# Enveloping actions for twisted partial actions

Shuangjian Guo1111Correspondence: Shuangjian Guo, School of Mathematics and
Statistics, Guizhou University of Finance and Economics, Guiyang 550025, P. R.
China; E-mail<EMAIL_ADDRESS>Shengxiang Wang2
1\. School of Mathematics and Statistics, Guizhou University of Finance and
Economics
Guiyang 550025, P. R. China
2\. School of Mathematics and Finance, Chuzhou University
Chuzhou 239000, P. R. China

Abstract Let $A\\#_{\alpha,\omega}H$ be a partial crossed product. In this
paper, we first generalize the theorem about the existence of an enveloping
action to twisted partial actions. Second, we construct a Morita context
between the partial crossed product and the crossed product related to the
enveloping action. Furthermore, we discuss equivalences of partial crossed
products Finally, we investigate when $A\subset A\\#_{\alpha,\omega}H$ becomes
a separable extension.

Key words Enveloping action, partial crossed product, Morita context,
separable extension.

AMS(2010) Subject Classification 16T05; 16T15

##  1  Introduction

Partial actions of groups have been introduced in the theory of operator
algebras as a general approach to study $C^{\ast}$-algebras by partial
isometries in [16]. A treatment from a purely algebraic point of view was
given recently in [9]-[13]. In particular, the Galois theory of partial group
actions developed in [9] inspired further Galois theoretic results in [6], as
well as the introduction and study of partial Hopf actions and coactions in
[7]. The latter paper became in turn the starting point for further
investigation of partial Hopf (co)actions in [1]-[4]. The general notion of a
(continuous) twisted partial action of a locally compact group on a
$C^{\ast}$-algebra (a twisted partial $C^{\ast}$-dynamical system) and the
corresponding crossed products were given by Exel in [17]. Twisted partial
actions of groups on abstract rings and corresponding crossed products were
recently introduced in [12]. More recent algebraic results on twisted partial
actions and corresponding crossed products were obtained in [13] and [5]. The
algebraic concept of twisted partial actions also motivated the study of
projective partial group representations, the corresponding partial Schur
Multiplier and the relation to partial group actions with K-valued twistings
in [14] and [15], contributing towards the elaboration of a background for a
general cohomological theory based on partial actions. Further information
around partial actions may be consulted in the survey [8]. As a unified
approach for twisted partial group actions, partial Hopf actions and twisted
actions of Hopf algebras, the notion of a twisted partial Hopf action is
introduced in [4].

Certainly, the theory of partial actions of Hopf algebras remains as a huge
landscape to be explored. Alves and Batista [2] generalized the theorem about
the existence of an enveloping action, also known as the globalization theorem
and they constructd a Morita context between the partial smash product and the
smash product related to the enveloping action, and this present work intends
to generalize some results for partial group actions and partial Hopf actions
to the context of twisted partial actions.

This paper is organized as follows:

In Section 3, we prove the existence of an enveloping action for such a
twisted partial actions. In Section 4, we construct a Morita context between
the partial crossed product $A\\#_{\alpha,\omega}H$ and the crossed product
$B\\#_{u}H$, where $H$ is a Hopf algebra which acts partially on the unital
algebra $A$, $B$ is an enveloping action for twisted partial actions. This
result can also be found in [2] for the context of partial group actions. In
Section 5, we shall discuss equivalences of partial crossed products and this
result is similar to [4]. In Section 6, we investigate when $A\subset
A\\#_{\alpha,\omega}H$ becomes a separable extension.

##  2 Preliminaries

Throughout the paper, let $k$ be a fixed field and all algebraic systems are
supposed to be over $k$. Let $M$ be a vector space over $k$ and let $id_{M}$
the usual identity map. For the comultiplication $\Delta$ in a coalgebra $C$
with a counit $\varepsilon_{C}$, we use the Sweedler-Heyneman’s notation (see
Sweedler [20]): $\Delta(c)=c_{1}\otimes c_{2}$, for any $c\in C$.

We first recall some basic results and propositions that we will need later
from Alves and Batista [1],[2].

2.1. Partial module algebra Let $H$ be a Hopf algerba and $A$ an algebra. $A$
is said to be a partial $H$-module algebra if there exists a $k$-linear map
$\cdot\\{\cdot:H\otimes A\rightarrow A\\}$ satisfying the following
conditions:

$\displaystyle h\cdot(ab)=(h_{(1)})\cdot a)(h_{(2)}\cdot b),$ $\displaystyle
1_{H}\cdot a=a,$ $\displaystyle h\cdot(g\cdot a)=(h_{(1)}\cdot
1_{A})(h_{(2)}g\cdot a),$

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

##  3 Enveloping actions

Recall from [4] that the definition of the partial crossed product. A twisted
partial action of $H$ on $A$ is a pair $(\alpha,\omega)$, where
$\alpha:H\otimes A\rightarrow A,\alpha(h\otimes a)=h\cdot a$ and
$\omega:H\otimes H\rightarrow A,\omega(h\otimes g)=\omega(h,g)$ be two linear
maps such that the following conditions hold:

