MORITA EQUIVALENCE AND GLOBALIZATION FOR PARTIAL HOPF ACTIONS ON NONUNITAL ALGEBRAS MARCELO MUNIZ ALVES AND TIAGO LUIZ FERRAZZA

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MORITA EQUIVALENCE AND GLOBALIZATION FOR PARTIAL HOPF

ACTIONS ON NONUNITAL ALGEBRAS

MARCELO MUNIZ ALVES AND TIAGO LUIZ FERRAZZA

Abstract. In this work we investigate partial actions of a Hopf algebra Hon nonunital alge-

bras and the associated partial smash products. We show that our partial actions correspond

to nonunital algebras in the category of partial representations of H. The central problem of

existence of a globalization for a partial action is studied in detail, and we provide suﬃcient con-

ditions for the existence (and uniqueness) of a minimal globalization for associative algebras in

general. Extending previous results by Abadie, Dokuchaev, Exel and Simon, we deﬁne Morita

equivalence for partial Hopf actions, and we show that if two symmetrical partial actions are

Morita equivalent then their standard globalizations are also Morita equivalent. Particularizing

to the case of a partial action on an algebra with local units, we obtain several strong results on

equivalences of categories of modules of partial smash products of algebras and partial smash

products of k-categories.

1. Introduction

A partial action of a Hopf algebra Hon a unital algebra Ais a weakened version of the well-

known concept of H-module algebra. A main motivation for introducing this subject was the

previous development of a theory of partial group actions, which had then recently culminated

in the Galois theory for partial group actions obtained by Dokuchaev, Ferrero and Paques in

[19].

Partial group actions on algebras were originally introduced by Exel in the area of C∗-algebras

[20], and later were being investigated from a more purely algebraic point of view. Some

important features are the construction of a partial skew group algebra A∗Gassociated to a

partial action of Gon A, which is a G-graded algebra, which is well-suited for the computation

of cohomology invariants [20, 3]; the existence, with mild hypotheses, of a globalization or

enveloping action of Gon an algebra Bwhich contains Aas an ideal, with partial action

induced by the G-action on B[17]; a Morita context between the skew group algebra A∗G

and the skew group algebra B∗Gof its globalization [17]; a Galois theory for partial actions

[19].

Partial actions of Hopf algebras on unital algebras ﬁrst appear in [10]. In this paper,

Caenepeel and Jansen show that to a (unital) partial H-module algebra Athere corresponds a

nonunital algebra structure on the tensor product A⊗H, and the idempotent 1A#1Hgenerates

a unital ideal called the partial smash product A#H. It was shown in [4] that there is always

an H-module algebra Bwhich globalizes A, and that there is a Morita context between A#H

and B#H. There is a Hopf-Galois theory as well for partial actions and coactions [10], and

partial H-module algebras may be characterized as algebras in an appropriate category, the

category of partial representations [6]; (see the survey [15]).

From this point on the theory was extended in the direction of generalizing the objects

that act, the ones that are acted upon, or both. In the ﬁrst case, there are partial actions of

Date: October 31, 2022.

2020 Mathematics Subject Classiﬁcation. Primary 16T05, Secondary 16S40.

Key words and phrases. Partial action; partial representation; nonunital algebra; globalization; Morita equiva-

lence.

This study was ﬁnanced in part by the Coordena¸c˜ao de Aperfei¸coamento de Pessoal de N´ıvel Superior – Brasil

(CAPES) – Finance Code 001. The ﬁrst author was partially supported by Conselho Nacional de Desenvolvi-

mento Cient´ıﬁco e Tecnol´ogico - CNPq (project 309469/2019-8).

arXiv:2210.15403v2 [math.RA] 28 Oct 2022

weak Hopf algebras on unital algebras [11]; in the second case, there are the partial actions of

Hopf algebras on k-categories [2]; and there are partial actions of multiplier Hopf algebras on

nondegenerate nonunital algebras, which fall in the third case [9, 21].

In the present work we investigate partial actions of Hopf algebras on nonunital algebras

in the broadest sense possible, including the degenerate ones, but we also study in depth the

case of partial actions on algebras with local units. This approach was motivated by partial

Hopf actions on k-categories, which were in turn inspired by group actions and Hopf actions

on categories [13, 23].

