a_{(2,\alpha )}\right)\right)=-h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}{\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)S_{\alpha }\left(a_{(3,\alpha )}\right),$$

It follows that

$$\begin{array}{cc}\hfill R_{{\alpha }^{-1}}=& -h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}{\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)S_{\alpha }\left(a_{(3,\alpha )}\right)\hfill \\ \hfill & +\sum {}_i\chi \left(a_{(1,1)}\right)w_{{\alpha }^{-1}}^{\left(i\right)}{S}_{\alpha }\left(w_{\alpha }^{\left(i\right)}\right)S_{\alpha }\left(a_{(2,\alpha )}\right).\hfill \end{array}$$

Therefore, $L_{{\alpha }^{-1}}=R_{{\alpha }^{-1}}$, namely, Equation (31) holds.

In summary, H becomes a Hopf group-coalgebra which satisfies Equation (17) and items (a), (b), (c) of (i), and hence H is a generalized Hopf group-coalgebra Ore extension of A.    □

Now, we study the relations of generalized Hopf group-coalgebra Ore extensions of two Hopf group-coalgebras. We first give some definitions and symbols. Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ and $A^{\prime }={\left\{A_{\alpha }^{\prime }\right\}}_{\alpha \in \pi }$ be two Hopf group-coalgebras. We call $f:A\to A^{\prime }$, where $f={\{f_{\alpha }:A_{\alpha }\to A_{\alpha }^{\prime }\}}_{\alpha \in \pi }$, a Hopf group-coalgebra morphism if for all $\alpha ,\beta \in \pi$, $f_{\alpha }$ is an algebra morphism, ${\Delta }_{\alpha ,\beta }^{\prime }{f}_{\alpha \beta }=(f_{\alpha }\otimes f_{\beta }){\Delta }_{\alpha ,\beta }$ and $S_{\alpha }^{\prime }{f}_{\alpha }=f_{{\alpha }^{-1}}{S}_{\alpha }$, where $\Delta$, ${\Delta }^{\prime }$, S, $S^{\prime }$ are the comultiplications and antipodes of A and $A^{\prime }$, respectively.

Definition 6.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra. Denote $h=\left\{h_{\alpha }\right\}$ and $\sigma =\left\{{\sigma }_{\alpha }\right\}$. A family of mappings $\delta =\left\{{\delta }_{\alpha }\right\}$ satisfying Equation (22) is called ah-coderivation. If δ is also a σ-derivation where σ is an algebra morphism satisfying Equation (21), then δ is called a$⟨\chi ,h⟩$-derivation.

Notation 1.

Denote the generalized Hopf group-coalgebra Ore extension $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ of $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ by $H={\{H_{\alpha }=A_{\alpha }(\chi ,h_{\alpha },{\delta }_{\alpha })\}}_{\alpha \in \pi },$ where $\chi :A_1\to \mathbb{k}$ is a character, $h={\left\{h_{\alpha }\right\}}_{\alpha \in \pi }$ is a π-grouplike element and $\delta ={\left\{{\delta }_{\alpha }\right\}}_{\alpha \in \pi }$ is a $⟨\chi ,h⟩$-derivation.

Definition 7.

Two generalized Hopf group-coalgebra Ore extensions

$$H={\{H_{\alpha }=A_{\alpha }(\chi ,h_{\alpha },{\delta }_{\alpha })\}}_{\alpha \in \pi }\mathrm{and}{H}^{\prime }={\{H_{\alpha }^{\prime }=A_{\alpha }^{\prime }({\chi }^{\prime },h_{\alpha }^{\prime },{\delta }_{\alpha }^{\prime })\}}_{\alpha \in \pi }$$

of Hopf group-coalgebras A and $A^{\prime }$ are said to be isomorphic if there is an isomorphism of Hopf group-coalgebras $\Phi :H\to H^{\prime }$ such that $\Phi \left(A\right)=A^{\prime }$.

Definition 8.

A $⟨\chi ,h⟩$-derivation $\delta ={\left\{{\delta }_{\alpha }\right\}}_{\alpha \in \pi }$ is inner, where $h={\left\{h_{\alpha }\right\}}_{\alpha \in \pi }$, if there is a family of elements ${\left\{d_{\alpha }\right\}}_{\alpha \in \pi }\in A$ such that for all $a_{\alpha }\in A_{\alpha }$, ${\delta }_{\alpha }\left(a_{\alpha }\right)={\sigma }_{\alpha }\left(a_{\alpha }\right)d_{\alpha }-d_{\alpha }{a}_{\alpha }$ and ${\Delta }_{\alpha ,\beta }\left(d_{\alpha \beta }\right)=d_{\alpha }\otimes 1_{\beta }+h_{\alpha }\otimes d_{\beta }$.

Proposition 2.

Let $H={\{H_{\alpha }=A_{\alpha }(\chi ,h_{\alpha },{\delta }_{\alpha })\}}_{\alpha \in \pi }$ be a generalized Hopf group-coalgebra Ore extension of a Hopf group-coalgebra A. If $\delta ={\left\{{\delta }_{\alpha }\right\}}_{\alpha \in \pi }$ is an inner $⟨\chi ,h⟩$-derivation, then $H={\{H_{\alpha }=A_{\alpha }(\chi ,h_{\alpha },{\delta }_{\alpha })\}}_{\alpha \in \pi }$ is isomorphic to the generalized Hopf group-coalgebra Ore extension $H^{\prime }={\{H_{\alpha }^{\prime }=A_{\alpha }(\chi ,h_{\alpha },0)\}}_{\alpha \in \pi }$.

Proof. 

The proof is similar to [29] (Proposition 4.8) and will not be repeated.    □

Proposition 3.

Let $H={\{H_{\alpha }=A_{\alpha }(\chi ,h_{\alpha },{\delta }_{\alpha })\}}_{\alpha \in \pi }$ and $H^{\prime }={\{H_{\alpha }=A_{\alpha }^{\prime }({\chi }^{\prime },h_{\alpha }^{\prime },{\delta }_{\alpha }^{\prime })\}}_{\alpha \in \pi }$ be two generalized Hopf group-coalgebra Ore extensions. If there exists an isomorphism of Hopf group-coalgebra $\Phi :H\to H^{\prime }$ such that ${\chi }^{\prime }=\chi {\Phi }^{-1}$, $h^{\prime }=\Phi \left(h\right)$, ${\delta }^{\prime }=\Phi \delta {\Phi }^{-1}+{\delta }^{″}$ and $w^{\prime }=(\Phi \otimes \Phi )\left(w\right)$, where $h=\left\{h_{\alpha }\right\}$, $h^{\prime }=\left\{h_{\alpha }^{\prime }\right\}$, $\delta =\left\{{\delta }_{\alpha }\right\}$, ${\delta }^{\prime }=\left\{{\delta }_{\alpha }^{\prime }\right\}$, $w=\left\{w_{\alpha ,\beta }\right\}$, $w^{\prime }=\left\{w_{\alpha ,\beta }^{\prime }\right\}$, ${\delta }^{″}$ is an inner $⟨{\chi }^{\prime },h^{\prime }⟩$-derivation of $A^{\prime }$, then $H\cong H^{\prime }$ as the generalized Hopf group-coalgebra Ore extensions.

Proof. 

The proof is similar to [29] (Theorem 4.9) and will not be repeated.    □

We end this section with an example.

Example 1.

Let $\mathbb{k}$ be a field with $\mathrm{Char}\left(\mathbb{k}\right)=p>0$. For all $\alpha \in \pi$, let $A_{\alpha }$ be the $\mathbb{k}$-algebra generated by $x_{\alpha }$ and $y_{\alpha }$ subject to the relations:

$$x_{\alpha }{y}_{\alpha }=y_{\alpha }{x}_{\alpha },x_{\alpha }^p=0,y_{\alpha }^p=0.$$

Then, $A_{\alpha }$ has a $\mathbb{k}$-basis $\left\{x_{\alpha }^i{y}_{\alpha }^j\right|0\le i,j<p\}$. It is easy to check that $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ is a Hopf group-coalgebra with the comultiplication Δ, counit ε and antipode S given by

$$\begin{array}{c}{\Delta }_{\alpha ,\beta }\left(x_{\alpha \beta }\right)=x_{\alpha }\otimes 1_{\beta }+1_{\alpha }\otimes x_{\beta },{\Delta }_{\alpha ,\beta }\left(y_{\alpha \beta }\right)=y_{\alpha }\otimes 1_{\beta }+1_{\alpha }\otimes y_{\beta },\\ \epsilon \left(x_1\right)=0,\epsilon \left(y_1\right)=0,S_{\alpha }\left(x_{\alpha }\right)=-x_{{\alpha }^{-1}},S_{\alpha }\left(y_{\alpha }\right)=-y_{{\alpha }^{-1}}.\end{array}$$

