Characterization of Automorphisms of (θ,ω)-Twisted Radford’s Hom-Biproduct

Abstract

In this paper, we study the Hom–Hopf algebra automorphism group of a $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct, which satisfies certain conditions. First, we study the endomorphism monoid and automorphism group of $(\theta ,\omega )$-twisted Radford’s Hom-biproducts, and show that the endomorphism has a factorization closely related to the factors $(A,\alpha )$ and $(H,\beta )$. Then, we consider $(\theta ,\omega )$-twisted Radford’s Hom-biproduct automorphism group ${\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,p)$ as a subgroup of some semidirect product $\mathcal{U}{(C,A)}^{\mathrm{op}}{⋊}_{\phi }\mathcal{G}\left(A\right)$. Finally, we characterize the automorphisms of a concrete example.

1. Introduction

The automorphism group of an algebra has a long history and extensive study, and determining the fully automorphic group of an algebra is usually a very difficult problem. For instance, the automorphism group of the polynomial ring of one or two variables has been understood. Jung [1] in 1942 first proved the case of two variables for characteristic zero and then van der Kulk [2] in 1953 for an arbitrary characteristic. However, the automorphism group of the polynomial ring of three variables is not yet fully understood, and a remarkable result is given by Shestakov and Umirbaev [3] in 2004, which shows that the Nagata automorphism is a wild automorphism. Many researchers have been successfully computing the automorphism groups of interesting infinite-dimensional noncommutative algebras, including certain quantum groups, generalized quantum Weyl algebras, skew polynomial rings, quantized algebras and many more—see [4,5,6,7,8,9,10,11,12,13,14,15], among others. The theory of algebra authomorphisms is very deep and related with mainstream, such as the Jacobian Conjecture [16] and Holonomic $\mathcal{D}$-modules [17,18].

In the theory of the classical Hopf algebras, Radford’s biproducts are very important Hopf algebras, which play an important role in the theory of classification of pointed Hopf algebra [19,20], and account for many examples of semisimple Hopf algebra. Therefore, the study of the automorphism group of Radford’s biproducts is very interesting, and Radford characterized Hopf algebra automorphisms in [21]. There are many generalizations of Radford’s biproducts, such as [22] for quasi-Hopf algebras, [23] for multiplier Hopf algebras, Ref. [24], for monoidal Hom–Hopf algebras, Ref. [25], for Hom–Hopf algebras, and recently [26] for $(\theta ,\omega )$-twisted-Radford’s Hom-biproducts.

Let $A\times H$ be the Radford’s biproduct, where A is both a left H-module algebra and a left H-comodule coalgebra. Define $\pi :A\times H\to H$, $a\times h\mapsto {\epsilon }_A\left(a\right)h$, let ${\mathrm{Aut}}_{\mathrm{Hopf}}(A\times H,\pi )$ be the set of Hopf algebra automorphisms F of $A\times H$ satisfying $\pi \circ F=\pi$. Radford [21] characterized these Hopf algebra automorphisms, and factorized F into two suitable maps. Inspired by Radford’s idea in [21], the authors of [27] discuss the automorphisms of twisted tensor biproducts, and reveal the relationships between results and Radford’s results, and the authors of [28] study automorphisms of Radford’s Hom-biproducts introduced in [24]. Motivated by these works, we want to characterize the Hom–Hopf algebra automorphisms of $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct introduced in [26].

This paper is organized as follows. In Section 2, we recall some definitions and basic results related to Hom-(co)algebras, Hom-bialgebras, Hom–Hopf algebras, Hom-(co)module, Hom-(co)module (co)algebras, $\theta$-twisted Hom-smash products, $\omega$-twisted Hom-smash coproducts and $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct. In Section 3, we study the endomorphisms and automorphisms of $(\theta ,\omega )$-twisted-Radford’s Hom-biproducts, and show that the automorphism has a factorization closely related to the factors $(A,\alpha )$ and $(H,\beta )$ of the $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct $(A{\times }_{\omega }^{\theta }H,\alpha \otimes \beta )$ in [26]. Finally, we characterize the automorphisms of a concrete example in Section 4.

2. Preliminaries

We always assume that $\mathbb{k}$ is a base field. All algebras, linear spaces, etc. will be over $\mathbb{k}$; the subscript of ${\otimes }_{\mathbb{k}}$ is omitted for simplicity. Let ${\mathrm{id}}_V$ denote the identity map on a $\mathbb{k}$-space V. We now recall some useful definitions and terminology as follows; see [25,29,30,31,32,33,34].

Definition 1

([29]). A Hom-algebra is a quadruple $(A,\mu ,\eta ,\alpha )$ (abbr. $(A,\alpha )$), where A is a $\mathbb{k}$-linear space, $\mu :A\times A\to A$ and $\eta :\mathbb{k}\to A$ are linear maps with notation $\mu (a\otimes b)=ab$, $\eta \left(1_{\mathbb{k}}\right)=1_A$, and α is an automorphism of A, satisfying the following conditions for all $a,b,c\in A$

(HA1)

$\alpha \left(ab\right)=\alpha \left(a\right)\alpha \left(b\right)$,

(HA2)

$\alpha \left(a\right)\left(bc\right)=\left(ab\right)\alpha \left(c\right)$,

(HA3)

$\alpha \left(1_A\right)=1_A$,

(HA4)

$a{1}_A=1_{A}a=\alpha \left(a\right)$.

Let $(A,{\mu }_A,{\eta }_A,{\alpha }_A)$ and $(B,{\mu }_B,{\eta }_B,{\alpha }_B)$ be two Hom-algebras. Then, a morphism $f:A\to B$ is called a Hom-algebra morphism if ${\alpha }_B\circ f=f\circ {\alpha }_A$, $f\left(1_A\right)=1_B$ and ${\mu }_B\circ (f\otimes f)=f\circ {\mu }_A$.

Definition 2

([30,32,34]). A Hom-coalgebra is a quadruple $(C,\Delta ,\epsilon ,\alpha )$ (abbr. $(C,\alpha )$), where C is a $\mathbb{k}$-linear space, $\Delta :C\to C\times C$ and $\eta :C\to \mathbb{k}$ are linear maps with notation $\Delta \left(c\right)=c_{\left(1\right)}\otimes c_{\left(2\right)}$ and α is an automorphism of C, satisfying the following conditions for all $c\in C$

(HC1)

$\alpha {\left(c\right)}_{\left(1\right)}\otimes \alpha {\left(c\right)}_{\left(2\right)}=\alpha \left(c_{\left(1\right)}\right)\otimes \alpha \left(c_{\left(2\right)}\right)$,

(HC2)

$\alpha \left(c_{\left(1\right)}\right)\otimes c_{\left(2\right)\left(1\right)}\otimes c_{\left(2\right)\left(2\right)}=c_{\left(1\right)\left(1\right)}\otimes c_{\left(1\right)\left(2\right)}\otimes \alpha \left(c_{\left(2\right)}\right)$,

(HC3)

$\epsilon \circ \alpha =\epsilon$,

(HC4)

$\epsilon \left(c_{\left(1\right)}\right)c_{\left(2\right)}=c_{\left(1\right)}\epsilon \left(c_{\left(2\right)}\right)=\alpha \left(c\right)$.

Let $(C,{\Delta }_C,{\epsilon }_C,{\alpha }_C)$ and $(D,{\Delta }_D,{\epsilon }_D,{\alpha }_D)$ be two Hom-coalgebras. Then, a morphism $f:C\to D$ is called a Hom-coalgebra morphism if ${\alpha }_D\circ f=f\circ {\alpha }_C$, ${\epsilon }_D\circ f={\epsilon }_C$ and ${\Delta }_D\circ f=(f\otimes f)\circ {\Delta }_C$.

Definition 3

([31,32]). A Hom-bialgebra is a sextuple $(H,\mu ,\eta ,\Delta ,\epsilon ,\beta )$ (abbr. $(H,\beta )$), where $(H,\mu ,\eta ,\beta )$ is a Hom-algebra and $(H,\Delta ,\epsilon ,\beta )$ is a Hom-coalgebra, such that Δ and ε are Hom-algebra morphisms, i.e.,

(HB1)

$\Delta \left(h{h}^{\prime }\right)=h_{\left(1\right)}{h}_{\left(1\right)}^{\prime }\otimes h_{\left(2\right)}{h}_{\left(2\right)}^{\prime }$,

(HB2)

$\Delta \left(1_H\right)=1_H\otimes 1_H$,

(HB3)

$\epsilon \left(h{h}^{\prime }\right)=\epsilon \left(h\right)\epsilon \left(h^{\prime }\right)$,

(HB4)

$\epsilon \left(1_H\right)=1_{\mathbb{k}}$.

Furthermore, if there is a linear map $S:H\to H$ such that

(HS1)

$S\left(h_{\left(1\right)}\right)h_{\left(2\right)}=h_{\left(1\right)}S\left(h_{\left(2\right)}\right)=\epsilon \left(h\right)1_H$,

(HS2)

$S\left(\beta \right(h\left)\right)=\beta \left(S\right(h\left)\right)$.

Then, we call $(H,\mu ,\eta ,\Delta ,\epsilon ,\beta ,S)$ (abbr. $(H,\beta ,S)$) a Hom–Hopf algebra.

The antipode S is both a Hom-anti-algebra morphism and a Hom-anti-coalgebra morphism. A morphism between two Hom-bialgebras is called a Hom-bialgebra morphism if it is both a Hom-algebra and a Hom-coalgebra morphism. A morphism between two Hom–Hopf algebras is called a Hom–Hopf algebra morphism if it is a Hom-bialgebra morphism and compatible with the antipodes. A useful remark is that Hom-bialgebra morphisms of Hom–Hopf algebras are Hom–Hopf algebra morphisms.

Definition 4

([32]). Let $(A,\beta )$ be a Hom-algebra. A left $(A,\beta )$-Hom-module is a triple $(X,\theta ,\alpha )$, where X is a $\mathbb{k}$-linear space, $\theta :A\otimes X\to X$ (write $\theta (a\otimes x)=a▹x$, $\forall a\in A,x\in X$) is a linear map, and α is an automorphism of X, such that

(HM1)

$\alpha (a▹x)=\beta \left(a\right)▹\alpha \left(x\right)$,

(HM2)

$\beta \left(a\right)▹(b▹x)=\left(ab\right)▹\alpha \left(x\right)$,

(HM3)

$1_A▹x=\alpha \left(x\right)$

are satisfied for $a,b\in A$ and $x\in X$.

Let $(X,{\theta }_X,{\alpha }_X)$ and $(Y,{\theta }_Y,{\alpha }_Y)$ be two left $(A,\beta )$-Hom-modules. Then, a morphism $f:X\to Y$ is called a $(A,\beta )$-Hom-module morphism if ${\alpha }_Y\circ f=f\circ {\alpha }_X$ and ${\theta }_Y\circ ({\mathrm{id}}_A\otimes f)=f\circ {\theta }_X$.

Definition 5

([32]). Let $(C,\beta )$ be a Hom-coalgebra. A left $(C,\beta )$-Hom-comodule is a triple $(X,\rho ,\alpha )$, where X is a $\mathbb{k}$-linear space, $\rho :X\to C\otimes X$ (write $\rho \left(x\right)=x_{[-1]}\otimes x_{\left[0\right]}$, $\forall x\in X$) is a linear map, and α is an automorphism of X, such that

(HCM1)

$\alpha {\left(x\right)}_{[-1]}\otimes \alpha {\left(x\right)}_{\left[0\right]}=\beta \left(x_{[-1]}\right)\otimes \alpha \left(x_{\left[0\right]}\right)$,

(HCM2)

$\beta \left(x_{[-1]}\right)\otimes x_{\left[0\right][-1]}\otimes x_{\left[0\right]\left[0\right]}=x_{[-1]\left(1\right)}\otimes x_{[-1]\left(2\right)}\otimes \alpha \left(x_{\left[0\right]}\right)$,

(HCM3)

${\epsilon }_C\left(x_{[-1]}\right)x_{\left[0\right]}=\alpha \left(x\right)$

are satisfied for $x\in X$.

Let $(X,{\rho }^X,{\alpha }_X)$ and $(Y,{\rho }^Y,{\alpha }_Y)$ be two left $(C,\beta )$-Hom-comodules. Then, a morphism $f:X\to Y$ is called a $(C,\beta )$-Hom-comodule morphism if ${\alpha }_Y\circ f=f\circ {\alpha }_X$ and ${\rho }^Y\circ f=({\mathrm{id}}_C\otimes f)\circ {\rho }^X$.