$\displaystyle 1_{H}\cdot a=a,$ (3.1) $\displaystyle h\cdot(ab)=(h_{(1)}\cdot
a)(h_{(2)}\cdot b),$ (3.2) $\displaystyle(h_{(1)}\cdot(l_{(1)}\cdot
a))\omega(h_{(2)},l_{(2)})=\omega(h_{(1)},l_{(1)})(h_{(2)}l_{(2)}\cdot a),$
(3.3) $\displaystyle\omega(h,l)=\omega(h_{(1)},l_{(1)})(h_{(2)}l_{(2)}\cdot
1_{A}),$ (3.4)

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

Lemma 3.1.${}^{\cite[cite]{[\@@bibref{}{A14}{}{}]}}$ If $(\alpha,\omega)$ is a
twisted partial action, then the following identities hold:

$\displaystyle\omega(h,l)=(h_{(1)}\cdot(l_{(1)}\cdot
1_{A}))\omega(h_{(2)},l_{(2)})=(h_{(1)}\cdot 1_{A})\omega(h_{(2)},l),$ (3.5)

for all $h,l\in H$.

We say that the map $\omega$ is trivial, if the following condition holds:

$\displaystyle h\cdot(l\cdot 1_{A})=\omega(h,l)=(h_{(1)}\cdot
1_{A})(h_{(2)}l\cdot 1_{A}),$

for all $h,l\in H$. In this case, the twisted partial action $(\alpha,\omega)$
turns out to be a partial action of $H$ on $A$.

Given any twisted partial action $(\alpha,\omega)$ of $H$ on $A$, we can
define on the vector space $A\otimes H$ a product, given by the multiplication

$\displaystyle(a\otimes h)(b\otimes l)=a(h_{(1)}\cdot
b)\omega(h_{(2)},l_{(1)})\otimes h_{(3)}l_{(2)},$

for all $a,b\in A$ and $h,l\in H$. Denote $A\\#_{\alpha,\omega}H$ to be the
subspace of $A\otimes H$ generated by the elements of the form
$a\\#h=a(h_{(1)}\cdot 1_{A})\otimes h_{(2)}$, for all $a\in A$ and $h\in H$.
Recall from [] that $A\\#_{\alpha,\omega}H$ is associative via the
multiplication Eq.(3.7) and $A\\#_{\alpha,\omega}H$ is unital with
$1_{A}\\#1_{H}=1_{A}\otimes 1_{H}$ if and only if

$\displaystyle\omega(1_{H},h)=\omega(h,1_{H})=h\cdot 1_{A},$ (3.6)
$\displaystyle(h_{(1)}\cdot(l_{(1)}\cdot
a))\omega(h_{(2)},l_{(2)})=\omega(h_{(1)},l_{(1)})(h_{(2)}l_{(2)}\cdot a),$
(3.7)
$\displaystyle(h_{(1)}\cdot\omega(l_{(1)},m_{(1)}))\omega(h_{(2)},l_{(2)}m_{(2)})=\omega(h_{(1)},l_{(1)})\omega(h_{(2)}l_{(2)},m_{(2)}),$
(3.8)

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

Recall from [4] that let $B$ be a unital $k$-algebra measured by an action
$\beta:H\otimes B\rightarrow B$, denoted by $\beta(h\otimes b)=h\triangleright
b$, which is twisted by a map $u:H\otimes H\rightarrow B$, i.e.,

$\displaystyle 1_{H}\triangleright a=a,$ $\displaystyle
h\triangleright(ab)=(h_{(1)}\triangleright a)(h_{(2)}\triangleright b),$
$\displaystyle h\triangleright 1_{A}=\varepsilon(h)1_{A},$ $\displaystyle
u(1_{H},h)=u(h,1_{H})=\varepsilon(h)1_{A},$
$\displaystyle(h_{(1)}\triangleright(l_{(1)}\triangleright
a))u(h_{(2)},l_{(2)})=u(h_{(1)},l_{(1)})(h_{(2)}l_{(2)}\triangleright a),$
$\displaystyle u(h_{(1)},l_{(1)})u(h_{(2)}l_{(2)},m)=(h_{(1)}\triangleright
u(l_{(1)},m_{(1)}))u(h_{(2)},l_{(2)}m_{(2)}),$

for all $h,l,m\in H$ and $a\in A$. Suppose that $1_{A}$ is a non-trivial
central idempotent of $B$, and let $A$ be the ideal generated by $1_{A}$.
Given $a\in A$ and $h\in H$, define a map $\cdot:H\otimes A\rightarrow A$ by
$h\cdot a=1_{A}(h\triangleright a)$. We say that the partial action $h\cdot
a=1_{A}(h\triangleright a)$ and $\omega(h,l)=(h_{(1)}\cdot
1_{A})u(h_{(2)},l_{(1)})(h_{(3)}l_{(2)}\cdot 1_{A})$ is the twisted partial
action induced by $B$.

Definition 3.2.${}^{\cite[cite]{[\@@bibref{}{A14}{}{}]}}$ With notations as
above. A morphism of algebras $\theta:A\rightarrow B$ is said to be a morphism
of partial $H$-module algebras if $\theta(h\cdot a)=h\cdot\theta(a)$ for all
$h,k\in H$ and $a\in A$. If, in addition, $\theta$ is an isomorphism, the
partial actions are called equivalent.