Let $h_{\alpha }=1_{\alpha }$, $w_{\alpha ,\beta }=x_{\alpha }\otimes y_{\beta }-y_{\alpha }\otimes x_{\beta }$ and $\chi =\epsilon$. Then, by Theorem 1(d), the induced algebra automorphism σ of A is the identity map on A, i.e., ${\sigma }_{\alpha }={\mathrm{Id}}_{\alpha }$ for all $\alpha \in \pi$. For all ${\ell }_1,{\ell }_2\in \mathbb{k}$, let ${\delta }_{\alpha }\left(x_{\alpha }\right)={\ell }_1{x}_{\alpha }$ and ${\delta }_{\alpha }\left(y_{\alpha }\right)={\ell }_2{y}_{\alpha }$, for all $\alpha \in \pi$. A straightforward computation shows that $\delta ={\left\{{\delta }_{\alpha }\right\}}_{\alpha \in \pi }$ can be uniquely extended to a derivation of A, denoted by δ still. It is easy to check that Theorem 1(e) and (f) are satisfied. Hence, we have a generalized Hopf group-coalgebra Ore extension ${\left\{A_{\alpha }[z_{\alpha };{\mathrm{Id}}_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }$, and

$$\begin{array}{cc}\hfill z_{\alpha }{x}_{\alpha }& =x_{\alpha }{z}_{\alpha }+{\ell }_1{x}_{\alpha },z_{\alpha }{y}_{\alpha }=y_{\alpha }{z}_{\alpha }+{\ell }_2{y}_{\alpha },\hfill \\ \hfill {\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)& =z_{\alpha }\otimes 1_{\beta }+1_{\alpha }\otimes z_{\beta }+x_{\alpha }\otimes y_{\beta }-y_{\alpha }\otimes x_{\beta },\hfill \\ \hfill S_{\alpha }\left(z_{\alpha }\right)& =x_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}-y_{{\alpha }^{-1}}{x}_{{\alpha }^{-1}}-z_{{\alpha }^{-1}}.\hfill \end{array}$$

4. Special Type

In this section, we discuss an explicit case when $w_{\alpha ,\beta }=\sum {}_i{w}_{\alpha }^{\left(i\right)}\otimes w_{\beta }^{\left(i\right)}\in A_{\alpha }\otimes A_{\beta }$ is a simpler form $x_{\alpha }\otimes y_{\alpha }$ in Definition 5 and call it the special Hopf group-coalgebra Ore extension, which also may be regarded as generalizing the theory in [12] to Hopf group-coalgebras.

Definition 9.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra and $H={\left\{H_{\alpha }\right\}}_{\alpha \in \pi }$ be a generalized Hopf group-coalgebra Ore extension off A. Replace the relation (2) by

$${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)=z_{\alpha }\otimes g_{\beta }+h_{\alpha }\otimes z_{\beta }+x_{\alpha }\otimes y_{\beta }$$

(32)

for some $h_{\alpha },x_{\alpha }\in A_{\alpha }$, $g_{\beta },y_{\beta }\in A_{\beta }$, $\forall \alpha ,\beta \in \pi$. Then, H is called the special Hopf group-coalgebra Ore extension of A. Moreover, we also say that H has a Hopf group-coalgebra structure determined by $(g,h,x,y)$, where $g={\left\{g_{\alpha }\right\}}_{\alpha \in \pi }$, $h={\left\{h_{\alpha }\right\}}_{\alpha \in \pi }$, $x={\left\{x_{\alpha }\right\}}_{\alpha \in \pi }$ and $y={\left\{y_{\alpha }\right\}}_{\alpha \in \pi }$ in A.

Note that the Equation (32) is a special form of Equation (2) that $w_{\alpha ,\beta }=x_{\alpha }\otimes y_{\alpha }$ has only one addition item. When $\pi =\left\{1\right\}$, the special Hopf group-coalgebra Ore extension is just the generalized Hopf-Ore extension in [12]. Obviously, $g_1,h_1$ are nonzero.

Proposition 4.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra, $g,h,x,y\in A$ and $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$. If H has a Hopf group-coalgebra structure determined by $(g,h,x,y)$, then $g,h\in G\left(A\right)$ and one of the followings holds:

(i)

$x_1=0$ or $y_1=0$;

(ii)

$x_{\alpha }=p{h}_{\alpha }$ and $y_{\beta }=q{g}_{\beta }$ for some $p,q\in {\mathbb{k}}^{\times }$;

(iii)

$x\in P_{r,h}\left(A\right)$ and $y\in P_{g,r}\left(A\right)$ for some $r\in G\left(A\right)$.

Proof. 

It is clear that $H_{\alpha }$ is a free left $A_{\alpha }$-module under left multiplication with the basis $\left\{z_{\alpha }^i\right|i\ge 0\}$ for all $\alpha \in \pi$. Therefore, $H_{\alpha }\otimes H_{\beta }\otimes H_{\gamma }$ is a free left $A_{\alpha }\otimes A_{\beta }\otimes A_{\gamma }$-module with the basis $\{z_{\alpha }^i\otimes z_{\beta }^j\otimes z_{\gamma }^m|i,j,m\ge 0\}$ for all $\alpha ,\beta ,\gamma \in \pi$. By calculating, we have

$$\begin{array}{cc}\hfill ({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }}){\Delta }_{\alpha \beta ,\gamma }\left(z_{\alpha \beta \gamma }\right)& =z_{\alpha }\otimes g_{\beta }\otimes g_{\gamma }+h_{\alpha }\otimes z_{\beta }\otimes g_{\gamma }+x_{\alpha }\otimes y_{\beta }\otimes g_{\gamma }\hfill \\ \hfill & +{\Delta }_{\alpha ,\beta }\left(h_{\alpha \beta }\right)\otimes z_{\gamma }+{\Delta }_{\alpha ,\beta }\left(x_{\alpha \beta }\right)\otimes y_{\gamma }\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma }){\Delta }_{\alpha ,\beta \gamma }\left(z_{\alpha \beta \gamma }\right)& =z_{\alpha }\otimes {\Delta }_{\beta ,\gamma }\left(g_{\beta \gamma }\right)+h_{\alpha }\otimes z_{\beta }\otimes g_{\gamma }+h_{\alpha }\otimes h_{\beta }\otimes z_{\gamma }\hfill \\ \hfill & +h_{\alpha }\otimes x_{\beta }\otimes y_{\gamma }+x_{\alpha }\otimes {\Delta }_{\beta ,\gamma }\left(y_{\beta \gamma }\right).\hfill \end{array}$$

According to coassociativity, we have

$${\Delta }_{\beta ,\gamma }\left(g_{\beta \gamma }\right)=g_{\beta }\otimes g_{\gamma },{\Delta }_{\alpha ,\beta }\left(h_{\alpha \beta }\right)=h_{\alpha }\otimes h_{\beta },$$

(33)

$$x_{\alpha }\otimes y_{\beta }\otimes g_{\gamma }+{\Delta }_{\alpha ,\beta }\left(x_{\alpha \beta }\right)\otimes y_{\gamma }=h_{\alpha }\otimes x_{\beta }\otimes y_{\gamma }+x_{\alpha }\otimes {\Delta }_{\beta ,\gamma }\left(y_{\beta \gamma }\right).$$

(34)

Since $g_1\ne 0$ and $h_1\ne 0$, it follows from Equation (33) that $g=\left\{g_{\alpha }\right\}$ and $h=\left\{h_{\alpha }\right\}$ are both $\pi$-grouplike elements. We have $x_1=0$ or $y_1=0$ or $x_1,y_1\ne 0$.

Case 1: $x_1,y_1\ne 0$ and $x_1=p{h}_1$ for some $p\in {\mathbb{k}}^{\times }$. Letting $\alpha =\beta =1$, the Equation (34) becomes $p{h}_1\otimes y_1\otimes g_{\gamma }+p{h}_1\otimes h_1\otimes \otimes y_{\gamma }=p{h}_1\otimes h_1\otimes y_{\gamma }+p{h}_1\otimes {\Delta }_{1,\gamma }\left(y_{\gamma }\right)$. Applying $\epsilon \otimes \epsilon \otimes {\mathrm{Id}}_{A_{\gamma }}$ to both sides, we have $y_{\gamma }=\epsilon \left(y_1\right)g_{\gamma }$ for all $\gamma \in \pi$. Let $q=\epsilon \left(y_1\right)$, and Equation (34) becomes $q{x}_{\alpha }\otimes g_{\beta }\otimes g_{\gamma }+q{\Delta }_{\alpha ,\beta }\left(x_{\alpha \beta }\right)\otimes g_{\gamma }=q{h}_{\alpha }\otimes x_{\beta }\otimes g_{\gamma }+q{x}_{\alpha }\otimes g_{\beta }\otimes g_{\gamma }$. Letting $\beta =\gamma =1$ and applying ${\mathrm{Id}}_{A_{\alpha }}\otimes \epsilon \otimes \epsilon$ to both sides, we obtain that $x_{\alpha }=p{h}_{\alpha }$ for all $\alpha \in \pi$.

Case 2: $x_1,y_1\ne 0$ and $x_1\ne p{h}_1$ for all $p\in {\mathbb{k}}^{\times }$. Since $x_1$ and $h_1$ are linearly independent, we have ${\Delta }_{\alpha ,\beta }\left(x_{\alpha \beta }\right)=h_{\alpha }\otimes x_{\beta }+u_{\alpha ,\beta }$ and ${\Delta }_{\beta ,\gamma }\left(y_{\beta \gamma }\right)=y_{\beta }\otimes g_{\gamma }+v_{\beta ,\gamma }$ for some $u_{\alpha ,\beta }\in A_{\alpha }\otimes A_{\beta }$, $v_{\beta ,\gamma }\in A_{\beta }\otimes A_{\gamma }$. It follows from Equation (34) that $u_{\alpha ,\beta }\otimes y_{\gamma }=x_{\alpha }\otimes v_{\beta ,\gamma }$, and hence $u_{\alpha ,\beta }=x_{\alpha }\otimes u_{\beta }$ and $v_{\beta ,\gamma }=v_{\beta }\otimes y_{\gamma }$ for some $u_{\beta }\ne 0,v_{\beta }\in A_{\beta }$. and so $u_{\beta }=v_{\beta }$. Thus, by ${\Delta }_{\alpha ,\beta }\left(x_{\alpha \beta }\right)=h_{\alpha }\otimes x_{\beta }+x_{\alpha }\otimes u_{\beta }$ and $({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{A_{\gamma }}){\Delta }_{\alpha \beta ,\gamma }\left(x_{\alpha \beta \gamma }\right)=({\mathrm{Id}}_{A_{\alpha }}\otimes {\Delta }_{\beta ,\gamma }){\Delta }_{\alpha ,\beta \gamma }\left(x_{\alpha \beta \gamma }\right)$, we have that ${\Delta }_{\beta ,\gamma }\left(u_{\beta \gamma }\right)=u_{\beta }\otimes u_{\gamma }$. Thus, $u={\left\{u_{\alpha }\right\}}_{\alpha \in \pi }$ is a $\pi$-grouplike element. This completes the proof by setting $r=u$.    □

Remark 1.

