Partial Hopf actions, partial invariants and a Morita context
Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions [1]. This allows to extend several results from the theory of partial grou...
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | Partial Hopf actions, partial invariants and a Morita context / Marcelo Muniz S. Alves, E. Batista // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 1–19. — Бібліогр.: 11 назв. — англ. |
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irk-123456789-1546162019-06-16T01:29:46Z Partial Hopf actions, partial invariants and a Morita context Marcelo Muniz S. Alves Batista, E. Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions [1]. This allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we explore some properties of the fixed point subalgebra with relation to a partial action of a Hopf algebra. We also construct, for partial actions of finite dimensional Hopf algebras a Morita context relating the fixed point subalgebra and the partial smash product. This is a generalization of a well known result in the theory of Hopf algebras [9] for the case of partial actions. Finally, we study Hopf-Galois extensions and reobtain some classical results in the partial case. 2009 Article Partial Hopf actions, partial invariants and a Morita context / Marcelo Muniz S. Alves, E. Batista // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 1–19. — Бібліогр.: 11 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:16W30; 57T05, 16S40, 16S35. http://dspace.nbuv.gov.ua/handle/123456789/154616 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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Partial actions of Hopf algebras can be considered as a generalization of partial actions of groups on algebras. Among important properties of partial Hopf actions, it is possible to assure the existence of enveloping actions [1]. This allows to extend several results from the theory of partial group actions to the Hopf algebraic setting. In this article, we explore some properties of the fixed point subalgebra with relation to a partial action of a Hopf algebra. We also construct, for partial actions of finite dimensional Hopf algebras a Morita context relating the fixed point subalgebra and the partial smash product. This is a generalization of a well known result in the theory of Hopf algebras [9] for the case of partial actions. Finally, we study Hopf-Galois extensions and reobtain some classical results in the partial case. |
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Marcelo Muniz S. Alves Batista, E. |
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Marcelo Muniz S. Alves Batista, E. Partial Hopf actions, partial invariants and a Morita context Algebra and Discrete Mathematics |
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Marcelo Muniz S. Alves Batista, E. |
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Marcelo Muniz S. Alves |
| title |
Partial Hopf actions, partial invariants and a Morita context |
| title_short |
Partial Hopf actions, partial invariants and a Morita context |
| title_full |
Partial Hopf actions, partial invariants and a Morita context |
| title_fullStr |
Partial Hopf actions, partial invariants and a Morita context |
| title_full_unstemmed |
Partial Hopf actions, partial invariants and a Morita context |
| title_sort |
partial hopf actions, partial invariants and a morita context |
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Інститут прикладної математики і механіки НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/154616 |
| citation_txt |
Partial Hopf actions, partial invariants and a Morita context / Marcelo Muniz S. Alves, E. Batista // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 3. — С. 1–19. — Бібліогр.: 11 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
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AT marcelomunizsalves partialhopfactionspartialinvariantsandamoritacontext AT batistae partialhopfactionspartialinvariantsandamoritacontext |
| first_indexed |
2025-07-14T06:39:42Z |
| last_indexed |
2025-07-14T06:39:42Z |
| _version_ |
1837603417467387904 |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 3. (2009). pp. 1 – 19
c⃝ Journal “Algebra and Discrete Mathematics”
Partial Hopf actions, partial invariants and a
Morita context
Marcelo Muniz S. Alves and Eliezer Batista
Communicated by guest editors
Abstract. Partial actions of Hopf algebras can be considered
as a generalization of partial actions of groups on algebras. Among
important properties of partial Hopf actions, it is possible to as-
sure the existence of enveloping actions [1]. This allows to extend
several results from the theory of partial group actions to the Hopf
algebraic setting. In this article, we explore some properties of the
fixed point subalgebra with relation to a partial action of a Hopf
algebra. We also construct, for partial actions of finite dimensional
Hopf algebras a Morita context relating the fixed point subalgebra
and the partial smash product. This is a generalization of a well
known result in the theory of Hopf algebras [9] for the case of par-
tial actions. Finally, we study Hopf-Galois extensions and reobtain
some classical results in the partial case.