Recall from [2] that if $B$ is an $H$-module algebra and $A$ is a right ideal
of $B$ with unity $1_{A}$, the induced partial action on $A$ is called
admissible if $B=H\rhd A$.

Definition 3.3. With notations as above. An enveloping action for $A$ is a
pair $(B,\theta),$ where

(a) $B$ is a unital $k$-algebra measured by an action $\beta:H\otimes
B\rightarrow B$;

(b) The map $\theta:A\rightarrow B$ is a monomorphism of algebras;

(c) The sub-algebra $\theta(A)$ is an ideal in $B$;

(d) The partial action on $A$ is equivalent to the induced partial action on
$\theta(A)$;

(e) The induced partial action on $\theta(A)$ is admissible.

(d) $\theta(a\omega(g,h))=\theta(a)u(g,h)$,
$\theta(\omega(g,h)a)=u(g,h)\theta(a)$, for any $g,h\in H$ and $a\in A$.

##  4 A Morita context

In this section, we will construct a Morita context between the partial
crossed product $A\\#_{\alpha,\omega}H$ and the crossed product $B\\#_{u}H$,
where $B$ is an enveloping action for the partial crossed product.

######  Lemma 4.1.

Let $(\alpha,\omega)$ be a twisted partial action and $(B,\theta)$ an
enveloping action, then there is an algebra monomorphism from the partial
crossed product $A\\#_{\alpha,\omega}H$ into the crossed product $B\\#_{u}H$.

Proof. Define $\Phi:A\\#_{\alpha,\omega}H\rightarrow B\\#_{u}H$ by $a\otimes
h\mapsto\theta(a)\otimes h$ for $h,g\in H$ and $a,b\in A$. We first check that
$\Phi$ is a morphism of algebras as follows:

$\displaystyle\Phi((a\otimes h)(b\otimes g))$ $\displaystyle=$
$\displaystyle\Phi(a(h_{(1)}\cdot b)\omega(h_{(2)},g_{(1)})\otimes
h_{(3)}g_{(2)}).$ $\displaystyle=$ $\displaystyle\theta(a(h_{(1)}\cdot
b)\omega(h_{(2)},g_{(1)}))\\#h_{(3)}g_{(2)}$ $\displaystyle=$
$\displaystyle\theta(a)(h_{(1)}\cdot\theta(b))u(h_{(2)},g_{(1)})\\#h_{(3)}g_{(2)}$
$\displaystyle=$ $\displaystyle(\theta(a)\\#h)(\theta(b)\\#g)$
$\displaystyle=$ $\displaystyle\Phi(a\otimes h)\Phi(b\otimes g).$

Next, we will verify that $\Phi$ is injective. For this purpose, take
$x=\sum_{i=1}^{n}a_{i}\otimes h_{i}\in ker\Phi$ and choose
$\\{a_{i}\\}^{n}_{i=1}$ to be linearly independent. Since $\theta$ is
injective, we conclude that $\theta(a_{i})$ are linearly independent. For each
$f\in H^{\ast}$, $\sum_{i=1}^{n}\theta(a_{i})f(h_{i})=0,$ it follows that
$f(h_{i})=0,$ so $h_{i}=0$. Therefore we have $x=0$ and $\Phi$ is injective,
as desired.

Since the partial crossed product $A\\#_{\alpha,\omega}H$ is a subalgebra of
$A\otimes H$, it is injectively mapped into $B\\#_{u}H$ by $\Phi.$ A typical
element of the image of the partial crossed product is

$\displaystyle\Phi((a\otimes h)(1_{A}\otimes 1_{H}))$ $\displaystyle=$
$\displaystyle\Phi(a\otimes h)\Phi(1_{A}\otimes 1_{H})$ $\displaystyle=$
$\displaystyle(\theta(a)\\#h)(\theta(1_{A})\\#1_{H})$ $\displaystyle=$
$\displaystyle\theta(a(h_{(1)}\cdot\theta(1_{A})))\\#h_{(2)}.$

And this completes the proof. $\hfill\Box$

Take $M=\Phi(A\otimes H)=\\{\sum_{i=1}^{n}\theta(a_{i})\\#h_{i};a_{i}\in A\\}$
and take $N$ as the subspace of $B\\#_{u}H$ generated by the elements
$h_{(1)}\cdot\theta(a)\otimes h_{(2)}$ with $h\in H$ and $a\in A.$

Proposition 4.2. Let $(\alpha,\omega)$ be a twisted partial action and
$(B,\theta)$ an enveloping action. Suppose that $\theta(A)$ is an ideal of
$B$, then $M$ is a right $B\\#_{u}H$ module and $N$ is a left $B\\#_{u}H$
module.

Proof. In order to prove $M$ is a right $B\\#_{u}H$ module, let
$\theta(a)\\#h\in M$ and $b\\#k\in B\\#_{u}H$, then

$(\theta(a)\\#h)(b\\#k)=\theta(a)(h_{(1)}\triangleright
b)u(h_{(2)},k_{(1)})\\#h_{(3)}k_{(2)}.$

Which lies in $\Phi(A\otimes H)$ because $\theta(A)$ is an ideal in $B$.