We relate partial H-module nonunital algebras and nonunital algebras in the category of

partial representations of H(Theorem 22). Constructing the partial smash product A#H, we

relate its left modules with the (A, H)-modules, extending a result from [12]. We introduce

globalizations and we establish suﬃcient conditions for the existence and uniqueness of minimal

globalizations (Thm. 41 and Thm. 43). It is also proved that there is a strict Morita context

between A#Hand B#H, where Bis a globalization of the partial action of Hon A.

In [2] it was shown that every partial action on a k-category Cinduces a partial action on

the “matrix algebra” a(C), which is an algebra with local units. Conversely, given an algebra

Awith a system of local units S, one may construct a k-category CS(A), and we study partial

actions on Athat induce partial actions on this category. We prove that the category of the

left unital modules of Ais equivalent to the category of the left modules of the category CS(A).

Also, if Cis a k-category, we proved that the category of the left C-modules and the category

of the left unital a(C)-modules are equivalent.

We develop a theory of Morita equivalence of partial Hopf actions, extending the concept

and results of Morita equivalence of partial group actions presented in [1]. Following the

ﬁrst results of Abadie, Dokuchaev, Exel and Sim´on, in [1], we deﬁne the concept of Morita

equivalence of partial H-actions, and with this we prove that every symmetric partial action

on an idempotent algebra is Morita equivalent to a partial action where the algebra has trivial

right (or left) annihilator (Thm. 86). In [1], the authors constructed a canonical globalization

for a regular partial group action and proved that whenever two regular partial G-actions are

Morita equivalent, the global actions of its canonical globalizations are also Morita equivalent;

we have obtained a similar result for partial H-actions (Thm. 87).

Finally, it also holds that if His a Hopf algebra and Ais a partial H-module algebra with a

system of local units S, then the categories of modules over the algebras A#Hand a(CS(A))#H,

and over the k-category CS(A)#H, are all equivalent (Corollaries 84 and 85).

Throughout this work, all the linear structures will be considered over a ﬁeld k; for instance,

algebra means k-algebra. Unless otherwise stated, “module” stands for “left module” and

“partial action” stands for “left partial action”.

2. Partial Hopf Actions

The intent of the next three subsections is to relate the suggested deﬁnition of partial Hopf

actions on associative algebras with previous results found in the literature.

2.1. Partial Hopf Actions and the Smash Product. In [10], Caenepeel and Jansen intro-

duced the concept of a Hopf algebra Hacting partially on a unital algebra Aas a linear map

from H⊗Ato Awhich satisﬁes some necessary and suﬃcient conditions for the respective

smash product A#Hto be a unital, associative algebra.

Deﬁnition 1 ([10]).Let Hbe a Hopf algebra and Aan algebra with unit 1A. A linear map

·:H⊗A−→ A,h⊗a7→ h·ais called a partial action of Hon Aif, for all a, b ∈A,h, g ∈H,

(1) 1H·a=a;

(2) h·(ab) = P(h(1) ·a)(h(2) ·b);

(3) h·(g·a) = P(h(1) ·1A)(h(2)g·a),

where ∆(h) = Ph(1) ⊗h(2). In this case we say that Ais a partial H-module algebra with

unity. If the equality

(4) h·(g·a) = P(h(1)g·a)(h(2) ·1A)

also holds, then we say that this partial action is symmetrical [2] .

Inspired by [2] and, mainly, by [10], we introduce the following deﬁnition of a partial action

on associative algebras in general.

Deﬁnition 2. Let Abe an associative algebra. A linear map ·:H⊗A−→ A,h⊗a7−→ h·a

will be called a partial action of Hon Aif, for all a, b ∈A,h, k ∈H,

(1) 1H·a=a;

(2) h·(a(k·b)) = P(h(1) ·a)(h(2)k·b).

In this case, Awill be called a partial H-module algebra. We will say that the partial action is

symmetrical if, additionally,

(3) h·((k·b)a) = P(h(1)k·b)(h(2) ·a)

for every a, b ∈A,h, k ∈H.

In [9] the authors considered partial actions of multiplier Hopf algebras on nondegenerate

algebras; here we are considering partial actions of Hopf algebras on all kind of associative

algebras, introducing additional nondegeneracy properties only when needed. At the end of

this section we will show that when His a Hopf algebra and Ais a nondegenerate algebra,

then both deﬁnitions of partial action coincide.

Recall that the right annihilator of an algebra Ais the ideal

r(A) = {a∈A|ba = 0,∀b∈A}.