Dedicated to Professor Miguel Ferrero
on occasion of his 70-th anniversary
Introduction
Partial group actions were first defined by R. Exel in the context of op-
erator algebras and they turned out to be a powerful tool in the study of
C∗-algebras generated by partial isometries on a Hilbert space [6]. The
developments originated by the definition of partial group actions, soon
became an independent topic of interest in ring theory [4]. Now, the
2000 Mathematics Subject Classification: 16W30; 57T05, 16S40, 16S35.
Key words and phrases: partial Hopf action, partial group action, partial smash
product, Hopf-Galois extensions.
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.2 Partial Hopf actions and partial invariants
results are formulated in a purely algebraic way, independent of the C∗
algebraic techniques which originated them.
A partial action � of a group G on a (possibly non-unital) k-algebra A
is a pair of families of subsets of A and maps indexed by G,
� = ({�g}g∈G, {Dg}g∈G), where each Dg is an ideal of A and each �g
is an algebra isomorphism �g : Dg−1 → Dg satisfying the following con-
ditions:
(i) De = A and �e = IA;
(ii) �g(Dg−1 ∩Dℎ) = Dg ∩Dgℎ for every g, ℎ ∈ G;
(iii) �g(�ℎ(x)) = �gℎ(x) for every x ∈ Dg−1 ∩D(gℎ)−1 .
A first example of partial action is the following: If G acts on a algebra
B by automorphisms and A is an ideal of B, then we have a partial action
� on A in the following manner: letting �g stand for the automorphism
corresponding to g, take Dg = A ∩ �g(A), and define �g : Dg−1 → Dg as
the restriction of the automorphism �g to Dg.
Partial Hopf actions were motivated by an attempt to generalize the
notion of partial Galois extensions of commutative rings [5] to a broader
context. The definition of partial Hopf actions and co-actions were intro-
duced by S. Caenepeel and K. Janssen in [2], using the notions of partial
entwining structures. In particular, partial actions of G determine partial
actions of the group algebra kG in a natural way. In the same article,
the authors also introduced the concept of partial smash product, which
in the case of the group algebra kG, turns out to be the crossed product
by a partial action A ⋊� G. Further developments in the theory of par-
tial Hopf actions were done by C. Lomp in [8], where the author pushed
forward classical results of Hopf algebras concerning smash products, like
the Blattner-Montgomery and Cohen-Montgomery theorems [9].
In [1], we proved the theorem of existence of an enveloping action for
a partial Hopf action, that is, if H is a Hopf algebra which acts partially
on a unital algebra A, then there exists an H-module algebra B such
that A is isomorphic to a right ideal of B, and the restriction of the
action of H to this ideal is equivalent to the partial action of H on A.
Basically, the same ideas for the proof of the existence of an enveloping
action for a partial group action [4] are present in the Hopf algebraic
case. In the same article, we also proved many results related to the
enveloping action: The existence of a Morita context between the partial
smash product A#H, where H is a Hopf algebra which acts partially
on the unital algebra A, and the smash product B#H, where B is an
enveloping action of A. The conditions for the existence of an enveloping
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.M. M. S. Alves, E. Batista 3
co-action associated to a partial co-action of a Hopf algebra H on a unital
algebra A. Finally, we introduced the notion of partial representation of
a Hopf algebra and showed that, under certain conditions on the algebra
H, the partial smash product A#H carries a partial representation of H.
In this work, we push forward the results obtained in [1]. First, we
define the invariant sub-algebra AH ⊆ A, where H is a Hopf algebra
acting partially on a unital algebra A, and explore some of its properties.
In what follows, we prove the existence of a Morita context between the
invariant sub-algebra AH and the partial smash product A#H, general-
izing a classical result in the theory of Hopf algebras [9]. We also study
Hopf-Galois extensions when H is finite-dimensional and reobtain, in the
partial case, classical results such as: if the trace mapping is surjective,
then the Morita context is strict if and only if the extension AH ⊂ A is
Hopf-Galois.
It is worth to mention that in the paper [2], the authors considered
a partial coaction of a Hopf algebra H on a unital algebra A and estab-
lished a Morita context relating the subalgebra of coinvariants ACoH and
the dual smash product #(H,A); here we use the fact that H is finite-
dimensional to build the Morita context relating directly AH and A#H,
and to study Hopf-Galois extensions with a more elementary approach.
1. Partial Hopf actions
We recall that a left action of a Hopf algebra H on a unital algebra A is a
linear mapping � : H⊗A→ A, which we will denote by �(ℎ⊗a) = ℎ⊳a,
such that
(i) ℎ⊳ (ab) =
∑
(ℎ(1) ⊳ a)(ℎ(2) ⊳ b),
(ii) 1⊳ a = a
(iii) ℎ⊳ (k ⊳ a) = ℎk ⊳ a
(iv) ℎ⊳ 1A = �(ℎ)1A.