Definition 6

([33]). Let $(H,\beta )$ be a Hom-bialgebra and $(A,\alpha )$ be a Hom-algebra. If $(A,▹,\alpha )$ is a left $(H,\beta )$-Hom-module, and for all $h\in H$ and $a,b\in A$,

(HMA1)

${\beta }^2\left(h\right)▹\left(ab\right)=(h_{\left(1\right)}▹a)(h_{\left(2\right)}▹b)$,

(HMA2)

$h▹1_A={\epsilon }_H\left(h\right)1_A$,

then, $(A,▹,\alpha )$ is called a left $(H,\beta )$-Hom-module Hom-algebra.

Definition 7

([25,34]). Let $(H,\beta )$ be a Hom-bialgebra and $(A,\alpha )$ a Hom-algebra. If $(A,\rho ,\alpha )$ is a left $(H,\beta )$-Hom-comodule and for all $a,b\in A$,

(HCMA1)

$\rho \left(ab\right)=a_{[-1]}{b}_{[-1]}\otimes a_{\left[0\right]}{b}_{\left[0\right]}$,

(HCMA2)

$\rho \left(1_A\right)=1_H\otimes 1_A$,

then, $(A,\rho ,\alpha )$ is called a left $(H,\beta )$-Hom-comodule Hom-algebra.

Definition 8

([25]). Let $(H,\beta )$ be a Hom-bialgebra and $(C,\alpha )$ be a Hom-coalgebra. If $(C,▹,\alpha )$ is a left $(H,\beta )$-Hom-module and for all $h\in H$ and $c\in C$,

(HMC1)

${(h▹c)}_{\left(1\right)}\otimes {(h▹c)}_{\left(2\right)}=(h_{\left(1\right)}▹c_{\left(1\right)})\otimes (h_{\left(2\right)}▹c_{\left(2\right)})$,

(HMC2)

${\epsilon }_C(h▹c)={\epsilon }_H\left(h\right){\epsilon }_C\left(c\right)$,

then $(C,▹,\alpha )$ is called a left $(H,\beta )$-Hom-module Hom-coalgebra.

Definition 9

([25]). Let $(H,\beta )$ be a Hom-bialgebra and $(C,\alpha )$ be a Hom-coalgebra. If $(C,\rho ,\alpha )$ is a left $(H,\beta )$-Hom-comodule and for all $c\in C$,

(HCMC1)

${\beta }^2\left(c_{[-1]}\right)\otimes c_{\left[0\right]\left[1\right]}\otimes c_{\left[0\right]\left[2\right]}=c_{\left(1\right)[-1]}{c}_{\left(2\right)[-1]}\otimes c_{\left(1\right)\left[0\right]}\otimes c_{\left(2\right)\left[0\right]}$,

(HCMC2)

$c_{[-1]}{\epsilon }_C\left(c_{\left[0\right]}\right)=1_H{\epsilon }_C\left(c\right)$,

then $(C,\rho ,\alpha )$ is called a left $(H,\beta )$-Hom-comodule Hom-coalgebra.

Now, we recall the definition of $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct from [26]. First, if $(H,\beta )$ is a Hom-bialgebra, we denote the group of all Hom-bialgebra automorphisms of H by ${\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{bialg}}\left(H\right)$.

Definition 10

([26]). Let $(H,\beta )$ be a Hom-bialgebra, $\theta \in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{bialg}}\left(H\right)$ and $(A,▹,\alpha )$ be a left $(H,\beta )$-Hom-module Hom-algebra. A θ-twisted smash Hom-product $(A{♯}^{\theta }H,\alpha \otimes \beta )$ of $(A,\alpha )$ and $(H,\beta )$ is defined as follows. For all $a,b\in A,h,k\in H$:

(i)

as a vector space, $A{♯}^{\theta }H=A\otimes H$,

(ii)

the multiplication is given by

$$\left(a{♯}^{\theta }h\right)\left(b{♯}^{\theta }k\right)=a(\theta \left(h_{\left(1\right)}\right)▹{\alpha }^{-1}\left(b\right)){♯}^{\theta }{\beta }^{-1}\left(h_{\left(2\right)}\right)k.$$

(1)

The θ-twisted smash Hom-product $(A{♯}^{\theta }H,\alpha \otimes \beta )$ is a Hom-algebra with the unit $1_A{♯}^{\theta }{1}_H$.

Definition 11

([26]). Let $(H,\beta )$ be a Hom-bialgebra, $\omega \in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{bialg}}\left(H\right)$ and $(A,\alpha )$ be a left $(H,\beta )$-Hom-comodule Hom-coalgebra. A ω-twisted smash Hom-coproduct $(A{♮}^{\omega }H,\alpha \otimes \beta )$ of $(A,\alpha )$ and $(H,\beta )$ is defined as follows. For all $a\in A,h\in H$:

(i)

as a vector space, $A{♮}^{\omega }H=A\otimes H$,

(ii)

the comultiplication is given by

$$\Delta \left(a{♮}^{\omega }h\right)=\left(a_{\left(1\right)}{♮}^{\omega }\omega \left(a_{\left(2\right)[-1]}\right){\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\otimes \left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right){♮}^{\omega }{h}_{\left(2\right)}\right).$$

(2)

The ω-twisted smash Hom-coproduct $(A{♮}^{\omega }H,\alpha \otimes \beta )$ is a Hom-coalgebra with the counit ${\epsilon }_A{♮}^{\theta }{\epsilon }_H$.

Theorem 1

([26]). Let $(H,\beta )$ be a Hom-bialgebra, $(A,\alpha )$ be a left $(H,\beta )$-Hom-module Hom-algebra with Hom-module structure ▹, and a left $(H,\beta )$-Hom-comodule Hom-coalgebra with Hom-comodule structure ρ, and $\theta ,\omega \in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{bialg}}\left(H\right)$, $(A{♯}^{\theta }H,\alpha \otimes \beta )$ be a θ-twisted smash Hom-product, and $(A{♮}^{\omega }H,\alpha \otimes \beta )$ be a ω-twisted smash Hom-coproduct. Then, the following two conditions are equivalent:

(i)

$(A{}_{{♮}^{\omega }}^{{♯}^{\theta }}H,\alpha \otimes \beta )$ is a Hom-bialgebra.

(ii)

The following conditions hold:

(T1)

${\epsilon }_A$ is a Hom-algebra map and ${\Delta }_A\left(1_A\right)=1_A\otimes 1_A$,

(T2)

$(A,\alpha )$ is a left $(H,\beta )$-Hom-module Hom-coalgebra,

(T3)

$(A,\alpha )$ is a left $(H,\beta )$-Hom-comodule Hom-algebra,

(T4)

${\Delta }_A\left(ab\right)=a_{\left(1\right)}(\theta \omega {\beta }^2\left(a_{\left(2\right)[-1]}\right)▹{\alpha }^{-1}\left(b_{\left(1\right)}\right))\otimes {\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)b_{\left(2\right)}$,

(T5)

$$\begin{gathered} {(h_{\left(1\right)}▹b)}_{[-1]}{\omega }^{-1}{\theta }^{-1}{\beta }^{-2}\left(h_{\left(2\right)}\right)\otimes {(h_{\left(1\right)}▹b)}_{\left[0\right]} \\ ={\omega }^{-1}{\theta }^{-1}{\beta }^{-2}\left(h_{\left(1\right)}\right)\beta \left(b_{[-1]}\right)\otimes \beta \left(h_{\left(2\right)}\right)▹b_{\left[0\right]} \end{gathered}$$,

for all $a,b\in A$ and $h\in H$.

Theorem 2

([26]). Suppose $(A{}_{{♮}^{\omega }}^{{♯}^{\theta }}H,\alpha \otimes \beta )$ is a $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct. If H is a Hom–Hopf algebra with antipode $S_H$ and $S_A:A\to A$ is a linear map, such that $S_A$ is a convolution inverse of ${\mathrm{id}}_A$ and $S_A\circ \alpha =\alpha \circ S_A$, then $(A{}_{{♮}^{\omega }}^{{♯}^{\theta }}H,\alpha \otimes \beta )$ is a Hom–Hopf algebra with antipode S described by

$$\begin{array}{cc}{S}_{A{}_{{♮}^{\omega }}^{{♯}^{\theta }}H}(a\otimes h)=\hfill & \theta \left(S_H{\left(\omega \left(a_{[-1]}\right){\beta }^{-1}\left(h\right)\right)}_{\left(1\right)}\right)▹S_A\left({\alpha }^{-2}\left(a_{\left[0\right]}\right)\right)\hfill \\ \hfill & \otimes {\beta }^{-1}\left(S_H{\left(\omega \left(a_{[-1]}\right){\beta }^{-1}\left(h\right)\right)}_{\left(2\right)}\right).\hfill \end{array}$$

(3)

The $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct $(A{}_{{♮}^{\omega }}^{{♯}^{\theta }}H,\alpha \otimes \beta )$ is a Hom–Hopf algebra and is denoted by $(A{\times }_{\omega }^{\theta }H,\alpha \otimes \beta )$, for short, and write elements $a\times h\in A{\times }_{\omega }^{\theta }H$.

3. Factorization of Certain Twisted Hom-Biproduct Endomorphisms

Let $(A{\times }_{\omega }^{\theta }H,\alpha \otimes \beta )$ be a $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct. We define $\pi :A{\times }_{\omega }^{\theta }H\to H$ by $\pi (a\times h)={\epsilon }_A\left(a\right)h$ for all $a\in A,h\in H$ and $j:H\to A{\times }_{\omega }^{\theta }H$ by $j\left(h\right)=1_A\times h$ for all $h\in H$, are Hom–Hopf algebra maps which satisfy $\pi \circ j={\mathrm{id}}_H$. Let ${\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$ be the set of all Hom–Hopf algebra endomorphisms F of $(A{\times }_{\omega }^{\theta }H,\alpha \otimes \beta )$ such that $\pi \circ F=\pi$ and let ${\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$ be its set of units, which is the group of Hom–Hopf algebra automorphisms F of $(A{\times }_{\omega }^{\theta }H,\alpha \otimes \beta )$ such that $\pi \circ F=\pi$ under composition. The purpose of this section is to show that F has a factorization closely related to the factors $(A,\alpha )$ and $(H,\beta )$ of $(A{\times }_{\omega }^{\theta }H,\alpha \otimes \beta )$.

We define $\Pi :A{\times }_{\omega }^{\theta }H\to A$ and $J:A\to A{\times }_{\omega }^{\theta }H$ by $\Pi (a\times h)=a{\epsilon }_H\left(h\right)$, for all $a\in A,h\in H$ and $J\left(a\right)=a\times 1_H$, for all $a\in A$, respectively. There is a fundamental relationship between these four maps given by:

$$J\circ \mathsf{\Pi }\circ ({\alpha }^2\otimes {\beta }^2)={\mathrm{id}}_{A{\times }_{\omega }^{\theta }H}\ast (j\circ S_H\circ \pi )$$

(4)

The factorization of F is given in terms of $F_L:A\to A$ and $F_R:H\to A$ defined by:

$$F_L=\mathsf{\Pi }\circ F\circ J\mathrm{and}{F}_R=\mathsf{\Pi }\circ F\circ j.$$

(5)

First, we shall reveal the relationships among F, $F_L$ and $F_R$ in the following lemma.

Lemma 1.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then:

$$F_L\circ \alpha =\alpha \circ F_L\mathrm{and}{F}_R\circ \beta =\alpha \circ F_R,$$

(6)

$$F_L\left(a\right)\times 1_H=F(a\times 1_H),$$

(7)

$$F_R\left({\beta }^2\left(h\right)\right)\times 1_H=F(1_A\times h_{\left(1\right)})(1_A\times S_H\left(h_{\left(2\right)}\right)),$$

(8)

$$F(a\times h)=F_L\left({\alpha }^{-1}\left(a\right)\right)F_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right).$$

(9)

for all $a\in A$ and $h\in H$.

Proof. 