If $x_{\alpha }=0$ or $y_{\beta }=0$ for all $\alpha ,\beta \in \pi$, then $x_{\alpha }\otimes y_{\beta }=0$, and hence all Hopf group-coalgebra structures on H determined by $(g,h,x,y)$ are the same. Therefore, we can assume $x_{\alpha }=y_{\beta }=0$ at this point. If $x_{\alpha }=p{h}_{\alpha }$ and $y_{\beta }=q{g}_{\beta }$ for some $p,q\in {\mathbb{k}}^{\times }$, let $z_{\alpha }^{\prime }=z_{\alpha }+pq{h}_{\alpha }$, ${\delta }_{\alpha }^{\prime }\left(a\right)={\delta }_{\alpha }\left(a\right)+pq(h_{\alpha }a-{\sigma }_{\alpha }\left(a\right)h_{\alpha })$ for $a\in A_{\alpha }$. Then, ${\delta }_{\alpha }^{\prime }$ is a ${\sigma }_{\alpha }$-derivation of A, $A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }^{\prime }]=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]$ as algebra, and ${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }^{\prime }\right)=z_{\alpha }^{\prime }\otimes g_{\beta }+h_{\alpha }\otimes z_{\beta }^{\prime }$. Thence, ${\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }$ has a Hopf group-coalgebra structure determined by $(g,h,x,y)$ if and only if ${\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }^{\prime }]\right\}}_{\alpha \in \pi }$ has a Hopf group-coalgebra structure determined by $(g,h,0,0)$. In this view, we have ${\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }^{\prime }]\right\}}_{\alpha \in \pi }={\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }$ as Hopf group-coalgebras. In the above proposition, item (b) reduces to a special case in item (a) via suitable changes of $z_{\alpha }$, $\alpha \in \pi$. Thus, we assume that either $x_1=y_1=0$ or x is $(r,h)$-primitive and y is $(g,r)$-primitive for some π-grouplike element r. We especially assume that $x_{\alpha }=0$ if and only if $y_{\beta }=0$, $\alpha ,\beta \in \pi$.

Corollary 1.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra, $g,h,x,y\in A$ and $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$. If H has a Hopf group-coalgebra structure determined by (g, h, x, y), then

$$\epsilon \left(z_1\right)=0$$

(35)

$$S_{\alpha }\left(z_{\alpha }\right)=h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{r}_{{\alpha }^{-1}}^{-1}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}-h_{{\alpha }^{-1}}^{-1}{z}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}$$

(36)

Proof. 

By Lemma 1, we have

$$\begin{array}{c}\epsilon \left(g_1\right)=\epsilon \left(h_1\right)=1,\epsilon \left(x_1\right)=\epsilon \left(y_1\right)=0,S_{\alpha }\left(g_{\alpha }\right)=g_{{\alpha }^{-1}}^{-1},S_{\alpha }\left(h_{\alpha }\right)=h_{{\alpha }^{-1}}^{-1},\\ S_{\alpha }\left(x_{\alpha }\right)=-h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{r}_{{\alpha }^{-1}}^{-1},S_{\alpha }\left(y_{\alpha }\right)=-r_{{\alpha }^{-1}}^{-1}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}\end{array}$$

Applying ${\mathrm{Id}}_{\alpha }\otimes \epsilon$ to ${\Delta }_{\alpha ,1}\left(z_{\alpha }\right)=z_{\alpha }\otimes g_1+h_{\alpha }\otimes z_1+x_{\alpha }\otimes y_1$, we obtain

$$z_{\alpha }=z_{\alpha }\epsilon \left(g_1\right)+h_{\alpha }\epsilon \left(z_1\right)+x_{\alpha }\epsilon \left(y_1\right)=z_{\alpha }+h_{\alpha }\epsilon \left(z_1\right).$$

Hence, $\epsilon \left(z_1\right)=0$. Then, applying $S_{\alpha }\otimes {\mathrm{Id}}_{{\alpha }^{-1}}$ to ${\Delta }_{\alpha ,{\alpha }^{-1}}\left(z_1\right)=z_{\alpha }\otimes g_{{\alpha }^{-1}}+h_{\alpha }\otimes z_{{\alpha }^{-1}}+x_{\alpha }\otimes y_{{\alpha }^{-1}}$, we obtain

$$\begin{array}{cc}\hfill 0& =\epsilon \left(z_1\right)=S_{\alpha }\left(z_{\alpha }\right)g_{{\alpha }^{-1}}+S_{\alpha }\left(h_{\alpha }\right)z_{{\alpha }^{-1}}+S_{\alpha }\left(x_{\alpha }\right)y_{{\alpha }^{-1}}\hfill \\ \hfill & =S_{\alpha }\left(z_{\alpha }\right)g_{{\alpha }^{-1}}+h_{{\alpha }^{-1}}^{-1}{z}_{{\alpha }^{-1}}-h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{r}_{{\alpha }^{-1}}^{-1}{y}_{{\alpha }^{-1}}.\hfill \end{array}$$

Thus, $S_{\alpha }\left(z_{\alpha }\right)=h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{r}_{{\alpha }^{-1}}^{-1}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}-h_{{\alpha }^{-1}}^{-1}{z}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}$.    □

By Remark 1, we have ${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)=z_{\alpha }\otimes g_{\beta }+h_{\alpha }\otimes z_{\beta }+x_{\alpha }\otimes y_{\beta }$, ${\Delta }_{\alpha ,\beta }\left(x_{\alpha \beta }\right)=x_{\alpha }\otimes r_{\beta }+h_{\alpha }\otimes x_{\beta }$ and ${\Delta }_{\alpha ,\beta }\left(y_{\alpha \beta }\right)=y_{\alpha }\otimes g_{\beta }+r_{\alpha }\otimes y_{\beta }$. Replacing $z_{\alpha }$ by $z_{\alpha }^{\prime }=r_{\alpha }^{-1}{z}_{\alpha }$, $(g,h,x,y)$ by $(g^{\prime },h^{\prime },x^{\prime },y^{\prime })=(r^{-1}g,r^{-1}h,r^{-1}x,r^{-1}y)$, where $r^{-1}g={\left\{r_{\alpha }^{-1}{g}_{\alpha }\right\}}_{\alpha \in \pi }$, and others are similar, we obtain that ${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }^{\prime }\right)=z_{\alpha }^{\prime }\otimes g_{\beta }^{\prime }+h_{\alpha }^{\prime }\otimes z_{\beta }^{\prime }+x_{\alpha }^{\prime }\otimes y_{\beta }^{\prime }$, ${\Delta }_{\alpha ,\beta }\left(x_{\alpha \beta }^{\prime }\right)=x_{\alpha }^{\prime }\otimes 1_{\beta }+h_{\alpha }^{\prime }\otimes x_{\beta }^{\prime }$ and ${\Delta }_{\alpha ,\beta }\left(y_{\alpha \beta }^{\prime }\right)=y_{\alpha }^{\prime }\otimes g_{\beta }^{\prime }+1_{\alpha }\otimes y_{\beta }^{\prime }$.

Notation 2.

Continuing the previous Notation, let us assume that if $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ has a Hopf group-coalgebra structure determined by $(g,h,x,y)$, then $g,h\in G\left(A\right)$, $x\in P_{1,h}\left(A\right)$, $y\in P_{g,1}\left(A\right)$ and the element z satisfies the relation (32) for all $\alpha ,\beta \in \pi$. Under this hypothesis, Equation (36) becomes

$$S_{\alpha }\left(z_{\alpha }\right)=h_{{\alpha }^{-1}}^{-1}(x_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}-z_{{\alpha }^{-1}})g_{{\alpha }^{-1}}^{-1}$$

(37)

The following theorem discusses the sufficient and necessary condition for an Ore extension of a Hopf group-coalgebra to have a special Hopf group-coalgebra Ore extension structure, which generalizes [12] (Theorem 1.6) and is a special case of Theorem 1.

Theorem 2.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra, $g,h\in G\left(A\right)$, $x\in P_{1,h}\left(A\right)$, $y\in P_{g,1}\left(A\right)$ and $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$. Then, H has a Hopf group-coalgebra structure determined by $(g,h,x,y)$ if and only if the following conditions are satisfied:

(i)

There is a character $\chi :A_1\to \mathbb{k}$ such that for all $\alpha \in \pi ,a\in A_{\alpha }$

$${\sigma }_{\alpha }\left(a\right)=\chi \left(a_{(1,1)}\right){Ad}_{g_{\alpha }}\left(a_{(2,\alpha )}\right),$$

(ii)

$\chi \left(a_{(1,1)}\right){Ad}_{g_{\alpha }}\left(a_{(2,\alpha )}\right)={Ad}_{h_{\alpha }}\left(a_{(1,\alpha )}\right)\chi \left(a_{(2,1)}\right)$, $\forall \alpha \in \pi ,a\in A_{\alpha }$

(iii)

The σ-derivation δ satisfies the relation for all $\alpha \in \pi ,a\in A_{\alpha }$

$$\begin{array}{c}{\displaystyle {\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right)+{\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)(x_{\alpha }\otimes y_{\beta })}\hfill \\ =(x_{\alpha }\otimes y_{\beta }){\Delta }_{\alpha ,\beta }\left(a\right)+{\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes g_{\beta }{a}_{(2,\beta )}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{\beta }\left(a_{(2,\beta )}\right).\hfill \end{array}$$

Proof. 

