1. Introduction
Generalized matrix algebras were first introduced in the literature by Brown in [Reference Brown9], where the author explored matrix rings associated with orthogonal groups. These matrix rings form a large class of rings that arise in various areas of algebra and module theory and have been extensively studied. Broadly speaking, a generalized matrix algebra is an algebra formed as a direct sum of
$\Bbbk$
-vector spaces,
\begin{align*} R = \bigoplus \limits _{i,j=1}^n {}_i M_j, \end{align*}
where the diagonal components
$R_i = {}_i M_i$
are
$\Bbbk$
-algebras, and the off-diagonal components
${}_i M_j$
are
$(R_i, R_j)$
-bimodules. The algebra structure of
$R$
emulates the usual structure of a matrix algebra by employing a series of bimodule morphisms to define the product between entries residing solely in the bimodules.
Notably, the class of
$2 \times 2$
generalized matrix algebras has been shown to correspond one-to-one with Morita contexts [Reference Morita15]. The study of the so-called Morita ring has gained significant attention in recent years, as it encapsulates key algebraic properties of the Morita context itself [Reference Haghany and Varadarajan13, Reference Krylov and Tuganbaev14, Reference Tang, Li and Zhou16]. Moreover, Morita rings were used to define the concept of Morita equivalence of partial actions initially for groups and later for Hopf algebras [Reference Abadie, Dokuchaev, Exel and Simón1, Reference Alves and Ferraza5]. Another context in which generalized matrix algebras arise is in the study of idempotents. Specifically, any algebra
$A$
equipped with a complete set of orthogonal idempotents
$\{e_i\}_{i=1}^n$
can be decomposed using Peirce decomposition. This decomposition can be organized into an
$n \times n$
generalized matrix algebra
$A^\pi$
, which is isomorphic to
$A$
. For further details, see, for instance, [Reference nh, Birkenmeyer and van Wik6].
Partial actions of Hopf algebras were introduced in [Reference Caenepeel and Janssen10], motivated by the Galois theory of partial group actions studied in [Reference Dokuchaev, Ferrero and Paques12]. Since then, numerous works on partial actions of Hopf algebras have been developed, exploring several problems that were initially addressed from a global perspective through a partial framework. For a detailed account of this topic, the interested reader is referred to [Reference Batista8].
The main objective of this work is to establish a connection between two significant topics: partial actions of Hopf algebras and generalized matrix algebras. Specifically, we aim to determine the necessary and sufficient conditions for a Hopf algebra to act partially on a generalized matrix algebra. The analogous problem was considered in [Reference Bagio and Pinedo7] for the case of partial group actions.
This paper is structured as follows. In Section 2, we review the fundamental concepts of generalized matrix algebras, partial actions, partial representations of Hopf algebras, and partial smash products. We also introduce the notion of an opposite covariant pair and demonstrate that it satisfies a universal property. In Section 3, we prove Theorem 3.3, which is the main result of this work. In this theorem, we provide necessary and sufficient conditions for the existence of a partial action
$\rhd$
of a Hopf algebra
$H$
on a generalized matrix algebra such that each
${}_i M_j$
is invariant under the action
$\rhd$
. Conditions for the existence of a partial action of a group
$G$
on
$R$
were previously discussed in [Reference Bagio and Pinedo7]. The relationship between these results and those of Theorem 3.4 is analyzed in Section 4, focusing on the specific case where the Hopf algebra is the group algebra
$\Bbbk G$
, where
$\Bbbk$
is a field. Finally, in Section 5, we discuss Morita-equivalent partial actions of a Hopf algebra. Additionally, we address an inconsistency in Proposition 73 of [Reference Alves and Ferraza5].
Conventions
Throughout this work,
$\Bbbk$
is a field, by an algebra we mean an associative unital
$\Bbbk$
-algebra, and vector spaces and tensor product are over
$\Bbbk$
. Also, all modules will be unital modules. Given an algebra
$A$
, the opposite algebra
$A^{\textrm {op}}$
is the vector space
$A$
with the opposite multiplication, that is
$a\cdot _{\textrm {op}} b=ba$
, for all
$a,b\in A$
. Given a Hopf algebra
$H$
and
$h\in H$
, we use the simplified Sweedler’s notation
$\Delta (h)=h_{(1)}\otimes h_{(2)}$
to denote the comultiplication of
$h$
. We will assume that every Hopf algebra
$H$
has a bijective antipode. For a coalgebra
$C$
, the coopposite coalgebra
$C^{\textrm {cop}}$
is the vector space
$C$
with comultiplication
$\Delta _{\textrm { cop}}(c)=c_{(2)}\otimes c_{(1)}$
, where
$\Delta (c)=c_{(1)}\otimes c_{(2)}$
. In order to avoid confusion, we will denote
$\Delta _{\textrm {cop}}(c)=c_{[1]}\otimes c_{[2]}$
, for all
$c\in C^{\textrm {cop}}$
. Finally, the set
$\{1,\ldots ,n\}$
will be denoted by
$\mathbb{I}_n.$
2. Preliminaries
In this section, we review definitions and results that will be used in the rest of the work. We introduce the concept of opposite covariant pair and prove that it satisfies a certain universal property.
2.1. Generalized matrix algebras
We begin with the following definition, which will allow us to introduce one of the key concepts of this work, namely, the notion of a generalized matrix ring.
Definition 2.1.
Let
$n$
be a positive integer. A generalized matrix datum of order
$n$
over
$\Bbbk$
is a triple
\begin{align*} \mathcal{R}= \left ( \{ {}_i M_j \}_{i,j\in \mathbb{I}_n} , \{ \theta _{ijk} \}_{ i,j,k \in \mathbb{I}_n} ,\{ \eta _i \}_{i \in \mathbb{I}_n} \right ) , \end{align*}
such that, for every
$i,j,k\in \mathbb{I}_n$
,
-
(GMD1)
${}_i M_j$
is a
$\Bbbk$
-vector space,
-
(GMD2)
$\theta _{ijk} \, : \, {}_i M_j \otimes {}_j M_k \rightarrow {}_i M_k$
and
$\eta _i \, : \, \Bbbk \rightarrow {}_i M_i$
are
$\Bbbk$
-linear maps,
-
(GMD3)
$\theta _{ijk}$
and
$\eta _i$
satisfy the following compatibility identities
\begin{align*} \theta _{ikl} \circ \left ( \theta _{ijk} \otimes \textrm {id}_{{}_kM_l} \right ) &=\theta _{ijl} \circ \left ( \textrm {id}_{{}_iM_j} \otimes \theta _{jkl} \right ),\\[4pt] \theta _{iij} \circ (\eta _i \otimes \textrm {id}_{{}_i M_j}) =l_{{}_i M_j},\quad &\quad \theta _{ijj} \circ (\textrm {id}_{{}_i M_j} \otimes \eta _j )=r_{{}_i M_j}, \end{align*}
where
$l_{{}_i M_j}$
(resp.
$r_{{}_i M_j}$
) is the natural
$\Bbbk$
-linear isomorphism from
$\Bbbk \otimes {}_i M_j$
(resp.
${}_i M_j\otimes \Bbbk$
) to
${}_i M_j$
.
The identities of (GMD3) can be interpreted by requiring that the following diagrams be commutative:
and
Proposition 2.4.
Let
$R$
be a generalized matrix datum of order
$n$
over
$\Bbbk$
. Then, for every
$i,j,k\in I_n$
, we have that:
-
(i) the vector spaces
${}_i M_i$
are unital
$\Bbbk$
-algebras, henceforth denoted as
$R_i$
, -
(ii) the vector spaces
${}_i M_j$
are unital
$(R_i , R_j)$
-bimodules,
-
(iii) the linear maps
$\theta _{ijk}$
are morphisms of
$(R_i ,R_k)$
-bimodules and are balanced over
$R_j$
, that is,
$\theta _{ijk}$
determines a well-defined mapping from
${}_i M_j \otimes _{R_j} {}_j M_k$
to
${}_i M_k$
.
Proof.
For (i), notice that
$R_i \, :\!= \, {}_i M_i$
is an algebra with multiplication and unity given, respectively, by
$\theta _{iii}$
and
$\eta _i$
. Also,
${}_i M_j$
has structure of left (resp. right)
$R_i$
-module (resp.
$R_j$
-module) given by
$\theta _{iij}$
(resp.
$\theta _{ijj}$
). Substituting
$j$
by
$i$
,
$k$
by
$j$
and
$l$
by
$j$
in (2.2), we obtain that
${}_i M_j$
is a
$(R_i , R_j)$
-bimodule. From (2.3) follows that
${}_i M_j$
is unital and (ii) is proved. The statement (iii) follows from (2.2) setting
$k=j$
and
$l=k$
.
From the previous proof, we have that for all
$i,j\in I_n$
,
-
⋄
$\,\,\theta _{iii}:\!_i{M}_{i}\otimes _{R_i}\,\!\!_{i}M_{i}\to \!_{i}M_{i}$
is simply the multiplication in
$R_i$
, -
⋄
$\,\,\theta _{iij}:\!_i{M}_{i}\otimes _{R_i}\,\!\!_{i}M_{j}\to \!_{i}M_{j}$
is the left
$R_i$
-module structure on
${}_iM_j$
, -
⋄
$\,\,\theta _{ijj}:\!_i{M}_{j}\otimes _{R_j}\,\!\!_{j}M_{j}\to \!_{i}M_{j}$
is the right
$R_j$
-module structure on
${}_iM_j$
.
Given a generalized matrix datum, one can associate to the vector space
\begin{align*} R=\bigoplus _{i,j=1}^n {}_iM_j \end{align*}
an algebra structure as follows. For all
$i,j,k,l \in \mathbb{I}_n$
, define the linear map
$\mu _{ij,kl} \, : \, {}_i M_j \otimes {}_k M_l \rightarrow {}_i M_l$
by
\begin{align*} \mu _{ij,kl}=\left \{ \begin{array}{ll} \theta _{ijl}, & \text{ if } j=k ,\\[3pt] 0 ,& \text{ otherwise}.\end{array}\right . \end{align*}
Hence, we have linear maps
$\iota _{il}\circ \mu _{ij,kl} \, : \, {}_i M_j \otimes {}_k M_l \to R$
, where
$\imath _{il} \, : \, {}_iM_l \rightarrow R$
is the inclusion map. Using the universal property of the direct sum
\begin{align*} R\otimes R \cong \bigoplus _{i,j,k,l =1}^n {}_iM_j \otimes {}_kM_l, \end{align*}
we obtain a linear map
$\mu \, : \, R\otimes R \rightarrow R$
. We also define the linear map
$\eta \, : \, \Bbbk \rightarrow R$
by
\begin{align*} \eta =\sum _{i=1}^n \imath _{ii} \circ \eta _i. \end{align*}
Proposition 2.5. Given a generalized matrix datum
\begin{align*} \mathcal{R}= \left ( \{ {}_i M_j \}_{i,j\in \mathbb{I}_n} , \{ \theta _{ijk} \}_{ i,j,k \in \mathbb{I}_n} ,\{ \eta _i \}_{i \in \mathbb{I}_n} \right ) , \end{align*}
the triple
$(R, \mu , \eta )$
defines a unital algebra over the field
$\Bbbk$
.
Proof.
One can write a generalized matrix algebra of order
$n$
over
$\Bbbk$
as a square array
\begin{align} R = \left ( \begin{matrix} R_1 &\quad _{1}M_{2} &\quad \ldots &\quad _{1}M_{n} \\ _{2}M_{1} &\quad R_2 &\quad \ddots &\quad \vdots \\ \vdots &\quad \ddots &\quad \ddots &\quad _{(n-1)}M_{n} \\ _{n}M_{1} &\quad \ldots &\quad _{n}M_{(n-1)} &\quad R_n \end{matrix} \right ) . \end{align}
A typical element of
$R$
can be written as a matrix
$x=(x_{ij})$
. Considering the vector space structure on the direct sum
$R$
one can see that the addition is done componentwise and the multiplication
$\mu$
on
$R$
is the row-column matrix multiplication,
\begin{align*} \mu (\left ( x_{ij} \right ) \otimes \left ( y_{ij} \right )) = \left ( x_{ij} \right ) \left ( y_{ij} \right )= \left ( \sum _{k=1}^n x_{ik} y_{kj}\right ) =\left ( \sum _{k=1}^n \theta _{ikj} (x_{ik} \otimes y_{kj}) \right ) . \end{align*}
The associativity relations of the maps
$\theta _{ijk}$
given in (GMD3) imply that the matrix multiplication is associative, and the linear map
$\eta$
is indeed a unit map. Explicitly,
\begin{align*} \eta (\lambda )= \lambda \left ( \begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1_{1} & 0 & \cdots & 0 \\ 0 & 1_{2} & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1_{n} \end{array} \right ) , \quad \lambda \in \Bbbk , \end{align*}
in which
$1_i$
denotes the unit of the corresponding algebra
$R_i$
.
Definition 2.7.
The algebra constructed in Proposition 2.5 will be called the generalized matrix algebra defined by the generalized matrix datum
$\mathcal{R}$
.
In what follows, a generalized matrix algebra
$R$
as defined above will be denoted by
$R=(_{i}M_{j})_{i,j\in \mathbb{I}_n}$
. Also, a diagonal matrix in
$R$
will be denoted by
$r=\operatorname {diag}(r_{11},\ldots ,r_{nn})$
, that is,
$r=(r_{ij})$
is the matrix such that
$r_{ij}=0$
if
$i\neq j$
.
Remark 2.8.
Let
$R=(_{i}M_{j})_{i,j\in \mathbb{I}_n}$
be a generalized matrix algebra and
$k,l\in \mathbb{I}_n$
. Notice that
$R$
is a left
$R_k$
-module with the following structure: for all
$r_k\in R_k$
and
$(m_{ij})\in R$
,
\begin{align*} r_k\cdot (m_{ij})=(\tilde {m}_{ij}),\quad \text{ where }\, \tilde {m}_{ij}=\begin{cases} 0,& \text{if } i\neq k,\\ r_k\cdot m_{ij},& \text{if } i=k.\\ \end{cases} \end{align*}
Similarly,
$R$
is a right
$R_l$
-module via: for all
$r_l\in R_l$
and
$(m_{ij})\in R$
,
\begin{align*} (m_{ij})\cdot r_l=(\tilde {m}_{ij}),\quad \text{ where }\, \tilde {m}_{ij}=\begin{cases} 0,& \text{if } j\neq l,\\ m_{ij}\cdot r_l,& \text{if } j=l.\\ \end{cases} \end{align*}
Example 2.9.