According to the definition of these four maps and F, we have

$$F_L\circ \alpha =\mathsf{\Pi }\circ F\circ J\circ \alpha =\mathsf{\Pi }\circ F\circ (\alpha \otimes \beta )\circ J=\mathsf{\Pi }\circ (\alpha \otimes \beta )\circ F\circ J=\alpha \circ \mathsf{\Pi }\circ F\circ J=\alpha \circ F_L,$$

and

$$F_R\circ \beta =\mathsf{\Pi }\circ F\circ j\circ \alpha =\mathsf{\Pi }\circ F\circ (\alpha \otimes \beta )\circ j=\mathsf{\Pi }\circ (\alpha \otimes \beta )\circ F\circ j=\alpha \circ \mathsf{\Pi }\circ F\circ j=\alpha \circ F_L$$

Next, we need to calculate $J\circ \mathsf{\Pi }\circ ({\alpha }^2\otimes {\beta }^2)\circ F$. For $a\in A$ and $h\in H$, we use (4) to compute

$$\begin{array}{ccc}J\circ & \mathsf{\Pi }& \circ ({\alpha }^2\otimes {\beta }^2)\circ F(a\times h)\\ & =& F\left({(a\times h)}_{\left(1\right)}\right)(j\circ S_H\circ \pi )F\left({(a\times h)}_{\left(2\right)}\right)\\ & =& F\left({(a\times h)}_{\left(1\right)}\right)(j\circ S_H\circ \pi )\left({(a\times h)}_{\left(2\right)}\right)\\ & =& F(a_{\left(1\right)}\times \omega \left(a_{\left(2\right)[-1]}\right){\beta }^{-1}\left(h_{\left(1\right)}\right))(j\circ S_H\circ \pi )({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\times h_{\left(2\right)})\\ & =& F(\alpha \left(a\right)\times h_{\left(1\right)})(1_A\times S_H\left(h_{\left(2\right)}\right)).\end{array}$$

It follows that

$$J\circ \mathsf{\Pi }\circ F({\alpha }^2\left(a\right)\times {\beta }^2\left(h\right))=F(\alpha \left(a\right)\times h_{\left(1\right)})(1_A\times S_H\left(h_{\left(2\right)}\right)),$$

for $a\in A$ and $h\in H$. Equations (7) and (8) follow from the above equation via setting $h=1_H$ and $a=1_A$ respectively. As for (9), we calculate

$$\begin{array}{ccc}F\hfill & (a\times h)& \\ & =& F({\alpha }^{-1}\left(a\right)\times 1_H)F(1_A\times {\beta }^{-1}\left(h\right))\hfill \\ & =& F({\alpha }^{-1}\left(a\right)\times 1_H)\left(F(1_A\times {\beta }^{-3}\left(h_{\left(1\right)}\right))\left(\left(1_A\times S_H\left({\beta }^{-5}\left(h_{\left(2\right)\left(1\right)}\right)\right)\right)\left(1_A\times {\beta }^{-5}\left(h_{\left(2\right)\left(2\right)}\right)\right)\right)\right)\hfill \\ & =& F({\alpha }^{-1}\left(a\right)\times 1_H)\left(\left(F(1_A\times {\beta }^{-4}\left(h_{\left(1\right)}\right))\left(1_A\times S_H\left({\beta }^{-5}\left(h_{\left(2\right)\left(1\right)}\right)\right)\right)\right)\left(1_A\times {\beta }^{-4}\left(h_{\left(2\right)\left(2\right)}\right)\right)\right)\hfill \\ & =& F({\alpha }^{-1}\left(a\right)\times 1_H)\left(\left(F(1_A\times {\beta }^{-5}\left(h_{\left(1\right)\left(1\right)}\right))\left(1_A\times S_H\left({\beta }^{-5}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\right)\left(1_A\times {\beta }^{-3}\left(h_{\left(2\right)}\right)\right)\right)\hfill \\ & =& F({\alpha }^{-1}\left(a\right)\times 1_H)\left((F_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\times 1_H)(1_A\times {\beta }^{-3}\left(h_{\left(2\right)}\right))\right)\hfill \\ & =& \left(F({\alpha }^{-2}\left(a\right)\times 1_H)(F_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\times 1_H)\right)(1_A\times {\beta }^{-2}\left(h_{\left(2\right)}\right))\hfill \\ & =& \left((F_L\left({\alpha }^{-2}\left(a\right)\right)\times 1_H)(F_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\times 1_H)\right)(1_A\times {\beta }^{-2}\left(h_{\left(2\right)}\right))\hfill \\ & =& (F_L\left({\alpha }^{-2}\left(a\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\times 1_H)(1_A\times {\beta }^{-2}\left(h_{\left(2\right)}\right))\hfill \\ & =& F_L\left({\alpha }^{-1}\left(a\right)\right)F_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\hfill \end{array}$$

as desired. □

By (7) and (8) of the above Lemma:

$${\left({\mathrm{id}}_{A{\times }_{\omega }^{\theta }H}\right)}_L={\mathrm{id}}_A\mathrm{and}{\left({\mathrm{id}}_{A{\times }_{\omega }^{\theta }H}\right)}_R={\eta }_A\circ {\epsilon }_H$$

(10)

Since $F_L\left(1_A\right)=1_A$ by Equation (7). By (9) of Lemma 1:

$$F(1_A\times h)=F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)$$

(11)

for all $h\in H$. Now, we can calculate the factors of a composite.

Corollary 1.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then:

(i)

${(F\circ G)}_L=F_L\circ G_L$,

(ii)

${(F\circ G)}_R\circ {\beta }^2=(F_L\circ G_R)\ast F_R$.

Proof. 

(i) For any $a\in A$, by (7) of Lemma 1, we have

$${(F\circ G)}_L\left(a\right)\times 1_H=(F\circ G)(a\times 1_H)=F(G_L\left(a\right)\times 1_H)=F_L\circ G_L\left(a\right)\times 1_H.$$

(ii) For all $h\in H$. Using Equation (11), the fact F is multiplicative and Equation (7), we obtain that

$$\begin{array}{ccc}\hfill (F& {\circ G)}_R& \left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\hfill \\ & =& (F\circ G)(1_A\times h)\hfill \\ & =& F\left(G_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\hfill \\ & =& F\left(G_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times 1_H\right)F\left(1_A\times {\beta }^{-2}\left(h_{\left(2\right)}\right)\right)\hfill \\ & =& \left(F_L{G}_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times 1_H\right)\left(F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-3}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & =& F_L{G}_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\left(\theta \left(1_H\right)▹F_R\left({\beta }^{-4}\left(h_{\left(2\right)\left(1\right)}\right)\right)\right)\times 1_H{\beta }^{-3}\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ & =& F_L{G}_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ & =& F_L{G}_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(2\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\hfill \end{array}$$

i.e.,

$${(F\circ G)}_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)=F_L{G}_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(2\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)$$

Applying ${\mathrm{id}}_A\otimes {\epsilon }_H$ to both sides of the above equation, and replacing h with ${\beta }^2\left(h\right)$, we can obtain part (ii). □

According to Lemma 1, in order to characterize F, we must characterize $F_L$ and $F_R$. First, we shall characterize $F_L$ in the following lemma.

Lemma 2.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then:

(i)

$F_L:A\to A$ is a Hom-algebra endomorphism.

(ii)

${\epsilon }_A\circ F_L={\epsilon }_A$.

(iii)

For all $a\in A$,

$$\Delta \left(F_L\left(a\right)\right)=F_L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)F_R\left(\omega \left(a_{\left(2\right)[-1]}\right)\right)\otimes F_L\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right).$$

(12)

(iv)

For all $a\in A$,

$$\rho \left(F_L\left(a\right)\right)=a_{[-1]}\otimes F_L\left(a_{\left[0\right]}\right).$$

(13)

(v)

For all $a\in A$ and $h\in H$,

$$F_L(\theta \left(h_{\left(1\right)}\right)▹a)F_R\left({\beta }^{-1}\left(h_{\left(2\right)}\right)\right)=F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)(\theta \left(h_{\left(2\right)}\right)▹F_L\left(a\right)).$$

(14)

Proof. 

(i) We need to check three aspects. From the above discussion, we have known that $F_L\left(1_A\right)=1_A$ and $F_L\circ \alpha =\alpha \circ F_L$ have checked in Lemma 1. Finally, we shall check that $F_L$ preserves the multiplication. In fact, for $a,b\in A$, we have

$$\begin{array}{ccc}{F}_L\left(ab\right)\hfill & =& (\mathsf{\Pi }\circ F\circ J)\left(ab\right)\hfill \\ & =& \mathsf{\Pi }\left(F(ab\times 1_H)\right)\hfill \\ & =& \mathsf{\Pi }\left(F(a\times 1_H)F(b\times 1_H)\right)\hfill \\ & =& \mathsf{\Pi }\left((F_L\left(a\right)\times 1_H)(F_L\left(b\right)\times 1_H)\right)\hfill \\ & =& F_L\left(a\right)F_L\left(b\right).\hfill \end{array}$$

It is easy to check part (ii), since ${\epsilon }_A\circ \mathsf{\Pi }=\mathsf{\Pi }\circ {\epsilon }_{A{\times }_{\omega }^{\theta }H}$, ${\epsilon }_{A{\times }_{\omega }^{\theta }H}\circ J={\epsilon }_A$ and F is a Hom-coalgebra morphism.

Next, we will check that parts (iii) and (iv) hold. As a matter of fact, we compute the coproduct of $F_L\left(a\right)\times 1_H=F(a\times 1_H)$ in two ways. Firstly, we have

$$\Delta (F_L\left(a\right)\times 1_H)=\left(F_L{\left(a\right)}_{\left(1\right)}\times \beta \omega \left(F_L{\left(a\right)}_{\left(2\right)[-1]}\right)\right)\otimes \left({\alpha }^{-1}\left(F_L{\left(a\right)}_{\left(2\right)\left[0\right]}\right)\times 1_H\right)$$

and secondly, since F is a Hom-coalgebra map, we have

$$\begin{array}{ccc}\Delta (\hfill & F(a& \times 1_H))\hfill \\ & =& F\left({(a\times 1_H)}_{\left(1\right)}\right)F\left({(a\times 1_H)}_{\left(2\right)}\right)\hfill \\ & =& F(a_{\left(1\right)}\times \beta \omega \left(a_{\left(2\right)[-1]}\right))\otimes F({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\times 1_H)\hfill \\ & \stackrel{\left(9\right)}{=}& \left(F_L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)F_R\left({\beta }^{-1}\omega \left(a_{\left(2\right)[-1]\left(1\right)}\right)\right)\times \omega \left(a_{\left(2\right)[-1]\left(2\right)}\right)\right)\otimes \left(F_L\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)\times 1_H\right).\hfill \end{array}$$

It follows that

$$\begin{array}{c}{\displaystyle \left(F_L{\left(a\right)}_{\left(1\right)}\times \beta \omega \left(F_L{\left(a\right)}_{\left(2\right)[-1]}\right)\right)\otimes \left({\alpha }^{-1}\left(F_L{\left(a\right)}_{\left(2\right)\left[0\right]}\right)\times 1_H\right)}\hfill \\ =\left(F_L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)F_R\left({\beta }^{-1}\omega \left(a_{\left(2\right)[-1]\left(1\right)}\right)\right)\times \omega \left(a_{\left(2\right)[-1]\left(2\right)}\right)\right)\otimes \left(F_L\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)\times 1_H\right).\end{array}$$

Applying ${\mathrm{id}}_A\otimes {\epsilon }_H\otimes {\mathrm{id}}_A\otimes {\epsilon }_H$ to both sides of the above equation yields (12). It follows easily that ${\epsilon }_A\circ F_R={\epsilon }_H$ from (11). Applying ${\epsilon }_A\otimes {\mathrm{id}}_H\otimes {\mathrm{id}}_A\otimes {\epsilon }_H$ to both sides of the above equation again, we can gain

$$\omega {\beta }^2\left(F_L{\left(a\right)}_{[-1]}\right)\otimes F_L{\left(a\right)}_{\left[0\right]}=\omega {\beta }^2\left(a_{[-1]}\right)\otimes F_L\left(a_{\left[0\right]}\right),$$

and applying ${\beta }^{-2}{\omega }^{-1}\otimes {\mathrm{id}}_A$ to both sides, we obtain (13).