Similar to Theorem 1, the proof consists of three steps. At Step 1, we prove that the comultiplication $\Delta$ of A can be extended to $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ by the relation (32) if and only if the conditions (i)–(iii) are satisfied. At Step 2, we show that if the conditions (i)–(iii) are satisfied, then H admits a counit extending the counit of A by Equation (5). At Step 3, we prove that if the conditions (i)–(iii) are satisfied, then H has an extension of antipode from A by Equation (7).

Step 1. Assume that the comultiplication $\Delta$ of A can be extended to H by the relation (32). Then, the comultiplication $\Delta$ preserve the relation $z_{\alpha }a={\sigma }_{\alpha }\left(a\right)z_{\alpha }+{\delta }_a\left(a\right)$ for all $a\in A_{\alpha },\alpha \in \pi$, i.e., ${\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right){\Delta }_{\alpha ,\beta }\left(a\right)={\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right){\Delta }_{\alpha ,\beta }\left(z_{(}\alpha \beta \right))+{\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right)$ for all $\alpha ,\beta \in \pi ,a\in A_{\alpha \beta }$. Then, a straightforward calculation shows that

$$\begin{array}{c}{\displaystyle {\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right){\Delta }_{\alpha ,\beta }\left(a\right)}\hfill \\ =({\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes g_{\beta }{a}_{(2,\beta )})(z_{\alpha }\otimes 1_{\beta })+(h_{\alpha }{a}_{(1,\alpha )}\otimes {\sigma }_{\beta }\left(a_{(2,\beta )}\right))(1_{\alpha }\otimes z_{\beta })\hfill \\ +{\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes g_{\beta }{a}_{(2,\beta )}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{\beta }\left(a_{(2,\beta )}\right)+(x_{\alpha }\otimes y_{\beta })(a_{(1,\alpha )}\otimes a_{(2,\beta )})\hfill \end{array}$$

and

$$\begin{array}{c}{\displaystyle {\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right){\Delta }_{\alpha ,\beta }\left(z_{(}\alpha \beta \right))+{\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right)}\hfill \\ =({\sigma }_{\alpha \beta }{\left(a\right)}_{(1,\alpha )}\otimes {\sigma }_{\alpha \beta }{\left(a\right)}_{(2,\beta )}{g}_{\beta })(z_{\alpha }\otimes 1_{\beta })+({\sigma }_{\alpha \beta }{\left(a\right)}_{(1,\alpha )}{h}_{\alpha }\otimes {\sigma }_{\alpha \beta }{\left(a\right)}_{(2,\beta )})(1_{\alpha }\otimes z_{\beta })\hfill \\ +({\sigma }_{\alpha \beta }{\left(a\right)}_{(1,\alpha )}\otimes {\sigma }_{\alpha \beta }{\left(a\right)}_{(2,\beta )})(x_{\alpha }\otimes y_{\beta })+{\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right)\hfill \end{array}$$

It follows that

$${\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)={\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes {Ad}_{g_{\beta }}\left(a_{(2,\beta )}\right)$$

(38)

$${\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)={Ad}_{h_{\alpha }}\left(a_{(1,\alpha )}\right)\otimes {\sigma }_{\beta }\left(a_{(2,\beta )}\right)$$

(39)

$$\begin{array}{c}{\displaystyle {\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right)+{\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)(x_{\alpha }\otimes y_{\beta })}\hfill \\ =(x_{\alpha }\otimes y_{\beta }){\Delta }_{\alpha ,\beta }\left(a\right)+{\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes g_{\beta }{a}_{(2,\beta )}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{\beta }\left(a_{(2,\beta )}\right).\hfill \end{array}$$

for all $a\in A_{\alpha \beta },\alpha ,\beta \in \pi$. The last equation coincides with that in (iii). Define a family of maps ${\{{\chi }_{\alpha }:A_1\to A_{\alpha }\}}_{\alpha \in \pi }$ by ${\chi }_{\alpha }\left(a\right)=\sum {\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)g_{\alpha }{S}_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)g_{\alpha }^{-1}$ for all $a\in A_1$. Obviously, we have ${\Delta }_{\alpha ,1}\left({\chi }_{\alpha }\left(a\right)\right)={\chi }_{\alpha }\left(a\right)\otimes 1_1$, ${\Delta }_{1,\alpha }\left({\chi }_{\alpha }\left(a\right)\right)={\chi }_1\left(a\right)\otimes 1_{\alpha }$ and ${\chi }_{\alpha }\left(a\right)=\epsilon \left({\chi }_1\left(a\right)\right)1_{\alpha }$ for all $a\in A_{\alpha },\alpha \in \pi$. Then, the proof of parts (i) and (ii) via Equations (38) and (39) is the same as the proof of Theorem 1. On the other hand, if conditions (i)–(iii) hold, then we have

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)& =\chi \left(a_{(1,1)}\right){\Delta }_{\alpha ,\beta }\left({Ad}_{g_{\alpha \beta }}\left(a_{(2,\alpha \beta )}\right)\right)\hfill \\ \hfill & =\chi \left(a_{(1,1)}\right){Ad}_{g_{\alpha }}\left(a_{(2,\alpha )}\right)\otimes {Ad}_{g_{\beta }}\left(a_{(3,\beta )}\right)\hfill \\ \hfill & ={\sigma }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes {Ad}_{g_{\beta }}\left(a_{(2,\beta )}\right)\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)& ={\Delta }_{\alpha ,\beta }\left({Ad}_{h_{\alpha \beta }}\left(a_{(1,\alpha \beta )}\right)\right)\chi \left(a_{(2,1)}\right)\hfill \\ \hfill & ={Ad}_{h_{\alpha }}\left(a_{(1,\alpha )}\right)\otimes {Ad}_{h_{\beta }}\left(a_{(2,\beta )}\right)\chi \left(a_{(3,1)}\right)\hfill \\ \hfill & ={Ad}_{h_{\alpha }}\left(a_{(1,\alpha )}\right)\otimes {\sigma }_{\beta }\left(a_{(2,\beta )}\right)\hfill \end{array}$$

for all $a\in A_{\alpha \beta },\alpha \beta \in \pi$. This proves the relations (38) and (39) hold. Hence, the comultiplication $\Delta$ of A can be extended to a homomorphism $\Delta :H\to H\otimes H$. Thus, this mapping $\Delta :H\to H\otimes H$ is comultiplication of H, since $({\Delta }_{\alpha ,\beta }\otimes {\mathrm{Id}}_{H_{\gamma }}){\Delta }_{\alpha \beta ,\gamma }\left(a\right)=({\mathrm{Id}}_{H_{\alpha }}\otimes {\Delta }_{\beta ,\gamma }){\Delta }_{\alpha ,\beta \gamma }\left(a\right)$ for all $\alpha ,\beta ,\gamma \in \pi ,a\in H_{\alpha \beta \gamma }$.

Step 2. Assume that the conditions (i)–(iii) are satisfied. Note that $\epsilon \left(x_1\right)=\epsilon \left(y_1\right)=0$ and $\epsilon \left(g_1\right)=\epsilon \left(h_1\right)=1$. Letting $\alpha =\beta =1$ and applying $\epsilon \otimes \epsilon$ to the equation in (iii), we obtain that $\epsilon \left({\delta }_1\left(a\right)\right)=\epsilon \left({\delta }_1\left(a\right)\right)+\epsilon \left({\delta }_1\left(a\right)\right)$, and hence $\epsilon \left({\delta }_1\left(a\right)\right)=0$ for all $a\in A_1$. According to [7], we know that $\epsilon$ admits an extension to $H_1$ if and only if $\epsilon \left({\delta }_1\left(a\right)\right)=0$ for all $a\in A_1$.

Step 3. Assume that the conditions (i)–(iii) are satisfied. In order to show that H has antipode S extending the antipode of A by Equation (37), then S preserves $z_{\alpha }a={\sigma }_{\alpha }\left(a\right)+{\delta }_{\alpha }\left(a\right)$ for all $a\in A_{\alpha },\alpha \in \pi$. This means that for all $\alpha \in \pi ,a\in A_{\alpha }$

$$S_{\alpha }\left(a\right)S_{\alpha }\left(z_{\alpha }\right)=S_{\alpha }\left(z_{\alpha }\right)S_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right).$$

(40)

On the other hand, if the above equation holds, then S can be extended as an antipode from A.