Let
$\mathcal{C}$
be a
$\Bbbk$
-linear category with finitely many objects, that is, the morphism sets of
$\mathcal{C}$
are
$\Bbbk$
-vector spaces and the composition
$\text{Hom}_{\mathcal{C}}(Y, X)\times \text{Hom}_{\mathcal{C}}(X, Y)\to \text{Hom}_{\mathcal{C}}(X, Z)$
is
$\Bbbk$
-bilinear, for all objects
$X,Y,Z$
of
$\mathcal{C}$
. For each pair of objects
$X_i , X_j \in \mathcal{C}^{(0)}$
, denote
${}_i M_j \, :\!= \, \text{Hom}_{\mathcal{C}}(X_j , X_i)$
. The total space
\begin{align*} A(\mathcal{C}) =\bigoplus _{i,\,j} {}_i M_j \end{align*}
is a generalized matrix algebra with the multiplication given by the ordinary composition of morphisms in the category
$\mathcal{C}$
.
Example 2.10.
Let
$(A,B,M,N,\mu ,\nu )$
be a Morita context between the algebras
$A$
and
$B$
. Then
\begin{align*} R = \left ( \begin{matrix} A &\quad M \\ N &\quad B \end{matrix} \right ) \end{align*}
is a
$2$
-by-
$2$
generalized matrix algebra which is called a Morita ring. Conversely, each
$2$
-by-
$2$
generalized matrix algebra determines a Morita context [Reference Haghany and Varadarajan13].
Example 2.11.
Let
$A$
be an algebra and
$e\in A$
be an idempotent element. One can construct the following generalized matrix algebra:
\begin{align*} R=\left ( \begin{array}{c@{\quad}c} eAe & eA(1-e) \\ (1-e)Ae & (1-e)A(1-e) \end{array} \right ) . \end{align*}
2.2. Partial actions and partial representations of Hopf algebras
Left partial actions of bialgebras on algebras were introduced in [Reference Caenepeel and Janssen10]. Below we give it with an additional symmetric condition as it appears in [Reference Alves and Batista4].
Definition 2.12.
A symmetric left partial action of a Hopf algebra
$H$
over a
$\Bbbk$
-algebra
$A$
is a linear map
\begin{align*} \begin{array}{rccl} \cdot \, : \, & H\otimes A & \rightarrow & A \\ \, & h\otimes a & \mapsto & h\cdot a \end{array} \end{align*}
satisfying, for all
$h,k\in H$
and
$a,b\in A$
,
-
(LPA1)
$1_H \cdot a =a$
, -
(LPA2)
$h\cdot (ab)=(h_{(1)} \cdot a)(h_{(2)}\cdot b)$
, -
(LPA3)
$h\cdot (k\cdot a)=(h_{(1)}\cdot 1_A )(h_{(2)}k\cdot a)=(h_{(1)}k\cdot a)(h_{(2)}\cdot 1_A)$
.
It follows from (LPA2) and (LPA3) that for all
$h,k\in H$
and
$a,b\in A$
,
\begin{align} & \left (h_{(1)}\cdot (k\cdot a)\right )(h_{(2)}\cdot b)=(h_{(1)}k\cdot a)(h_{(2)}\cdot 1_A)(h_{(3)}\cdot b)= (h_{(1)}k\cdot a)(h_{(2)}\cdot b), \\[-10pt] \nonumber \end{align}
\begin{align} &(h_{(1)}\cdot a)\left (h_{(2)}\cdot (k\cdot b)\right )=(h_{(1)}\cdot a)(h_{(2)}\cdot 1_A)\left (h_{(3)}k\cdot b\right )=(h_{(1)}\cdot a)(h_{(2)}k\cdot b). \\[8pt] \nonumber \end{align}
Similarly, one can give the notion of a right partial action of a Hopf algebra as follows.
Definition 2.15.
A symmetric right partial action of a Hopf algebra
$H$
over a
$\Bbbk$
-algebra
$A$
is a linear map
\begin{align*} \begin{array}{rccl} \cdot \, : \, & A\otimes H & \rightarrow & A \\ \, & a\otimes h & \mapsto & a\cdot h \end{array} \end{align*}
satisfying, for all
$h,k\in H$
and
$a,b\in A$
,
-
(RPA1)
$a\cdot 1_H =a$
, -
(RPA2)
$(ab)\cdot h=(a\cdot h_{(1)} )(b\cdot h_{(2)})$
, -
(RPA3)
$(a\cdot k)\cdot h=(a\cdot kh_{(1)})(1_A \cdot h_{(2)})=(1_A\cdot h_{(1)})(a\cdot kh_{(2)})$
.
Remark 2.16.
From now on in this work by left (right) partial action, we mean a symmetric left (right) partial action. Also, we say that an algebra
$A$
is a left (right) partial
$H$
-module algebra if it is endowed with a left (right) partial action.
Example 2.17.
Given a unital partial action
$ \alpha =\left ( A_g, \alpha _g \right )_{g\in G}$
of a group
$G$
on a unital algebra
$A$
(see Definition 1.1 in [11]) in which each ideal
$A_g \trianglelefteq A$
is generated by a central idempotent element
$1_g \in A$
, one can define a left partial action of the group algebra
$\Bbbk G$
on
$A$
by
\begin{align*} g \cdot a= \alpha _g (a1_{g^{-1}}),\,\, a\in A,\,\, g\in G. \end{align*}
Conversely, let
$H=\Bbbk G$
and
$\cdot \, : \, H\otimes A\to A$
a left partial action of
$H$
on
$A$
. For each
$g\in G$
, consider
$1_g \, :\!= \, g\cdot 1_A$
. Notice that
$1_g$
is a central idempotent of
$A$
. Indeed,
$1_g1_g=(g\cdot 1_A)(g\cdot 1_A)=g\cdot (1_A1_A)=g\cdot 1_A=1_g$
. Also, for all
$a\in A$
,
\begin{align*} 1_ga&=(g\cdot 1_A)(1_{\Bbbk G}\cdot a)=(g\cdot 1_A)(gg^{-1}\cdot a)\\ &=g\cdot (g^{-1}\cdot a)=(gg^{-1} \cdot a)(g\cdot 1_A) \qquad \quad \big (\textrm {by (LPA3)} \big )\\ &=a1_g. \end{align*}
Then
$\alpha =\big (D_g,\alpha _g\big )_{g\in G}$
is a unital partial action of
$G$
on
$A$
, where
$D_g=A1_g,$
and
$\alpha _g(a)=g\cdot a, a\in D_{g^{-1}}.$
Example 2.18.
Let
$\cdot \, : \, H\otimes A\to A$
be a left partial action of a Hopf algebra
$H$
over an algebra
$A$
. Consider the linear map
$\triangleleft \, : \, A\otimes H^{\textrm {op}}\to A$
defined by
\begin{equation} a\triangleleft h=h\cdot a,\quad a\in {A},\,\,h\in H^{\textrm {op}}. \end{equation}
It is straightforward to verify that
$\triangleleft$
is a right partial action of
$H^{\textrm {op}}$
over
$ A$
.
Remark 2.20.
According to [Reference Alves, Batista and Vercruysse2], if
$A$
and
$B$
are partial left
$H$
-module algebras, one can define a morphism of partial
$H$
-module algebras as an algebra morphism
$f \, : \, A \to B$
such that
$f(h \cdot a) =h \cdot f(a)$
, for all
$a\in A$
and
$h\in H$
. The category of all symmetric partial left
$H$
-module algebras and the morphisms of partial
$H$
-module algebras between them is denoted as LParAct
$H.$
Analogously, one defines the category RParAct
$H$
of symmetric right partial
$H$
-module algebras. By the previous example, the categories LParAct
$H$
and RParAct
$H^{\textrm {op}}$
are isomorphic.
We proceed with the next.
Definition 2.21.
Let
$H$
be a Hopf algebra and
$B$
be a unital algebra. A partial representation of
$H$
in
$B$
is a linear map
$\pi \, : \, H\to B$
such that
-
(PR1)
$\pi (1_H) =1_B$
, -
(PR2)
$ \pi (h)\pi (k_{(1)})\pi (S(k_{(2)}))=\pi (hk_{(1)})\pi (S(k_{(2)}))$
, -
(PR3)
$\pi (h_{(1)})\pi (S(h_{(2)}))\pi (k)=\pi (h_{(1)})\pi (S(k_{(2)})k)$
,
for any
$h,k \in H.$
Remark 2.22.
Let
$H$
be a Hopf algebra,
$B$
be a unital algebra, and
$\pi \,:\,\,H\to B$
be a partial representation. It follows by Lemma 2.11 of [Reference Alves, Batista, Castro, Quadros and Vercruysse3] that the following assertions also hold: for all
$h,k \in H$
,
-
(PR4)
$ \pi (h)\pi (S(k_{(1)}))\pi (k_{(2)})=\pi (hS(k_{(1)}))\pi (k_{(2)})$
, -
(PR5)
$ \pi (S(h_{(1)}))\pi (h_{(2)})\pi (k)=\pi (S(h_{(1)}))\pi (h_{2}k)$
.
Example 2.23.
Let
$H$
be a Hopf algebra,
$B$
be a unital algebra and
$\pi \, : \, H\to B$
be a partial representation. Consider the linear map
$\overline {\pi } \, : \, H^{\textrm {opcop}}\to B^{\textrm {op}}$
given by
\begin{align*} \overline {\pi }(h)=\pi (h),\quad h\in H^{\textrm {opcop}}. \end{align*}
We claim that
$\overline {\pi }$
is a partial representation. Notice that (PR1) is clear. To show (PR2), consider
$h,k\in H^{\textrm {opcop}}$
. Then
\begin{align*} \overline \pi (h)\cdot _{\textrm {op}}\overline \pi (k_{[1]})\cdot _{\textrm { op}}\overline \pi (S(k_{[2]}))&= \pi (S(k_{(1)}))\pi (k_{(2)})\pi (h) \\ & \overset {{\scriptscriptstyle {\textrm {(PR5)}}}}{=}\,\,\pi (S(k_{(1)}))\pi (k_{(2)}h) \\ &= \overline \pi (h\cdot _{\textrm {op}}k_{[1]})\cdot _{\textrm {op}}\overline \pi (S(k_{[2]})). \end{align*}
Similarly, using that
$\pi$
satisfies (PR4), we obtain that
$\overline {\pi }$
satisfies (PR3).
Example 2.24.
Let
$H$
be a Hopf algebra and
$\cdot \, : \, H\otimes A\to A$
be a left partial action of
$H$
on an algebra
$A$
. By Example 3.5 in [Reference Alves, Batista and Vercruysse2], the linear map
\begin{align} \pi \, : \, H\to \textrm {End}(A),\quad \pi (h)(a)=h\cdot a,\quad h\in H,\,\, a\in A, \end{align}
is a partial representation of
$H$
in
$\textrm {End}(A)$
. It follows from the previous example that the linear map
$\overline {\pi} \, : \, H^{\textrm {opcop}}\to \textrm {End}(A)^{\textrm {op}}$
given by
\begin{align*} \overline {\pi }(h)=\pi (h),\quad h\in H^{\textrm {opcop}}, \end{align*}
is a partial representation.
2.3. The partial smash product
Let
$\cdot \, : \, H\otimes A\to A$
be a partial action of a Hopf algebra
$H$
over an algebra
$A$
. On the vector space
$A\otimes H$
, one can define the following associative product
\begin{align*} (a\otimes h)(b\otimes k)=a(h_{(1)}\cdot b)\otimes (h_{(2)}k),\quad a,b\in A,\,\,h,k\in H. \end{align*}
Following [Reference Caenepeel and Janssen10], the (left) partial smash product is the vector subspace
\begin{align*} \underline {A\# H} \, :\!= \, (A\otimes H)(1_A\otimes 1_H) \end{align*}
of
$A\otimes H$
which is generated by elements of the form
\begin{align*} a\#h=a(h_{(1)}\cdot 1_A)\otimes h_{(2)}, \quad a\in A,\,h\in H. \end{align*}
Notice that, by construction,
$\underline {A\# H}$
is a unital algebra and
$1_{\underline {A\# H}}=1_A\#1_H$
.
We need the following notion to recall the universal property of the partial smash product proved in [Reference Alves, Batista and Vercruysse2].
Definition 2.26.
Let
$A$
and
$B$
be algebras,
$H$
be a Hopf algebra and
$\cdot \, : \, H\otimes A\to A$
be a partial action of
$H$
on
$A$
. A left covariant pair associated to these data is a pair of maps
$(\psi , \pi ),$
where
$\psi \, : \, A\to B$
is an algebra morphism and
$\pi \, : \, H\to B$
is a partial representation, that satisfies, for all
$h\in H$
and
$a\in A$
,
-
(CP1)
$\psi (h\cdot a) =\pi (h_{(1)})\psi (a)\pi (S(h_{(2)}))$
, -
(CP2)
$\psi (a)\pi (S(h_{(1)}))\pi (h_{(2)})=\pi (S(h_{(1)}))\pi (h_{(2)})\psi (a).$
Consider the following linear maps
\begin{align*} &\phi _0 \, : \, A\to \underline {A\# H},\quad a\mapsto a\#1_H,& &\pi _0 \, : \, H\to \underline {A\# H},\quad h\mapsto 1_A\#h.& \end{align*}
It was proved in [Reference Alves, Batista and Vercruysse2] that
$\phi _0$
is an algebra monomorphism and
$\pi _0$
is a partial representation. The next result is Theorem 3.9 of [Reference Alves, Batista and Vercruysse2].
Theorem 2.27.
Let
$A$
and
$B$
be unital algebras and
$\cdot \, : \, H\otimes A\to A$
be a left partial action of the Hopf algebra
$H$
on
$A$
. Suppose that
$(\psi ,\pi )$
is a covariant pair associated to these data. Then the map
\begin{align*} \Phi \, : \, \underline {A\# H}\to B,\,\,\, a\#h\to \psi (a)\pi (h),\,\,\, a\in A,\,\,\, h\in H, \end{align*}
is the unique algebra morphism such that
$\psi =\Phi \circ \phi _0$
and
$\pi =\Phi \circ \pi _0.$
Consider now a right partial action
$\cdot \, : \, A\otimes H\to A$
of a Hopf algebra
$H$
on an algebra
$A$
. As in the left side case, one can define the right partial smash product. Precisely, one endows the vector space
$H\otimes A$
with an associative product
\begin{align*} (k\otimes b)(h\otimes a)=kh_{(1)}\otimes (b\cdot h_{(2)})a,\quad a,b\in A,\,\,h,k\in H. \\[-35pt] \nonumber \end{align*}
Definition 2.28.