(v) Finally, it is left to us to check (14). Indeed, for $a\in A$ and $h\in H$, we have

$$\begin{array}{ccc}F\left((1_A\times h)(a\times 1_H)\right)\hfill & =& F((\beta \theta \left(h_{\left(1\right)}\right)▹a)\times h_{\left(2\right)})\hfill \\ & =& F_L(\theta \left(h_{\left(1\right)}\right)▹{\alpha }^{-1}\left(a\right))F_R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right).\hfill \end{array}$$

On the other hand, Since F preserves the multiplication, we compute:

$$\begin{array}{ccc}F\left((1_A\times h)(a\times 1_H)\right)\hfill & =& F(1_A\times h)F(a\times 1_H)\hfill \\ & =& \left(F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\right)(F_L\left(a\right)\times 1_H)\hfill \\ & =& F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\left(\theta {\beta }^{-1}\left(h_{\left(2\right)\left(1\right)}\right)▹{\alpha }^{-1}{F}_L\left(a\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right).\hfill \end{array}$$

Thus,

$$\begin{array}{c}{\displaystyle F_L(\theta \left(h_{\left(1\right)}\right)▹{\alpha }^{-1}\left(a\right))F_R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)}\hfill \\ =F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\left(\theta {\beta }^{-1}\left(h_{\left(2\right)\left(1\right)}\right)▹{\alpha }^{-1}{F}_L\left(a\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right).\end{array}$$

Applying ${\mathrm{id}}_A\times {\epsilon }_H$ to the above equation and replacing a with $\alpha \left(a\right)$, we obtain (14). □

As the reader might suspect, whether or not $F_L$ is a Hom-coalgebra morphism is explained in terms of $F_R$.

Corollary 2.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then, $F_L$ is a Hom-coalgebra morphism if, and only if, $F_R\left(\omega \left(c_{[-1]}\right)\right)\otimes c_{\left[0\right]}=1_A\otimes \alpha \left(c\right)$, for any $c\in \mathrm{im}\left(F_L\right)$.

Proof. 

Suppose $F_R\left(\omega \left(c_{[-1]}\right)\right)\otimes c_{\left[0\right]}=1_A\otimes \alpha \left(c_{\left[0\right]}\right)$, for any $c\in \mathrm{im}\left(F_L\right)$. Then, we have

$$\begin{array}{ccc}\Delta \left(F_L\left(a\right)\right)\hfill & =& F_L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)F_R\left(\omega \left(a_{\left(2\right)[-1]}\right)\right)\otimes F_L\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)\hfill \\ & \stackrel{\left(13\right)}{=}& F_L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)F_R\left(\omega \left(F_L{\left(a_{\left(2\right)}\right)}_{[-1]}\right)\right)\otimes {\alpha }^{-1}\left(F_L{\left(a_{\left(2\right)}\right)}_{\left[0\right]}\right)\hfill \\ & =& F_L\left(a_{\left(1\right)}\right)\otimes F_L\left(a_{\left(2\right)}\right)\hfill \end{array}$$

Conversely, suppose that $F_L$ is a Hom-coalgebra morphism. For all $a\in A$, we compute

$$\begin{array}{ccc}{F}_R\hfill & (\omega (F_L& {\left(a\right)}_{[-1]}\left)\right)\otimes F_L{\left(a\right)}_{\left[0\right]}\hfill \\ & \stackrel{\left(13\right)}{=}& F_R\left(\omega \left(a_{[-1]}\right)\right)\otimes F_L\left(a_{\left[0\right]}\right)\hfill \\ & \stackrel{\left(\mathrm{HC}4\right)}{=}& F_L\left({\epsilon }_A\left({\alpha }^{-5}\left(a_{\left(1\right)}\right)\right)1_A\right)F_R\left(\omega {\beta }^{-2}\left(a_{\left(2\right)[-1]}\right)\right)\otimes F_L\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HS}1\right)}{=}& \left(F_L\left(S_A\left({\alpha }^{-5}\left(a_{\left(1\right)\left(1\right)}\right)\right)\right)F_L\left({\alpha }^{-5}\left(a_{\left(1\right)\left(2\right)}\right)\right)\right)F_R\left(\omega {\beta }^{-2}\left(a_{\left(2\right)[-1]}\right)\right)\otimes F_L\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& \left(F_L\left(S_A\left({\alpha }^{-4}\left(a_{\left(1\right)}\right)\right)\right)F_L\left({\alpha }^{-5}\left(a_{\left(2\right)\left(1\right)}\right)\right)\right)F_R\left(\omega {\beta }^{-3}\left(a_{\left(2\right)\left(2\right)[-1]}\right)\right)\otimes F_L\left({\alpha }^{-2}\left(a_{\left(2\right)\left(2\right)\left[0\right]}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& F_L\left(S_A\left({\alpha }^{-3}\left(a_{\left(1\right)}\right)\right)\right)\left(F_L\left({\alpha }^{-5}\left(a_{\left(2\right)\left(1\right)}\right)\right)F_R\left(\omega {\beta }^{-4}\left(a_{\left(2\right)\left(2\right)[-1]}\right)\right)\right)\hfill \\ & & \otimes {\alpha }^3\left(F_L\left({\alpha }^{-5}\left(a_{\left(2\right)\left(2\right)\left[0\right]}\right)\right)\right)\hfill \\ & \stackrel{\left(12\right)}{=}& F_L\left(S_A\left({\alpha }^{-3}\left(a_{\left(1\right)}\right)\right)\right)F_L{\left({\alpha }^{-4}\left(a_{\left(2\right)}\right)\right)}_{\left(1\right)}\otimes {\alpha }^3\left(F_L{\left({\alpha }^{-4}\left(a_{\left(2\right)}\right)\right)}_{\left(2\right)}\right)\hfill \\ & =& F_L\left(S_A\left({\alpha }^{-3}\left(a_{\left(1\right)}\right)\right)\right)F_L\left({\alpha }^{-4}\left(a_{\left(2\right)\left(1\right)}\right)\right)\otimes F_L\left({\alpha }^{-1}\left(a_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& F_L\left(S_A\left({\alpha }^{-4}\left(a_{\left(1\right)\left(1\right)}\right)\right)\right)F_L\left({\alpha }^{-4}\left(a_{\left(1\right)\left(2\right)}\right)\right)\otimes F_L\left(a_{\left(2\right)}\right)\hfill \\ & =& 1_A\otimes \alpha \left(F_L\left(a\right)\right),\hfill \end{array}$$

as desired. □

From Lemma 2, we have characterized the conditions that $F_L$ satisfies. Pay special attention to its part (v) describing a commutation relation between $F_L$ and $F_R$. It is left to us to characterize $F_R$ as follows.

Lemma 3.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then:

(i)

$F_R\left(1_H\right)=1_A$.

(ii)

For all $h,k\in H$,

$$F_R\left(hk\right)=F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\left(\theta \left(h_{\left(2\right)}\right)▹F_R\left({\beta }^{-1}\left(k\right)\right)\right).$$

(15)

(iii)

$F_R:H\to A$ is a Hom-coalgebra morphism.

(iv)

For all $h\in H$,

$$\rho \left(F_R\left(h\right)\right)={\omega }^{-1}\left({\beta }^{-4}\left(h_{\left(1\right)\left(1\right)}\right)S_H\left({\beta }^{-3}\left(h_{\left(2\right)}\right)\right)\right)\otimes F_R\left({\beta }^{-1}\left(h_{\left(1\right)\left(2\right)}\right)\right).$$

(16)

Proof. 

(i) By (11), we have $F(1_A\times h)=F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)$ for $h\in H$. Since F is a Hom-algebra morphism and $\beta$ is an automorphism, $1_A\times 1_H=F(1_A\times 1_H)=F_R\left(1_A\right)\times 1_H$, which implies $F_R\left(1_H\right)=1_A$.

(ii) For $h,k\in H$, we have

$$F(1_A\times hk)=F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right){\beta }^{-1}\left(k_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right).$$

and on the other hand,

$$\begin{array}{ccc}F\hfill & (1_A& \times hk)\hfill \\ & =& F(1_A\times h)F(1_A\times k)\hfill \\ & =& \left(F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\left(F_R\left({\beta }^{-1}\left(k_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(k_{\left(2\right)}\right)\right)\hfill \\ & =& F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\left(\theta {\beta }^{-1}\left(h_{\left(2\right)\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(k_{\left(1\right)}\right)\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \end{array}$$

Applying ${\mathrm{id}}_A\otimes {\epsilon }_H$ to both expressions for $F(1_A\times hk)$, it follows that (15) holds.

To prove parts (iii) and (iv), we calculate $\Delta \left(F(1_A\times h)\right)$, for all $h\in H$, in two ways as follows.

$$\begin{array}{ccc}\Delta \hfill & (F\hfill & (1_A\times h))\hfill \\ & =& F(1_A\times h_{\left(1\right)})\otimes F(1_A\times h_{\left(2\right)})\hfill \\ & =& \left(F_R\left({\beta }^{-1}\left(h_{\left(1\right)\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(1\right)\left(2\right)}\right)\right)\otimes \left(F_R\left({\beta }^{-1}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right).\hfill \end{array}$$

On the other hand,

$$\begin{array}{ccc}\Delta \left(F(1_A\times h)\right)\hfill & =& \Delta (F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right))\hfill \\ & =& \left(F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{\left(1\right)}\times \omega \left(F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{\left(2\right)[-1]}\right){\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right)\right)\hfill \\ & & \otimes \left({\alpha }^{-1}\left(F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{\left(2\right)\left[0\right]}\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \end{array}$$

Applying ${\mathrm{id}}_A\otimes {\epsilon }_H\otimes {\mathrm{id}}_A\otimes {\epsilon }_H$ to the expressions for $\Delta \left(F(1_A\times h)\right)$ gives $F_R{\left(h\right)}_{\left(1\right)}\otimes F_R{\left(h\right)}_{\left(2\right)}=F_R\left(h_{\left(1\right)}\right)\otimes F_R\left(h_{\left(2\right)}\right)$. since $\alpha \circ F_R=F_R\circ \beta$ and ${\epsilon }_A\circ F_R={\epsilon }_H$ have been discussed before, and hence, part (iii) holds.

Applying ${\epsilon }_A\otimes {\mathrm{id}}_H\otimes {\mathrm{id}}_A\otimes {\epsilon }_H$ to the expressions for $\Delta \left(F(1_A\times h)\right)$ again yields

$$h_{\left(1\right)}\otimes F_R\left(h_{\left(2\right)}\right)=\omega \beta \left(F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{[-1]}\right){\beta }^{-1}\left(h_{\left(2\right)}\right)\otimes F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{\left[0\right]}$$

(17)

Therefore,

$$\begin{array}{ccc}(\omega \hfill & \otimes {\mathrm{id}}_A)\hfill & \left(\rho \left(F_R\left(h\right)\right)\right)\hfill \\ & =& {\beta }^{-1}\omega \left(F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{[-1]}\right){\epsilon }_H\left({\beta }^{-5}\left(h_{\left(2\right)}\right)\right)1_H\otimes F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{\left[0\right]}\hfill \\ & \stackrel{\left(\mathrm{HS}1\right)}{=}& {\beta }^{-1}\omega \left(F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{[-1]}\right)\left({\beta }^{-5}\left(h_{\left(2\right)\left(1\right)}\right)S_H\left({\beta }^{-5}\left(h_{\left(2\right)\left(2\right)}\right)\right)\right)\otimes F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{\left[0\right]}\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& \left({\beta }^{-2}\omega \left(F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{[-1]}\right){\beta }^{-5}\left(h_{\left(2\right)\left(1\right)}\right)\right)S_H\left({\beta }^{-4}\left(h_{\left(2\right)\left(2\right)}\right)\right)\otimes F_R{\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)}_{\left[0\right]}\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& \left({\beta }^{-2}\omega \left(F_R{\left({\beta }^{-2}\left(h_{\left(1\right)\left(1\right)}\right)\right)}_{[-1]}\right){\beta }^{-5}\left(h_{\left(1\right)\left(2\right)}\right)\right)S_H\left({\beta }^{-3}\left(h_{\left(2\right)}\right)\right)\otimes F_R{\left({\beta }^{-2}\left(h_{\left(1\right)\left(1\right)}\right)\right)}_{\left[0\right]}\hfill \\ & =& {\beta }^{-2}\left(\omega \beta \left(F_R{\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)}_{[-1]}\right){\beta }^{-3}\left(h_{\left(1\right)\left(2\right)}\right)\right)S_H\left({\beta }^{-3}\left(h_{\left(2\right)}\right)\right)\hfill \\ & & \otimes \alpha \left(F_R{\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)}_{\left[0\right]}\right)\hfill \\ & \stackrel{\left(17\right)}{=}& {\beta }^{-4}\left(h_{\left(1\right)\left(1\right)}\right)S_H\left({\beta }^{-3}\left(h_{\left(2\right)}\right)\right)\otimes F_R\left({\beta }^{-1}\left(h_{\left(1\right)\left(2\right)}\right)\right).\hfill \end{array}$$

it follows that

$$(\omega \otimes {\mathrm{id}}_A)\left(\rho \left(F_R\left(h\right)\right)\right)={\beta }^{-4}\left(h_{\left(1\right)\left(1\right)}\right)S_H\left({\beta }^{-3}\left(h_{\left(2\right)}\right)\right)\otimes F_R\left({\beta }^{-1}\left(h_{\left(1\right)\left(2\right)}\right)\right).$$

Applying ${\omega }^{-1}\otimes {\mathrm{id}}_A$ to both sides of the equation gives part (iv). □

Corollary 3.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then, $F_L$ is a left $(H,\beta )$-Hom-module morphism if and only if the condition $F_L(\theta \left(h_{\left(1\right)}\right)▹a)F_R\left({\beta }^{-1}\left(h_{\left(2\right)}\right)\right)=F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)F_L(\theta \left(h_{\left(2\right)}\right)▹a)$ holds for all $a\in A$ and $h\in H$.