Now we show that $N$ is a left $B\\#H$ module is similar to [2]. $\hfill\Box$

Now, the left $A\\#_{\alpha,\omega}H$ module structure on $M$ and a right
$A\\#_{\alpha,\omega}H$ module structure on $N$ induced by the monomorphism
$\Phi$ is in [2].

Proposition 4.3. Under the same hypotheses of Proposition 4.2, $M$ is indeed a
left $A\\#_{\alpha,\omega}H$ module with the map $\blacktriangleright$ and N
is a right $A\\#_{\alpha,\omega}H$ module with the map $\blacktriangleleft$.

Proof. We first claim that $A\\#_{\alpha,\omega}H\blacktriangleright
M\subseteq M.$ In fact,

$\displaystyle(a(h_{(1)}\cdot 1_{A})\otimes
h_{(2)})\blacktriangleright(\theta(b))\\#k)$ $\displaystyle=$
$\displaystyle(\theta(a)(h_{(1)}\cdot\theta(1_{A}))\\#h_{(2)})(\theta(b))\\#k)$
$\displaystyle=$
$\displaystyle\theta(a)(h_{(1)}\cdot\theta(1_{A}))(h_{(2)}\triangleright\theta(b))u(h_{(3)},k_{(1)})\\#h_{(4)}k_{(2)}$
$\displaystyle=$
$\displaystyle\theta(a)(h_{(1)}\cdot\theta(1_{A}))(h_{(2)}\cdot\theta(b))u(h_{(3)},k_{(1)})\\#h_{(4)}k_{(2)}$
$\displaystyle=$
$\displaystyle\theta(a)(h_{(1)}\cdot\theta(b))u(h_{(2)},k_{(1)})\\#h_{(3)}k_{(2)}.$

Which lies inside $M$ because $\theta(A)$ is an ideal of $B$. $\hfill\Box$

Next, we verify that $N\blacktriangleleft A\\#_{\alpha,\omega}H\subseteq N$,
which is similarly to $N$ is a left $B\\#_{u}H$ module. which holds because
$\theta(1_{A})$ is a central idempotent.

The last ingredient for a Morita context is to define two bimodule morphisms

$\sigma:N\otimes_{A\\#_{\alpha,\omega}H}M\rightarrow B\\#_{u}H~{}~{}\mbox{and
}$ $\tau:M\otimes_{B\\#_{u}H}N\rightarrow
A\\#_{\alpha,\omega}H\cong\Phi(A\\#_{\alpha,\omega}H).$

As $M,N$ and $A\\#_{\alpha,\omega}H$ are viewed as subalgebras of $B\\#H,$
these two maps can be taken as the usual multiplication on $B\\#_{u}H.$ The
associativity of the product assures us that these maps are bimodule morphisms
and satisfy the associativity conditions.

Proposition 4.4. The partial crossed product $A\\#_{\alpha,\omega}H$ is Morita
equivalent to the crossed product $B\\#_{u}H$.

##  5 Equivalences of partial crossed products

In this section, we shall discuss equivalences of partial crossed products and
this result is similar to [4]. From now on, unless explicitly stated, we
always assume partial actions of a Hopf algebra $H$ over an algebra $A$ such
that the map $e\in$ Hom$(H,A)$ , given by $e(h)=h\cdot 1_{A}$, is central with
respect to the convolution product. Given a twisted partial H-module algebra
$A=(A,\alpha,\omega)$, we define two linear maps $f_{1},f_{2}:H\otimes
H\rightarrow A$ as follows:

$\displaystyle f_{1}(h,k)=(h\cdot
1_{A})\varepsilon(k),~{}~{}~{}~{}~{}f_{2}(h,k)=hk\cdot 1_{A}.$

Definition 5.1.${}^{\cite[cite]{[\@@bibref{}{A14}{}{}]}}$ Let
$A=(A,\alpha,\omega)$ be a twisted partial $H$-module algebra. We will say
that the partial action is symmetric, if
(1) $f_{1},f_{2}$ are central in Hom ($H\otimes H,A$),
(2) $\omega$ satisfies the conditions Eq.(3.7) and Eq.(3.8) and has a
convolution inverse $\omega^{\prime}$ in the ideal $<f_{1}\ast f_{2}>\in$ Hom
($H\otimes H,A$),
(3) $h\cdot(k\cdot 1_{A})=(h_{(1)}\cdot 1_{A})(h_{(2)}k\cdot 1_{A})$ for any
$h,k\in H$.

Definition 5.2. If there exists linear maps $u,v\in$ Hom$(H,A)$ satisfy
$(u\ast v)(h)=(v\ast u)(h)=h\cdot 1_{A}$, $u(h)=u(h_{(1)})(h_{(2)}\cdot
1_{A})=(h_{(1)}\cdot 1_{A})u(h_{(2)})$ and $u(1_{H})=v(1_{H})=1_{A}$. Then we
call $v$ is a weak convolution-invertible linear map and $u=v^{-1}$.

Let $H$ be a Hopf algebra and $(A,\alpha,\omega)$ a symmetric twisted partial
$H$-module algebra, and $v\in$ Hom$(H,A)$ a weak convolution-invertible linear
map. Define $\omega^{v}:H\otimes H\rightarrow A$ and weakly action of $H$ on
$A$ by

$\omega^{v}(h,g)=v(h_{(1)})(h_{(2)}\cdot
v(g_{(1)}))\omega(h_{(3)},g_{(2)})v^{-1}(h_{(4)}g_{(3)})$

and

$h\cdot^{v}a=v(h_{(1)})(h_{(2)}\cdot a)v^{-1}(h_{(3)})$

for any $h,g\in H$ and $a\in A$.