Let
$\cdot \, : \, A\otimes H\to A$
be a right partial action of a Hopf algebra
$H$
on an algebra
$A$
. The (right) partial smash product is the vector subspace
\begin{align*} \underline {H\# A} \, :\!= \, (1_H\otimes 1_A)(H\otimes A) \end{align*}
of
$H\otimes A$
which is generated by typical elements of the form
\begin{equation} h\#a=h_{(1)}\otimes (1_A\cdot h_{(2)})a, \,\,\, a\in A,\,h\in H. \end{equation}
The right partial smash product
$\underline {H\# A}$
is an associative unital algebra and its unity element is
$1_{\underline {H\# A}}=1_H\#1_A$
.
Example 2.30.
Suppose that
$ \alpha =\left ( A_g, \alpha _g \right )_{g\in G}$
is a unital partial action of a group
$G$
on an algebra
$A$
. As we saw in Example 2.17
, we have a left partial action of
$\Bbbk G$
on
$A$
given by
$g\cdot a=\alpha _g(a1_{g^{-1}})$
, for all
$a\in A$
and
$g\in G$
. We recall from [11] that the partial crossed product by
$\alpha$
is the associative algebra
$A\rtimes _{\alpha } G$
whose elements are finite formal sums in the form
$\sum _{g\in G}a_g\delta _g$
, where
$a_g\in A_g$
, and the multiplication is given by
\begin{align*} a_g\delta _g\cdot a_h\delta _h=a_g\alpha _g(a_h1_{g^{-1}})\delta _{gh}, \,\,\,g,h\in G, \,\,\,a_g\in A_g,\,\,a_h\in A_h. \end{align*}
Using ( 2.29 ) we get that the map
\begin{equation} \eta \, : \, \underline {A\# \Bbbk G}\to A\rtimes _{\alpha } G, \,\,a\# g\mapsto a1_{g}\delta _{g},\quad g\in G,\,\,a\in A, \end{equation}
is an algebra isomorphism with inverse
$b\delta _g\mapsto b\# g, \,\,b\in A_g, \, g\in G.$
Moreover, it follows by Example 2.18
, that the partial action
$\alpha$
of
$G$
on
$A$
induces a right partial action of
$(\Bbbk G)^{\textrm {op}}=(\Bbbk G)^{\textrm {opcop}}$
on
$A^{\textrm {op}}$
via
$a\cdot g=\alpha _g(a1_{g^{-1}})$
, for all
$a\in A$
and
$g\in G$
. Also, by (
2.29
) it is not difficult to verify that
\begin{equation} \lambda \, : \, \underline {(\Bbbk G)^{\textrm {op}}\# A}\to ( A^{\textrm {op}}\ltimes _{\alpha } G)^{\textrm {op}},\,\,\, g\# a\mapsto a1_{g}\delta _{g},\quad g\in G,\,\,a\in A, \end{equation}
is an algebra isomorphism.
Definition 2.33.
Let
$A$
and
$B$
be algebras,
$H$
be a Hopf algebra and
$\cdot \, : \, A\otimes H^{\textrm {op}}\to A$
be a right partial action of
$H^{\textrm {op}}$
on
$A$
. A opposite covariant pair associated to these data is a pair of maps
$(\varphi , \gamma ),$
where
$\varphi \, : \,A\to {B}$
is an algebra morphism and
$\gamma \, : \, H^{\textrm {opcop}}\to {B}$
is a partial representation, that satisfies, for all
$h\in H^{\textrm {opcop}}$
and
$a\in A$
,
-
(OCP1)
$\varphi (a\cdot h) =\gamma (S^{-1}(h_{[2]}))\varphi (a)\gamma (h_{[1]}))$
, -
(OCP2)
$\varphi (a)\gamma ({h}_{[2]})\gamma (S^{-1}(h_{[1]}))=\gamma (h_{[2]})\gamma (S^{-1}(h_{[1]}))\varphi (a)$
.
Lemma 2.34.
Let
$H$
be a Hopf algebra and
$A$
be a right partial
$H^{\textrm {op}}$
-module. Then the following assertions hold:
-
(i) the linear map
$\varphi _0 \, : \, A\to \underline {H^{\textrm {op}}\# A},\,\, a\mapsto 1_H\#a$
, is an algebra monomorphism,
-
(ii) the linear map
$\gamma _0 \, : \, H^{\textrm {opcop}}\to \underline {H^{\textrm {op}}\# A},\,\, h\mapsto h\#1_A$
, is a partial representation,
-
(iii)
$(\varphi _0, \gamma _0)$
is an opposite covariant pair.
Proof.
It is straightforward to verify that (i) is true. To prove (ii), we observe that
$\gamma _0(1_H)=1_H\#1_A=1_{\underline {H^{\textrm {op}}\# A}}$
. Also, given
$h,k\in H^{\textrm {opcop}}$
, we have that
\begin{align*} \gamma _0(h)\gamma _0(k_{[1]})\gamma _0(S(k_{[2]}))&=(h\#1_A)({k}_{(2)} \# 1_A)(S({k}_{(1)}) \# 1_A ) \\[2pt] \nonumber&=({k}_{(2)}h \# 1_A\cdot {k}_{(3)})(S({k}_{(1)}) \# 1_A ) \\[2pt]\nonumber&=S({k}_{(2)}){k}_{(3)}h \# (1_A\cdot {k}_{(4)})\cdot S({k}_{(1)}) \\[2pt]\nonumber&=h \# (1_A\cdot {k}_{(2)})\cdot S({k}_{(1)}) \\[2pt]\nonumber&=h \# (1_A\cdot S({k}_{(2)}){k}_{(3)})(1_A\cdot S({k}_{(1)})) \qquad (\text{by } {\textrm {(RPA3)}}) \\[2pt]\nonumber&=h \# 1_A\cdot S(k). \end{align*}
On the other hand,
\begin{align*} \gamma _0(h\cdot _{op} k_{[1]})\gamma _0(S(k_{[2]}))&=({k}_{(2)}h\#1_A)(S(k_{(1)})\#1_A) \\[2pt] &=S(k_{(2)}) {k}_{(3)}h\#1_A\cdot S(k_{(1)}) \\[2pt]&=h\#1_A\cdot S(k), \end{align*}
and consequently (PR2) holds. Finally, notice that
\begin{align*} \gamma _0({h}_{[1]})\gamma _0(S({h}_{[2]}))\gamma _0(k)&= ({h}_{(2)} \# 1_A)(S({h}_{(1)}) \# 1_A )(k\#1_A) \\[2pt]&= (S ({h}_{(2)}){h}_{(3)}\# 1_A\cdot S ({h}_{(2)}))(k\#1_A) \\[2pt]&= (1_H\# 1_A\cdot S (h))(k\#1_A) \\[2pt]&= {k}_{(1)}\#( 1_A\cdot S (h))\cdot {k}_{(2)}, \end{align*}
and
\begin{align*} \gamma _0({h}_{[1]})\gamma _0(S({h}_{[2]}) \cdot _{op} k)&= ( {h}_{(2)} \# 1_A)(kS({h}_{(1)} ) \# 1_A) \\[2pt]&=k_{(1)}S({h}_{(2)} ){h}_{(3)}\# 1_A\cdot k_{(2)}S({h}_{(1)} ) \\[2pt]&=k_{(1)}\# 1_A\cdot k_{(2)}S(h) \\[2pt]&=k_{(1)}\# 1_A\cdot (S(h)\cdot _{\textrm {op}}k_{(2)}) \\[2pt]&=k_{(1)}\# (1_A\cdot k_{(2)})(1_A\cdot (S(h)\cdot _{\textrm {op}} k_{(3)}) \,\qquad \text{(by (2.29))} \\[2pt]&=k_{(1)}\# (1_A\cdot S (h))\cdot {k}_{(2)},\qquad \quad \qquad \qquad (\text{by } {\textrm {(RPA3)}}) \end{align*}
which implies that (PR3) holds, and hence (ii) is proved. To prove (iii) we note that
\begin{align*} \gamma _0(S^{-1}(h_{[2]}))\varphi _0(a)\gamma _0(h_{[1]}))&=(S^{-1}(h_{(1)})\#1_A)(1_H\# a) (h_{(2)}\#1_A) \\[2pt] &=(S^{-1}(h_{(1)})\# a)(h_{(2)}\#1_A) \\[2pt] &=h_{(2)}S^{-1}(h_{(1)})\# a\cdot h_{(3)} \\[2pt] &=1_H\# a\cdot h \\[2pt] &=\varphi _0(a\cdot h). \end{align*}
Also,
\begin{align*} \varphi _0(a)\gamma _0({h}_{[2]})\gamma _0(S^{-1}(h_{[1]}))\,\,&=\,\,(1_H\# a)(h_{(1)}\#1_A)(S^{-1}(h_{(2)})\#1_A) \\[2pt] &=\,\,(h_{(1)}\# a\cdot h_{(2)})(S^{-1}(h_{(3)})\#1_A) \\[2pt] &= \,\,S^{-1}(h_{(4)})h_{(1)}\# (a\cdot h_{(2)})\cdot S^{-1}(h_{(3)}) \\[2pt] &\overset {{\scriptscriptstyle {\textrm {(RPA3)}}}}{=}\,\, S^{-1}(h_{(5)})h_{(1)}\# (1_A\cdot S^{-1}(h_{(4)}))(a\cdot (h_ {(2)}\cdot _{\textrm {op}}S^{-1}(h_{(3)})) ) \\[2pt] &=\,\, S^{-1}(h_{(5)})h_{(1)}\# (1_A\cdot S^{-1}(h_{(4)}))(a\cdot S^{-1}(h_{(3)})h_ {(2)} ) \\[2pt] &=\,\, S^{-1}(h_{(3)})h_{(1)}\# (1_A\cdot S^{-1}(h_{(2)}))a, \end{align*}
and
\begin{align*} \gamma _0(h_{[2]})\gamma _0(S^{-1}(h_{[1]}))\varphi _0(a)&=(h_{(1)}\#1_A)(S^{-1}(h_{(2)})\#1_A)(1_H\# a) \\[2pt] &=(h_{(1)}\#1_A)(S^{-1}(h_{(2)})\#a) \\[2pt] &=S^{-1}(h_{(3)})h_{(1)}\#(1_A\cdot S^{-1}(h_{(2)}))a, \end{align*}
then
$\gamma _0(h_{[2]})\gamma _0(S^{-1}(h_{[1]}))\varphi _0(a)=\varphi _0(a)\gamma _0({h}_{[2]})\gamma _0(S^{-1}(h_{[1]})),$
which finishes the proof.
We now establish a universal property for opposite covariant pairs, analogous to the version for covariant pairs presented in Theorem 2.25.
Theorem 2.35.
Let
$A$
and
$B$
be unital algebras and
$\cdot \, : \, A\otimes H^{\textrm {op}}\to A$
be a right partial action of the Hopf algebra
$H$
on
$A$
. Suppose that
$(\varphi ,\gamma )$
is an opposite covariant pair associated with these data. Then there exists a unique algebra morphism
$\Gamma \, : \,\underline {H^{\textrm {op}}\#A} \to B$
such that
$\varphi =\Gamma \circ \varphi _0$
and
$\gamma =\Gamma \circ \gamma _0$
.
Proof. Consider the linear map
\begin{equation} \Gamma \, : \, \underline {H^{\textrm {op}}\#A} \to B,\qquad \Gamma (h\#a)=\gamma (h)\varphi (a) , \quad h\in H,\,\,a\in A. \end{equation}
It is straightforward to see that
$\varphi =\Gamma \circ \varphi _0$
and
$\gamma =\Gamma \circ \gamma _0$
. Given
$k,h\in H$
and
$a,b\in A$
, we have
\begin{align*} \Gamma ((k\#b)(h\# a))&=\Gamma (k\cdot _{\textrm {op}} h_{(1)}\# (b\cdot h_{(2)})a) \\[2pt] &= \gamma (k\cdot _{\textrm {op}} h_{(1)})\varphi ((b\cdot h_{(2)})a) \\[2pt] &= \gamma (k\cdot _{\textrm {op}} h_{[2]})\varphi (b\cdot h_{[1]})\varphi (a) \\[2pt] &=\gamma (k\cdot _{\textrm {op}} h_{[3]})\gamma (S^{-1}(h_{[2]})) \varphi (b)\gamma (h_{[1]})\varphi (a)\qquad \quad \qquad \, (\text{by (OCP1)}) \\&=\gamma (k)\gamma ( h_{[3]})\gamma (S^{-1}(h_{[2]})) \varphi (b)\gamma (h_{[1]})\varphi (a)\qquad \quad \qquad \,\, (\text{by (PR4)}) \\[2pt] &=\gamma (k)\varphi (b) \gamma (h_{[3]})\gamma (S^{-1}(h_{[2]}))\gamma (h_{[1]})\varphi (a) \qquad \quad \qquad \,\, (\text{by (OCP2)}) \\&=\gamma (k)\varphi (b) \gamma (h_{[3]})\gamma (S^{-1}(h_{[2]}))\gamma (S(S^{-1}(h_{[1]})))\varphi (a) \\[2pt] &=\gamma (k)\varphi (b) \gamma (h_{[3]})\gamma (S^{-1}(h)_{[1]})\gamma (S(S^{-1}(h)_{[2]}))\varphi (a) \\[2pt] &=\gamma (k)\varphi (b) \gamma (h_{[3]}\cdot _{\textrm {op}}S^{-1}(h_{[2]}))\gamma (h_{[1]})\varphi (a)\qquad \quad \qquad \,\, (\text{by (PR2)}) \\[2pt] &=\gamma (k)\varphi (b)\varphi (a)\gamma (h)\varphi (a) \\[2pt] &=\Gamma (k\#b)\circ \Gamma (h\#a). \end{align*}
Since
$\Gamma (1_H\#1_A)=1_B$
it follows that
$\Gamma$
is a morphism of algebras. The proof of the uniqueness of
$\Gamma$
is similar to the proof given in [Reference Alves, Batista and Vercruysse2, Theorem 3.9].