Proof. 

The necessary condition can be followed easily from (14) of Lemma 2. Now, we shall prove a sufficient part. Suppose that the condition holds. Note that $F_R$ is a Hom-coalgebra morphism by Lemma 3(iii). Using this fact and (14), for all $a\in A$ and $h\in H$, we have

$$\begin{array}{ccc}\theta \left(h\right)\hfill & ▹\hfill & F_L\left(a\right)\hfill \\ & =& \left(S_H\left(F_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\left(\theta {\beta }^{-2}\left(h_{\left(2\right)}\right)▹F_L\left({\alpha }^{-1}\left(a\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& S_H\left(F_R\left({\beta }^{-2}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)\left(F_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(2\right)}\right)\right)\left(\theta {\beta }^{-3}\left(h_{\left(2\right)}\right)▹F_L\left({\alpha }^{-2}\left(a\right)\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& S_H\left(F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\right)\left(F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\left(\theta {\beta }^{-4}\left(h_{\left(2\right)\left(2\right)}\right)▹F_L\left({\alpha }^{-2}\left(a\right)\right)\right)\right)\hfill \\ & \stackrel{\left(14\right)}{=}& S_H\left(F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\right)\left(F_L(\theta {\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right)▹{\alpha }^{-2}\left(a\right))F_R\left({\beta }^{-5}\left(h_{\left(2\right)\left(2\right)}\right)\right)\right)\hfill \\ & =& S_H\left(F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\right)\left(F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)F_L(\theta {\beta }^{-4}\left(h_{\left(2\right)\left(2\right)}\right)▹{\alpha }^{-2}\left(a\right))\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& \left(S_H\left(F_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\right)F_L(\theta {\beta }^{-3}\left(h_{\left(2\right)\left(2\right)}\right)▹{\alpha }^{-1}\left(a\right))\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& \left(S_H\left(F_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)F_L(\theta {\beta }^{-2}\left(h_{\left(2\right)}\right)▹{\alpha }^{-1}\left(a\right))\hfill \\ & =& {\epsilon }_A\left(F_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\right)1_A{F}_L(\theta {\beta }^{-2}\left(h_{\left(2\right)}\right)▹{\alpha }^{-1}\left(a\right))\hfill \\ & =& F_L(\theta \left(h\right)▹a),\hfill \end{array}$$

and since $\theta \in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}\left(H\right)$, we show that $F_L$ is a left $(H,\beta )$-Hom-module morphism. □

Lemma 4.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then, $F_R$ is a Hom-algebra morphism if, and only if, $h▹F_R\left(k\right)={\epsilon }_H\left(h\right)F_R\left(\beta \left(k\right)\right)$ for all $h,k\in H$.

Proof. 

Suppose that $F_R$ is a Hom-algebra morphism. Using (ii) and (iii) of Lemma 3, for $h,k\in H$, we have

$$\begin{array}{ccc}\theta {\beta }^3\hfill & \left(h\right)▹\hfill & F_R\left(k\right)\hfill \\ & =& {\epsilon }_H\left(h_{\left(1\right)}\right)1_A\left(\theta \beta \left(h_{\left(2\right)}\right)▹F_R\left({\beta }^{-1}\left(k\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HS}1\right)}{=}& \left(S_A\left(F_R\left({\beta }^{-2}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)F_R\left({\beta }^{-2}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\left(\theta \beta \left(h_{\left(2\right)}\right)▹F_R\left({\beta }^{-1}\left(k\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& S_A\left(F_R\left({\beta }^{-1}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)\left(F_R\left({\beta }^{-2}\left(h_{\left(1\right)\left(2\right)}\right)\right)\left(\theta \left(h_{\left(2\right)}\right)▹F_R\left({\beta }^{-2}\left(k\right)\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& S_A\left(F_R\left(h_{\left(1\right)}\right)\right)\left(F_R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right)\right)\left(\theta {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)▹F_R\left({\beta }^{-2}\left(k\right)\right)\right)\right)\hfill \\ & \stackrel{\left(15\right)}{=}& S_A\left(F_R\left(h_{\left(1\right)}\right)\right)F_R\left({\beta }^{-1}\left(h_{\left(2\right)}k\right)\right)\hfill \\ & =& S_A\left(F_R\left(h_{\left(1\right)}\right)\right)\left(F_R\left({\beta }^{-1}\left(h_{\left(2\right)}\right)\right)F_R\left({\beta }^{-1}\left(k\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& \left(S_A\left(F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\right)F_R\left({\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\right)F_R\left(k\right)\hfill \\ & =& {\epsilon }_H\left(h\right)F_R\left(\beta \left(k\right)\right)\hfill \end{array}$$

i.e., $\theta {\beta }^3\left(h\right)▹F_R\left(k\right)={\epsilon }_H\left(h\right)F_R\left(\beta \left(k\right)\right)$. Replacing h by ${\beta }^{-3}{\theta }^{-1}\left(h\right)$ yields the condition.

Conversely, if $h▹F_R\left(k\right)={\epsilon }_H\left(h\right)F_R\left(\beta \left(k\right)\right)$ holds, by using Equation (15), we have

$$\begin{array}{ccc}{F}_R\left(hk\right)\hfill & =& F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right)\left(\theta \left(h_{\left(2\right)}\right)▹F_R\left({\beta }^{-1}\left(k\right)\right)\right)\hfill \\ & =& F_R\left({\beta }^{-1}\left(h_{\left(1\right)}\right)\right){\epsilon }_H\left(h_{\left(2\right)}\right)F_R\left(k\right)\hfill \\ & =& F_R\left(h\right)F_R\left(k\right).\hfill \end{array}$$

The proof is completed. □

Corollary 4.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then, $F_R$ has a convolution inverse $J_R$ defined by $J_R\left(h\right)=\theta \left(h_{\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(S_H\left(h_{\left(2\right)}\right)\right)\right)$ for $h\in H$.

Proof. 

Let $h\in H$. Then, by parts (i) and (ii) of Lemma 3, we have

$$\begin{array}{ccc}{F}_R\ast J_R\left(h\right)\hfill & =& F_R\left(h_{\left(1\right)}\right)J_R\left(h_{\left(2\right)}\right)\hfill \\ & =& F_R\left(h_{\left(1\right)}\right)\left(\theta \left(h_{\left(2\right)\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(S_H\left(h_{\left(2\right)\left(2\right)}\right)\right)\right)\right)\hfill \\ & =& F_R\left({\beta }^{-1}\left(h_{\left(1\right)\left(1\right)}\right)\right)\left(\theta \left(h_{\left(1\right)\left(2\right)}\right)▹F_R\left({\beta }^{-1}\left(S_H\left(h_{\left(2\right)}\right)\right)\right)\right)\hfill \\ & \stackrel{\left(15\right)}{=}& F_R\left(h_{\left(1\right)}{S}_H\left(h_{\left(2\right)}\right)\right)\hfill \\ & =& {\epsilon }_H\left(h\right)1_A,\hfill \end{array}$$

and using the fact that $(A,\alpha )$ is a left $(H,\beta )$-Hom-module algebra, we have

$$\begin{array}{ccc}{J}_R\ast \hfill & F_R\left(h\right)\hfill & \\ & =& \left(\theta \left(h_{\left(1\right)\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(S_H\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\right)F_R\left(h_{\left(2\right)}\right)\hfill \\ & =& \left(\theta {\beta }^{-1}\left(h_{\left(1\right)\left(1\right)\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(S_H\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\right)\hfill \\ & & \left(\left(\theta {\beta }^{-3}\left(h_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\right)\theta S_H{\beta }^{-3}\left(h_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}\right)\right)▹F_R\left({\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HM}2\right)}{=}& \left(\theta {\beta }^{-1}\left(h_{\left(1\right)\left(1\right)\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(S_H\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\right)\hfill \\ & & \left(\theta {\beta }^{-2}\left(h_{\left(1\right)\left(1\right)\left(2\right)\left(1\right)}\right)▹\left(\theta S_H{\beta }^{-3}\left(h_{\left(1\right)\left(1\right)\left(2\right)\left(2\right)}\right)▹F_R\left({\beta }^{-2}\left(h_{\left(2\right)}\right)\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& \left(\theta {\beta }^{-2}\left(h_{\left(1\right)\left(1\right)\left(1\right)\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(S_H\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\right)\hfill \\ & & \left(\theta {\beta }^{-2}\left(h_{\left(1\right)\left(1\right)\left(1\right)\left(2\right)}\right)▹\left(\theta S_H{\beta }^{-2}\left(h_{\left(1\right)\left(1\right)\left(2\right)}\right)▹F_R\left({\beta }^{-2}\left(h_{\left(2\right)}\right)\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HM}1\right)}{=}& \theta \left(h_{\left(1\right)\left(1\right)\left(1\right)}\right)▹\left(F_R\left({\beta }^{-2}\left(S_H\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\left(\theta S_H{\beta }^{-2}\left(h_{\left(1\right)\left(1\right)\left(2\right)}\right)▹F_R\left({\beta }^{-2}\left(h_{\left(2\right)}\right)\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& \theta \beta \left(h_{\left(1\right)\left(1\right)}\right)▹\left(F_R\left({\beta }^{-3}\left(S_H\left(h_{\left(1\right)\left(2\right)\left(2\right)}\right)\right)\right)\left(\theta S_H{\beta }^{-2}\left(h_{\left(1\right)\left(2\right)\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(h_{\left(2\right)}\right)\right)\right)\right)\hfill \\ & =& \theta \beta \left(h_{\left(1\right)\left(1\right)}\right)▹\left(F_R\left({\beta }^{-3}\left(S_H{\left(h_{\left(1\right)\left(2\right)}\right)}_{\left(1\right)}\right)\right)\left(\theta {\beta }^{-2}\left(S_H{\left(h_{\left(1\right)\left(2\right)}\right)}_{\left(2\right)}\right)▹F_R\left({\beta }^{-2}\left(h_{\left(2\right)}\right)\right)\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& \theta {\beta }^2\left(h_{\left(1\right)}\right)▹\left(F_R\left({\beta }^{-3}\left(S_H{\left(h_{\left(2\right)\left(1\right)}\right)}_{\left(1\right)}\right)\right)\left(\theta {\beta }^{-2}\left(S_H{\left(h_{\left(2\right)\left(1\right)}\right)}_{\left(2\right)}\right)▹F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(2\right)}\right)\right)\right)\right)\hfill \\ & \stackrel{\left(15\right)}{=}& \theta {\beta }^2\left(h_{\left(1\right)}\right)▹F_R\left({\beta }^{-2}\left(S_H\left(h_{\left(2\right)\left(1\right)}\right)\right){\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & =& \theta {\beta }^3\left(h\right)▹F_R\left(1_H\right)\hfill \\ & =& {\epsilon }_H\left(h\right)1_A.\hfill \end{array}$$

The proof is completed. □

Using the above lemmas and corollaries that we have, we can gain the main result.

Theorem 3.