We also say that A is an H module algebra. Note that (ii) and (iii) say
that A is a left H-module.
Definition 1. A partial action of the Hopf algebra H on the algebra A
is a linear mapping � : H ⊗ A → A, denoted here by �(ℎ ⊗ a) = ℎ ⋅ a,
such that
(i) ℎ ⋅ (ab) =
∑
(ℎ(1) ⋅ a)(ℎ(2) ⋅ b),
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.4 Partial Hopf actions and partial invariants
(ii) 1 ⋅ a = a,
(iii) ℎ ⋅ (g ⋅ a) =
∑
(ℎ(1) ⋅ 1A)((ℎ(2)g) ⋅ a).
In this case, we call A a partial H module algebra.
Because of the good dual properties of Hopf algebras, one can also
define the concept of partial (right) coaction of a Hopf algebra H on a
unital algebra A, which will be important later.
Definition 2. A partial coaction of the Hopf algebra H on the algebra A
is a linear mapping � : A→ H ⊗A, denoted by �(a) =
∑
a[0] ⊗ a[1], such
that
(i) �(a.b) = �(a).�(b), ∀a, b ∈ A,
(ii) (I ⊗ �)�(a) = a, ∀a ∈ A,
(iii) (�⊗ I)�(a) = (�(1A)⊗ 1H)((I ⊗Δ)�(a)).
It is easy to see that every action is also a partial action (and the
same holds for coactions).
As a basic example, consider a partial action � of a group G on an
unital algebra A. Suppose that each Dg is also a unital algebra, that is,
Dg is of the form Dg = A1g then there is a partial action of the group
algebra kG on A defined on the elements of the basis by
g ⋅ a = �g(a1g−1), (1)
and extended linearly to all elements of kG.
There is an important class of examples of partial Hopf actions: these
induced by total actions. Basically, the induced partial actions can be
described by the following result.
Proposition 1. Let H be a Hopf algebra, B a H-module algebra and let
A be a right ideal of B with unity 1A. Then H acts partially on A by
ℎ ⋅ a = 1A(ℎ⊳ a)
The proof of this proposition is a straightforward calculation and can
be seen in [1].
If the algebra A is a bilateral ideal of B, then its unit 1A is a cen-
tral idempotent of B and the induced partial action has a symmetric
formulation, satisfying the additional relation [1],
(iv) ℎ ⋅ (g ⋅ a) =
∑
((ℎ(1)g) ⋅ a)(ℎ(2) ⋅ 1A).
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.M. M. S. Alves, E. Batista 5
This property is satisfied by partial actions of group algebras kG.
As a nontrivial example of partial Hopf action, we can consider the
restriction of the action of the dual group algebra kG∗ of a finite group
G on the group algebra kG. Let {pg}g∈G be the dual basis for kG∗, the
(global) action of kG∗ on kG is given by pg ⊳ ℎ = �g,ℎℎ. Consider now a
normal subgroup N ⊴ G, N ∕= {1}, such that char(k) ∤ ∣N ∣; let eN ∈ kG
be the central idempotent
eN =
1
∣N ∣
∑
n∈N
n,
and let A be the ideal A = eNkG, which is also a unital algebra with
1A = eN . It is possible to restrict the action of kG∗ on kG to a partial
action on A. Given x ∈ G and pg ∈ kG∗, note that
pg ⊳ (eNx) =
∑
ℎ∈G
(pgℎ−1 ⊳ eN )(pℎ ⊳ x)
=
∑
ℎ∈G
(pgℎ−1 ⊳ eN )pℎ(x)x
= (pgx−1 ⊳ eN )x
=
1
∣N ∣
∑
n∈N
pgx−1(n)nx
which is equal to (1/∣N ∣)g if gx−1 ∈ N , and is zero otherwise. Hence, if
gx−1 ∈ N ,
pg ⋅ (eNx) = eN (pg ⊳ eNx) = (1/∣N ∣)eNg = (1/∣N ∣)eNx
and pg ⋅ (eNx) = 0 otherwise. Therefore, pg ⋅ eN ∕= �(pg)eN when g ∈ N
and the action is really partial.