3. The main result
Throughout this section,
$H$
denotes a Hopf algebra with inversible antipode
$S$
and
$R=(_{i}M_{j})_{i,j\in \mathbb{I}_n}$
denotes a generalized matrix algebra such that
$_{i}M_{i}=R_i$
is a unital algebra. The identity element of
$R_i$
is denoted by
$1_i$
. This section is dedicated to prove the main result of this work, which provides necessary and sufficient conditions for the existence of a partial action of
$H$
on
$R$
.
Given
$i,j\in \mathbb{I}_n$
, we denote by
$\iota _{ij}:\,\,\!\!_{i}M_{j}\to R$
the natural inclusion of
$_{i}M_{j}$
in
$R$
, that is,
$\iota _{ij}(m)$
is the matrix whose entries are all zero except the
$(i,j)$
-entry which is equal to
$m$
, for all
$m\in \, \!_{i}M_{j}$
. By Remark 2.8,
$R$
is an
$(R_i,R_j)$
-bimodule, and it is easy to see that
$\iota _{ij}$
(resp.
$\iota _{ii}$
) is a monomorphism of
$(R_i,R_j)$
-bimodules (resp. of algebras). From now on, we will denote by
$_{i}\tilde {M}_{j}=\iota _{ij}(_{i}M_{j})$
the
$(R_i,R_j)$
-subbimodule of
$R$
which is isomorphic to
$_{i}M_{j}$
.
Definition 3.1.
Let
$\rhd \, : \, H\otimes R\to R$
be a left partial action of
$H$
on
$R$
and
$i,j\in \mathbb{I}_n$
. The subbimodule
$_{i}\tilde {M}_{j}$
of
$R$
is said to be
$H$
-invariant if
$h\rhd m\in \,\! _{i}\tilde {M}_{j}$
, for all
$h\in H$
and
$m\in \,\! _{i}\tilde {M}_{j}$
.
Lemma 3.2.
Let
$\rhd \, : \, H\otimes R\to R$
be a left partial action of
$H$
on
$R$
such that
$_{i}\tilde {M}_{j}$
is invariant, for all
$i,j\in \mathbb{I}_n$
. Then:
-
(i) the linear map
$ \rightharpoonup _i \, : \, H\otimes R_i \rightarrow R_i$
given by
is a partial action,
\begin{align*} h\rightharpoonup _i r=\iota ^{-1}_{ii}\big ( h\triangleright \iota _{ii}(r)\big ), \quad h\in H,\,\,r\in R_i, \end{align*}
-
(ii) the linear map
$\pi _{ij}\, :\, H\to \operatorname {End}(_{i}M_{j})$
given by
is a partial representation,
\begin{align*} \pi _{ij}(h)(m)=\iota ^{-1}_{ij}\big (h\rhd \iota _{ij}(m)\big ),\quad h\in H,\,\,\,m\in \,\!_{i}M_{j}, \end{align*}
-
(iii) the linear map
$\gamma _{ij}\, :\, H^{\textrm {opcop}}\to \operatorname {End}(_{i}M_{j})^{\textrm {op}}$
given by
is a partial representation.
\begin{align*} \gamma _{ij}(h)(m)=\iota ^{-1}_{ij}\big (h\rhd \iota _{ij}(m)\big ),\quad h\in H,\,\,\,m\in \,\!_{i}M_{j}, \end{align*}
Proof.
(i) Let
$r\in R_i$
. Then
\begin{align*} 1_H \rightharpoonup _i r =\iota ^{-1}_{ii}\big ( 1_H \triangleright \iota _{ii}(r)\big ) =\iota ^{-1}_{ii}\big ( \iota _{ii}(r)\big ) =r, \end{align*}
which implies (LPA1). For (LPA2), consider
$r,s \in R_i$
and
$h\in H$
. Thus,
\begin{align*} h \rightharpoonup _i (rs) & = \iota ^{-1}_{ii}\big ( h \triangleright \iota _{ii}(rs)\big ) \\[2pt] & = \iota ^{-1}_{ii}\big ( h \triangleright (\iota _{ii}(r) \iota _{ii}(s))\big ) \\[2pt] & = \iota ^{-1}_{ii}\big ( \left (h_{(1)} \triangleright \iota _{ii}(r)\right ) \left (h_{(2)} \triangleright \iota _{ii}(s)\right )\big ) \\[2pt] & = \iota ^{-1}_{ii} \left (h_{(1)} \triangleright \iota _{ii}(r)\right ) \iota ^{-1}_{ii}\left (h_{(2)} \triangleright \iota _{ii}(s)\right ) \\[2pt] & = \left ( h_{(1)} \rightharpoonup _i r \right ) \left ( h_{(2)} \rightharpoonup _i r \right ) . \end{align*}
Finally, in order to verify (LPA3), consider
$i\in \mathbb{I}_n$
,
$h\in H$
and
$r\in R_i$
. Then,
\begin{eqnarray*} (h\triangleright 1_R)\iota _{ii}(r) = (h\triangleright \iota _{ii}(1_i))\iota _{ii} (r) ,\qquad \iota _{ii}(r)(h\triangleright 1_R) = \iota _{ii}(r)(h\triangleright \iota _{ii}(1_i)) . \end{eqnarray*}
Then, for every
$h,k\in H$
and
$r\in R_i$
\begin{align*} h\rightharpoonup _i (k\rightharpoonup _i r) & =\,\, \iota ^{-1}_{ii}\big ( h \triangleright \big ( k\triangleright \iota _{ii}(r)\big ) \big ) \\[2pt] & = \,\, \iota ^{-1}_{ii}\big ( \left ( h_{(1)} \triangleright 1_R \right ) \left ( h_{(2)}k\triangleright \iota _{ii}(r)\right ) \big ) \\[2pt] & \stackrel {(\ast )}{=} \iota ^{-1}_{ii}\big ( \left ( h_{(1)} \triangleright \iota _{ii}(1_i) \right ) \left ( h_{(2)}k\triangleright \iota _{ii}(r)\right ) \big ) \\[2pt] & = \,\, \iota ^{-1}_{ii} \left ( h_{(1)} \triangleright \iota _{ii}(1_i) \right ) \iota ^{-1}_{ii}\left ( h_{(2)}k\triangleright \iota _{ii}(r)\right ) \\[2pt] & = \,\left ( h_{(1)} \rightharpoonup _i 1_i \right )\left ( h_{(2)}k \rightharpoonup _i r \right ), \end{align*}
where (
$\ast$
) follows from the
$H$
-invariance of
$\tilde {R}_i$
. Similarly, one verifies that
\begin{align*} h\rightharpoonup _i (k\rightharpoonup _i r)=\left ( h_{(1)}k \rightharpoonup _i r \right )\left ( h_{(2)} \rightharpoonup _i 1_i \right ) \end{align*}
(ii) For any
$m\in {}_i M_j$
, we have that
\begin{align*} \pi _{ij}(1_H)(m)=\iota ^{-1}_{ij}\big (1_H\rhd \iota _{ij}(m)\big )=\iota ^{-1}_{ij}\big (\iota _{ij}(m)\big )=m, \end{align*}
and hence (PR1) holds. Also, given
$h,k\in H$
, we have that
\begin{align*} \pi _{ij}(h)\pi _{ij}(k_{(1)})\pi _{ij}(S(k_{(2)}))(m)&=\iota ^{-1}_{ij}\big (h\rhd \big (k_{(1)}\rhd (S(k_{(2)})\rhd \iota _{ij}(m))\big )\big )\\[2pt] &=\iota ^{-1}_{ij}\big (h\rhd \big ((k_{(1)}\rhd 1_R)(k_{(2)}S(k_{(3)})\rhd \iota _{ij}(m))\big ) \big )\, (\text{by {(LPA3)}})\\[2pt] &=\iota ^{-1}_{ij}\big (h\rhd \big ((k\rhd 1_R)\iota _{ij}(m)\big )\big ), \end{align*}
and, on the other hand,
\begin{align*} \pi _{ij}(hk_{(1)})\pi _{ij}(S(k_{(2)}))(m)&=\iota ^{-1}_{ij}\big (hk_{(1)}\rhd (S(k_{(2)})\rhd \iota _{ij}(m))\big )\\[2pt] &=\iota ^{-1}_{ij}\big ((h_{(1)}k_{(1)}\rhd 1_R)(h_{(2)}k_{(2)}S(k_{(3)})\rhd \iota _{ij}(m))\big )\quad \,(\text{by (LPA3)})\\[2pt] &=\iota ^{-1}_{ij}\big ((h_{(1)}k\rhd 1_R)(h_{(2)}\rhd \iota _{ij}(m))\big )\\[2pt] &=\iota ^{-1}_{ij}\big ((h_{(1)}k\rhd 1_R)(h_{(2)}\rhd ( 1_R \cdot \iota _{ij}(m)))\big )\\[2pt] &= \iota ^{-1}_{ij}\big ((h_{(1)}k\rhd 1_R)(h_{(2)}\rhd 1_R)(h_{(3)}\rhd \iota _{ij}(m))\big ) \quad \,(\text{by (LPA2)}) \\[2pt] &=\iota ^{-1}_{ij}\big ((h_{(1)}\rhd (k\rhd 1_R))(h_{(2)}\rhd \iota _{ij}(m))\big ) \quad \,(\text{by (LPA3)})\\[2pt] &=\iota ^{-1}_{ij}\big (h\rhd \big ( (k\rhd 1_R)\iota _{ij}(m)\big )\big ) \quad \,(\text{by (LPA2)}). \end{align*}
Thus, (PR2) is true. Similarly, we can verify that (PR3) also holds, and consequently (ii) is proved. Item (iii) follows from (ii) and Example 2.23.
Theorem 3.3. The following assertions are equivalent:
-
(i) there exists a left partial action
$\rhd \, : \, H\otimes R\to R$
of
$H$
on
$R$
such that
$_{i}\tilde {M}_{j}$
is invariant, for all
$i,j\in \mathbb{I}_n$
, -
(ii) the following conditions hold:
-
(a) for each
$i\in \mathbb{I}_n$
, there is a left partial action
$\rightharpoonup _{i} \, : \, H\otimes R_i\to R_i$
of
$H$
on
$R_i$
, -
(b)
$_{i}M_{j}$
is a left
$\underline {R_i\# H}$
-module and a right
$\underline {H^{\textrm {op}}\# R_j}$
-module such that
for all
\begin{align*} & (1_i\#h)\cdot m=m\cdot (h\# 1_j),& &\,\,(1_i\#h)\cdot r= h\rightharpoonup _{i} r,&\\ &\,\,\,(r_i\# 1_H)\cdot m=r_i\cdot m,& &m\cdot (1_H\#r_j)=m\cdot r_j,& \end{align*}
$h\in H$
,
$m\in \,\! _{i}M_{j}$
,
$r,r_i\in R_i$
,
$r_j\in R_j$
and
$i,j\in \mathbb{I}_n$
,
-
(c)
$(1_i\#h)\cdot mn=\big ((1_i\#h_{(1)})\cdot m\big )\big ((1_j\#h_{(2)})\cdot n\big ),\,\,\, \text{for all }\,h\in H,\,m\in \,\!\! _{i}M_{j},\,n\in \,\!\! _{j}M_{k}$
.
-
Proof.