Analogously, the left annihilator of Ais the ideal

l(A) = {b∈A|ba = 0,∀a∈A}.

The following lemmas follow closely the equivalence presented in [10] mentioned in the be-

ginning of this section. For this, we will assume that His a Hopf algebra, Ais an associative

algebra with r(A) = 0, ·:H⊗A→Ais a linear map given by ·(h⊗a) = h·aand A#His its

associated smash product, which is the vector space A⊗Hendowed with the product

(a#h)(b#k) = a(h(1) ·b)#h(2)k.

As it is customary, to avoid confusion with the tensor algebra A⊗H, we will use the notation

a#hfor the vector a⊗has an element of the algebra A#H.

Lemma 3. A#His an associative algebra if and only if

h·(a(k·b)) = X(h(1) ·a)(h(2)k·b)

for every h, k ∈H,a, b ∈A.

Proof. In fact, A#His an associative algebra if and only if for every a, b, c ∈A,h, k, l ∈H,

((c#l)(a#h))(b#k)=(c#l)((a#h)(b#k))

Xc(l(1) ·a)(l(2)h(1) ·b)#l(3)h(2)k=Xc(l(1) ·(a(h(1) ·b)))#l(2)h(2)k

⇓I⊗ε, k=1H

Xc(l(1) ·a)(l(2)h·b)=c(l·(a(h·b)))

c[X(l(1) ·a)(l(2)h·b)−(l·(a(h·b)))] = 0.

Since r(A) = 0, we have the required equality. Conversely, if h·(a(k·b)) = P(h(1) ·a)(h(2)k·b),

clearly A#His associative. 

The next two lemmas are proved similarly, using a“support element” cas before.

Lemma 4. A#His an A-bimodule with structure given by

b(a#h)b0=Xba(h(1) ·b0)#h(2)

if and only if

h·ab =X(h(1) ·a)(h(2) ·b),

for every h∈H,a, b ∈A.

Lemma 5. The linear map ι:A→A#H,a7→ a#1H, is a right A-linear morphism if and

only if 1H·a=a.

In the previous three lemmas, even if r(A)6= 0, the axioms of a partial H-module algebra

guarantee that the respective smash product will be an associative algebra, an A-bimodule and

that the inclusion ι:A→A#His a right A-linear morphism. We only need r(A) = 0 for the

converse.

Proposition 6. Let Hbe a cocommutative Hopf algebra. If Aand Bare both (symmetrical)

partial H-module algebras, then A⊗Bis a (symmetrical) partial H-module algebra via

h·(a⊗b) = Ph(1) ·a⊗h(2) ·b.

In [2], this result was obtained (for unital algebras) as a consequence of a similar result for

partial Hopf actions on categories.

2.2. Partial H-module Algebras and Algebras in HMpar.In [6], Alves, Batista e Ver-

cruysse proved that, when the antipode of His bijective, there is a bijective correspondence

between partial H-module algebras with symmetrical partial actions and unital algebras in the

category of the partial H-modules. We will show that an analogous correspondence still holds

for nonunital algebras Athat satisfy at least one of the following properties:

(1) A2=A;

(2) l(A) = 0;

(3) r(A) = 0.

In order to do so, we consider the category of (A, H)-modules associated to a partial H-

module Aintroduced in [12]. We will show that, under the same conditions above, the category

of (A, H)-modules is equivalent to the category of A#H-modules, and then we use the fact that

the partial H-modules are the same as modules over an appropriate partial smash product.

We begin by recalling the deﬁnition of a partial representation of a Hopf algebra from [8],

which is a shorter version of the one originally introduced in [6].

Proposition 7. [8, Lemma 2.11] Let Hbe a Hopf k-algebra, Bbe a unital k-algebra and

π:H→Bbe a linear map. The following are equivalent:

(1) π:H→Bsatisﬁes

(PR1) π(1H) = 1B;

(PR2) π(h)π(k(1))π(S(k(2))) = π(hk(1))π(S(k(2))), for every h, k ∈H;

(PR3) π(h(1))π(S(h(2)))π(k) = π(h(1))π(S(h(2))k), for every h, k ∈H.