Let $(A{\times }_{\omega }^{\theta }H,\alpha \otimes \beta )$ be the $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct, let $\pi :A{\times }_{\omega }^{\theta }H\to H$ be the projection from $A{\times }_{\omega }^{\theta }H$ onto H, and let ${\mathcal{F}}_{A,H}$ be the set of pairs $(L,R)$, where $L:A\to A$, $R:H\to A$ are morphisms which satisfy the conclusions of Lemmas 2 and 3 for $F_L$ and $F_R$, respectively. Then,

(i)

The function $\Phi :{\mathcal{F}}_{A,H}\to {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$, described by $(L,R)\mapsto F$, where $F(a\times h)=L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)$, for all $a\in H$ and $h\in H$, is a bijection. Furthermore, $F_L=L$ and $F_R=R$.

(ii)

Suppose $(L,R)\in {\mathcal{F}}_{A,H}$, then, $F\in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$ if, and only if, L is a bijection.

Proof. 

(i) Assume that the function is well-defined. We define $\Psi :{\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )\to {\mathcal{F}}_{A,H}$ by $\Psi \left(F\right)=(\mathsf{\Pi }\circ F\circ J,\mathsf{\Pi }\circ F\circ j)$. It is easily proven that $\Phi$ and $\Psi$ are mutually inverse; see Equations (5) and (9). According to the previous results, to complete the proof of part (a), we only need to prove that elements of ${\mathcal{F}}_{A,H}$ give rise to elements of ${\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$, as indicated. As the reader might suspect, the proof that $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$ is somewhat tedious. We will use Lemmas 1–3, which do not need special citations initially.

It is easy to see that $\pi \circ F=\pi$. Obviously, $(\alpha \otimes \beta )\circ F=F\circ (\alpha \otimes \beta )$. Note that $F(1_A\times 1_H)=1_A\times 1_H$ and

$$\begin{array}{ccc}\epsilon \left(F\right(a\times h\left)\right)\hfill & =& \epsilon \left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\hfill \\ & =& {\epsilon }_A\left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right){\epsilon }_H\left({\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\hfill \\ & =& {\epsilon }_A\left(L\left({\alpha }^{-1}\left(a\right)\right)\right){\epsilon }_A\left(R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right){\epsilon }_H\left({\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\hfill \\ & =& {\epsilon }_A\left(a\right){\epsilon }_H\left(h\right),\hfill \end{array}$$

for $a\in A$ and $h\in H$, which means that $\epsilon \circ F=\epsilon$.

Let $a,b\in A$ and $h,k\in H$. Then,

$$\begin{array}{ccc}F\left(\right(a& \times & h\left)\right(b\times k\left)\right)\\ & =& F(a(\theta \left(h_{\left(1\right)}\right)▹{\alpha }^{-1}\left(b\right))\times {\beta }^{-1}\left(h_{\left(2\right)}\right)k)\hfill \\ & =& \left(L\left({\alpha }^{-1}\left(a\right)\right)L(\theta {\beta }^{-1}\left(h_{\left(1\right)}\right)▹{\alpha }^{-2}\left(b\right))\right)R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right){\beta }^{-2}\left(k_{\left(1\right)}\right)\right)\hfill \\ & & \times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & \stackrel{\left(15\right)}{=}& (\left(L\left({\alpha }^{-1}\left(a\right)\right)L(\theta {\beta }^{-1}\left(h_{\left(1\right)}\right)▹{\alpha }^{-2}\left(b\right))\right)\hfill \\ & & \left(R\left({\beta }^{-4}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\right)\left(\theta {\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)▹R\left({\beta }^{-3}\left(k_{\left(1\right)}\right)\right)\right)\right))\hfill \\ & & \times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& (\left(L\left({\alpha }^{-1}\left(a\right)\right)\left(L(\theta {\beta }^{-2}\left(h_{\left(1\right)}\right)▹{\alpha }^{-3}\left(b\right))R\left({\beta }^{-5}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\right)\right)\right)\hfill \\ & & \left(\theta {\beta }^{-2}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)▹R\left({\beta }^{-2}\left(k_{\left(1\right)}\right)\right)\right))\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& (\left(L\left({\alpha }^{-1}\left(a\right)\right)\left(L(\theta {\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)▹{\alpha }^{-3}\left(b\right))R\left({\beta }^{-4}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\right)\hfill \\ & & \left(\theta {\beta }^{-1}\left(h_{\left(2\right)\left(1\right)}\right)▹R\left({\beta }^{-2}\left(k_{\left(1\right)}\right)\right)\right))\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & \stackrel{\left(14\right)}{=}& (\left(L\left({\alpha }^{-1}\left(a\right)\right)\left(R\left({\beta }^{-4}\left(h_{\left(1\right)\left(1\right)}\right)\right)(\theta {\beta }^{-3}\left(h_{\left(1\right)\left(2\right)}\right)▹L\left({\alpha }^{-3}\left(b\right)\right))\right)\right)\hfill \\ & & \left(\theta {\beta }^{-1}\left(h_{\left(2\right)\left(1\right)}\right)▹R\left({\beta }^{-2}\left(k_{\left(1\right)}\right)\right)\right))\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& (\left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)\hfill \\ & & \left(\left(\theta {\beta }^{-2}\left(h_{\left(1\right)\left(2\right)}\right)▹L\left({\alpha }^{-2}\left(b\right)\right)\right)\left(\theta {\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right)▹R\left({\beta }^{-3}\left(k_{\left(1\right)}\right)\right)\right)\right))\hfill \\ & & \times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& (\left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)\hfill \\ & & \left(\left(\theta {\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)▹L\left({\alpha }^{-2}\left(b\right)\right)\right)\left(\theta {\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)▹R\left({\beta }^{-3}\left(k_{\left(1\right)}\right)\right)\right)\right))\hfill \\ & & \times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & \stackrel{\left(\mathrm{HMA}1\right)}{=}& \left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)\left(\theta {\beta }^{-1}\left(h_{\left(2\right)\left(1\right)}\right)▹L\left({\alpha }^{-2}\left(b\right)\right)R\left({\beta }^{-3}\left(k_{\left(1\right)}\right)\right)\right)\hfill \\ & & \times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & =& \left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)\left(\theta {\beta }^{-1}\left(h_{\left(2\right)\left(1\right)}\right)▹{\alpha }^{-1}\left(L\left({\alpha }^{-1}\left(b\right)\right)R\left({\beta }^{-2}\left(k_{\left(1\right)}\right)\right)\right)\right)\hfill \\ & & \times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right){\beta }^{-1}\left(k_{\left(2\right)}\right)\hfill \\ & =& \left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\left(L\left({\alpha }^{-1}\left(b\right)\right)R\left({\beta }^{-2}\left(k_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(k_{\left(2\right)}\right)\right)\hfill \\ & =& F(a\times h)F(b\times k).\hfill \end{array}$$

Therefore, F is a Hom-algebra morphism.

Next, we shall check that $\Delta \circ F=(F\otimes F)\circ \Delta$. Indeed, for all $a\in A$, $h\in H$,

$$\begin{array}{ccc}\Delta \left(F\right(a\hfill & \times h\left)\right)\hfill & \\ & =& \Delta (L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right))\hfill \\ & =& ({\left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)}_{\left(1\right)}\hfill \\ & & \times \omega \left({\left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)}_{\left(2\right)[-1]}\right){\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\left(L\left({\alpha }^{-1}\left(a\right)\right)R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)}_{\left(2\right)\left[0\right]}\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{T}4\right)}{=}& (L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(1\right)}\left(\theta \omega {\beta }^2\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)[-1]}\right)▹{\alpha }^{-1}\left(R{\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)}_{\left(1\right)}\right)\right)\hfill \\ & & \times \omega \left({\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]}\right)R{\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)}_{\left(2\right)}\right)}_{[-1]}\right){\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]}\right)R{\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)}_{\left(2\right)}\right)}_{\left[0\right]}\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{{}_{\left(\mathrm{Lemma}\right(\mathrm{iii}\left)\right)}}{=}& (L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(1\right)}\left(\theta \omega {\beta }^2\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)[-1]}\right)▹R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)\hfill \\ & & \times \omega \left({\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)}_{[-1]}\right){\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)}_{\left[0\right]}\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HCMA}1\right)}{=}& (L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(1\right)}\left(\theta \omega {\beta }^2\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)[-1]}\right)▹R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)\hfill \\ & & \times \omega \left({\beta }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right][-1]}\right)R{\left({\beta }^{-2}\left(h_{\left(1\right)\left(2\right)}\right)\right)}_{[-1]}\right){\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]\left[0\right]}\right)R{\left({\beta }^{-2}\left(h_{\left(1\right)\left(2\right)}\right)\right)}_{\left[0\right]}\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(16\right)}{=}& (L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(1\right)}\left(\theta \omega {\beta }^2\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)[-1]}\right)▹R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)\hfill \\ & & \times \omega \left({\beta }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right][-1]}\right){\omega }^{-1}\right({\beta }^{-6}\left(h_{\left(1\right)\left(2\right)\left(1\right)\left(1\right)}\right)\hfill \\ & & S_H\left({\beta }^{-5}\left(h_{\left(1\right)\left(2\right)\left(2\right)}\right)\right)\left)\right){\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]\left[0\right]}\right)R\left({\beta }^{-3}\left(h_{\left(1\right)\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& (L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(1\right)}\left(\theta \omega {\beta }^2\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)[-1]}\right)▹R\left({\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)\right)\hfill \\ & & \times \left(\omega {\beta }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right][-1]}\right){\beta }^{-5}\left(h_{\left(1\right)\left(2\right)\left(1\right)\left(1\right)}\right)\right)\hfill \\ & & \left(S_H\left({\beta }^{-4}\left(h_{\left(1\right)\left(2\right)\left(2\right)}\right)\right){\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]\left[0\right]}\right)R\left({\beta }^{-3}\left(h_{\left(1\right)\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& (L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(1\right)}\left(\theta \omega {\beta }^2\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)[-1]}\right)▹R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)\hfill \\ & & \times \left(\omega {\beta }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right][-1]}\right){\beta }^{-4}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\right)\hfill \\ & & \left(S_H\left({\beta }^{-5}\left(h_{\left(2\right)\left(2\right)\left(1\right)\left(1\right)}\right)\right){\beta }^{-5}\left(h_{\left(2\right)\left(2\right)\left(1\right)\left(2\right)}\right)\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HS}1\right)}{=}& (L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(1\right)}\left(\theta \omega {\beta }^2\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)[-1]}\right)▹R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)\hfill \\ & & \times \omega \left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right][-1]}\right){\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a\right)\right)}_{\left(2\right)\left[0\right]\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(12\right)}{=}& (\left(L\left({\alpha }^{-2}\left(a_{\left(1\right)}\right)\right)R\left(\omega {\beta }^{-1}\left(a_{\left(2\right)[-1]}\right)\right)\right)\left(\theta \omega {\beta }^2\left(L{\left({\alpha }^{-2}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{[-1]}\right)▹R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)\hfill \\ & & \times \omega \left(L{\left({\alpha }^{-2}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{\left[0\right][-1]}\right){\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-2}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{\left[0\right]\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(13\right)}{=}& (\left(L\left({\alpha }^{-2}\left(a_{\left(1\right)}\right)\right)R\left(\omega {\beta }^{-1}\left(a_{\left(2\right)[-1]}\right)\right)\right)\left(\theta \omega \left(a_{\left(2\right)\left[0\right][-1]}\right)▹R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)\hfill \\ & & \times \omega \left(L{\left({\alpha }^{-2}\left(a_{\left(2\right)\left[0\right]\left[0\right]}\right)\right)}_{[-1]}\right){\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-2}\left(a_{\left(2\right)\left[0\right]\left[0\right]}\right)\right)}_{\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HCM}2\right)}{=}& (\left(L\left({\alpha }^{-2}\left(a_{\left(1\right)}\right)\right)R\left(\omega {\beta }^{-2}\left(a_{\left(2\right)[-1]\left(1\right)}\right)\right)\right)\left(\theta \omega \left(a_{\left(2\right)[-1]\left(2\right)}\right)▹R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\right)\hfill \\ & & \times \omega \left(L{\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{[-1]}\right){\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& (L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)\left(R\left(\omega {\beta }^{-2}\left(a_{\left(2\right)[-1]\left(1\right)}\right)\right)\left(\theta \omega {\beta }^{-1}\left(a_{\left(2\right)[-1]\left(2\right)}\right)▹R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\right)\right)\hfill \\ & & \times \omega \left(L{\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{[-1]}\right){\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(15\right)}{=}& (L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)R\left(\omega {\beta }^{-1}\left(a_{\left(2\right)[-1]}\right){\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\hfill \\ & & \times \omega \left(L{\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{[-1]}\right){\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right))\hfill \\ & & \otimes \left({\alpha }^{-1}\left({\alpha }^{-1}\left(L{\left({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\right)}_{\left[0\right]}\right)R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(13\right)}{=}& \left(L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)R\left(\omega {\beta }^{-1}\left(a_{\left(2\right)[-1]}\right){\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times \omega {\beta }^{-1}\left(a_{\left(2\right)\left[0\right][-1]}\right){\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right)\right)\hfill \\ & & \otimes \left(\left(L\left({\alpha }^{-3}\left(a_{\left(2\right)\left[0\right]\left[0\right]}\right)\right)R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HCM}2\right)}{=}& (L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)R\left(\omega {\beta }^{-2}\left(a_{\left(2\right)[-1]\left(1\right)}\right){\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times \omega {\beta }^{-1}\left(a_{\left(2\right)[-1]\left(2\right)}\right)\hfill \\ & & {\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(1\right)}\right))\otimes \left(\left(L\left({\alpha }^{-2}\left(a_{\left(2\right)\left[0\right]}\right)\right)R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& (L\left({\alpha }^{-1}\left(a_{\left(1\right)}\right)\right)R\left(\omega {\beta }^{-2}\left(a_{\left(2\right)[-1]\left(1\right)}\right){\beta }^{-3}\left(h_{\left(1\right)\left(1\right)}\right)\right)\times \omega {\beta }^{-1}\left(a_{\left(2\right)[-1]\left(2\right)}\right)\hfill \\ & & {\beta }^{-2}\left(h_{\left(1\right)\left(2\right)}\right))\otimes \left(\left(L\left({\alpha }^{-2}\left(a_{\left(2\right)\left[0\right]}\right)\right)R\left({\beta }^{-2}\left(h_{\left(2\right)\left(1\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & =& F((a_{\left(1\right)}\times \omega \left(a_{\left(2\right)[-1]}\right){\beta }^{-1}\left(h_{\left(1\right)}\right))\otimes F({\alpha }^{-1}\left(a_{\left(2\right)\left[0\right]}\right)\times h_{\left(2\right)})\hfill \\ & =& F\left({(a\times h)}_{\left(1\right)}\right)\otimes F\left({(a\times h)}_{\left(1\right)}\right).\hfill \end{array}$$