(i)
$\Rightarrow$
(ii) Let
$i\in \mathbb{I}_n$
. Since
$_{i}\tilde {M}_{i}$
is invariant by the partial action of
$H$
on
$R$
, it follows from Lemma 3.2 (i) that the restriction
$\rightharpoonup _{i}\, :\, H\otimes R_i\to R_i$
is a partial action of
$H$
on
$R_i$
and hence (a) is proved. To prove (b), we recall that
$_{i}M_{j}$
an
$(R_i,R_j)$
-bimodule, for all
$i,j\in \mathbb{I}_n$
. In particular,
$_{i}M_{j}$
is a left
$R_i$
-module and consequently there is an algebra morphism
$\psi _{ij}\,:\, R_i\to \operatorname {End}(_{i}M_{j})$
, given by
\begin{align*} \psi _{ij} (r)(m) =\iota ^{-1}_{ij}\big ( \iota _{ii}(r)\iota _{ij}(m) \big )=\iota ^{-1}_{ij}\big ( \iota _{ij}(r\cdot m)\big )=r\cdot m, \quad \text{for all } r\in R_i,\,\, m\in {}_i M_j . \end{align*}
Moreover, by Lemma 3.2 (ii),
$\pi _{ij}\, :\, H\to \operatorname {End}(_{i}M_{j})$
given by
\begin{align*} \pi _{ij}(h)(m)=\iota ^{-1}_{ij}\big (h\rhd \iota _{ij}(m)\big ),\qquad h\in H,\,\,\,m\in \,\!_{i}M_{j}, \end{align*}
is a partial representation. One needs to verify that
$(\psi _{ij},\pi _{ij})$
is a covariant pair. In order to prove (CP1), take
$r\in R_i$
,
$h\in H$
and
$m\in {}_i M_j$
. Then
\begin{align*} \psi _{ij} (h\rightharpoonup _i r)(m) & = \iota ^{-1}_{ij}\left ( \iota _{ii}(h\rightharpoonup _i r)\iota _{ij} (m) \right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( \iota _{ii} \iota ^{-1}_{ii} (h\triangleright \iota _{ii}(r)) \iota _{ij} (m)\right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( (h\triangleright \iota _{ii}(r)) \iota _{ij} (m)\right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( (h_{(1)}\triangleright \iota _{ii}(r)) (h_{(2)}S(h_{(3)})\triangleright \iota _{ij} (m))\right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( (h_{(1)}\triangleright \iota _{ii}(r)) (h_{(2)}\triangleright (S(h_{(3)})\triangleright \iota _{ij} (m)))\right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( h_{(1)}\triangleright \left ( \iota _{ii}(r) (S(h_{(2)})\triangleright \iota _{ij} (m))\right )\right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( h_{(1)}\triangleright \left ( \iota _{ii}(r) \iota _{ij}\iota ^{-1}_{ij}(S(h_{(2)})\triangleright \iota _{ij} (m))\right )\right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( h_{(1)}\triangleright \left ( \iota _{ii}(r) \iota _{ij}(\pi _{ij}(S(h_{(2)}))(m))\right )\right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( h_{(1)}\triangleright \iota _{ij}\iota ^{-1}_{ij}\left ( \iota _{ii}(r) \iota _{ij}(\pi _{ij}(S(h_{(2)}))(m))\right )\right ) \\[2pt] & = \iota ^{-1}_{ij}\left ( h_{(1)}\triangleright \iota _{ij}\left ( \psi _{ij}(r) (\pi _{ij}(S(h_{(2)}))(m))\right )\right ) \\[2pt] & = \pi _{ij} (h_{(1)})\left ( \psi _{ij}(r) (\pi _{ij}(S(h_{(2)}))(m))\right ) \\[2pt] & = \pi _{ij} (h_{(1)})\circ \psi _{ij}(r) \circ \pi _{ij}(S(h_{(2)}))(m) . \end{align*}
For (CP2), on the one hand we have
\begin{eqnarray*} \psi _{ij} (r) \circ \pi _{ij} (S(h_{(1)})) \circ \pi _{ij} (h_{(2)}) (m) & = & \psi _{ij} (r) \circ \pi _{ij} (S(h_{(1)})) \left ( \iota ^{-1}_{ij} ( h_{(2)}\triangleright \iota _{ij} (m))\right ) \\[2pt] & = & \psi _{ij} (r) \left ( \iota ^{-1}_{ij} \left ( S(h_{(1)})\triangleright ( h_{(2)}\triangleright \iota _{ij} (m))\right ) \right )\\[2pt] & = & \psi _{ij} (r) \left ( \iota ^{-1}_{ij} \left (\left ( S(h_{(2)})h_{(3)}\triangleright \iota _{ij} (m)\right )\left ( S(h_{(1)}) \triangleright 1_R \right ) \right )\right ) \\[2pt] & = & \psi _{ij} (r) \left ( \iota ^{-1}_{ij} \left ( \iota _{ij} (m)\left ( S(h) \triangleright 1_R \right ) \right )\right )\\[2pt] & = & \iota ^{-1}_{ij} \left (\iota _{ii} (r) \iota _{ij} (m)\left ( S(h) \triangleright 1_R \right ) \right ) . \end{eqnarray*}
On the other hand,
\begin{eqnarray*} \pi _{ij} (S(h_{(1)})) \circ \pi _{ij} (h_{(2)}) \circ \psi _{ij} (r) (m) & = & \pi _{ij} (S(h_{(1)})) \circ \pi _{ij} (h_{(2)}) \left ( \iota ^{-1}_{ij} (\iota _{ii}(r) \iota _{ij}(m)) \right ) \\[2pt] & = & \pi _{ij} (S(h_{(1)})) \left ( \iota ^{-1}_{ij} \left ( h_{(2)}\triangleright (\iota _{ii}(r) \iota _{ij}(m)) \right )\right ) \\[2pt] & = & \iota ^{-1}_{ij} \left ( S(h_{(1)})\triangleright \left ( h_{(2)}\triangleright (\iota _{ii}(r) \iota _{ij}(m)) \right )\right ) \\[2pt] & = & \iota ^{-1}_{ij} \left (\left ( S(h_{(1)})h_{(2)}\triangleright (\iota _{ii}(r) \iota _{ij}(m)) \right ) \left ( S(h_{(1)})\triangleright 1_R \right )\right ) \\[2pt] & = & \iota ^{-1}_{ij} \left ( \iota _{ii}(r) \iota _{ij}(m) \left ( S(h)\triangleright 1_R \right )\right ) . \end{eqnarray*}
Therefore,
$(\psi _{ij} , \pi _{ij})$
is a covariant pair. It follows from Theorem 2.27 that
\begin{align*} \Phi _{ij} \, : \, R_i\#H\to \operatorname {End}(_{i}M_{j}),\quad \Phi _{ij}(r\#h)=\psi _{ij}(r)\pi _{ij}(h), \,\,r\in R_i,\,\,h\in H, \end{align*}
is an algebra morphism. Therefore,
$_{i}M_{j}$
is a left
$R_i\#H$
-module.
Now, we will denote by
$\cdot \, : \,\,\!_{i}M_{j}\otimes R_j\to \,\!_{i}M_{j}$
the right action of
$R_j$
on
$_{i}M_{j}$
. Then,
$\varphi _{ij}\, :\, R_j\to \operatorname {End}(_{i}M_{j})^{\textrm {op}}$
given by
$\varphi _{ij}(r)(m)=m\cdot r$
,
$r\in R_j$
and
$m\in \,\! _{i}M_{j}$
, is an algebra morphism. Also, by Lemma 3.2 (ii), the map
$\gamma _{ij}\, :\, H^{\textrm {opcop}}\to \operatorname {End}(_{i}M_{j})^{\textrm {op}}$
given by
\begin{align*} \gamma _{ij}(h)(m)=\iota ^{-1}_{ij}\big (h\rhd \iota _{ij}(m)\big ),\qquad h\in H,\,\,\,m\in \,\!_{i}M_{j}, \end{align*}
is a partial representation. We claim that
$(\varphi _{ij},\gamma _{ij})$
is an opposite covariant pair. Firstly, by Example 2.18, the linear map
$\leftharpoonup _{j}\, :\, R_j\otimes H^{\textrm {op}}\to R_j,$
given by
\begin{align*} r \leftharpoonup _{j} h \, :\!= \, h\rightharpoonup _{j}r=\iota ^{-1}_{jj}(h\rhd \iota _{jj}(r)),\,\,\, \text{for all}\,\, r\in R_j,\,\, h\in H^{\textrm {op}}, \end{align*}
defines a right partial action of
$H^{\textrm {op}}$
in
$R_j$
. For all
$r\in R_j$
and
$m\in \!_{i}M_{j}$
, we will denote by
$u(m,r) \, :\!= \, \gamma _{ij}(S^{-1}(h_{[2]}))\cdot _{\textrm {op}}\varphi _{i,j}(r)\cdot _{\textrm {op}}\gamma _{ij}(h_{[1]})(m)$
the element of
$_{i}M_{j}$
obtained using the opposite product of
$\operatorname {End}(_{i}M_{j})$
. Then
\begin{align*} u(m,r)&=\,\,\gamma _{ij}(h_{[1]})\varphi _{i,j}(r)\gamma _{ij}(S^{-1}(h_{[2]}))(m)\\[2pt] &=\,\,\iota ^{-1}_{ij}\Big (h_{(2)}\rhd \iota _{ij}\big (\big (\iota ^{-1}_{ij} \big (S^{-1}(h_{(1)})\rhd \iota _{ij}(m)\big )\cdot r\big ) \big )\Big ) \\[2pt] &=\,\,\iota ^{-1}_{ij}\big (h_{(2)}\rhd \big (S^{-1}(h_{(1)})\rhd \iota _{ij}(m)\big )\cdot r \big )\\[2pt] &=\,\,\iota ^{-1}_{ij}\big (h_{(2)}\rhd \big (S^{-1}(h_{(1)})\rhd \iota _{ij}(m)\big )\iota _{jj}(r) \big )\\[2pt] &=\,\,\iota ^{-1}_{ij}\Big (\big (h_{(2)}\rhd \big ( S^{-1}(h_{(1)})\rhd \iota _{ij}(m)\big )(h_{(3)}\rhd \iota _{jj}(r) \big )\Big )\\[2pt] &\overset {(2.13)}{=}\,\,\iota ^{-1}_{ij}\big (\big (h_{(2)} S^{-1}(h_{(1)})\rhd \iota _{ij}(m)\big )(h_{(3)}\rhd \iota _{jj}(r) \big )\big )\\[2pt] &=\,\,\iota ^{-1}_{ij}\big (\iota _{ij}(m)\big (h\rhd \iota _{jj}(r) \big )\big )\\[2pt] &\stackrel {(*)}=\,\,m\cdot \iota ^{-1}_{jj}(h\rhd \iota _{jj}(r) \big )\big )\\[2pt] &=\,\, \varphi _{ij}( r\leftharpoonup _{j} h)(m), \end{align*}
where
$(*)$
holds because there exists a unique
$r'\in R_j$
such that
$h\rhd \iota _{jj}(r)=\iota _{jj}(r')$
and consequently
\begin{align*} \iota ^{-1}_{ij}\big (\iota _{ij}(m)\big (h\rhd \iota _{jj}(r) \big )\big )=\iota ^{-1}_{ij}\big (\iota _{ij}(m)\iota _{jj}(r')\big )=\iota ^{-1}_{ij}\big (\iota _{ij}(m\cdot r')\big )=m\cdot r'. \end{align*}
Thus, (OCP1) is true. In order to prove (OCP2) , given
$r\in R_j$
and
$m\in \!_{i}M_{j}$
we will denote
$v(m,r)=\varphi _{ij}(r)\cdot _{\textrm {op}}\gamma _{ij}({h}_{[2]})\cdot _{\textrm {op}}\gamma _{ij}(S^{-1}(h_{[1]})(m)\in \!_{i}M_{j}$
. Notice that
\begin{align*} v(m,r) &=\,\,\gamma _{ij}(S^{-1}(h_{[1]}))\gamma _{ij}({h}_{[2]})\varphi _{ij}(r)(m) \\[2pt] &=\,\,\iota ^{-1}_{ij}\big (S^{-1}(h_{(2)})\rhd (h_{(1)}\rhd \iota _{ij}(m\cdot r))\big ) \\[2pt] &=\,\,\iota ^{-1}_{ij}\big (S^{-1}(h_{(2)})\rhd (h_{(1)}\rhd \iota _{ij}(m)\iota _{jj}(r))\big ) \\[2pt] &=\,\,\iota ^{-1}_{ij}\Big (S^{-1}(h_{(3)})\rhd \big ((h_{(1)}\rhd \iota _{ij}(m))(h_{(2)}\rhd \iota _{jj}(r))\big )\Big ) \\[2pt] &=\,\,\iota ^{-1}_{ij}\Big (\big (S^{-1}(h_{(4)})\rhd (h_{(1)}\rhd \iota _{ij}(m))\big )\big (S^{-1}(h_{(3)})\rhd (h_{(2)}\rhd \iota _{jj}(r))\big )\Big ) \\[2pt] &\overset {(2.14)}{=}\,\,\iota ^{-1}_{ij}\Big (\big (S^{-1}(h_{(4)})\rhd (h_{(1)}\rhd \iota _{ij}(m))\big )\big (h_{(2)}S^{-1}(h_{(3)})\rhd \iota _{jj}(r)\big )\Big ) \\[2pt] &=\,\,\iota ^{-1}_{ij}\Big (\big (S^{-1}(h_{(2)})\rhd (h_{(1)}\rhd \iota _{ij}(m))\big )\iota _{jj}(r)\Big ) \\[2pt] &=\,\,\iota ^{-1}_{ij}\Big (\big (S^{-1}(h_{[1]})\rhd (h_{[2]}\rhd \iota _{ij}(m))\big )\iota _{jj}(r)\Big ) \\[2pt] &=\,\,\varphi _{ij}(r) \gamma _{ij}(S^{-1}(h_{[1]}) \gamma _{ij}({h}_{[2]})(m) \\[2pt] &=\,\,\gamma _{ij}({h}_{[2]})\cdot _{\textrm {op}} \gamma _{ij}(S^{-1}(h_{[1]}))\cdot _{\textrm {op}} \varphi _{ij}(r)(m). \end{align*}
Hence, by Theorem 2.35, the map
$\Gamma _{ij}\,:\, \underline {H^{\textrm {op}}\#R_j} \to \operatorname {End}(_{i}M_{j})^{\textrm {op}}$
given by
\begin{align*} \Gamma _{ij}(h\#r)=\gamma _{ij}(h)\cdot _{\textrm {op}}\varphi _{ij}(r) , \quad h\in H,\,\,r\in R_j, \end{align*}
is an algebra morphism. Thus,
$_{i}M_{j}$
is a right
$\underline {H^{\textrm {op}}\#R_j}$
-module. Now for
$h\in H$
,
$m\in \!_{i}M_{j}$
, and
$r\in R_i$
, we have
\begin{align*} (1_i\#h)\cdot m&=\Phi _{ij}(1_i\#h)(m)=(\psi _{ij}(1_i)\pi _{ij}(h))(m)\\[2pt] &=\iota ^{-1}_{ij}\big (h\rhd \iota _{ij}(m)\big )=\gamma _{ij}(h)(m)\\[2pt] &=(\gamma _{ij}(h)\cdot _{\textrm {op}}\varphi _{ij}(1_j))(m)=\Gamma _{ij}(h\#1_j)(m)\\[2pt] &=m\cdot (h\#1_j), \end{align*}
and
\begin{align*} (1_i\#h)\cdot r =\iota ^{-1}_{ii}\big (h\rhd \iota _{ii}(r)\big )=h\rightharpoonup _{i} r. \end{align*}
The other two identities of (b) follow directly from the definitions of
$\Phi _{ij}$
and
$\Gamma _{ij}$
. Finally,
\begin{align*} (1_i\#h)\cdot mn&=\iota ^{-1}_{ik}\big (h\rhd \iota _{ik}(mn)\big ) \\[2pt] &=\iota ^{-1}_{ik}\big (h\rhd \iota _{ij}(m)\iota _{jk}(n)\big ) \\[2pt] &=\iota ^{-1}_{ik}\big ((h_{(1)}\rhd \iota _{ij}(m))(h_{(2)}\rhd \iota _{jk}(n))\big ) \\[2pt] &=\iota ^{-1}_{ij}(h_{(1)}\rhd \iota _{ij}(m))\iota ^{-1}_{jk}(h_{(2)}\rhd \iota _{jk}(n)) \\[2pt] &= \big ((1_i\#h_{(1)})\cdot m\big )\big ((1_j\#h_{(2)})\cdot n\big ), \end{align*}
for all
$h\in H$
,
$m\in \,\! _{i}M_{j}$
and
$n\in \,\! _{j}M_{k}$
. Hence, (c) is true and the implication (i)
$\Rightarrow$
(ii) is proved.