Now, by $\epsilon \left(a\right)1_{{\alpha }^{-1}}=a_{(1,{\alpha }^{-1})}{S}_{\alpha }\left(a_{(2,\alpha )}\right)$, we have

$$\begin{array}{cc}\hfill 0& ={\delta }_{{\alpha }^{-1}}\left(\epsilon \left(a\right)1_{{\alpha }^{-1}}\right)\hfill \\ \hfill & ={\delta }_{{\alpha }^{-1}}\left(a_{(1,{\alpha }^{-1})}{S}_{\alpha }\left(a_{(2,\alpha )}\right)\right)\hfill \\ \hfill & ={\sigma }_{\alpha }^{-1}\left(a_{(1,{\alpha }^{-1})}\right){\delta }_{{\alpha }^{-1}}\left(S_{\alpha }\left(a_{(2,\alpha )}\right)\right)+{\delta }_{{\alpha }^{-1}}\left(a_{(1,{\alpha }^{-1})}\right)S_{\alpha }\left(a_{(2,\alpha )}\right)\hfill \end{array}$$

(41)

Applying $m_{{\alpha }^{-1}}({\mathrm{Id}}_{A_{\alpha }^{-1}}\otimes S_{\alpha })$ to the equation in (iii), we obtain

$$\begin{array}{c}{\displaystyle \epsilon \left(a\right)x_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}}\hfill \\ ={\sigma }_1{\left(a\right)}_{(1,{\alpha }^{-1})}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_1{\left(a\right)}_{(2,\alpha )}\right)\hfill \\ +{\delta }_{a^{-1}}\left(a_{(1,{\alpha }^{-1})}\right)S_{\alpha }\left(a_{(2,\alpha )}\right)g_{{\alpha }^{-1}}^{-1}+h_{{\alpha }^{-1}}{a}_{(1,{\alpha }^{-1})}{S}_{\alpha }\left({\delta }_{\alpha }\left(a_{(2,\alpha )}\right)\right)\hfill \end{array}$$

(42)

Then, we have

$$\begin{array}{c}{\displaystyle S_{\alpha }\left(a\right)h_{a^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}}\hfill \\ =S_{\alpha }\left(a_{(1,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}\epsilon \left(a_{(2,1)}\right)x_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}\hfill \\ \stackrel{\left(42\right)}{=}{S}_{\alpha }\left(a_{(1,\alpha )}\right)h_{a^{-1}}^{-1}{\sigma }_1{\left(a_{(2,1)}\right)}_{(1,{\alpha }^{-1})}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_1{\left(a_{(2,1)}\right)}_{(2,\alpha )}\right)\hfill \\ +S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}{\delta }_{{\alpha }^{-1}}\left(a_{(2,a^{-1})}\right)S_{\alpha }\left(a_{(3,\alpha )}\right)g_{{\alpha }^{-1}}^{-1}\hfill \\ +S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}{h}_{{\alpha }^{-1}}{a}_{(2,{\alpha }^{-1})}{S}_{\alpha }\left({\delta }_{\alpha }\left(a_{(3,\alpha )}\right)\right)\hfill \\ \stackrel{\left(39\right)}{=}{S}_{\alpha }\left(a_{(1,\alpha )}\right)a_{(2,{\alpha }^{-1})}{h}_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a_{(3,\alpha )}\right)\right)\hfill \\ +S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}{\delta }_{{\alpha }^{-1}}\left(a_{(2,{\alpha }^{-1})}\right)S_{\alpha }\left(a_{(3,\alpha )}\right)g_{{\alpha }^{-1}}^{-1}+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)\hfill \\ \stackrel{\left(41\right)}{=}{h}_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)\hfill \\ -S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}{\sigma }_{a^{-1}}\left(a_{(2,{\alpha }^{-1})}\right){\delta }_{a^{-1}}\left(S_{\alpha }\left(a_{(3,\alpha )}\right)\right)g_{{\alpha }^{-1}}^{-1}\hfill \\ \stackrel{(i,ii)}{=}{h}_{a^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)\hfill \\ -S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}{h}_{{\alpha }^{-1}}{a}_{(2,{\alpha }^{-1})}{h}_{{\alpha }^{-1}}^{-1}\chi \left(a_{(3,1)}\right){\delta }_{{\alpha }^{-1}}\left(S_{\alpha }\left(a_{(4,\alpha )}\right)\right)g_{a^{-1}}^{-1}\hfill \\ =h_{a^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)\hfill \\ -h_{{\alpha }^{-1}}^{-1}\chi \left(a_{(1,1)}\right){\delta }_{{\alpha }^{-1}}\left(S_{\alpha }\left(a_{(2,\alpha )}\right)\right)g_{a^{-1}}^{-1}\hfill \end{array}$$

Again, by (i) and (ii), we have

$$\begin{array}{cc}\hfill S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}& =h_{{\alpha }^{-1}}^{-1}\chi \left(a_{(1,1)}\right)h_{{\alpha }^{-1}}{S}_{\alpha }\left(a_{(3,\alpha )}\right)h_{{\alpha }^{-1}}^{-1}\chi \left(S_1\left(a_{(2,1)}\right)\right)\hfill \\ \hfill & =h_{{\alpha }^{-1}}^{-1}\chi \left(a_{(1,1)}\right){\sigma }_{{\alpha }^{-1}}\left(S_{\alpha }\left(a_{(2,\alpha )}\right)\right)\hfill \end{array}$$

Now, substituting Equation (32) into Equation (40), we obtain

$$\begin{array}{c}{\displaystyle S_{\alpha }\left(a\right)S_{\alpha }\left(z_{\alpha }\right)}\hfill \\ =S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}-S_{\alpha }\left(a\right)h_{{\alpha }^{-1}}^{-1}{z}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}\hfill \\ =h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)-h_{{\alpha }^{-1}}^{-1}\chi \left(a_{(1,1)}\right){\delta }_{{\alpha }^{-1}}\left(S_{\alpha }\left(a_{(2,\alpha )}\right)\right)g_{a^{-1}}^{-1}\hfill \\ +S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)-h_{{\alpha }^{-1}}^{-1}\chi \left(a_{(1,1)}\right){\sigma }_{{\alpha }^{-1}}\left(S_{\alpha }\left(a_{(2,\alpha )}\right)\right)g_{a^{-1}}^{-1}\hfill \\ =h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)-h_{{\alpha }^{-1}}^{-1}\chi \left(a_{(1,1)}\right)S_{\alpha }\left(a_{(2,\alpha )}\right)g_{a^{-1}}^{-1}+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)\hfill \\ =h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)-h_{{\alpha }^{-1}}^{-1}{z}_{{\alpha }^{-1}}{g}_{a^{-1}}^{-1}{S}_{\alpha }\left(\chi \left(a_{(1,1)}\right)g_{a^{-1}}{a}_{(2,\alpha )}{g}_{a^{-1}}^{-1}\right)+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)\hfill \\ =h_{{\alpha }^{-1}}^{-1}{x}_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}{g}_{{\alpha }^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)-h_{{\alpha }^{-1}}^{-1}{z}_{{\alpha }^{-1}}{g}_{a^{-1}}^{-1}{S}_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)\hfill \\ =S_{\alpha }\left(z_{\alpha }\right)S_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)+S_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)\hfill \end{array}$$

for all $a\in A_{\alpha },\alpha \in \pi$. The proof is over. □

Corollary 2.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf group-coalgebra, $g,h\in G\left(A\right)$, $x\in P_{1,h}\left(A\right)$, $y\in P_{g,1}\left(A\right)$ and $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$. If H has a Hopf group-coalgebra structure determined by $(g,h,x,y)$, then

(i)

χ is invertible in $A^{*}$ with ${\chi }^{-1}=\chi \circ S_1$, where χ is the character determined by σ as inTheorem 2,

(ii)

${\sigma }_{\alpha }$ is an algebra automorphism with

$${\sigma }_{\alpha }^{-1}\left(a\right)={\chi }^{-1}\left(a_{(1,1)}\right)g_{\alpha }^{-1}{a}_{(2,\alpha )}{g}_{\alpha }=h_{\alpha }^{-1}{a}_{(1,\alpha )}{h}_{\alpha }{\chi }^{-1}\left(a_{(2,1)}\right)$$

for all $a\in A_{\alpha },\alpha \in \pi$,

(iii)

$g_{\alpha }{h}_{\alpha }=h_{\alpha }{g}_{\alpha }$ for all $\alpha \in \pi$.

Proof. 

It is easy to see that parts (i) and (ii) hold. According to Theorem 2(ii), we have $\chi \left(a_{(1,1)}\right)g_{\alpha }{a}_{(2,\alpha )}{g}_{\alpha }^{-1}=h_{\alpha }{a}_{(1,\alpha )}{h}_{\alpha }^{-1}\chi \left(a_{(2,1)}\right)$, and hence $\chi \left(g_1\right)g_{\alpha }{g}_{\alpha }{g}_{\alpha }^{-1}=h_{\alpha }{g}_{\alpha }{h}_{\alpha }^{-1}\chi \left(g_1\right)$ by taking $a=g$. Then, the part (iii) follows for $\chi \left(g_1\right)\ne 0$. □

Corollary 3.

If $x_{\alpha }\otimes y_{\beta }$ satisfies ${\Delta }_{\alpha ,\beta }\left({\sigma }_{\alpha \beta }\left(a\right)\right)(x_{\alpha }\otimes y_{\beta })=(x_{\alpha }\otimes y_{\beta }){\Delta }_{\alpha ,\beta }\left(a\right)$ for all $\alpha ,\beta \in \pi ,a\in A_{\alpha \beta }$, then the equation inTheorem 2(iii)becomes

$${\Delta }_{\alpha ,\beta }\left({\delta }_{\alpha \beta }\left(a\right)\right)={\delta }_{\alpha }\left(a_{(1,\alpha )}\right)\otimes g_{\beta }{a}_{(2,\beta )}+h_{\alpha }{a}_{(1,\alpha )}\otimes {\delta }_{\beta }\left(a_{(2,\beta )}\right).$$

(43)

Proof. 

The proof is straightforward. □

Notation 3.

Let $A={\left\{A_{\alpha }\right\}}_{\alpha \in \pi }$ be a Hopf π-coalgebra and $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ has a Hopf group-coalgebra structure determined by $(g,h,x,y)$ for some $g,h,x,y\in A$. Denote the special Hopf group-coalgebra Ore extension H by $H=A(\chi ,g,h,x,y,\delta )$, where $\chi :A_1\to k$ is a character such that ${\sigma }_{\alpha }\left(a\right)=\chi \left(a_{(1,1)}\right){Ad}_{g_{\alpha }}\left(a_{(2,\alpha )}\right)$ for $\alpha \in \pi$, $g,h\in G\left(A\right)$, $x\in P_{1,h}\left(A\right)$, $y\in P_{g,1}\left(A\right)$, and parts (ii) and (iii) in Theorem 2 are satisfied for $\{\chi ,g,h,x,y,\delta \}$.