We have shown that F preserves comultiplication. Therefore, F is a Hom-coalgebra morphism, and hence, is a Hom-bialgebra morphism. Since Hom-bialgebra morphisms of Hom–Hopf algebras are Hom-Hopf algebra morphisms, this completes the proof of part (i).

(ii) Suppose $F\in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then, $F_L$ and ${\left(F^{-1}\right)}_L$ are inverse by (10) and Corollary 1(i). Thus, $F_L$ is bijective and ${\left(F_L\right)}^{-1}={\left(F^{-1}\right)}_L$.

Conversely, suppose that $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$ and $F_L$ are bijective. Set $G_L={\left(F_L\right)}^{-1}$. From Corollary 4, $F_R$ has a convolution inverse $J_R$. Set $G_R=G_L\circ J_R={\left(F_L\right)}^{-1}\circ J_R$ and define $G(a\times h)=G_L\left({\alpha }^{-1}\left(a\right)\right)G_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)$ for all $a\in A$ and $h\in H$. Since $G_L$ is a Hom-algebra morphism, we compute

$$\begin{array}{ccc}G(F\hfill & (a\times h& \left)\right)\\ & =& G\left(F_L\left({\alpha }^{-1}\left(a\right)\right)F_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\hfill \\ & =& G_L\left(F_L\left({\alpha }^{-2}\left(a\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\right)G_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ & =& \left({\alpha }^{-2}\left(a\right)G_L\left(F_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\right)\right)G_L\left(J_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ & \stackrel{\left(\mathrm{HA}2\right)}{=}& {\alpha }^{-1}\left(a\right)G_L\left(F_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)J_R\left({\beta }^{-4}\left(h_{\left(2\right)\left(1\right)}\right)\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ & \stackrel{\left(\mathrm{HC}2\right)}{=}& {\alpha }^{-1}\left(a\right)G_L\left(F_R\left({\beta }^{-4}\left(h_{\left(1\right)\left(1\right)}\right)\right)J_R\left({\beta }^{-4}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\hfill \\ & =& {\alpha }^{-1}\left(a\right)1_A{\epsilon }_H\left(h_{\left(1\right)}\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\hfill \\ & =& a\times h\hfill \end{array}$$

and

$$\begin{array}{ccc}F(G\hfill & (a\times h& \left)\right)\\ & =& F\left(G_L\left({\alpha }^{-1}\left(a\right)\right)G_R\left({\beta }^{-2}\left(h_{\left(1\right)}\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\right)\hfill \\ & =& F_L\left(G_L\left({\alpha }^{-2}\left(a\right)\right)G_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ & =& \left({\alpha }^{-2}\left(a\right)F_L\left(G_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ & =& \left({\alpha }^{-2}\left(a\right)J_R\left({\beta }^{-3}\left(h_{\left(1\right)}\right)\right)\right)F_R\left({\beta }^{-3}\left(h_{\left(2\right)\left(1\right)}\right)\right)\times {\beta }^{-2}\left(h_{\left(2\right)\left(2\right)}\right)\hfill \\ & \underset{\left(\mathrm{HC}2\right)}{\stackrel{\left(\mathrm{HA}2\right)}{=}}& {\alpha }^{-1}\left(a\right)\left(J_R\left({\beta }^{-4}\left(h_{\left(1\right)\left(1\right)}\right)\right)F_R\left({\beta }^{-4}\left(h_{\left(1\right)\left(2\right)}\right)\right)\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\hfill \\ & =& {\alpha }^{-1}\left(a\right)1_A{\epsilon }_H\left(h_{\left(1\right)}\right)\times {\beta }^{-1}\left(h_{\left(2\right)}\right)\hfill \\ & =& a\times h.\hfill \end{array}$$

Thus, we have shown that $G\circ F={\mathrm{id}}_{A{\times }_{\omega }^{\theta }H}=F\circ G$. □

Now, let ${\mathcal{F}}_{A,H}^{•}$ denote the set of $(L,R)\in {\mathcal{F}}_{A,H}$, such that L is bijective. Then, the correspondence of the above theorem induces a bijection ${\mathcal{F}}_{A,H}^{•}\to {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$.

When $F(1_A\times H)\subseteq 1_A\times H$, the structure of F is particularly simple.

Proposition 1.

Let $F\in {\mathrm{End}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$. Then, the following conditions are equivalent:

(i)

$F(1_A\times H)\subseteq 1_A\times H$.

(ii)

$F_R\left(h\right)\in \mathbb{k}{1}_A$ for all $h\in H$.

(iii)

$F_R\left(h\right)={\epsilon }_H\left(h\right)1_A$ for all $h\in H$.

(iv)

$F_R={\eta }_A\circ {\epsilon }_H$.

(v)

$F(a\times h)=F_L\left(a\right)\times h$ for all $a\in A$ and $h\in H$.

(vi)

$F(1_A\times h)=1_A\times h$ for all $h\in H$.

Proof. 

Firstly, we prove that part (i) implies part (ii) implies part (iii). Suppose part (i) holds. Then, we have $F_R\left(H\right)\subseteq \mathbb{k}{1}_A$ by Equation (11), and hence, $F_R\left(h\right)={\epsilon }_H\left(h\right)1_A$ for all $h\in H$, because ${\epsilon }_A\circ F_R={\epsilon }_H$. Secondly, it is easy to see that parts (iii) and (iv) are equivalent. Thirdly, part (iv) implies part (v) by Equation (9), and part (v) implies part (vi) as $F_L\left(1_A\right)=1_A$. At last, part (vi) trivially implies part (i). □

Let $(A,\alpha )$ be a Hom-algebra and $(C,\beta )$ be a Hom-coalgebra over $\mathbb{k}$. Let $\mathrm{Hom}(C,A)$ be the set of $\mathbb{k}$-linear maps $f:C\to A$ satisfying $\alpha \circ f=f\circ \beta$. The group $\mathcal{G}\left(A\right)={\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Alg}}\left(A\right)$ acts on the (non-associative) convolution algebra $\mathrm{Hom}(C,A)$ by $f▸g=f\circ g$ for all $f\in \mathcal{G}\left(A\right))$ and $g\in \mathrm{Hom}(C,A)$. Obviously, this action satisfies:

$$f▸({\eta }_A\circ {\epsilon }_C)={\eta }_A\circ {\epsilon }_C\mathrm{and}f▸(g\ast g^{\prime })=(f▸g)\ast (f▸g^{\prime })$$

for all $f\in \mathcal{G}\left(A\right)$ and $g,g^{\prime }\in \mathrm{Hom}(C,A)$. Then, $(\mathrm{Hom}(C,A),{\alpha }^2)$ is a Hom-algebra with the unit ${\eta }_A\circ {\epsilon }_C$ under the convolution *. Indeed, for $f,g,g^{\prime }\in \mathrm{Hom}(C,A)$ and $c\in C$,

$$\begin{array}{ccc}(({\alpha }^2▸f)\ast (g\ast g^{\prime }))\left(c\right)\hfill & =& ({\alpha }^2▸f)\left(c_{\left(1\right)}\right)\left(g\left(c_{\left(2\right)\left(1\right)}\right)g^{\prime }\left(c_{\left(2\right)\left(2\right)}\right)\right)\hfill \\ & =& (\alpha ▸f)\left(c_{\left(1\right)\left(1\right)}\right)\left(g\left(c_{\left(1\right)\left(2\right)}\right)g^{\prime }\left(\beta \left(c_{\left(2\right)}\right)\right)\right)\hfill \\ & =& \left(f\left(c_{\left(1\right)\left(1\right)}\right)g\left(c_{\left(1\right)\left(2\right)}\right)\right)({\alpha }^2▸g^{\prime })\left(c_{\left(2\right)}\right)\hfill \\ & =& ((f\ast g)\ast ({\alpha }^2▸g^{\prime }))\left(c\right),\hfill \end{array}$$

and

$$(f\ast ({\eta }_A\circ {\epsilon }_C))\left(c\right)=f\left(c_{\left(1\right)}\right){\epsilon }_H\left(c_{\left(2\right)}\right)1_A=\alpha \left(f\left(\beta \left(c\right)\right)\right)=({\alpha }^2▸f)\left(c\right).$$

Thus, it follows that $({\alpha }^2▸f)\ast (g\ast g^{\prime })=(f\ast g)\ast ({\alpha }^2▸g^{\prime })$ and $f\ast ({\eta }_A\circ {\epsilon }_C)={\alpha }^2▸f$. Moreover, $({\eta }_A\circ {\epsilon }_C)\ast f={\alpha }^2▸f$ can be checked similarly.

It is easy to check that $(\mathrm{Hom}(C,A),{\alpha }^{-2}\circ \ast ,{\eta }_A\circ {\epsilon }_C)$ is an ordinary associative algebra. Let $\mathcal{U}(C,A)$ be the group of units of the algebra $(\mathrm{Hom}(C,A),{\alpha }^{-2}\circ \ast ,{\eta }_A\circ {\epsilon }_C)$. Then $\mathcal{G}\left(A\right)▸\mathcal{U}(C,A)\subseteq \mathcal{U}(C,A)$; thus, there is a group homomorphism

$$\phi :\mathcal{G}\left(A\right)\to {\mathrm{Aut}}_{\mathrm{Group}}\left(\mathcal{U}(C,A)\right)$$

given by $\phi \left(f\right)\left(g\right)=f▸g=f\circ g$ for all $f\in \mathcal{G}\left(A\right)$ and $g\in \mathcal{U}(C,A)$. The resulting group $\mathcal{U}(C,A){⋊}_{\phi }\mathcal{G}\left(A\right)$ has product given by

$$(g,f)(g^{\prime },f^{\prime })=({\alpha }^{-2}(g\ast (f\circ g^{\prime })),f\circ f^{\prime }).$$

Note that the action of $\mathcal{G}\left(A\right)$ on $\mathcal{U}(C,A)$ by group homomorphisms is also one on $\mathcal{U}{(C,A)}^{\mathrm{op}}$. For a group G, the group $G^{\mathrm{op}}$ is the group whose underlying set is G and the product is given by $g·g^{\prime }=g^{\prime }g$ for all $g,g^{\prime }\in G$. Therefore, we have the following result of Theorem 3 and Corollary 1.