(ii)
$\Rightarrow$
(i) Let
$h\in H$
and
$r=(_{i}m_{j})_{i,j\in \mathbb{I}_n}\in R$
. Using that
$_{i}M_{j}$
is a left
$\underline {R_i\# H}$
-module, we will define the linear map
$\rhd \, : \,H\otimes R\to R$
by
\begin{align*} h\rhd r=\big ( (1_i\#h)\cdot \,\! _{i} m_{j}\big )_{i,j\in \mathbb{I}_n}. \end{align*}
It is clear that (LPA1) is true. Also, the condition (LPA2) follows directly from (c). Now, consider
$h,k\in H$
and
$r=\big (_{i}m_{j}\big )_{i,j\in \mathbb{I}_n}\in \,\!_{i}M_{j}$
. Then
\begin{align*} h\rhd \big (k\rhd r\big )&=h\rhd \Big ( \big ((1_i\# k)\cdot \,\! _{i}m_{j}\big )_{i,j\in \mathbb{I}_n} \Big )\\[2pt] &=\Big ( (1_i\# h)\cdot \big ((1_i\# k)\cdot \,\! _{i}m_{j}\big )\Big )_{i,j\in \mathbb{I}_n} \\[2pt] &=\Big ( (1_i\# h)(1_i\# k)\cdot \,\! _{i}m_{j}\Big )_{i,j\in \mathbb{I}_n} \\[2pt] &=\Big ( \big ( h_{(1)}\rightharpoonup _{i} 1_i\# h_{(2)}k\big ) \cdot \,\! _{i}m_{j}\Big )_{i,j\in \mathbb{I}_n} , \end{align*}
\begin{align*} ( h_{(1)}\rhd 1_R)\big (h_{(2)}k\rhd r\big )& = \operatorname {diag}((1_1\# h_{(1)})\cdot 1_1,\ldots ,(1_n\# h_{(1)})\cdot 1_n)(h_{(2)}k\rhd r)\\[2pt] &\overset {{\textrm {(b)}}}{=}\operatorname {diag}(h_{(1)}\rightharpoonup _{1} 1_1,\ldots ,h_{(1)}\rightharpoonup _{n} 1_n)(h_{(2)}k\rhd r)\\[2pt] &= \Big (\big ((h_{(1)}\rightharpoonup _{i} 1_i)(1_i\#h_{(2)}k)\big )\cdot \,\! _{i}m_{j}\Big )_{i,j\in \mathbb{I}_n}\\[2pt] &=\Big ( \big ( h_{(1)}\rightharpoonup _{i} 1_i\# h_{(2)}k\big ) \cdot \,\! _{i}m_{j}\Big )_{i,j\in \mathbb{I}_n}. \end{align*}
Denote by
$D=\operatorname {diag}(h_{(2)}\rightharpoonup _{j} 1_1,\ldots ,h_{(2)}\rightharpoonup _{j} 1_n)\in R$
. Then
\begin{align*} \big ( h_{(1)}k\rhd (_{i}m_{j}) \big )_{i,j\in \mathbb{I}_n}(h_{(2)}\rhd 1_R)&\overset {{\textrm {(b)}}}{=} \big ( (1_i\# h_{(1)}k)\cdot \,\! _{i}m_{j}\big )_{i,j\in \mathbb{I}_n} D\\[2pt] &\overset {{\textrm {(b)}}}{=} \big ( _{i}m_{j}\cdot \,\! (h_{(1)}k\#1_j)\big )_{i,j\in \mathbb{I}_n} D\\[2pt] &= \big (_{i}m_{j}\cdot (h_{(1)}k\# h_{(2)}\rightharpoonup _{j} 1_j)\big )_{i,j\in \mathbb{I}_n}\\[2pt] &= \big (_{i}m_{j}\cdot (h_{[2]}k\# h_{[1]}\rightharpoonup _{j} 1_j)\big )_{i,j\in \mathbb{I}_n}\\[2pt] &= \big (_{i}m_{j}\cdot (k\# 1_j)(h\#1_j)\big )_{i,j\in \mathbb{I}_n}\\[2pt] &\stackrel {\textrm {(b)}}= \big ( (1_i\# h)(1_i\#k)\cdot \!_{i}m_{j}\big )_{i,j\in \mathbb{I}_n} \\[2pt] &=\big ( (h_{(1)}\rightharpoonup _{i} 1_i)\# h_{(2)}k\cdot \,\! _{i}m_{j}\big )_{i,j\in \mathbb{I}_n}, \end{align*}
which implies (LPA3). Thus, (ii)
$\Rightarrow$
(i) is proved.
We provide a couple of remarks related to Theorem 3.3.
Remark 3.4. We notice that conditions (b) and (c) of Theorem 3.3 are equivalent to (b) and (c′), where
\begin{align*} \text{(c}^{\prime}\text{)}\qquad mn\cdot (h\# 1_k) =\big( m\cdot (h_{(1)}\# 1_j ) \big) \big( n\cdot (h_{(2)}\# 1_k ) \big), \quad \text{for all }\, h\in H,\,m\in \,\!\! _{i}M_{j},\,n\in \,\!\! _{j}M_{k} . \end{align*}
Indeed, assume that (b) and (c) are hold. Then
\begin{align*} mn\cdot (h\# 1_k) &= (1_i\#h)\cdot mn \\[2pt] &= \big ((1_i\#h_{(1)})\cdot m\big )\big ((1_j\#h_{(2)})\cdot n\big ) \\[2pt] &= \big ( m\cdot (h_{(1)}\# 1_j ) \big )\big ( n\cdot (h_{(2)}\# 1_k ) \big ). \end{align*}
Analogously, it can be verified that (b) and (c′) imply (b) and (c).
Remark 3.5.
Let
$H$
be a Hopf algebra acting partially on the generalized matrix algebra
$R=(_{i}M_{j})_{i,j\in \mathbb{I}_n}$
. By Theorem 3.3
,
${}_iM_j$
is a left
$\underline {R_i \# H}$
-module and a right
$\underline {H^{op}\# R_j}$
-module. However,
${}_iM_j$
is not necessarily a
$(\underline {R_i \# H} ,\underline {H^{op}\# R_j})$
-bimodule. Indeed, since
$(1_i \# h) \cdot m =m\cdot (h\# 1_j)$
, for all
$h\in H$
and
$m\in {}_{i}M_{j}$
, then we have
\begin{align*} ((1_i \# h) \cdot m)\cdot (k\# 1_j)= (1_i \# k)\cdot ((1_i \# h) \cdot m)=((k_{(1)} \rightharpoonup _{i} 1_i) \# k_{(2)}h) \cdot m , \end{align*}
while
\begin{align*} (1_i \# h) \cdot (m\cdot (k\# 1_j))=(1_i \# h)\cdot ((1_i \# k) \cdot m)=((h_{(1)} \rightharpoonup _{i} 1_i) \# h_{(2)}k) \cdot m. \end{align*}
Example 3.6.
Let
$A$
be a unital algebra and suppose that
$e\in A$
is an idempotent such that
$A=AeA$
. It is easily seen that
$(A,eAe,Ae,eA,\mu ,\nu )$
, where
$\mu \, : \, Ae\otimes _{eAe} eA\to A$
and
$\nu \, : \, eA\otimes _A Ae\to eAe$
are the multiplication maps, is a strict Morita context. Thus, we have the generalized matrix algebra
\begin{align*} R = {\left ( \begin{matrix} A &\quad Ae\\ eA &\quad eAe \end{matrix} \right )}. \end{align*}
In this case,
$_{1}M_{1}=R_1=A$
,
$_{1}M_{2}=Ae$
,
$_{2}M_{1}=eA$
, and
$_{2}M_{2}=R_2=eAe$
. Let
$H$
be a Hopf algebra and
$\cdot \, : \, H\otimes A\to A$
a left partial action of
$H$
on
$A$
such that
$h\cdot e=\varepsilon (h)e$
, for all
$h\in H$
. Then
\begin{align*} h\cdot (ea)=(h_{(1)}\cdot e)(h_{(2)}\cdot a)=\varepsilon (h_{(1)})e(h_{(2)}\cdot a)=e(h\cdot a), \end{align*}
and similarly
$h\cdot (ae)=(h\cdot a)e$
, for all
$a\in A$
and
$h\in H$
. Thus,
$h\cdot (eae)=e(h\cdot a)e\in eAe$
, for all
$a\in A$
, and we have a partial action of
$H$
on
$eAe$
obtained by restriction. It is straightforward to check that
$A$
and
$Ae$
are left
$\underline {A\#H}$
-modules with actions given, respectively, by
\begin{align} a\#h\rightharpoonup b&=a(h\cdot b), &a,b\in A,\,\,h\in H, \\[-10pt] \nonumber \end{align}
\begin{align} a\#h\rightharpoonup be&=a(h\cdot b)e, &a,b\in A,\,\,h\in H. \\[8pt] \nonumber \end{align}
On the other hand,
$A$
is a right
$\underline {H^{\textrm {op}}\#A}$
-module and
$Ae$
is a right
$\underline {H^{\textrm {op}}\#(eAe)}$
-module via the actions given, respectively, by
\begin{align} b\leftharpoonup h\# a&=(h\cdot b)a, &a,b\in A,\,\,h\in H^{\textrm {op}}, \\[-10pt] \nonumber \end{align}
\begin{align} be\leftharpoonup h\# eae&=(h\cdot b)eae,&a,b\in A,\,\,h\in H^{\textrm {op}}. \\[8pt] \nonumber \end{align}
Observe that
\begin{equation*}1_A\#h\rightharpoonup be=(h\cdot b)e=be\leftharpoonup h\# e, \qquad 1_A\#h\rightharpoonup b=h\cdot b=b \leftharpoonup h\# 1_A,\quad b\in A,\,\,h\in H.\end{equation*}
Hence, the bimodules
$_{1}M_{1}=A$
and
$_{1}M_{2}=Ae$
satisfy the condition (b) of Theorem 3.3
. In a similar way,
$eA$
and
$eAe$
are left
$\underline {(eAe)\#H}$
-modules with structures
\begin{align} eae\#h\rightharpoonup eb&= eae(h\cdot b), &a,b\in A,\,\,h\in H, \\[-10pt] \nonumber \end{align}
\begin{align} eae\#h\rightharpoonup ebe&= eae(h\cdot b)e,&a,b\in A,\,\,h\in H. \\[8pt] \nonumber \end{align}
while
$eA$
is a right
$\underline {H^{\textrm {op}}\# A}$
-module and
$eAe$
is a right
$\underline {H^{\textrm {op}}\# eAe}$
-module with structures
\begin{align} eb\leftharpoonup h\# a&=e(h\cdot b)a,& a,b\in A,\,\,k\in H,\,\,h\in H^{\textrm {op}}, \\[-10pt] \nonumber \end{align}
\begin{align} ebe\leftharpoonup h\# eae&=e(h\cdot b)eae,& a,b\in A,\,\,k\in H,\,\,h\in H^{\textrm {op}}. \\[8pt] \nonumber \end{align}
The bimodules
$_{2}M_{1}=eA$
and
$_{2}M_{2}=eAe$
also satisfy the condition (b) of Theorem 3.3
. By a direct computation, we can verify that the condition (c) of Theorem 3.3 is valid. Thus, by Theorem 3.3
, the linear map
$\rhd \, : \, H\otimes R\to R$
given by
\begin{align*} h\rhd {\left ( {\begin{matrix} a &\quad be \\ ec &\quad ede \end{matrix}}\right )} ={\left ({ \begin{matrix} h\cdot a &\quad (h\cdot b)e \\ e(h\cdot c) &\quad e(h\cdot d)e \end{matrix}}\right )} \end{align*}
is a partial action of
$H$
on the Morita ring
$R$
.
4. The group case revisited
In what follows in this section,
$G$
is a group and
$R=(_{i}M_{j})_{i,j\in \mathbb{I}_n}$
is a generalized matrix algebra as in (2.6). It follows from [Reference Caenepeel and Janssen10] that having a left unital partial action of
$G$
on
$R$
is the same as having a left partial action of the Hopf algebra
$\Bbbk G$
on
$R$
. In Section 4 of [Reference Bagio and Pinedo7], the authors give sufficient conditions to define a unital partial action of
$G$
on
$R$
. In what follows we relate these conditions with the conditions (a), (b) and (c) in (ii) of Theorem 3.3. For the convenience of the reader, we will recall some definitions and results presented in [Reference Bagio and Pinedo7].
Definition 4.1.
Let
$n$
be a positive integer,
$R=(_{i}M_{j})_{i,j\in \mathbb{I}_n}$
be a generalized matrix algebra, and
${\mathcal{I}}=\{I_j\subset R_j\,:\,I_j \text{ is an ideal of }R_j,\text{ for all }j\in \mathbb{I}_n\}$
. The
$(R_i,R_j)$
-bimodule
$_{i}M_{j}$
will be called
$\mathcal{I}$
-symmetric if
$I_i\cdot \,\!_{i}M_{j}=\!_{i}M_{j}\cdot I_j$
. When
$_{i}M_{j}$
is
$\mathcal{I}$
-symmetric for all
$i,j\in \mathbb{I}_n$
, we say that
$R$
is
$\mathcal{I}$
-symmetric.
The next Lemma is Corollary 3.8 of [Reference Bagio and Pinedo7].
Lemma 4.2.
Let
$n$
be a positive integer,
$R=(_{i}M_{j})_{i,j\in \mathbb{I}_n}$
be a generalized matrix algebra, and
${\mathcal{I}}=\{I_j\subset R_j\,:\,I_j \text{ is an ideal of }R_j,\text{ for all }j\in \mathbb{I}_n\}$
be a finite family of ideals. If
$R$
is
$\mathcal{I}$
-symmetric, then
\begin{align*} I = \left ( \begin{matrix} I_{11} &\quad I_{12} &\quad \ldots &\quad I_{1n}\\ I_{21} &\quad I_{22} &\quad \ldots &\quad I_{2n} \\ \vdots &\quad \vdots &\quad \vdots &\quad \vdots \\ I_{n1} &\quad I_{n2} &\quad \ldots &\quad I_{nn}\end{matrix} \right ), \quad \text{where }\, I_{jk} \, : \!= \, I_j\cdot {}_{j}M_{k}+{}_{j}M_{k}\cdot I_k, \end{align*}
is an ideal of
$R$
.
Assume that, for each
$i\in \mathbb{I}_n$
, there is a partial action
$\alpha ^{(i)}=\big (D^{(i)}_g,\alpha ^{(i)}_g\big )_{g\in G}$
of
$G$
on
$R_i$
. Consider
${\mathcal{I}}_g \, :\!= \, \big \{D^{(i)}_g\,:\,i\in \mathbb{I}_n\big \}$
and suppose that
$R$
is
${\mathcal{I}}_g$
-symmetric, for all
$g\in G$
. By Lemma 4.2, the subset
$I_g$
of
$R$
given by
\begin{equation} I_{g}=\big (D^{(i)}_g\,_{i}M_{j}\big )_{i,j\in \mathbb{I}_n}, \end{equation}
is an ideal of
$R$
.