(2) π:H→Bsatisﬁes

(PR1) π(1H) = 1B;

(PR4) π(h)π(S(k(1)))π(k(2)) = π(hS(k(1)))π(k(2)), for every h, k ∈H;

(PR5) π(S(h(1)))π(h(2))π(k) = π(S(h(1)))π(h(2)k), for every h, k ∈H.

Deﬁnition 8. Let Hbe a Hopf k-algebra, and let Bbe a unital k-algebra. A partial repre-

sentation of Hin Bis a linear map π:H→Bwhich satisﬁes the equivalent conditions of

Proposition 7.

Remark 9 ([6]).If His cocommutative, then the items in the deﬁnition of a partial represen-

tation coalesce into (PR1),(PR2) and (PR5).

We will show in the following that, under mild conditions, a partial H-module algebra A

carries a partial representation π:H→End(A) (this is well-known when Ais unital).

First note that if Ais any partial H-module algebra, then for all x, y ∈A,h∈H, we have

that Xh(1) ·S(h(2))·xy =X(h(1) ·S(h(2))·x)y, (1)

because

Xh(1) ·S(h(2))·xy =Xh(1) ·[(S(h(3))·x)(S(h(2))·y)]

=X(h(1) ·S(h(4))·x)(h(2)S(h(3))·y)

=X(h(1) ·S(h(2))·x)y.

Equality (1) means that the linear map a7→ Ph(1) ·S(h(2))·ais a right A-module map from

Ato Afor every h∈H. Analogously, if the partial action is symmetrical, the equality

XS(h(1))·h(2) ·xy =x(XS(h(1))·h(2) ·y) (2)

means that the linear map a7→ PS(h(1))·h(2) ·ais a left A-module map from Ato Afor every

h∈H.

Lemma 10. Let Abe a partial H-module algebra with symmetrical partial action and Ha Hopf

algebra. If Ais either idempotent, l(A) = 0 or r(A) = 0 and the antipode of His bijective,

then the linear map π:H→End(A), deﬁned by π(h)(a) = h·a, is a partial representation.

Proof. (PR1) is straightforward: given a∈A,π(1H)(a) = 1H·a=a, therefore π(1H) = idA.Let

us proceed by proving (PR2) and (PR3) ﬁrst for an idempotent algebra A.

Since for any element a∈Athere exist b1, . . . , bn, c1, . . . , cn∈Asuch that a=Pn

i=1 biciand

πis a linear map, we only need to check the axioms of partial representation for elements of

the form xy ∈A. Let x, y ∈Aand h, k ∈H, and

Xπ(h)π(k(1))π(S(k(2)))(xy) = Xh·k(1) ·S(k(2))·(xy)

(1)

=Xh·((k(1) ·S(k(2))·x)y)

=X(h(1)k(1) ·S(k(2))·x)(h(2) ·y)

=X(h(1)k(1) ·S(k(4))·x)(h(2)k(2)S(k(3))·y)

=Xhk(1) ·[(S(k(3))·x)(S(k(2))·y)]

=Xhk(1) ·S(k(2))·(xy)

=Xπ(hk(1))π(S(k(2)))(xy).

This proves that item (PR2) holds. For item (PR3), we have that

Xπ(k(1))π(S(k(2)))π(h)(xy) = Xk(1) ·S(k(2))·h·(xy)

=Xk(1) ·S(k(2))·(h(1) ·x)(h(2) ·y)

=Xk(1) ·[(S(k(3))h(1) ·x)(S(k(2))·h(2) ·y)]

=X(k(1) ·S(k(2))h(1) ·x)(h(2) ·y)

=Xk(1) ·[(S(k(3))h(1) ·x)(S(k(2))h(2) ·y)]

=Xk(1) ·S(k(2))h·(xy)

=Xπ(k(1))π(S(k(2)h))(xy).

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MORITAEQUIVALENCEANDGLOBALIZATIONFORPARTIALHOPFACTIONSONNONUNITALALGEBRASMARCELOMUNIZALVESANDTIAGOLUIZFERRAZZAAbstract.InthisworkweinvestigatepartialactionsofaHopfalgebraHonnonunitalalge-brasandtheassociatedpartialsmashproducts.Weshowthatourpartialactionscorrespondtononunitalalgebrasinthecategoryofp...

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MORITA EQUIVALENCE AND GLOBALIZATION FOR PARTIAL HOPF ACTIONS ON NONUNITAL ALGEBRAS MARCELO MUNIZ ALVES AND TIAGO LUIZ FERRAZZA

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