Theorem 4.

Let $(A{\times }_{\omega }^{\theta }H,\alpha \otimes \beta )$ be the $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct, let $\pi :A{\times }_{\omega }^{\theta }H\to H$ be the projection from $A{\times }_{\omega }^{\theta }H$ onto H. Then, there is an injective group homomorphism $\psi :{\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )\to \mathcal{U}{(C,A)}^{\mathrm{op}}{⋊}_{\phi }\mathcal{G}\left(A\right)$, which is given by $F\mapsto (F_R,F_L)$ for all $F\in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$.

Proof. 

For $F\in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$, it is easy to see that $F_L$ and $F_R$ are bijective by Theorem 3(ii) and Equation (11). If $(F_R,F_L)=({\eta }_A\circ {\epsilon }_C,{\mathrm{id}}_A)$, then, using (9), we have $F(a\times h)=a\times h$ for all $a\in A$ and $h\in H$. Thus, it follows that $F={\mathrm{id}}_{A{\times }_{\omega }^{\theta }H}$, i.e., $\psi$ is injective.

For $F,G\in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$, since

$$\begin{array}{ccc}\psi (F\circ G)\hfill & =& ({(F\circ G)}_R,{(F\circ G)}_L)\hfill \\ & =& ({\alpha }^{-2}\left(F_R{\ast }^{\mathrm{op}}(F_L\circ G_R)\right),F_L\circ G_L)\hfill \\ & \stackrel{\mathrm{Cor}1}{=}& (F_R,F_L)(G_R,G_L)\hfill \\ & =& \psi \left(F\right)\psi \left(G\right),\hfill \end{array}$$

it follows that $\psi$ preserves the multiplication. Thus, $\psi$ is an injective group homomorphism. □

In conclusion, by Theorem 3, we obtain the characterization of the automorphism group ${\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$ of a $(\theta ,\omega )$-twisted-Radford’s Hom-biproduct $A{\times }_{\omega }^{\theta }H$ via the automorphisms of the factors $(A,\alpha )$ and $(H,\beta )$. Furthermore, by Theorem 4, the automorphism group ${\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\omega }^{\theta }H,\pi )$ can be regarded as a subgroup of some semidirect product $\mathcal{U}{(C,A)}^{\mathrm{op}}{⋊}_{\phi }\mathcal{G}\left(A\right)$, when $\pi$ is a projection. These are important for the study of automorphism groups of twisted-Radford’s Hom-biproducts.

4. Example

In this section, we consider an example in [26]. Let $A=\mathbb{k}\langle 1_A,x\rangle$ and the automorphism $\alpha :A\to A$ via $\alpha \left(1_A\right)=1_A$ and $\alpha \left(x\right)=-x$. Then, $(A,\alpha )$ is both a Hom-algebra with product: $1_A{1}_A=1_A$, $1_{A}x=x{1}_A=-x$, $x^2=0$, and a Hom-coalgebra with coproduct and counit given by

$$\begin{array}{c}{\Delta }_A\left(1_A\right)=1_A\otimes 1_A,{\Delta }_A\left(x\right)=(-x)\otimes 1_A+1_A\otimes (-x),\\ {\epsilon }_A\left(1_A\right)=1_A,{\epsilon }_A\left(x\right)=0.\end{array}$$

We define $S_A:A\to A$ by $S_A\left(1_A\right)=1_A$, $S_A\left(x\right)=-x$, which is the convolution inverse of ${\mathrm{id}}_A$.

Let $H=\mathbb{k}\langle 1_H,g,g^2\rangle$ be the group Hopf algebra with $g^3=1_H$, ${\Delta }_H\left(g\right)=g\otimes g$, $S_H\left(g\right)=g^2$ and $S_H\left(g^2\right)=g$. Define an automorphism $\beta :H\to H$ by $\beta \left(g\right)=g^2$, $\beta \left(g^2\right)=g$. Then, we can define a new product $m_{\beta }=\beta \circ m_H$ and a new coproduct ${\Delta }_{\beta }=\Delta \circ \beta$ on H. So, we obtain a Hom–Hopf algebra $H_{\beta }=(H,m_{\beta },{\Delta }_{\beta },S_H,\beta )$. Then, $(A,\alpha )$ is a left $H_{\beta }$-Hom-module Hom-algebra and Hom-module Hom-coalgebra with the action $▹:H\otimes A\to A$ given by

$$1_H▹1_A=1_A,1_H▹x=-x,g▹1_A=1_A,g▹x=-x,g^2▹1_A=1_A,g^2▹x=-x.$$

Furthermore, $(A,\alpha )$ is a left $H_{\beta }$-Hom-comodule Hom-algebra and Hom-comodule Hom-coalgebra with the coaction $\rho :A\to H\otimes A$ given by

$$\rho \left(1_A\right)=1_H\otimes 1_A,\rho \left(x\right)=1_H\otimes (-x).$$

Then, we can obtain many twisted Radford’s Hom-biproducts, such as $(\beta ,\beta )$-twisted Radford’s Hom-biproduct, $(\beta ,{\mathrm{id}}_H)$-twisted Radford’s Hom-biproduct, $({\mathrm{id}}_H,\beta )$-twisted Radford’s Hom-biproduct, $({\mathrm{id}}_H,{\mathrm{id}}_H)$-twisted Radford’s Hom-biproduct, and so on.

Now, we clearly write the structures of the $(\beta ,\beta )$-twisted Radford’s Hom-biproduct $(A{\times }_{\beta }^{\beta }{H}_{\beta }=\{1_A\times 1_H,1_A\times g,1_A\times g^2,x\times 1_H,x\times g,x\times g^2\},\alpha \otimes \beta )$ and characterize the automorphisms of it. Its product is defined as follows:

$$\begin{array}{ccccccc}m& 1_A\times 1_H& 1_A\times g& 1_A\times g^2& x\times 1_H& x\times g& x\times g^2\\ 1_A\times 1_H& 1_A\times 1_H& 1_A\times g^2& 1_A\times g& -x\times 1_H& -x\times g^2& -x\times g\\ 1_A\times g& 1_A\times g^2& 1_A\times g& 1_A\times 1_H& -x\times g^2& -x\times g& -x\times 1_H\\ 1_A\times g^2& 1_A\times g& 1_A\times 1_H& 1_A\times g^2& -x\times g& -x\times 1_H& -x\times g^2\\ x\times 1_H& -x\times 1_H& -x\times g^2& -x\times g& 0& 0& 0\\ x\times g& -x\times g^2& -x\times g& -x\times 1_H& 0& 0& 0\\ x\times g^2& -x\times g& -x\times 1_H& -x\times g^2& 0& 0& 0\end{array}$$

Its coproduct and counit are defined as follows:

$$\begin{array}{cc}\hfill & \Delta (1_A\times 1_H)=(1_A\times 1_H)\otimes (1_A\times 1_H),\epsilon (1_A\times 1_H)=1,\hfill \\ \hfill & \Delta (1_A\times g)=(1_A\times g^2)\otimes (1_A\times g^2),\epsilon (1_A\times g)=1,\hfill \\ \hfill & \Delta (1_A\times g^2)=(1_A\times g)\otimes (1_A\times g),\epsilon (1_A\times g^2)=1,\hfill \\ \hfill & \Delta (x\times 1_H)=(-x\times 1_H)\otimes (1_A\times 1_H)+(1_A\times 1_H)\otimes (x\times 1_H),\epsilon (x\times 1_H)=0,\hfill \\ \hfill & \Delta (x\times g)=(-x\times g^2)\otimes (1_A\times g^2)+(1_A\times g^2)\otimes (x\times g^2),\epsilon (x\times g)=0,\hfill \\ \hfill & \Delta (x\times g^2)=(-x\times g)\otimes (1_A\times g)+(1_A\times g)\otimes (x\times g),\epsilon (x\times g^2)=0,\hfill \end{array}$$

Its antipodes are as follows:

$$\begin{array}{ccc}S(1_A\times 1_H)=1_A\times 1_H,& S(1_A\times g)=1_A\times g^2,& S(1_A\times g^2)=1_A\times g,\\ S(x\times 1_H)=-x\times 1_H,& S(x\times g)=-x\times g^2,& S(x\times g^2)=-x\times g.\end{array}$$

Now, we compute the morphisms $L\in \mathrm{End}\left(A\right)$ satisfying the conclusions of Lemma 2. This is taking a base of $\mathrm{End}\left(A\right)$, $\{L_1,L_2,L_3,L_4\}$, given respectively by

$$\begin{array}{cc}{L}_1\hfill & :1_A\mapsto 1_A,x\mapsto 0,\hfill \\ L_2\hfill & :1_A\mapsto x,x\mapsto 0,\hfill \\ L_3\hfill & :1_A\mapsto 0,x\mapsto 1_A,\hfill \\ L_4\hfill & :1_A\mapsto 0,x\mapsto x.\hfill \end{array}$$

Let $L=t_1{L}_1+t_2{L}_2+t_3{L}_3+t_4{L}_4$, where $t_1,t_2,t_3,t_4\in \mathbb{k}$. If L satisfies Lemma 2(ii), it follows that $t_1=1$ and $t_3=0$. By part (iv) of Lemma 2, we can obtain $t_2=0$. Therefore, $L=L_1+t{L}_4$ for some $t\in \mathbb{k}$. So, there is a bijection between the set of the morphisms $L\in \mathrm{End}\left(A\right)$ satisfying the conclusions of Lemma 2 and the set $\left\{\left(\begin{array}{cc}1& 0\\ 0& t\end{array}\right)\right|t\in \mathbb{k}\}$.

Now, we will discuss the morphisms of $\mathrm{Hom}(H,A)$ which satisfy Lemma 3 in a similar way as above. Taking a base of $\mathrm{Hom}(H,A)$, $\{R_1,R_2,\cdots ,R_6\}$, given respectively by

$$\begin{array}{cc}{R}_1\hfill & :1_H\mapsto 1_A,g\mapsto 0,g^2\mapsto 0,\hfill \\ R_2\hfill & :1_H\mapsto x,g\mapsto 0,g^2\mapsto 0,\hfill \\ R_3\hfill & :1_H\mapsto 0,g\mapsto 1_A,g^2\mapsto 0,\hfill \\ R_4\hfill & :1_H\mapsto 0,g\mapsto x,g^2\mapsto 0,\hfill \\ R_5\hfill & :1_H\mapsto 0,g\mapsto 0,g^2\mapsto 1_A,\hfill \\ R_6\hfill & :1_H\mapsto 0,g\mapsto 0,g^2\mapsto x.\hfill \end{array}$$

Let $R={\sum }_{i=1}^6{k}_i{R}_i$, where $k_i\in \mathbb{k}$, $i=1,2,3,4,5,6$. Since R satisfies part (i) of Lemma 3, we have $k_1=1$ and $k_2=0$. By Equation (16), it follows that $k_3=k_5$ and $k_4=-k_6$. Thus, $R=R_1+k_3{R}_3+k_4{R}_4+k_3{R}_5-k_4{R}_6$. Next, we shall check part (ii) of Lemma 3, and we obtain $k_3=1$ and $k_4=0$. Therefore, $R=R_1+R_3+R_5$, i.e., $R\left(1_H\right)=1_A$, $R\left(g\right)=1_A$, $R\left(g^2\right)=1_A$. Hence, ${\mathcal{F}}_{A,H}^{•}\cong \left\{\left(\begin{array}{cc}1& 0\\ 0& t\end{array}\right)\right|0\ne t\in \mathbb{k}\}\cong {\mathbb{k}}^{\times }$. We can obtain the concrete characterization of ${\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\beta }^{\beta }{H}_{\beta },\pi )$. Let $F\in {\mathrm{Aut}}_{\mathrm{Hom}-\mathrm{Hopf}}(A{\times }_{\beta }^{\beta }{H}_{\beta },\pi )$. By Theorem 3, we have

$$\begin{array}{ccc}F(1_A\times 1_H)=1_A\times 1_H,& F(1_A\times g)=1_A\times g,& F(1_A\times g^2)=1_A\times g^2,\\ F(x\times 1_H)=tx\times 1_H,& F(x\times g)=tx\times g,& F(x\times g^2)=tx\times g^2,\end{array}$$

where $t\in {\mathbb{k}}^{\times }$.