On the other hand, given a partial action of a Hopf algebra H on a
unital algebra A, we can construct its enveloping action. For this intent,
we need some preliminary definitions.
Definition 3. Let A and B be two partial H-module algebras. We will
say that a morphism of algebras � : A → B is a morphism of partial
H-module algebras if �(ℎ ⋅ a) = ℎ ⋅ �(a) for all ℎ ∈ H and all a ∈ A. If �
is an isomorphism, we say that the partial actions are equivalent.
Definition 4. Let B be an H-module algebra and let A be a right ideal
of B with unity 1A. We will say that the induced partial action on A is
admissible if B = H ⊳A.
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.6 Partial Hopf actions and partial invariants
Definition 5. Let A be a partial H-module algebra. An enveloping action
for A is a pair (B, �), where
(i) B is a (not necessarily unital) H-module algebra, that is, the item
(iv) of the definition of total action doesn’t need to be satisfied.
(ii) The map � : A→ B is a monomorphism of algebras.
(iii) The sub-algebra �(A) is a right ideal in B.
(iv) The partial action on A is equivalent to the induced partial action
on �(A).
(v) The induced partial action on �(A) is admissible.
Then we have the following result [1].
Theorem 1. Let A be a partial H-module algebra and let ' : A →
Homk(H,A) be the map given by '(a)(ℎ) = ℎ ⋅a, and let B = H⊳'(A);
then (B,') is an enveloping action of A.
This enveloping action is called standard enveloping action, it is min-
imal in the sense that for every other enveloping action (B′, �), there is
a epimorphism of H module algebras Φ : B′ → B such that Φ(�(A)) =
'(A).
As an example of globalization of partial Hopf actions, let H4 be the
Sweedler 4-dimensional Hopf algebra, with � = {1, g, x, xg} as a basis
over the field k, where cℎar(k) ∕= 2. Another basis for H4 is given by the
elements
e1 = (1 + g)/2, e2 = (1− g)/2, ℎ1 = xe1, ℎ2 = xe2
where the ei’s form a complete system of primitive orthogonal idempo-
tents of H4, and the ideal generated by the ℎi’s is the radical of H4
(the structure constants for the product and coproduct with respect
to this basis can be found in [1], for instance). Taking the dual ba-
sis �∗ = {1∗, g∗, x∗, (xg)∗}, we obtain an isomorphism of Hopf algebras
: H∗
4 → H4 given by 1∗ 7→ e1, g
∗ 7→ e2, x
∗ 7→ ℎ1 and (xg)∗ 7→ ℎ2. In the
reference [2] the authors constructed a partial coaction � : k → k ⊗H4,
given by 1 7→ 1⊗e, where e = 1
2+
1
2g−�xg. This partial coaction induces
a partial H∗
4 action on k by ℎ ⋅ 1 = ℎ(e) for each ℎ ∈ H∗
4 . For the basis
elements of H∗
4 , considering the above isomorphism, we have
e1 ⋅ 1 =
1
2
, e2 ⋅ 1 =
1
2
, ℎ1 ⋅ 1 = 0, ℎ2 ⋅ 1 = −�.
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.M. M. S. Alves, E. Batista 7
Globalizing, we have the map ' : k → Homk(H
∗
4 , k)
∼= H4 defined by
'(1)(ℎ) = (ℎ ⋅ 1), for each ℎ ∈ H∗
4 . Then, identifying H∗
4 with H4 by ,
we have
e1⊳'(1) =
1
2
, e2⊳'(1) =
1
2
g−�xg, ℎ1⊳'(1) = 0, ℎ2⊳'(1) = −�1
and hence the minimal enveloping action is given by (B,'), where B =
⟨1, e⟩k (which is isomorphic to k × k as an algebra).
In what follows, unless stated otherwise, we will consider only the
case that A is a bilateral ideal of B, and then the partial action satisfies
the symmetric property (iv) above.
2. Partial invariant subalgebras
In the theory of partial group actions, one can define the invariant sub-
algebra in the following manner [5]: If � is a partial action of a group G
on a unital algebra A, such that each ideal Dg is unital, for every g ∈ G,
then the invariant subalgebra is the set
A� = {a ∈ A ∣�g(a1g−1) = a1g, ∀g ∈ G}.
It is an easy calculation to verify that A� is, indeed, a sub-algebra of A.
Motivated by this definition, we can define the invariant subalgebra by a
partial action of a Hopf algebra.