Now we recall the notion of datum for
$R$
introduced in Definition 4.2 of [Reference Bagio and Pinedo7].
Definition 4.4.
Let
$i\in \mathbb{I}_n$
and
$g\in G$
. Suppose that
$\alpha ^{(i)}$
and
${\mathcal{I}}_g$
are as above and that
$R$
is
${\mathcal{I}}_g$
-symmetric. For each
$g\in G$
, assume that there exists a collection of linear bijections
\begin{align*} \gamma _g=\left \{\gamma ^{(ij)}_g:D^{(i)}_{g^{-1}}\,_{i}M_{j}\to D^{(i)}_g\,_{i}M_{j}\right \}_{i,j\in \mathbb{I}_n}. \end{align*}
The pair
$\mathcal{D}=\big (\{\alpha ^{(i)}\}_{i\in \mathbb{I}_n}, \{\gamma _{g}\}_{g\in G}\big )$
is called a datum for
$R$
if the following statements hold: for all
$g,h\in G$
,
$i,j,k\in \mathbb{I}_n$
,
\begin{align} &\gamma ^{(ii)}_g=\alpha ^{(i)}_g,\quad \quad \gamma ^{(ij)}_e=\textrm {id}_{M_{ij}}, \end{align}
\begin{align} &\gamma ^{(ik)}_g(u)\gamma ^{(kj)}_g(v)=\gamma ^{(ij)}_g(uv),\,\,\,\text{for all } u\in D^{(i)}_{g^{-1}}\,_{i}M_{k},\,v\in D^{(k)}_{g^{-1}}\,_{k}M_{j}, \end{align}
\begin{align} &\big (\gamma ^{(ij)}_h\big )^{^{-1}}\!\!\left (D^{(i)}_{g^{-1}}\,_{i}M_{j}\cap D^{(i)}_{h}\,_{i}M_{j}\right ) \subset D^{(i)}_{(gh)^{-1}}\,_{i}M_{j}, \end{align}
\begin{align} &\gamma ^{(ij)}_g\left (\gamma ^{(ij)}_h(a)\right )=\gamma ^{(ij)}_{gh}(a),\,\,\,\text{for all } a\in \big (\gamma ^{(ij)}_h\big )^{^{-1}}\!\!\left (D^{(i)}_{g^{-1}}\,_{i}M_{j}\cap D^{(i)}_{h}\,_{i}M_{j}\right ). \\[8pt] \nonumber \end{align}
It was proved in Theorem 4.3 of [Reference Bagio and Pinedo7] that there is a partial action of
$G$
on
$R$
associated with each datum for
$R$
.
Remark 4.9.
Let
$\mathcal{D}=\big (\{\alpha ^{(i)}\}_{i\in \mathbb{I}_n}, \{\gamma _{g}\}_{g\in G}\big )$
be a datum for
$R$
. By Remark 4.10 of [Reference Bagio and Pinedo7], for all
$g,h\in G$
and
$i,j\in \mathbb{I}_n$
, we have that
-
(i)
$(\gamma ^{(ij)}_g)^{^{-1}}=\gamma ^{(ij)}_{g^{-1}}$
; -
(ii)
$\big (\gamma ^{(ij)}_h\big )^{^{-1}}\!\!\left (D^{(i)}_{g^{-1}}\,_{i}M_{j}\cap D^{(i)}_{h}\,_{i}M_{j}\right )=D^{(i)}_{(gh)^{-1}}\,_{i}M_{j}\cap D^{(i)}_{h^{-1}}\,_{i}M_{j}$
;
Remark 4.10.
Let
$\mathcal{D}=\big (\{\alpha ^{(i)}\}_{i\in \mathbb{I}_n}, \{\gamma _{g}\}_{g\in G}\big )$
be a datum for
$R$
. Assume that
$\alpha ^{(i)}$
is unital, for all
$i\in \mathbb{I}_n$
, and denote by
$1_g^{(i)}$
the identity element of
$D_{g}^{(i)}$
,
$g\in G$
. For each
$g\in G$
, consider the diagonal matrix
\begin{align} 1_g \, :\!= \, \operatorname {diag}\big (1^{(1)}_g,\ldots ,1^{(n)}_g\big )= \left ( \begin{matrix} 1^{(1)}_{g} &\quad 0 &\quad \ldots &\quad 0 \\ 0 &\quad 1^{(2)}_{g} &\quad \ddots &\quad \vdots \\ \vdots &\quad \ddots &\quad \ddots &\quad 0 \\ 0 &\quad \ldots &\quad 0 &\quad 1^{(n)}_{g} \end{matrix} \right ) \in R. \end{align}
It was proved in Lemma 4.5 of [Reference Bagio and Pinedo7 ] that
$1_g$
is a central idempotent of
$R$
if and only if
\begin{equation} 1^{(i)}_gm=m1^{(j)}_g, \quad \text{for all } m\in \!\,_{i}M_{j}\text{ and } i,j\in \mathbb{I}_n. \end{equation}
In this case, by Lemma 4.5 of [Reference Bagio and Pinedo7 ], we have that
$I_g=R1_g$
.
Definition 4.13.
A datum
$\mathcal{D}=\big (\{\alpha ^{(i)}\}_{i\in \mathbb{I}_n}, \{\gamma _{g}\}_{g\in G}\big )$
for
$R$
is called unital, if the ideal of
$R$
given in (
4.3
) is unital with identity element
$1_g$
given by (
4.11
), for all
$g\in G$
.
In order to relate the conditions of a datum for
$R$
with the conditions given in (ii) of Theorem 3.3, we consider the following statement:
-
(a)′ there is a left partial action
$\rightharpoonup _{i} \, : \, \Bbbk G\otimes R_i\to R_i$
of
$\Bbbk G$
on
$R_i$
such that
$g\rightharpoonup _{i} 1_{R_i}$
satisfies (4.12), for all
$g\in G$
and
$i\in \mathbb{I}_n$
.
Proposition 4.14.
Let
$\mathcal{D}=\big (\{\alpha ^{(i)}\}_{i\in \mathbb{I}_n}, \{\gamma _{g}\}_{g\in G}\big )$
be a unital datum for
$R$
. Then the Hopf algebra
$H=\Bbbk G$
satisfies the conditions (a)′ and the conditions (b)–(c) of Theorem 3.3 (ii).
Proof.
Let
$i\in \mathbb{I}_n$
. Consider the linear map
$\rightharpoonup _{i} \, : \, \Bbbk G\otimes R_i\to R_i$
given by
\begin{align*} g\rightharpoonup _{i}x=\gamma _g^{(ii)}(x1^{(i)}_{g^{-1}}), \quad g\in G,\,\,x\in R_i. \end{align*}
Since
$\gamma _g^{(ii)}=\alpha _g^{(i)}$
, it follows that
$\rightharpoonup _{i}$
is a left partial action of
$\Bbbk G$
on
$R_i$
and
$g\rightharpoonup _{i} 1_{R_i}=\alpha _g^{(i)}(1^{(i)}_{g^{-1}})=1_g^{(i)}$
. Hence, (a)′ is proved. To prove (b) of Theorem 3.3, we recall from (2.31) that
$\underline {}R_i\# \Bbbk G\simeq R_i\rtimes _{\alpha _i} G$
as algebras. We define the linear map
$\cdot \, : \,(R_i\rtimes _{\alpha _i} G)\otimes \,\! _{i}M_{j}\to \!\, _{i}M_{j}$
by
\begin{align*} (r\delta _g)\cdot m=r\gamma _g^{(ij)}(1^{(i)}_{g^{-1}}m)=r\gamma _g^{(ij)}(m1^{(j)}_{g^{-1}}), \quad r\in D_g^{(i)}, \,g\in G, \,m\in \!\, _{i}M_{j}. \\[-25pt] \nonumber \end{align*}
We claim that this linear map defines a left
$R_i\rtimes _{\alpha _i} G$
-module structure on
$_{i}M_{j}$
. Using that
$\gamma ^{(ij)}_e=\textrm {id}_{M_{ij}}$
it is immediate that
$(1_i\delta _e)\cdot m=m$
, for all
$m\in \!\, _{i}M_{j}$
. Moreover, for all
$g,h\in G$
,
$r\in D_g^{(i)}$
and
$s\in D_h^{(i)}$
, we have
\begin{align*} \big ((r\delta _g)(s\delta _h)\big )\cdot m&=\,r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})\delta _{gh}\cdot m\\[2pt] &=\,r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})\gamma _{gh}^{(ij)}(m1^{(j)}_{(gh)^{-1}})\\[2pt] &=\,r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})1_g^{(i)}\gamma _{gh}^{(ij)}(m1^{(j)}_{(gh)^{-1}})\\[2pt] &\overset {(4.12)}{=}\,r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})\gamma _{gh}^{(ij)}(m1^{(j)}_{(gh)^{-1}})1_g^{(j)} \\[2pt] &=\,r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})\gamma _{gh}^{(ij)}(m1^{(j)}_{(gh)^{-1}})1_g^{(j)}1_{gh}^{(j)} \\[2pt] &\overset {(4.5)}{=}\, r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})\gamma _{gh}^{(ij)}(m1^{(j)}_{(gh)^{-1}})\gamma _{gh}^{(jj)}( 1_{h^{-1}}^{(j)}1_{(gh)^{-1}}^{(j)})\\[2pt] &\overset {(4.6)}{=}\,r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})\gamma _{gh}^{(ij)}(m1^{(j)}_{(gh)^{-1}}1_{h^{-1}}^{(j)})\\[2pt] &\overset {(4.8)}{=}\, r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})\gamma _{g}^{(ij)}(\gamma _{h}^{(ij)}(m1^{(j)}_{(gh)^{-1}}1_{h^{-1}}^{(j)}))\\[2pt] &= r\alpha ^{(i)}_g(s1^{(i)}_{g^{-1}})\gamma _g^{(ij)}(\gamma _h^{(ij)}(m1^{(j)}_{h^{-1}} )1^{(j)}_{g^{-1}})\\[2pt] &=\,r\gamma ^{(ii)}_g(s1^{(i)}_{g^{-1}})\gamma _g^{(ij)}(\gamma _h^{(ij)}(m1^{(j)}_{h^{-1}} )1^{(j)}_{g^{-1}}) \\[2pt] &= \, r\gamma _g^{(ij)}(s\gamma _h^{(ij)}(m1^{(j)}_{h^{-1}} )1^{(j)}_{g^{-1}}) \\[2pt] &=\,(r\delta _g)\cdot ((s\delta _h)\cdot m). \end{align*}
To complete the proof of (b), we need to verify that
$_{i}M_{j}$
is a right
$\underline {\Bbbk G^{\textrm {op}}\# {R_j}}$
-module. By (2.32),
$\underline {\Bbbk G^{\textrm {op}}\# {R_j}}\simeq (R_j^{\textrm {op}}\ltimes _{\alpha _j} G)^{\textrm {op}}$
as algebras. Thus, we will prove that
$_{i}M_{j}$
is a left
$(R_j^{\textrm {op}}\rtimes _{\alpha _j} G)$
-module. For that, define the linear map
$\cdot \, : \, R_j^{\textrm {op}}\rtimes _{\alpha _j}\! G\otimes \,\! _{i}M_{j}\to \!\, _{i}M_{j}$
by
\begin{align*} (r\delta _g)\cdot m=\gamma _g^{(ij)}(m1^{(j)}_{g^{-1}})r,\quad r\in D_g^{(j)}, \,g\in G, \,m\in \!\, _{i}M_{j}. \end{align*}
Notice that by 4.12 this action is well defined. Also, by (4.5), we have
$(1_{R_j}\delta _e)\cdot m=\gamma _e^{(ij)}(m1^{(j)}_{e})1_{R_j}=\gamma _e^{(ij)}(m1_{R_j})1_{R_j}=m$
. Finally, observe that
\begin{align*} \big ( (r\delta _g)(s\delta _h)\big )\cdot m&=\alpha ^{(j)}_g(s1^{(j)}_{g^{-1}})r\delta _{gh}\cdot m\\[2pt] &=\gamma _{gh}^{(ij)}(m1^{(j)}_{(gh)^{-1}})\alpha ^{(j)}_g(s1^{(j)}_{g^{-1}})r\\[2pt] &=\gamma _{gh}^{(ij)}(m1^{(j)}_{(gh)^{-1}}) 1_g^{(j)}1_{gh}^{(j)}\alpha ^{(j)}_g(s1^{(j)}_{g^{-1}})r\\[2pt] &=\gamma _g^{(ij)}(\gamma _h^{(ij)}(m1^{(j)}_{h^{-1}} )1^{(j)}_{g^{-1}})\alpha ^{(j)}_g(s1^{(j)}_{g^{-1}})r \\[2pt] &=\gamma _g^{(ij)}(\gamma _h^{(ij)}(m1^{(j)}_{h^{-1}} )s1^{(j)}_{g^{-1}})r, \end{align*}
and
\begin{align*} (r\delta _g)\cdot \big ( (s\delta _h)\cdot m\big )&=\, (r\delta _g)\cdot \big (\gamma _h^{(ij)}(m1^{(j)}_{h^{-1}})s\big )\\ &=\,\gamma _g^{(ij)}\Big ( \big (\gamma _h^{(ij)}(m1^{(j)}_{h^{-1}})s\big ) 1^{(j)}_{g^{-1}}\Big )r\\ &=\,\gamma _g^{(ij)}\big ( \gamma _h^{(ij)}(m1^{(j)}_{h^{-1}})s 1^{(j)}_{g^{-1}}\big )r. \end{align*}
Given
$h\in G$
and
$m\in \!\, _{i}M_{j}$
, from (2.31) follows that
\begin{align*} (1_i\#h)\cdot m=(1_h^{(i)}\delta _h)\cdot m=\gamma _h^{(ij)}(m1^{(j)}_{h^{-1}}), \end{align*}
and (2.32) implies
\begin{align*} m\cdot (h\# 1_j)=(1^{(j)}_{h}\delta _{h})\cdot m=\gamma _{h}^{(ij)}(m1^{(j)}_{h^{-1}}). \end{align*}
Thus, (b) is proved.