Lemma 5.3. With the notations as above, let $H$ be a Hopf algebra and
$(A,\alpha,\omega)$ a symmetric twisted partial $H$-module algebra. Then
$\omega^{vu}=(\omega^{u})^{v}$ and $\cdot^{vu}=(\cdot^{u})^{v}$, where
$u,v\in$ Hom$(H,A)$ are weak convolution-invertible linear maps.

Proof. For any $h,g\in H,a\in A$, we have we have

$\displaystyle\omega^{vu}(h,g)$ $\displaystyle=$
$\displaystyle(vu(h_{(1)})(h_{(2)}\cdot(vu(g_{(1)})))\omega(h_{(3)},g_{(2)})(vu)^{-1}(h_{(4)}g_{(3)})$
$\displaystyle=$ $\displaystyle v(h_{(1)})u(h_{(2)})(h_{(3)}\cdot
v(g_{(1)}))(h_{(4)}\cdot
u(g_{(2)})\omega(h_{(5)},g_{(3)})u^{-1}(h_{(6)}g_{(4)})v^{-1}(h_{(7)}g_{(5)})$
$\displaystyle=$ $\displaystyle v(h_{(1)})u(h_{(2)})(h_{(3,1)}\cdot
v(g_{(1)}))u^{-1}(h_{(4)})u(h_{(5)})(h_{(6)}\cdot u(g_{(2)}))$
$\displaystyle\omega(h_{(7)},g_{(3)})u^{-1}(h_{(8)}g_{(4)})v^{-1}(h_{(9)}g_{(5)})$
$\displaystyle=$ $\displaystyle
v(h_{(1)})(h_{(2)}\cdot^{u}v(g_{(1)}))\omega^{u}(h_{(3)},g_{(2)})v^{-1}(h_{(4)}g_{(3)})$
$\displaystyle=$ $\displaystyle(\chi^{u})^{v}(h,g)$

and thus $\chi^{vu}=(\chi^{u})^{v}$.

Also,

$\displaystyle h(\cdot^{u})^{v}a$ $\displaystyle=$ $\displaystyle
v(h_{(1)})(h_{(2)}\cdot^{u}a)v^{-1}(h_{(3)})$ $\displaystyle=$ $\displaystyle
v(h_{(1)})u(h_{(2)})(h_{(3)}\cdot a)u^{-1}(h_{(4)})v^{-1}(h_{(5)})$
$\displaystyle=$ $\displaystyle v(h_{(1)})u(h_{(2)}))(h_{(3)}\cdot
a)u^{-1}(h_{(4)})v^{-1}(h_{(5)})$ $\displaystyle=$ $\displaystyle
h\cdot^{vu}a$

and so $\cdot^{vu}=(\cdot^{u})^{v}$

This completes the proof. $\hfill\Box$

Theorem 5.4. Let $H$ be a Hopf algebra and $(A,\alpha,\omega)$ a symmetric
twisted partial $H$-module algebra, and $v\in$ Hom$(H,A)$ a convolution-
invertible linear map, with the above notations $\omega^{v},\cdot^{v}$. Then
we have the following assertions:

(1) As algebras, $A\\#_{\alpha,\omega}H\cong A\\#_{\alpha,\omega^{v}}H$;

(2) $\omega$ satisfies Eq.(3.6) if and only if $\omega^{v}$ satisfies
Eq.(3.6);

(3) $(\omega,\cdot)$ satisfies Eq.(3.7) if and only if
$(\omega^{v},\cdot^{v})$ satisfies Eq.(3.7);

(4) If $(\omega,\cdot)$ satisfies Eq.(3.7), then $(\omega,\cdot)$ satisfies
Eq.(3.8) if and only if $(\omega^{v},\cdot^{v})$ satisfies Eq.(3.8);

(5) $A\\#_{\alpha,\omega}H$ is a partial crossed product algebra if and only
if $A\\#_{\alpha,\omega^{v}}H$ is a partial crossed product algebra , and they
are isomorphic.

Proof. (1) Define $\Phi:A\\#_{\alpha,\omega}H\mapsto
A\\#_{\alpha,\omega^{v}}H$ by $a\\#h\rightarrow av(h_{(1)})\\#h_{(2)}$, for
any $h,g\in H$, we have

$\displaystyle\Phi((a\\#h)(b\\#g))$ $\displaystyle=$
$\displaystyle\Phi(a(h_{(1)}\cdot^{v}b)\omega^{v}(h_{(2)},g_{(1)})\\#h_{(3)}g_{(2)})$
$\displaystyle=$ $\displaystyle
a(h_{(1)}\cdot^{v}b)\omega^{v}(h_{(2)},g_{(1)})v(h_{(3)}g_{(2)})\\#h_{(4)}g_{(3)})$
$\displaystyle=$ $\displaystyle av(h_{(1)})(h_{(2)}\cdot
b)v^{-1}h_{(3)}v(h_{(4)})h_{(5)}\cdot\omega(h_{(6)},g_{(2)})v^{-1}(h_{(7)}g_{(3)})v(h_{(8)}g_{(4)})\\#h_{(9)}g_{(5)}$
$\displaystyle=$ $\displaystyle av(h_{(1)})(h_{(2)}\cdot b)(h_{(3)}\cdot
v(g_{(1)})\omega(h_{(4)},g_{(2)})\\#h_{(5)}g_{(3)}$ $\displaystyle=$
$\displaystyle\Phi(a\\#h)\Phi(b\\#g).$