Definition 6. Let H be a Hopf algebra acting partially on a k-algebra A.
We define the set of invariants of the partial action as
AH = {a ∈ A;ℎ ⋅ a = a(ℎ ⋅ 1A)}.
It is easy to prove thatAH is a subalgebra of A. Indeed, take a, b ∈ AH
and ℎ ∈ H, then
ℎ ⋅ (ab) =
∑
(ℎ(1) ⋅ a)(ℎ(2) ⋅ b) =
∑
a(ℎ(1) ⋅ 1A)(ℎ(2) ⋅ b) =
= a(ℎ ⋅ b) = ab(ℎ ⋅ 1A).
In the case of partial group actions each idempotent 1g is central. More
generally, it can be shown that if H is cocommutative then each element
ℎ ⋅ 1A is central in A. Throughout this paper, we assume that ℎ ⋅ 1A lies
in the center of A for each ℎ ∈ H.
In the case of a partial coaction of a Hopf algebra H acting on a
k-algebra A, we have also the notion of subalgebra of coinvariants.
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.8 Partial Hopf actions and partial invariants
Definition 7. Let H be a Hopf algebra coacting partially on a k-algebra
A. We define the set of coinvariants of the partial coaction as
ACoH = {a ∈ A∣ �(a) = a�(1A)}
In the same manner, one can easily verify that ACoH is a subalgebra
of A.
Let us consider now an enveloping action (B, �) of the partial action
of H on A, or enveloping action of A, for short. If BH is the invariant
subalgebra of B with relation to the total action ⊳ of H on B, that is
BH = {b ∈ B ∣ℎ ⊳ b = �(ℎ)b ∀ℎ ∈ H},
it is easy to see that 1AB
H ⊆ AH , obviously, considering 1AB
H =
�(1A)B
H and AH = �(AH). That is because if b ∈ BH then
ℎ ⋅ 1Ab = 1A(ℎ ⊳ 1Ab) = 1A(
∑
(ℎ(1) ⊳ 1A)(ℎ(2) ⊳ b)) =
= 1A(
∑
(ℎ(1) ⊳ 1A)(�(ℎ(2))b)) = 1A(ℎ ⊳ 1A)b = (ℎ ⋅ 1A)(1Ab).
On the other hand, the conditions to be satisfied in order to fulfill the
equality are more restrictive, as we shall see later.
From now on, we will consider only the case where the Hopf algebra
H is finite dimensional and the partial action is symmetric, i.e., where
property (iv) holds. Since H is finite dimensional, there exists a nonzero
left integral t ∈ H. Define the partial trace map t̂ : A → A by t̂(a) =
t ⋅ a. In the case of total actions this trace reduces to the classical one
(see reference [9], definition 4.3.3). The following two results are quite
analogous to the classical results in the theory of Hopf algebras [3].
Lemma 1. t̂ is a AH −AH bimodule mapping from A into AH .
Proof. First, one need to check that t̂(A) ⊆ AH . Since t is a left integral,
given a ∈ A and ℎ ∈ H,
ℎ ⋅ t̂(a) = ℎ ⋅ (t ⋅ a) =
∑
(ℎ(1) ⋅ 1A)(ℎ(2)t ⋅ a) =
=
∑
(ℎ(1) ⋅ 1A)(�(ℎ(2))t ⋅ a) = (ℎ ⋅ 1A)t̂(a)
and therefore t̂(A) ⊂ AH .
Now the bimodule morphism property. If b ∈ AH and a ∈ A, then
t̂(ab) = t ⋅ ab =
∑
(t(1) ⋅ a)(t(2) ⋅ b) =
∑
(t(1) ⋅ a)(t(2) ⋅ 1A)b = (t ⋅ a)b
and
t̂(ba) = t⋅ba =
∑
(t(1)⋅b)(t(2)⋅a) =
∑
b(t(1)⋅1A)(t(2)⋅a) = b(t⋅a) = bt̂(a).
Note that in the third equality in the verification above we used the
symmetry property (iv).
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Proposition 2. If H is semisimple and unimodular, and 1AB
H = AH ,
then the partial trace mapping is surjective.