Similar to Definition 7, two special Hopf group-coalgebra Ore extensions $H=A(\chi ,g,$$h,x,y,\delta )$ and $H^{\prime }=A^{\prime }({\chi }^{\prime },g^{\prime },$$h^{\prime },x^{\prime },y^{\prime },{\delta }^{\prime })$ of Hopf group-coalgebras A and $A^{\prime }$, respectively, are said to be isomorphic if there is a Hopf group-coalgebra isomorphism $\Psi ={\{{\Psi }_{\alpha }:H_{\alpha }\to H_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ such that $\Psi \left(A\right)=A^{\prime }$, i.e., ${\Psi }_{\alpha }\left(A_{\alpha }\right)=A_{\alpha }^{\prime }$ for all $\alpha \in \pi$. To avoid ambiguity, the multiplication, the unit, the comultiplication, the counit, and the antipode of $H^{\prime }$ are written as $m^{\prime }$, 1, ${\Delta }^{\prime }$, ${\epsilon }^{\prime }$, and $S^{\prime }$, respectively.

Proposition 5.

Let $H=A(\chi ,g,h,x,y,\delta )$ and $H^{\prime }=A^{\prime }({\chi }^{\prime },g^{\prime },h^{\prime },x^{\prime },y^{\prime },{\delta }^{\prime })$ be two special Hopf group-coalgebra Ore extensions. If there is $\lambda \in {\mathbb{k}}^{\times }$, $r={\left\{r_{\alpha }\right\}}_{\alpha \in \pi }\in G\left(A^{\prime }\right)$, $b={\left\{b_{\alpha }\right\}}_{\alpha \in \pi }\in A^{\prime }$ and a Hopf group-coalgebra isomorphism $\Phi ={\{{\Phi }_{\alpha }:A_{\alpha }\to A_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ such that

(i)

${\Phi }_{\alpha }\left(g_{\alpha }\right)=r_{\alpha }{g}_{\alpha }^{\prime }$ and ${\Phi }_{\alpha }\left(h_{\alpha }\right)=r_{\alpha }{h}_{\alpha }^{\prime }$,

(ii)

${\Delta }_{\alpha ,\beta }^{\prime }\left(b_{\alpha \beta }\right)=b_{\alpha }\otimes g_{\beta }^{\prime }+h_{\alpha }^{\prime }\otimes b_{\beta }+\lambda r_{\alpha }^{-1}{\Phi }_{\alpha }\left(x_{\alpha }\right)\otimes r_{\beta }^{-1}{\Phi }_{\beta }\left(y_{\beta }\right)-x_{\alpha }^{\prime }\otimes y_b^{\prime }$, hence ${\epsilon }^{\prime }\left(b_1\right)=0$,

(iii)

${\chi }^{\prime }{\Phi }_1=\chi$,

(iv)

${\delta }_{\alpha }^{\prime }=\lambda r_{\alpha }^{-1}{\Phi }_{\alpha }{\delta }_{\alpha }{\Phi }_{\alpha }^{-1}+{\delta }_{\alpha }^{″}$,

for all $\alpha ,\beta \in \pi$, where ${\delta }_{\alpha }^{″}$ is an inner ${\sigma }_{\alpha }^{\prime }$-derivation of $A_{\alpha }^{\prime }$ defined by ${\delta }_{\alpha }^{\prime \prime }\left(a^{\prime }\right)={\sigma }_{\alpha }^{\prime }\left(a^{\prime }\right)b_{\alpha }-b_{\alpha }{a}^{\prime }$ for all $a^{\prime }\in A_{\alpha }^{\prime }$, ${\sigma }_{\alpha }^{\prime }$ is induced by ${\chi }^{\prime }$ as in Theorem 2(i), then $H\cong H^{\prime }$ as Hopf group-coalgebras. On the other hand, if A or $A^{\prime }$ has no zero-divisors, then the converse also holds.

Proof. 

Let $\sigma$ and ${\sigma }^{\prime }$ be the algebra endomorphisms of A and $A^{\prime }$ determined by $\chi$ and ${\chi }^{\prime }$ as in Theorem 2, respectively. It follows from Corollary 2(ii) that $\sigma$ and ${\sigma }^{\prime }$ are algebra automorphisms.

Assume that there is some $\lambda \in {\mathbb{k}}^{\times }$, $r\in G\left(A^{\prime }\right)$, $b={\left\{b_{\alpha }\right\}}_{\alpha \in \pi }\in A^{\prime }$ and a Hopf group-coalgebra isomorphism $\Phi ={\{{\Phi }_{\alpha }:A_{\alpha }\to A_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ such that the conditions (i)–(iv) are satisfied. Let ${\Psi }_{\alpha }\left(a\right)={\Phi }_{\alpha }\left(a\right)$ for all $a\in A_{\alpha },\alpha \in \pi$ and ${\Psi }_{\alpha }\left(z_{\alpha }\right)={\lambda }^{-1}{r}_{\alpha }(z_{\alpha }^{\prime }+b_{\alpha })$. Then, we have that $\Psi ={\left\{{\Psi }_{\alpha }\right\}}_{\alpha \in \pi }$ can be uniquely extended to an algebra isomorphism from H to $H^{\prime }$ by a straightforward calculation. Moreover, it is easy to check that ${\Delta }_{\alpha ,\beta }^{\prime }\left({\Psi }_{\alpha \beta }\left(z_{\alpha \beta }\right)\right)=({\Psi }_{\alpha }\otimes {\Psi }_{\beta }){\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)$ and ${\epsilon }^{\prime }\left({\Psi }_1\left(z_1\right)\right)=\epsilon \left(z_1\right)=0$. Therefore, $\Psi$ is a group bialgebra isomorphism, and it is clear that ${\Psi }_{{\alpha }^{-1}}\left(S_{\alpha }\left(z_{\alpha }\right)\right)=S_{\alpha }\left({\Psi }_{\alpha }\left(z_{\alpha }\right)\right)$. Consequently, $\Psi$ is a Hopf group-coalgebra isomorphism.

Conversely, we assume that A or $A^{\prime }$ has no zero-divisors and that there is a Hopf group-coalgebra isomorphism $\Psi ={\{{\Psi }_{\alpha }:H_{\alpha }\to H_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ such that ${\Psi }_{\alpha }\left(A_{\alpha }\right)=A_{\alpha }^{\prime }$ for all $\alpha \in \pi$. Then, ${\Phi =\Psi |}_A$ is a Hopf group-coalgebra isomorphism from A to $A^{\prime }$, and hence neither A and $A^{\prime }$ have zero-divisors. Thus, ${\Psi }_{\alpha }\left(z_{\alpha }\right)=c_{\alpha }^{\prime }{z}_{\alpha }^{\prime }+d_{\alpha }^{\prime }$ for some $c_{\alpha }^{\prime }\ne 0,d_{\alpha }^{\prime }\in A_{\alpha }^{\prime }$, $\forall \alpha \in \pi$. Similarly, ${\Psi }_{\alpha }^{-1}\left(z_{\alpha }^{\prime }\right)=c_{\alpha }{z}_{\alpha }+d_{\alpha }$ for some $c_{\alpha }\ne 0,d_{\alpha }\in A_{\alpha }$, $\forall \alpha \in \pi$. Therefore, $z_{\alpha }^{\prime }={\Psi }_{\alpha }{\Psi }_{\alpha }^{-1}\left(z_{\alpha }^{\prime }\right)={\Phi }_{\alpha }\left(c_{\alpha }\right)c_{\alpha }^{\prime }{z}_{\alpha }^{\prime }+{\Phi }_{\alpha }\left(c_{\alpha }\right)d_{\alpha }^{\prime }+{\Phi }_{\alpha }\left(d_{\alpha }\right)$. Then, we have ${\Phi }_{\alpha }\left(c_{\alpha }\right)c_{\alpha }^{\prime }=1$ via comparing the coefficients of $z^{\prime }$. Similarly, we have ${\Phi }_{\alpha }^{-1}\left(c_{\alpha }^{\prime }\right)c_{\alpha }=1$ from $z_{\alpha }={\Psi }_{\alpha }^{-1}{\Psi }_{\alpha }\left(z_{\alpha }\right)$, and hence $c_{\alpha }^{\prime }\Phi \left(c_{\alpha }\right)=1$. Thence, $c_{\alpha }^{\prime }$ is invertible with the inverse ${\Phi }_{\alpha }\left(c_{\alpha }\right)$.