Clearly $\Phi$ is bijective, $\Phi^{-1}(a\\#h)=\sum
av^{-1}(h_{(1)})\\#h_{(2)}$ $a,b\in A,h\in H$, since

$\displaystyle\Phi\Psi(a\\#h)$ $\displaystyle=$
$\displaystyle\Phi(av^{-1}(h_{(1)})\\#h_{(2)})$ $\displaystyle=$
$\displaystyle av^{-1}(h_{(1)})v(h_{(2)})\\#h_{(3)}$ $\displaystyle=$
$\displaystyle av^{-1}(h_{(1)})v(h_{(2)})\\#h_{(3)}$ $\displaystyle=$
$\displaystyle a\\#h.$

(2) Straightforward.

(3) If $(\omega,\cdot)$ satisfies Eq.(3.7), then

$\displaystyle(h_{(1)}\cdot^{v}(g_{(1)}\cdot^{v}(a)))\omega^{v}(h_{(2)},g_{(2)})$
$\displaystyle=$ $\displaystyle
v(h_{(1)})(h_{(2)}\cdot(v(g_{(1)}))(g_{(2)}\cdot
a)v^{-1}(g_{(3)})v^{-1}(h_{(3)})v(h_{(4)})(h_{(5)}\cdot v(g_{(4)})$
$\displaystyle\hskip 199.16928pt\omega(h_{(6)},g_{(5)})v^{-1}(h_{(7)}g_{(6)})$
$\displaystyle=$ $\displaystyle v(h_{(1)})(h_{(2)}\cdot
v(g_{(1)}))(h_{(3)}\cdot(g_{(2)}\cdot
a))\omega(h_{(4)},g_{(3)})v^{-1}(h_{(5)}g_{(4)})$ $\displaystyle=$
$\displaystyle v(h_{(1)})(h_{(2)}\cdot
v(g_{(1)}))\omega(h_{(3)},g_{(2)})(h_{(4)}g_{(3)}\cdot
a)v^{-1}(h_{(5)}g_{(4)})$ $\displaystyle=$
$\displaystyle\omega^{v}(h_{(1)},g_{(1)})(h_{(2)}g_{(2)}\cdot^{v}a).$

Conversely, we get it from Lemma 5.3.

(4) If $(\omega,\cdot)$ satisfies Eq.(3.7)and Eq.(3.8), then for $h,g,m\in H$,
we have

$\displaystyle(h_{(1)}\cdot^{v}\omega^{v}(g_{(1)},m_{(1)}))\omega^{v}(h_{(2)},g_{(2)}m_{(2)})$
$\displaystyle=$ $\displaystyle
v(h_{(1)})(h_{(2)}\cdot[(v(g_{(1)})(g_{(2)}\cdot^{v}(m_{(1)})))\omega(g_{(3)},m_{(2)})v^{-1}(g_{(4)}m_{(3)})])$
$\displaystyle v^{-1}(h_{(3)})v(h_{(4)})(h_{(5)}\cdot
v(g_{(5)}m_{(4)}))\omega(h_{(6)},g_{(6)}m_{(5)})v^{-1}(h_{(7)}g_{(7)}m_{(6)}$
$\displaystyle=$ $\displaystyle v(h_{(1)})(h_{(2)}\cdot
v(g_{(1)}))(h_{(3)}\cdot(g_{(2)}\cdot v(m_{(1)})))$
$\displaystyle(h_{(4)}\cdot\omega(g_{(3)}m_{(2)}))\omega(h_{(5)},g_{(4)}m_{(3,1)})v^{-1}(h_{(6)}g_{(5)}m_{(4)})$
$\displaystyle=$ $\displaystyle v(h_{(1)})(h_{(2)}\cdot
v(g_{(1)}))(h_{(3)}\cdot(g_{(2)}\cdot v(m_{(1)})))$
$\displaystyle\omega(h_{(4)},g_{(3)})\chi(h_{(5,1)},g_{(4,1)}m_{(2,1)})v^{-1}(h_{(6)}g_{(5)}m_{(4)})$
$\displaystyle=$ $\displaystyle v(h_{(1,\alpha)})(h_{(2)}\cdot
v(g_{(1)}))\omega(h_{(3)}g_{(2)})(h_{(4)}g_{(3)}\cdot v(m_{(1)}))$
$\displaystyle\omega(h_{(5)}g_{(4)},m_{(2)})v^{-1}(h_{(6)}g_{(5)}m_{(4)})$
$\displaystyle=$
$\displaystyle\omega^{v}(h_{(1)},g_{(1)})\omega(h_{(2)}g_{(2)},m).$

Conversely, we get it from Lemma 5.3.