Proof. In fact, if a ∈ AH then a = 1Ab where b ∈ BH ; since H is
semisimple, we have a nonzero left integral t ∈ H such that �(t) = 1, and
hence 1Ab = 1A�(t)b = 1A(t ⊳ b) (because b ∈ BH). Now b =
∑
ℎi ⊳ ai
for some ℎi ∈ H and ai ∈ A, and therefore, since t is also a right integral,
a = 1A(t⊳ b) = 1A(t⊳ (
∑
ℎi ⊳ ai)) =
= 1A(
∑
�(ℎi)t⊳ ai) = t ⋅ (
∑
�(ℎi)ai) ∈ t ⋅A.
Proposition 3. If the partial trace mapping is surjective then 1AB
H =
AH .
Proof. Let x ∈ AH . Since the partial trace is surjective, there is an
element y ∈ A such that x = t ⋅ y, then
x = t ⋅ y = 1A(t ⊳ y) ∈ 1AB
H .
Now, some words about the partial smash product. Let A be a partial
H module algebra, we can endow the tensor product A ⊗ H with an
associative algebra structure by
(a⊗ ℎ)(b⊗ k) =
∑
a(ℎ(1) ⋅ b)⊗ ℎ(2)k.
Define the partial smash product as the algebra
A#H = (A⊗H)(1A ⊗ 1H),
in other words, the smash product is the subalgebra generated by ele-
ments of the form
a#ℎ =
∑
a(ℎ(1) ⋅ 1A)⊗ ℎ(2).
One can easily verify that the product with the symbol # satisfies
(a#ℎ)(b#k) =
∑
a(ℎ(1) ⋅ b)#ℎ(2)k.
Note that the elements of A can be embedded into A#H by the map a 7→
a#1H = a⊗1H , this is an algebra map. On the other hand, the elements
of H can be written into the smash product as 1A#ℎ =
∑
(ℎ(1) ⋅1A)⊗ℎ(2),
but these elements do not form a subalgebra of A#H. Nevertheless, just
as in the global case, the partial action is now implemented internally,
since ∑
(1A#ℎ(1)) (a#1) (1A#S(ℎ(2))) = (ℎ ⋅ a)#1.
The partial smash product has a very interesting factorization prop-
erty, which will be useful later.
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.10 Partial Hopf actions and partial invariants
Proposition 4. Let H be a Hopf algebra with invertible antipode acting
partially on the algebra A. Then A#H = (1A ⊗H)(A⊗ 1).
Proof. In fact, consider a#ℎ ∈ A#H, then we have
a#ℎ =
∑
a(ℎ(1) ⋅ 1A)⊗ ℎ(2) =
∑
�(ℎ(1))a(ℎ(2) ⋅ 1A)⊗ ℎ(3) =
=
∑
(ℎ(2)S
−1(ℎ(1)) ⋅ a)(ℎ(3) ⋅ 1A)⊗ ℎ(4) =
=
∑
ℎ(2) ⋅ (S
−1(ℎ(1)) ⋅ a)⊗ ℎ(3) =
=
∑
(1A#ℎ(2))((S
−1(ℎ(1)) ⋅ a)#1H).
Proposition 5. If t̂ : A → AH is surjective, then there is a non-zero
idempotent e in A#H such that e(A#H)e = (AH#1H)e ≡ AH as k-
algebras.
Proof. Here, for sake of simplicity, if ℎ ∈ H and a ∈ A, we will write
a for a#1H and ℎ for 1A#ℎ. With this notation aℎ = a#ℎ and ℎa =∑
(ℎ(1) ⋅ a)ℎ(2), for every a ∈ A and ℎ ∈ H.
First, ℎat = (ℎ ⋅ a)t for any ℎ ∈ H and a ∈ A:
ℎat = (1A#ℎ)(a#1H)(1A#t) =
∑
(1A#ℎ)(a#t) =
=
∑
(ℎ(1) ⋅ a)#ℎ(2)t =
∑
(ℎ(1) ⋅ a)#�(ℎ(2))t =
= (ℎ ⋅ a)#t = (ℎ ⋅ a)t.
Assuming the surjectivity of the trace, let c ∈ A be such that t̂(c) =
1A, and consider e = tc ∈ A#H. This element is an idempotent, since
e2 = tctc = (tct)c = (t ⋅ c)tc = 1A#1H(tc) = tc = e.
Let us verify that e(A#H)e = (AH#1H)e:
e(aℎ)e = tcaℎtc = tcaℎ1Atc =
= t(ca(ℎ ⋅ 1A))tc = t ⋅ (ca(ℎ ⋅ 1A))tc =
= t ⋅ (ca(ℎ ⋅ 1A))e = t̂(ca(ℎ ⋅ 1A))e,
which lies in AHe.