We have ${\Delta }_{\alpha ,\beta }^{\prime }\left({\Psi }_{\alpha \beta }\left(z_{\alpha \beta }\right)\right)=({\Psi }_{\alpha }\otimes {\Psi }_{\beta })\left({\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)\right)$ and ${\epsilon }^{\prime }\left({\Psi }_1\left(z_1\right)\right)=\epsilon \left(z_1\right)$ because $\Psi$ is a $\pi$-coalgebra map. It follows that ${\Delta }_{\alpha ,\beta }^{\prime }\left(c_{\alpha \beta }^{\prime }\right)(z_{\alpha }^{\prime }\otimes g_{\beta }^{\prime }+h_{\alpha }^{\prime }\otimes z_{\beta }^{\prime }+x_{\alpha }^{\prime }\otimes y_{\beta }^{\prime })+{\Delta }_{\alpha ,\beta }^{\prime }\left(d_{\alpha \beta }^{\prime }\right)=(c_{\alpha }^{\prime }{z}_{\alpha }^{\prime }+d_{\alpha }^{\prime })\otimes {\Phi }_{\beta }\left(g_{\beta }\right)+{\Phi }_{\alpha }\left(h_{\alpha }\right)\otimes (c_{\beta }^{\prime }{z}_{\beta }^{\prime }+d_{\beta }^{\prime })+{\Phi }_{\alpha }\left(x_{\alpha }\right)\otimes {\Phi }_{\beta }\left(y_{\beta }\right)$ and ${\epsilon }^{\prime }\left(d_1^{\prime }\right)=0$. Then, we have

$$\begin{array}{c}{\Delta }_{\alpha ,\beta }^{\prime }\left(c_{\alpha \beta }^{\prime }\right)(1_{\alpha }\otimes g_{\beta }^{\prime })=c_{\alpha }^{\prime }\otimes {\Phi }_{\beta }\left(g_{\beta }\right),\end{array}$$

(44)

$$\begin{array}{c}{\Delta }_{\alpha ,\beta }^{\prime }\left(c_{\alpha \beta }^{\prime }\right)(h_{\alpha }^{\prime }\otimes 1_{\beta })={\Phi }_{\alpha }\left(h_{\alpha }\right)\otimes c_{\beta }^{\prime }\end{array}$$

(45)

and

$$\begin{array}{c}{\displaystyle {\Delta }_{\alpha ,\beta }^{\prime }\left(c_{\alpha \beta }^{\prime }\right)(x_{\alpha }^{\prime }\otimes y_{\beta }^{\prime })+{\Delta }_{\alpha ,\beta }^{\prime }\left(d_{\alpha \beta }^{\prime }\right)}\hfill \\ =d_{\alpha }^{\prime }\otimes {\Phi }_{\beta }\left(g_{\beta }\right)+{\Phi }_{\alpha }\left(h_{\alpha }\right)\otimes d_{\beta }^{\prime }+{\Phi }_{\alpha }\left(x_{\alpha }\right)\otimes {\Phi }_{\beta }\left(y_{\beta }\right).\hfill \end{array}$$

(46)

Setting $\alpha =\beta =1$ and applying ${\epsilon }^{\prime }\otimes i{d}_{A_{\alpha }}$ to both sides of Equation (44), we obtain that ${\Phi }_{\alpha }\left(g_{\alpha }\right)={\epsilon }^{\prime }{\left(c_1^{\prime }\right)}^{-1}{c}_{\alpha }^{\prime }{g}_{\alpha }^{\prime }$ because $c_1^{\prime }$ is invertible. Thus, ${\left\{{\epsilon }^{\prime }{\left(c_1^{\prime }\right)}^{-1}{c}_{\alpha }^{\prime }\right\}}_{\alpha \in \pi }$ is a $\pi$-grouplike element. Similarly, setting $\beta =1$ and applying $i{d}_{A_{\alpha }}\otimes {\epsilon }^{\prime }$ to both sides of Equation (45), we obtain ${\Phi }_{\alpha }\left(h_{\alpha }\right)={\epsilon }^{\prime }{\left(c_1^{\prime }\right)}^{-1}{c}_{\alpha }^{\prime }{h}_{\alpha }^{\prime }$. Let $\lambda ={\epsilon }^{\prime }{\left(c_1^{\prime }\right)}^{-1}$ and $r={\{r_{\alpha }=\lambda c_{\alpha }^{\prime }\}}_{\alpha \in \pi }$. Then, $\lambda \ne 0$, r is a $\pi$-grouplike element, ${\Phi }_{\alpha }\left(g_{\alpha }\right)=r_{\alpha }{g}_{\alpha }^{\prime }$ and ${\Phi }_{\alpha }\left(h_{\alpha }\right)=r_{\alpha }{h}_{\alpha }^{\prime }$ for all $\alpha \in \pi$. Let $b={\{b_{\alpha }={\left(c_{\alpha }^{\prime }\right)}^{-1}{d}_{\alpha }^{\prime }\}}_{\alpha \in \pi }$. Then, we obtain the first equation in part (ii) and ${\epsilon }^{\prime }\left(b_1\right)=0$ since ${\epsilon }^{\prime }\left(d_1^{\prime }\right)=0$.

Since $\Psi$ is an algebra map, we have ${\Psi }_{\alpha }\left(z_{\alpha }\right){\Psi }_{\alpha }\left(a\right)={\Psi }_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right){\Psi }_{\alpha }\left(z_{\alpha }\right)+{\Psi }_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right)$, $\forall \alpha \in \pi$, namely,

$$c_{\alpha }^{\prime }{\sigma }_{\alpha }^{\prime }\left({\Phi }_{\alpha }\left(a\right)\right)z_{\alpha }^{\prime }+c_{\alpha }^{\prime }{\delta }_{\alpha }^{\prime }\left({\Phi }_{\alpha }\left(a\right)\right)+d_{\alpha }^{\prime }{\Phi }_{\alpha }\left(a\right)={\Phi }_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)c_{\alpha }^{\prime }{z}_{\alpha }^{\prime }+{\Phi }_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)d_{\alpha }^{\prime }+{\Phi }_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right).$$

Thus, we have

$$c_{\alpha }^{\prime }{\sigma }_{\alpha }^{\prime }\left({\Phi }_{\alpha }\left(a\right)\right)={\Phi }_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right),$$

(47)

$$c_{\alpha }^{\prime }{\delta }_{\alpha }^{\prime }\left({\Phi }_{\alpha }\left(a\right)\right)+d_{\alpha }^{\prime }{\Phi }_{\alpha }\left(a\right)={\Phi }_{\alpha }\left({\sigma }_{\alpha }\left(a\right)\right)d_{\alpha }^{\prime }+{\Phi }_{\alpha }\left({\delta }_{\alpha }\left(a\right)\right).$$

(48)

According to Theorem 2(i) and Equation (47) and because $\Phi$ is a Hopf group-coalgebra isomorphism, we have

$$\begin{array}{cc}\hfill c_{\alpha }^{\prime }{\chi }^{\prime }\left({\Phi }_1\left(a_{(1,1)}\right)\right)g_{\alpha }^{\prime }{\Phi }_{\alpha }\left(a_{(2,\alpha )}\right){\left(g_{\alpha }^{\prime }\right)}^{-1}& =\chi \left(a_{(1,1)}\right){\Phi }_{\alpha }\left(g_{\alpha }\right){\Phi }_{\alpha }\left(a_{(2,\alpha )}\right){\Phi }_{\alpha }{\left(g_{\alpha }\right)}^{-1}{c}_{\alpha }^{\prime }\hfill \\ \hfill & =\chi \left(a_{(1,1)}\right)r_{\alpha }{g}_{\alpha }^{\prime }{\Phi }_{\alpha }\left(a_{(2,\alpha )}\right){\left(g_{\alpha }^{\prime }\right)}^{-1}{r}_{\alpha }^{-1}{c}_{\alpha }^{\prime }\hfill \\ \hfill & =\chi \left(a_{(1,1)}\right)c_{\alpha }^{\prime }{g}_{\alpha }^{\prime }{\Phi }_{\alpha }\left(a_{(2,\alpha )}\right){\left(g_{\alpha }^{\prime }\right)}^{-1}.\hfill \end{array}$$

Thus, ${\chi }^{\prime }\left({\Phi }_1\left(a_{(1,1)}\right)\right){\Phi }_{\alpha }\left(a_{(2,\alpha )}\right)=\chi \left(a_{(1,1)}\right){\Phi }_{\alpha }\left(a_{(2,\alpha )}\right)$. Then, setting $\alpha =1$ and applying ${\epsilon }^{\prime }$ on both sides, we obtain ${\chi }^{\prime }\left({\Phi }_1\left(a\right)\right)=\chi \left(a\right)$ for all $a\in A_1$, i.e., ${\chi }^{\prime }{\Phi }_1=\chi$. For every ${\Phi }_{\alpha }$ that is bijective, we may replace a by ${\Phi }_{\alpha }^{-1}\left(a^{\prime }\right)$ in Equation (48), where $a^{\prime }\in A_{\alpha }^{\prime }$. Then, we have $c_{\alpha }^{\prime }{\delta }_{\alpha }^{\prime }\left(a^{\prime }\right)+d_{\alpha }^{\prime }{a}^{\prime }=c_{\alpha }^{\prime }{\sigma }_{\alpha }^{\prime }\left(a^{\prime }\right){\left(c_{\alpha }^{\prime }\right)}^{-1}{d}_{\alpha }^{\prime }+{\Phi }_{\alpha }{\delta }_{\alpha }{\Phi }_{\alpha }^{-1}\left(a^{\prime }\right)$ by Equation (47). Define ${\delta }_{\alpha }^{″}:A_{\alpha }^{\prime }\to A_{\alpha }^{\prime }$ by ${\delta }_{\alpha }^{″}\left(a^{\prime }\right)={\sigma }_{\alpha }^{\prime }\left(a^{\prime }\right)b_{\alpha }-b_{\alpha }{a}^{\prime }$. Then, ${\delta }^{\prime }=\lambda r_{\alpha }^{-1}{\Phi }_{\alpha }{\delta }_{\alpha }{\Psi }_{\alpha }^{-1}+{\delta }_{\alpha }^{″}$ for all $\alpha \in \pi$. □

Corollary 4.

Let $A(\chi ,g,h,x,y,\delta )$ be a special Hopf group-coalgebra Ore extension of a Hopf group-coalgebra A. Then, we have

(i)

$A(\chi ,g,h,x,y,\delta )\cong A(\chi ,g,h,cx,dy,cd\delta )$ for all $c,d\in {\mathbb{k}}^{\times }$.

(ii)

$A(\chi ,g,h,c(1-h),y,\delta )\cong A(\chi ,g,h,0,0,\delta +{\delta }^{\prime })$, where $c\in \mathbb{k}$ and ${\delta }^{\prime }\left(a\right)=c({\sigma }_{\alpha }\left(a\right)y_{\alpha }-y_{\alpha }a)$ for all $a\in A_{\alpha },\alpha \in \pi$.