(5) Clearly. $\hfill\Box$

##  6 Separable extension for partial crossed products

In this section, we will investigate when $A\subset A\\#_{\alpha,\omega}H$ is
a separable extension.

Definition 6.1.${}^{\cite[cite]{[\@@bibref{}{A14}{}{}]}}$ Let $B$ be a right
$H$-comodule unital algebra with coaction given by $\rho:B\rightarrow B\otimes
H$ and let $A$ be a subalgebra of $B$. We will say that $A\subset B$ is an
$H$-extension if $A=B^{coH}$. An $H$-extension $A\subset B$ is partially cleft
if there is a pair of $k$-linear maps $\gamma,\gamma^{\prime}:H\rightarrow B$
such that
(i) $\gamma(1_{H})=1_{B},$
(ii)$\rho\circ\gamma=(\gamma\otimes id_{H})\Delta$ and
$\rho\circ\gamma^{\prime}=(\gamma^{\prime}\otimes S)\Delta^{cop}$,
(iii)$(\gamma\ast\gamma^{\prime})\circ M$ is a central element in the
convolution algebra Hom$(H\otimes H,A)$, where $M:H\otimes H\rightarrow H$ is
the multiplication in $H$, and $(\gamma\ast\gamma^{\prime})(h)$ commutes with
every element of $A$ for each $h\in H$.

Lemma 6.2 Let $c\in C_{A\\#_{\alpha,\omega}H}(A)$, the centralizer of $A$ in
$A\\#_{\alpha,\omega}H$, and assume that $H$ is cocommutative. Then for $h\in
H$,

$\displaystyle\gamma^{\prime}(h_{(1)})c(S(h_{(2)})\cdot 1_{A})\gamma(h_{(3)})$
$\displaystyle=$ $\displaystyle\gamma(S(h_{(2)}))c\gamma^{\prime}(S(h_{(1)}))$
$\displaystyle=$ $\displaystyle S(h)\cdot c.$

Proof. For any $h\in H$, we have

$\displaystyle\gamma^{\prime}(h_{(1)})c(S(h_{(2)})\cdot 1_{A})\gamma(h_{(3)})$
$\displaystyle=$
$\displaystyle\gamma^{\prime}(h_{(1)})c\gamma(h_{(4)})\gamma(S(h_{(3)}))\gamma^{\prime}(S(h_{(2)}))$
$\displaystyle=$
$\displaystyle\gamma^{\prime}(h_{(1)})c\gamma(h_{(2)})\gamma(S(h_{(3)}))\gamma^{\prime}(S(h_{(4)}))$
$\displaystyle=$
$\displaystyle\gamma^{\prime}(h_{(1)})\gamma(h_{(2)})\gamma(S(h_{(3)}))c\gamma^{\prime}(S(h_{(4)}))$
$\displaystyle=$ $\displaystyle(S(h_{(1)})\cdot
1_{A})\gamma(S(h_{(2)}))c\gamma^{\prime}(S(h_{(3)}))$ $\displaystyle=$
$\displaystyle(S(h_{(1)})\cdot 1_{A})(S(h_{(2)})\cdot c)$ $\displaystyle=$
$\displaystyle S(h)\cdot c$

Theorem 6.2. Let $H$ be a finite dimensional cocommutative Hopf algebra and
$(A,\cdot,(\omega,\omega^{\prime}))$ is a symmetric partial twisted $H$-module
algebra, and $t\neq 0$ be a left integral in $H$. Assume that there exists $c$
in the center of $A$ such that $t\cdot c=1_{A}$. Then $A\subset
A\\#_{\alpha,\omega}H$ is a separable extension.

Proof.  In order to show that $A\subset A\\#_{\alpha,\omega}H$ is separable,
it suffices to find a separability idempotent
$e\in(A\\#_{\alpha,\omega}H)\otimes_{A}(A\\#_{\alpha,\omega}H)$; this means
that
(1)
$((a\\#_{\alpha,\omega}h)\otimes_{A}(1\\#_{\alpha,\omega}1))e=e((1\\#_{\alpha,\omega}1)\otimes_{A}(a\\#_{\alpha,\omega}h))$
for all $a\\#_{\alpha,\omega}h\in A\\#_{\alpha,\omega}H$ and
(2) $m_{A\\#_{\alpha,\omega}H}(e)=1\\#_{\alpha,\omega}1$, where
$m_{A\\#_{\alpha,\omega}H}$ denotes multiplication
$(A\\#_{\alpha,\omega}H)\otimes(A\\#_{\alpha,\omega}H)\rightarrow(A\\#_{\alpha,\omega}H)$.