Conversely, if a ∈ AH , then
t ⋅ (ca) =
∑
(t(1) ⋅ c)(t(2) ⋅ a) =
=
∑
(t(1) ⋅ c)(t(2) ⋅ 1A)a =
= (t ⋅ c)a = a,
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and
ae = (t ⋅ (ca))tc = tcatc = eae.
Hence eAHe = AHe.
Finally, eAHe is isomorphic to AH , since for a, b ∈ AH
(ae)(be) = at(cb)tc = a(t ⋅ cb)tc = a(t ⋅ c)btc = abtc = abe.
(where we used t ⋅ c = 1A).
3. A Morita context
In what follows we show that, just as in the case of global actions (see
reference [9] paragraphs 4.4 and 4.5), there is a Morita context connecting
the algebras AH and A#H. From now on, we are assuming that H is
a Hopf algebra with invertible antipode. Let us begin by recalling the
definition of a Morita context between two rings.
Definition 8. A Morita context is a six-tuple (R,S,M,N, [⋅, ⋅], ⟨⋅, ⋅⟩)
where
1. R and S are rings,
2. M is an R− S bimodule,
3. N is an S −R bimodule,
4. [⋅, ⋅] :M ⊗S N → R is a bimodule morphism,
5. ⟨⋅, ⋅⟩ : N ⊗R M → S is a bimodule morphism,
such that
[m,n]m′ = m⟨n,m′⟩, ∀m,m′ ∈M, ∀n ∈ N, (2)
and
⟨n,m⟩n′ = n[m,n′], ∀m ∈M, ∀n, n′ ∈ N. (3)
By a fundamental theorem due to Morita (see, for example [7] on
pages 167-170), if the morphisms [, ] and ⟨, ⟩ are surjective, then the
categories RMod and SMod are equivalent. In this case, we say that
R and S are Morita equivalent.
For the Morita context between AH and A#H, the bimodules M and
N will both have A as subjacent vector space; since A already has a
canonical AH -bimodule structure, the trouble lies in defining right and
left A#H-module structures on A.
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.12 Partial Hopf actions and partial invariants
Let
∫ l
H
be the subspace generated by left integrals in H. We remind
the reader that if t ∈
∫ l
H
then so does tℎ for every ℎ ∈ H. Since dim
∫ l
H
=
1, tℎ = �(ℎ)t for some �(ℎ) ∈ k. This defines a map � : H → k, which
is an algebra morphism.
Lemma 2. Given b ∈ A and a#ℎ =
∑
a(ℎ(1) ⋅ 1A) ⊗ ℎ(2) ∈ A#H, the
mappings
(a#ℎ)⊳ b = a(ℎ ⋅ b)
and
b⊲ (a#ℎ) =
∑
�(ℎ(2))S
−1(ℎ(1)) ⋅ (ba)
define left and right A#H-module structures on A. Furthermore, if we
consider the canonical left and right AH− module structures on A, then
A is both an AH −A#H and A#H −AH bimodule.
Proof. Let us begin with the left A#H module structure
((a#ℎ)(b#k))⊳ c =
∑
(a(ℎ(1) ⋅ b)#ℎ(2)k)⊳ c =
= (a(ℎ(1) ⋅ b)(ℎ(2)k ⋅ c) =
= (a(ℎ ⋅ (b(k ⋅ c)) =
= (a#ℎ)⊳ (b(k ⋅ c)) =
= (a#ℎ)⊳ ((b#k)⊳ c)
On the right side we have
∑
�(ℎ(2))(S
−1(ℎ(1)) ⋅ ab)⊲ (c#k) =
= (
∑
�(ℎ(2))(S
−1(ℎ(1)) ⋅ ab)⊲ (c#k) =
=
∑
�(k(2))S
−1(k(1)) ⋅ ((
∑
�(ℎ(2))(S
−1(ℎ(1)) ⋅ ab)c)) =
=
∑
�(ℎ(2)k(3))(S
−1(k(2)) ⋅ (S
−1(ℎ(1)) ⋅ ab))(S
−1(k(1)) ⋅ c) =
=
∑
�(ℎ(2)k(4))(S
−1(k(3))(S
−1(ℎ(1)) ⋅ ab))(S
−1(k(2)) ⋅ 1A)(S
−1(k(1)) ⋅ c) =
=
∑
�(ℎ(2)k(3))(S
−1(ℎ(1)k(2)) ⋅ ab)(S
−1(k(1)) ⋅ c) =
=
∑
�(ℎ(4)k(3))(S
−1(ℎ(3)k2) ⋅ ab)(S
−1(k(1))S
−1(ℎ(2))ℎ(1) ⋅ c) =
=
∑
�(ℎ(4)k(3))(S
−1(ℎ(3)k(2)) ⋅ ab)(S
−1(ℎ(2)k(1))ℎ(1) ⋅ c) =
=
∑
�(ℎ(3)k(2))(S
−1(ℎ(2)k(1)) ⋅ (ab(ℎ(1) ⋅ c)) =
= a⊲ (
∑
b(ℎ(1) ⋅ c)#ℎ(2)k) =
= a⊲ ((b#ℎ)(c#k)).