(iii)

$A(\chi ,g,h,x,d(1-g),\delta )\cong A(\chi ,g,h,0,0,\delta +{\delta }^{\prime \prime })$, where $d\in \mathbb{k}$ and ${\delta }^{″}\left(a\right)=d({\sigma }_{\alpha }\left(a\right)x_{\alpha }-x_{\alpha }a)$ for all $a\in A_{\alpha },\alpha \in \pi$.

(iv)

Assume that $y=cx$ (namely, $y_{\alpha }=c{x}_{\alpha }$, $\forall \alpha \in \pi$), $c\in \mathbb{k}$ and $x\notin \mathbb{k}G\left(A\right)$. Then, $g=h=1$. Moreover, if $\mathrm{Char}\left(\mathbb{k}\right)\ne 2$, then $A(\chi ,g,h,x,cx,\delta )\cong A(\chi ,1,1,0,0,\delta +{\delta }^{\prime \prime \prime })$, where $c\in \mathbb{k}$ and ${\delta }_{\alpha }^{\prime \prime \prime }\left(a\right)=\frac{1}{2}c({\sigma }_{\alpha }\left(a\right)x_{\alpha }^2-x_{\alpha }^{2}a)$ for all $a\in A_{\alpha },\alpha \in \pi$.

(v)

$A(\chi ,g,h,x,y,\delta )\cong A(\chi {\Phi }_1^{-1},\Phi \left(g\right),\Phi \left(h\right),\Phi \left(x\right),\Phi \left(y\right),\Phi \delta {\Phi }^{-1})$, where $\Phi ={\{{\Phi }_{\alpha }:A_{\alpha }\to A_{\alpha }^{\prime }\}}_{\alpha \in \pi }$ is a Hopf group-coalgebra isomorphism.

Proof. 

(i) Let

$$A(\chi ,g,h,x,y,\delta )={\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }$$

and

$$A(\chi ,g,h,cx,dy,cd\delta )={\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },cd{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }.$$

An isomorphism $\Psi :{\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }\to {\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },cd{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }$ is defined by ${\Psi }_{\alpha }\left(a\right)=a$ for all $a\in A_{\alpha }$ and ${\Psi }_{\alpha }\left(z_{\alpha }\right)=c^{-1}{d}^{-1}{z}_{\alpha }^{\prime }$, $\alpha \in \pi$.

(ii) Let

$$A(\chi ,g,h,c(1-h),y,\delta )={\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }$$

and

$$A(\chi ,g,h,0,0,\delta +{\delta }^{\prime })={\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }+{\delta }_{\alpha }^{\prime }]\right\}}_{\alpha \in \pi }.$$

An isomorphism $\Psi :{\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }\to {\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }+{\delta }_{\alpha }^{\prime }]\right\}}_{\alpha \in \pi }$ is defined by ${\Psi }_{\alpha }\left(a\right)=a$ for all $a\in A_{\alpha }$ and ${\Psi }_{\alpha }\left(z_{\alpha }\right)=z_{\alpha }^{\prime }+c{y}_{\alpha }$, $\alpha \in \pi$.

(iii) Let

$$A(\chi ,g,h,x,d(1-g),\delta )={\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }$$

and

$$A(\chi ,g,h,0,0,\delta +{\delta }^{\prime \prime })={\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }+{\delta }_{\alpha }^{″}]\right\}}_{\alpha \in \pi }.$$

An isomorphism $\Psi :{\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }\to {\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }+{\delta }_{\alpha }^{″}]\right\}}_{\alpha \in \pi }$ is defined by ${\Psi }_{\alpha }\left(a\right)=a$ for all $a\in A_{\alpha }$ and ${\Psi }_{\alpha }\left(z_{\alpha }\right)=z_{\alpha }^{\prime }+d{x}_{\alpha }$, $\alpha \in \pi$.

(iv) According to Notation 2, $x\in P_{1,h}\left(A\right)$ and $y\in P_{g,1}\left(A\right)$. Thus, $g=h=1$ for $y=cx$ and $x\notin \mathbb{k}G\left(A\right)$. Assume that $\mathrm{Char}\left(\mathbb{k}\right)\ne 0$. Let $A(\chi ,g,h,x,cx,\delta )={\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }$ and $A(\chi ,1,1,0,0,\delta +{\delta }^{\prime \prime \prime })={\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }+{\delta }_{\alpha }^{\prime \prime \prime }]\right\}}_{\alpha \in \pi }$. An isomorphism $\Psi :{\left\{A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }\to {\left\{A_{\alpha }[z_{\alpha }^{\prime };{\sigma }_{\alpha },{\delta }_{\alpha }+{\delta }_{\alpha }^{\prime \prime \prime }]\right\}}_{\alpha \in \pi }$ is defined by ${\Psi }_{\alpha }\left(a\right)=a$ for all $a\in A_{\alpha }$ and ${\Psi }_{\alpha }\left(z_{\alpha }\right)=z_{\alpha }^{\prime }+\frac{1}{2}c{x}_{\alpha }^2$, $\alpha \in \pi$.

(v) It follows from Proposition 5 by setting $\lambda =1,r=1,b=0$. □

Now, we give some examples.

Example 2.

For all $\alpha \in \pi$, let $A_{\alpha }=\mathbb{k}G$ be the group algebra of a group G over a field $\mathbb{k}$. If $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha };{\sigma }_{\alpha },{\delta }_{\alpha }]\}}_{\alpha \in \pi }$ has a Hopf group-coalgebra structure determined by $(g,h,x,y)$, then $x=c(1-h)$, $y=d(1-g)$ for some $c,d\in {\mathbb{k}}^{\times }$. According to Corollary 4(ii) and (iii), $H\cong A(\chi ,g,h,0,0,{\delta }^{\prime })$. Therefore, H is a usual Hopf group-coalgebra Ore extension.

Example 3.

Let $A_{\alpha }$ be the same as in Example 1. Let $g=h={\left\{1_{\alpha }\right\}}_{\alpha \in \pi }$ and $\chi =\epsilon$. By Theorem 2(i), the induced algebra automorphism ${\sigma }_{\alpha }$ of $A_{\alpha }$ is the identity map on $A_{\alpha }$, i.e., ${\sigma }_{\alpha }={\mathrm{Id}}_{\alpha }$, $\forall \alpha \in \pi$. Obviously, Theorem 2(ii) is satisfied in this case. For all ${\lambda }_{ij}\in \mathbb{k}$, $1\le i,j\le 2$, let ${\delta }_{\alpha }\left(x_{\alpha }\right)={\lambda }_{11}{x}_{\alpha }+{\lambda }_{21}{y}_{\alpha }$ and ${\delta }_{\alpha }\left(y_{\alpha }\right)={\lambda }_{12}{x}_{\alpha }+{\lambda }_{22}{y}_{\alpha }$ for all $\alpha \in \pi$. A straightforward calculation shows that $\delta ={\left\{{\delta }_{\alpha }\right\}}_{\alpha \in \pi }$ may be uniquely extended to a derivation of A, denoted by δ still. Then, it is clear that Theorem 2(iii) is satisfied. Thus, we have a special Hopf group-coalgebra Ore extension ${\left\{A_{\alpha }[z_{\alpha };i{d}_{\alpha },{\delta }_{\alpha }]\right\}}_{\alpha \in \pi }=A(\epsilon ,1,1,x,y,\delta )$ by Theorem 2. In $A(\epsilon ,1,1,x,y,\delta )$, we have

$$\begin{array}{c}{z}_{\alpha }{x}_{\alpha }=x_{\alpha }{z}_{\alpha }+{\lambda }_{11}{x}_{\alpha }+{\lambda }_{21}{y}_{\alpha },z_{\alpha }{y}_{\alpha }=y_{\alpha }{z}_{\alpha }+{\lambda }_{12}{x}_{\alpha }+{\lambda }_{22}{y}_{\alpha },\\ {\Delta }_{\alpha ,\beta }\left(z_{\alpha \beta }\right)=z_{\alpha }\otimes 1_{\beta }+1_{\alpha }\otimes z_{\beta }+x_{\alpha }\otimes y_{\beta },S_{\alpha }\left(z_{\alpha }\right)=x_{{\alpha }^{-1}}{y}_{{\alpha }^{-1}}-z_{{\alpha }^{-1}}.\end{array}$$

In conclusion, the Hopf group-coalgebra Ore extensions have nice properties, and some examples may be regarded as generalized or special Hopf group-coalgebra Ore extensions. We can construct more examples of Hopf group-coalgebras by the method of Hopf group-coalgebra Ore extension. The readers may study more complex cases, through iterated extensions, $H={\{H_{\alpha }=A_{\alpha }[z_{\alpha }^{\left(1\right)};{\sigma }_{\alpha }^{\left(1\right)},{\delta }_{\alpha }^{\left(1\right)}][z_{\alpha }^{\left(2\right)};{\sigma }_{\alpha }^{\left(2\right)},{\delta }_{\alpha }^{\left(2\right)}]\cdots [z_{\alpha }^{\left(i\right)};{\sigma }_{\alpha }^{\left(i\right)},{\delta }_{\alpha }^{\left(i\right)}]\}}_{\alpha \in \pi }$ for positive integer i.

It is well known that ribbon Hopf algebras and quasitriangular Hopf algebras give rise to invariants of links and solutions of quantum Yang–Baxter equations. The ribbon and quasitriangular Hopf group-coalgebra were introduced by V. Turaev [18], partly for reasons of homotopy field theory. A. Virelizier [20,21] generalized the main properties of quasitriangular and ribbon Hopf algebras to the setting of Hopf group-coalgebras, which he used to develop certain invariants of principal group bundles over link complements and over three-manifolds. Therefore, the following question needs future research: When the Hopf group-coalgebra is quasitriangular or ribbon and gives the necessary and sufficient conditions when the generalized or special Hopf group-coalgebra Ore extensions will preserve the quasitriangular structure.