Now since $t$ is a left integral in $H$, $u=S(t)$ is a right integral in $H$.
We claim that
$f=\gamma^{\prime}(u_{(1)})\otimes_{A}\gamma(u_{(2)})\in(A\\#_{\alpha,\omega}H)\otimes_{A}(A\\#_{\alpha,\omega}H)$
satisfies condition (1) above. By [7], one can get the canonical map can:

$\displaystyle
A\\#_{\alpha,\omega}H)\otimes_{A}(A\\#_{\alpha,\omega}H)\rightarrow(A\\#_{\alpha,\omega}H)\otimes
H$
$\displaystyle(a\\#_{\alpha,\omega}h)(b\\#_{\alpha,\omega}g)\mapsto(a\\#_{\alpha,\omega}h)(b\\#_{\alpha,\omega}g_{(1)})\otimes
g_{(2)}$

is a bijection, since by [4] $A\subset A\\#_{\alpha,\omega}H$ is a partially
cleft extension, we can check $A\subset A\\#_{\alpha,\omega}H$ also is partial
$H$-Hopf Galois, we claim that both
$((a\\#_{\alpha,\omega}h)\otimes_{A}(1\\#_{\alpha,\omega}1))f$ and
$f((1\\#_{\alpha,\omega}1)\otimes_{A}(a\\#_{\alpha,\omega}h))$ go to
$(a\\#_{\alpha,\omega}h)(S(u_{(1)})\cdot 1_{A})\otimes u_{(2)}$. As a matter
of fact,

$\displaystyle
can(((a\\#_{\alpha,\omega}h)\otimes_{A}(1\\#_{\alpha,\omega}1))f)$
$\displaystyle=$ $\displaystyle
can((a\\#_{\alpha,\omega}h)\gamma^{\prime}(u_{(1)})\otimes_{A}\gamma(u_{(2)}))$
$\displaystyle=$
$\displaystyle(a\\#_{\alpha,\omega}h)\gamma^{\prime}(u_{(1)})\gamma(u_{(2)})\otimes
u_{(3)}$ $\displaystyle=$
$\displaystyle(a\\#_{\alpha,\omega}h)(S(u_{(1)})\cdot 1_{A})\otimes u_{(2)},$

and

$\displaystyle
can((f(a\\#_{\alpha,\omega}h)\otimes_{A}(1\\#_{\alpha,\omega}1)))$
$\displaystyle=$ $\displaystyle
can(\gamma^{\prime}(u_{(1)})\otimes_{A}\gamma(u_{(2)})(a\\#_{\alpha,\omega}h))$
$\displaystyle=$ $\displaystyle
can(\gamma^{\prime}(u_{(1)})\otimes_{A}(u_{(2)}\cdot
a)\omega(u_{(3)},h_{(1)})\gamma(u_{(4)}h_{(2)}))$ $\displaystyle=$
$\displaystyle\gamma^{\prime}(u_{(1)})(u_{(2)}\cdot
a)\omega(u_{(3)},h_{(1)})\gamma(u_{(4)}h_{(2)})\otimes u_{(5)}h_{(3)}$
$\displaystyle=$
$\displaystyle\gamma^{\prime}(u_{(1)})\gamma(u_{(2)})a\gamma^{\prime}(u_{(3)})\omega(u_{(4)},h_{(1)})\gamma(u_{(5)}h_{(2)})\otimes
u_{(6)}h_{(3)}$ $\displaystyle=$ $\displaystyle(S(u_{(1)})\cdot
1_{A})a\gamma^{\prime}(u_{(2)})\omega(u_{(3)},h_{(1)})\gamma(u_{(4)}h_{(2)})\otimes
u_{(5)}h_{(3)}$ $\displaystyle=$ $\displaystyle(S(u_{(1)})\cdot
1_{A})a\gamma(h_{(1)})\gamma^{\prime}(u_{(2)}h_{(2)})\gamma(u_{(3)}h_{(3)})\otimes
u_{(4)}h_{(4)}$ $\displaystyle=$ $\displaystyle(S(u_{(1)})\cdot
1_{A})a\gamma(h_{(1)})\gamma^{\prime}(u_{(2)})\gamma(u_{(3)})\otimes u_{(4)}$
$\displaystyle=$
$\displaystyle(a\\#_{\alpha,\omega}h)(a\\#_{\alpha,\omega}h)(S(u_{(1)})\cdot
1_{A})\otimes u_{(2)}.$

Now since $c$ is in the center of $A$, the map

$\displaystyle(A\\#_{\alpha,\omega}H)\otimes_{A}(A\\#_{\alpha,\omega}H)\rightarrow(A\\#_{\alpha,\omega}H)\otimes_{A}(A\\#_{\alpha,\omega}H)$
$\displaystyle(a\\#_{\alpha,\omega}h)\otimes_{A}(b\\#_{\alpha,\omega}g)\mapsto(a\\#_{\alpha,\omega}h)c\otimes_{A}(b\\#_{\alpha,\omega}g)$

is well-defined. Thus $e=\gamma^{\prime}(u_{(1)})c(S(u_{(2)})\cdot
1_{A})\otimes_{A}\gamma(u_{(3)})$ also satisfies (1).

We also claim that $m(e)=\gamma^{\prime}(u_{(1)})c(S(u_{(2)})\cdot
1_{A})\gamma(u_{(3)})=1_{A}$. For, by Lemma 6.1,

$\displaystyle\gamma^{\prime}(u_{(1)})c(S(u_{(2)})\cdot
1_{A})\gamma(u_{(3)})=S(u)\cdot c=S^{2}(t)\cdot c=t\cdot c=1_{A}.$

It follows that $A\subset A\\#_{\alpha,\omega}H$ is a separable extension.

## Acknowledgement

The work was supported by the TianYuan Special Funds of the National Natural
Science Foundation of China (11426073), the NSF of Jiangsu Province
(BK2012736) and the Fund of Science and Technology Department of Guizhou
Province (2014GZ81365).

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