Now, we will see that (c) holds. In fact, if
$n\in \!\, _{j}M_{k}$
, then
\begin{align*} (1_i\#h)\cdot mn&=\,\,\gamma _h^{(ik)}(mn1_{h^{-1}}^{(k)})\\ &\overset {(4.12)}{=}\,\,\gamma _h^{(ik)}(m1_{h^{-1}}^{(j)}n1_{h^{-1}}^{(k)})\\ &\overset {(4.6)}{=}\,\,\gamma _h^{(ij)}(m1_{h^{-1}}^{(j)})\gamma _h^{(jk)}(n1_{h^{-1}}^{(k)})\\ &=\,\,\big ((1_i\#h)\cdot m\big )\big ((1_j\#h)\cdot n\big ), \end{align*}
which finishes the proof.
We proceed with the next.
Proposition 4.15.
Assume that
$\Bbbk G$
satisfies the conditions (a)′ and the conditions (b)–(c) of Theorem 3.3 (ii). Then:
-
° given
$i\in \mathbb{I}_n$
, there exists a unital partial action
$\alpha ^{(i)}=\big (D^{(i)}_g,\alpha ^{(i)}_g\big )_{g\in G}$
of
$G$
on
$R_i$
, -
° there exists a family of
$\Bbbk$
-linear bijections
$\gamma _g=\left \{\gamma ^{(ij)}_g:D^{(i)}_{g^{-1}}\,_{i}M_{j}\to D^{(i)}_g\,_{i}M_{j}\right \}_{i,j\in \mathbb{I}_n}$
, for all
$g\in G$
,
such that
$\mathcal{D}=\big (\{\alpha ^{(i)}\}_{i\in \mathbb{I}_n}, \{\gamma _{g}\}_{g\in G}\big )$
is a unital datum for
$R$
.
Proof.
From Example 2.17, we obtain that
$\alpha ^{(i)}=\big (D^{(i)}_g,\alpha ^{(i)}_g\big )_{g\in G}$
is a unital partial action of
$G$
on
$R_i$
, where
\begin{align*} D^{(i)}_g=R_i1^{(i)}_g, \qquad 1^{(i)}_g \, :\!= \, g\rightharpoonup _{i} 1_{R_i},\qquad \alpha ^{(i)}_g(y)=g\rightharpoonup _{i} y,\quad y\in D^{(i)}_{g^{-1}}. \end{align*}
Notice that (4.12) implies that
$D^{(i)}_{g}\,_{i}M_{j}=\,\!_{i}M_{j}\,D^{(j)}_{g}$
, for all
$i,j\in \mathbb{I}_n$
and
$g\in G$
. Hence,
$R$
is
${\mathcal{I}}_g$
-symmetric, where
${\mathcal{I}}_g=\big \{D^{(i)}_g\,:\,i\in \mathbb{I}_n\big \}$
. Also, by (b) in (ii) of Theorem 3.3, for each
$(i,j)\in \mathbb{I}_n\times \mathbb{I}_n$
, we have that
$_{i}M_{j}$
is a left
$R_i\rtimes _{\alpha _i} G$
-module. Using this left action, we define the following
$\Bbbk$
-linear morphisms
\begin{equation} \gamma ^{(ij)}_g \, : \, D^{(i)}_{g^{-1}}\,_{i}M_{j}\to D^{(i)}_g\,_{i}M_{j},\quad \gamma _g^{(ij)}(x)= (1^{(i)}_g\delta _g)\cdot x,\quad x\in D^{(i)}_{g^{-1}}\,_{i}M_{j}. \end{equation}
Notice that
$\gamma ^{(ij)}_g$
is well defined. In fact, it follows from (b) of Theorem 3.3 (ii) that
$r_i\delta _e\cdot m=r_i\cdot m$
, for all
$r_i\in R_i$
and
$m\in \,\! _{i}M_{j}$
. Hence, given
$x\in D^{(i)}_{g^{-1}}\,_{i}M_{j}$
,
\begin{align*} (1^{(i)}_g\delta _g)\cdot x= ((1^{(i)}_g\delta _e)\cdot (1^{(i)}_g\delta _g))\cdot x= (1^{(i)}_g\delta _e)\cdot [\big (1^{(i)}_g\delta _g)\cdot x]= 1^{(i)}_g\cdot \big ( (1^{(i)}_g\delta _g)\cdot x\big ), \end{align*}
which implies that
$(1^{(i)}_g\delta _g)\cdot x\in D^{(i)}_g\,_{i}M_{j}$
. Also,
\begin{align*} \gamma ^{(ij)}_{g^{-1}}\gamma ^{(ij)}_g(x)=1^{(i)}_{g^{-1}}\delta _e\cdot x=1^{(i)}_{g^{-1}}\cdot x=x. \end{align*}
Similarly,
$\gamma ^{(ij)}_g\gamma ^{(ij)}_{g^{-1}}=\textrm {id}_{D^{(i)}_{g}\,_{i}M_{j}}$
and consequently
$\gamma ^{(ij)}_{g}$
is a
$\Bbbk$
-linear isomorphism with inverse
$\gamma ^{(ij)}_{g^{-1}}$
. Clearly,
$\gamma _e^{(ij)}={\textrm {id}}_{_{i}M_{j}}$
. Moreover, if
$\Phi _{ii}$
,
$\psi _{ii}$
and
$\pi _{ii}$
are the maps defined in the proof of (i)
$\Rightarrow$
(ii) of Theorem 3.3, then
\begin{align*} \gamma ^{(ii)}_g(a)=(1_g\delta _g)\cdot a=\Phi _{ii}(1_g\#g)(a)=(\psi _{ii}(1_g)\circ \pi _{ii}(g))(a) =\alpha _g(a), \end{align*}
for all
$a\in D^{(i)}_{g^{-1}}$
. Hence, (4.5) is proved. In order to prove (4.6), consider
$u\in D^{(i)}_{g^{-1}}\,_{i}M_{j}$
and
$v\in D^{(j)}_{g^{-1}}\,_{j}M_{k}$
. We recall that
$1_i\#g=1_i(g\rightharpoonup _{i} 1_i)\otimes g=1^{(i)}_g\#g$
. Then, it follows from (c) of Theorem 3.3 (ii) that
\begin{align*} \gamma ^{(ij)}_g(u)\gamma ^{(jk)}_g(v)&=\big ((1^{(i)}_g\delta _g)\cdot u\big )\big ((1^{(j)}_g\delta _g)\cdot v\big )=\big ((1^{(i)}_g\# g)\cdot u\big )\big ((1^{(j)}_g\# g)\cdot v\big )\\ &=\big ((1_i\# g)\cdot u\big )\big (1_j\# g)\cdot v\big )=(1_i\# g)\cdot (uv)\\ &=(1^{(i)}_g\# g)\cdot (uv)=(1^{(i)}_g\delta _g)\cdot (uv)=\gamma ^{(ik)}_g(uv). \end{align*}
Now, we prove (4.7). Consider
$ x\in (\gamma ^{(ij)}_h\big )^{^{-1}}\!\!\left (D^{(i)}_{g^{-1}}\,_{i}M_{j}\cap D^{(i)}_{h}\,_{i}M_{j}\right )$
. Then, observe that
$y=\gamma ^{(ij)}_{h}(x)\in D^{(i)}_{g^{-1}}\,_{i}M_{j}\cap D^{(i)}_{h}\,_{i}M_{j}$
. Since
$y=1^{(i)}_{g^{-1}}1^{(i)}_hy$
, it follows that
\begin{align*} x&=\gamma ^{(ij)}_{h^{-1}}(y)= \gamma ^{(ij)}_{h^{-1}}(1^{(i)}_{g^{-1}}1^{(i)}_hy)=\alpha _{h^{-1}}(1^{(i)}_{g^{-1}}1^{(i)}_h)\gamma ^{(ij)}_{h^{-1}}(y)=1^{(i)}_{(gh)^{-1}}1^{(i)}_{h^{-1}}\gamma ^{(ij)}_{h^{-1}}(y), \end{align*}
and hence
$x\in D^{(i)}_{h^{-1}}\,_{i}M_{j}\cap D^{(i)}_{(gh)^{-1}}\,_{i}M_{j}\subseteq D^{(i)}_{(gh)^{-1}}\,_{i}M_{j}$
. Finally, we prove (4.8). Let
$a\in \gamma ^{(ij)}_{h^{^{-1}}}( D^{(i)}_{g^{-1}}\,_{i}M_{j}\cap D^{(i)}_{h}\,_{i}M_{j})$
. Then
\begin{align*} \gamma ^{(ij)}_g\left (\gamma ^{(ij)}_h(a)\right )&=(1^{(i)}_g\delta _g)\cdot \big ((1^{(i)}_h\delta _h)\cdot a\big )=(1^{(i)}_g1^{(i)}_{gh}\delta _{gh})\cdot a=1^{(i)}_g\gamma ^{(ij)}_{gh}(a). \end{align*}
However, using that
$a\in D^{(i)}_{h^{-1}}\,_{i}M_{j}$
, we obtain
\begin{align*} \gamma ^{(ij)}_{gh}(a)&= \gamma ^{(ij)}_{gh}( 1^{(i)}_{h^{-1}}a)=\gamma ^{(ii)}_{gh}(1^{(i)}_{h^{-1}}1^{(i)}_{(gh)^{-1}})\gamma ^{(ij)}_{gh}(a) \\ &=\alpha ^{(i)}_{gh}(1^{(i)}_{h^{-1}}1^{(i)}_{(gh)^{-1}})\gamma ^{(ij)}_{gh}(a)=1^{(i)}_g\gamma ^{(ij)}_{gh}(a). \end{align*}
Thus,
$\gamma ^{(ij)}_g\left (\gamma ^{(ij)}_h(a)\right )=\gamma ^{(ij)}_{gh}(a)$
. Hence,
$\mathcal{D}=\big (\{\alpha ^{(i)}\}_{i\in \mathbb{I}_n}, \{\gamma _{g}\}_{g\in G}\big )$
is a unital datum for
$R$
.
We finish this section with the following result.
Theorem 4.17. The following assertions are equivalent:
-
(i) There is a unital datum
$\mathcal{D}=\big (\{\alpha ^{(i)}\}_{i\in \mathbb{I}_n}, \{\gamma _{g}\}_{g\in G}\big )$
for
$R$
. -
(ii) The Hopf algebra
$\Bbbk G$
satisfies the conditions (a)′ and the conditions (b)-(c) of Theorem 3.3 (ii).
5. Morita equivalent partial Hopf actions
We start by recalling from [Reference Alves and Ferraza5] the notion of Morita equivalence for partial actions of Hopf algebras.
Definition 5.1. ([Reference Alves and Ferraza5], Definition 69) Let
$H$
be a Hopf algebra and
$A$
and
$B$
two partial
$H$
-module algebras with partial actions
$\rightharpoonup _{A}$
and
$\rightharpoonup _{B}$
, respectively. We say that these two partial actions are Morita equivalent if
-
(i) there exists a strict Morita context
$(A,B, {}_AM_B ,{}_BN_A , \tau , \sigma )$
between
$A$
and
$B$
, -
(ii) there exists a partial action of
$H$
on the Morita ring
$\left ( \begin{matrix} A &\quad M\\ N \quad& B \end{matrix} \right )$
such that the restrictions to
$\left ( \begin{matrix} A &\quad 0\\ 0 &\quad 0 \end{matrix} \right )$
and to
$\left ( \begin{matrix} 0 &\quad 0\\ 0 &\quad B \end{matrix} \right )$
coincide with
$\rightharpoonup _{A}$
and
$\rightharpoonup _{B}$
, respectively.
The next result characterizes Morita equivalent Hopf partial actions.
Proposition 5.2.
Let
$H$
be a Hopf algebra and
$A$
and
$B$
two partial
$H$
-module algebras with partial actions
$\rightharpoonup _{A}$
and
$\rightharpoonup _{B}$
, respectively. The following statements are equivalent:
-
(1)
$\rightharpoonup _{A}$
and
$\rightharpoonup _{B}$
are Morita equivalent,
-
(2) The following assertions hold:
-
(i) there exists a strict Morita context
$(A,B,M,N,\mu ,\nu )$
, -
(ii)
$M$
is a left
$\underline {A\# H}$
-module and a right
$\underline {H^{\textrm {op}}\# B}$
-module such that
for all
\begin{align*} & (1_A\#h)\cdot m=m\cdot (h\# 1_B),& &\,\,(1_A\#h)\cdot a= h\rightharpoonup _{A} a,&\\ &\,\,\,(a'\# 1_H)\cdot m=a'm,& &m\cdot (1_H\#b)=mb,& \end{align*}
$h\in H$
,
$m\in M$
,
$a,a'\in A$
,
$b\in B$
,
-
(iii)
$N$
is a left
$\underline {B\# H}$
-module and a right
$\underline {H^{\textrm {op}}\# A}$
-module such that
for all
\begin{align*} & (1_B\#h)\cdot n=n\cdot (h\# 1_A),& &\,\,(1_B\#h)\cdot b= h\rightharpoonup _{B} b,&\\ &\,\,\,(b'\# 1_H)\cdot n=b'n,& &n\cdot (1_H\#a)=na,& \end{align*}
$h\in H$
,
$n\in N$
,
$b,b'\in B$
,
$a\in A$
.
-
Proof.
The proof follows applying Theorem 3.3 to the Morita ring
$ R = \left ( \begin{matrix} A &\quad M \\ N &\quad B \end{matrix} \right ).$
Now we recall the following notion.
Definition 5.3.
([Reference Alves and Ferraza5], Definition 72) Let
$H$
be a Hopf algebra and
$B$
be a right partial
$H$
-module algebra. A vector space
$N$
is a right partial
$(B,H)$
-module if
$N$
is a right
$B$
-module together with a linear map
$N\otimes H\to N$
,
$n\otimes h\mapsto nh$
such that
-
(i)
$n1_H=n$
, -
(ii)
$((nk)b)h=(n(kh_{(1)}))(b\cdot h_{(2)})$
,
for all
$h,k\in H$
,
$b\in B$
and
$n\in N$
.
We conclude this work by highlighting an inconsistency in the statement of Proposition 73 of [Reference Alves and Ferraza5]. In fact, the authors assume that
$H$
partially acts on
$B$
from the left. However, since
$N$
is a right
$(B,H)$
-module,
$H$
indeed acts partially on
$B$
from the right.
Funding statement
D. Bagio was partially supported by CAPES-PRINT 88887.894056/2023-00. H. Pinedo was partially supported by CAPES-PRINT 88887.895167/2023-00 and by FAPESP, process n∘ 2023/14066-5.
Competing interests
The authors have no conflict of interest to declare that are relevant to this article.