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Hence A is a left and right A#H module.
A is an AH − A#H− bimodule. Given a ∈ A, b ∈ AH and c#ℎ ∈
A#H, we have
(ba)⊲ (c#ℎ) =
∑
�(ℎ(2))S
−1(ℎ(1)) ⋅ (bac) =
=
∑
�(ℎ(3))(S
−1(ℎ(2)) ⋅ b)(S
−1(ℎ(1)) ⋅ ac) =
=
∑
�(ℎ(3))b(S
−1(ℎ(2)) ⋅ 1A)(S
−1(ℎ(1)) ⋅ ac) =
=
∑
�(ℎ(2))b(S
−1(ℎ(1)) ⋅ ac)
= b(a⊲ (c#ℎ))
and
((c#ℎ)⊳ a)b = c(ℎ ⋅ a)b =
∑
c(ℎ(1) ⋅ a)(ℎ(2) ⋅ 1A)b =
= c(ℎ ⋅ (ab)) = (c#ℎ)⊳ (ab).
For the Morita context we define the maps
[⋅, ⋅] : A⊗AH A→ A#H
a⊗ b 7→ [a, b] = atb
and
⟨⋅, ⋅⟩ : A⊗A#H A → AH
a⊗ b 7→ ⟨a, b⟩ = t̂(ab) = t ⋅ ab
Remember that atb = (a#1H)(1A#t)(b#1H) is indeed an element of
A#H.
We must check that these maps are well-defined, i.e., that [, ] is AH -
balanced and ⟨, ⟩ is A#H-balanced. Both are clearly k-linear maps from
A ⊗k A to A#H and AH respectively. Then, we need to check only
whether these maps are balanced or not.
First for the map [⋅, ⋅]: if a, b ∈ A and c ∈ AH then
[a, cb] = atcb = (a#1H)(
∑
(t(1) ⋅ c)#t(2))(b#1) =
= (a#1H)(
∑
c(t(1) ⋅ 1A)#t(2))(b#1H) =
= (a#1H)(c#1H)(1A#t)(b#1H) = actb = [ac, b]
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.14 Partial Hopf actions and partial invariants
Now, for the second map, ⟨⋅, ⋅⟩: if a, b ∈ A and c#ℎ ∈ A#H then
⟨a⊲ (c#ℎ), b⟩ =
= t ⋅ ((
∑
�(ℎ(2))S
−1(ℎ(1)) ⋅ (ac))b) =
=
∑
�(ℎ(2))(t(1) ⋅ (S
−1(ℎ(1)) ⋅ ac))(t(2) ⋅ b) =
=
∑
�(ℎ(2))(t(1)S
−1(ℎ(1)) ⋅ ac)(t(2) ⋅ 1A)(t(3) ⋅ b) =
=
∑
�(ℎ(2))(t(1)S
−1(ℎ(1)) ⋅ ac)(t(2) ⋅ b)
=
∑
�(ℎ(3))(t(1)S
−1(ℎ(2)) ⋅ ac)(t(2)�(ℎ(1)) ⋅ b)
=
∑
�(ℎ(4))(t(1)S
−1(ℎ(3)) ⋅ ac)(t(2)S
−1(ℎ(2))ℎ(1) ⋅ b) =
=
∑
�(ℎ(3))(tS
−1(ℎ(2))) ⋅ (ac(ℎ(1) ⋅ b)) =
=
∑
(tℎ(3)S
−1(ℎ(2))) ⋅ (ac(ℎ(1) ⋅ b)) =
=
∑
(t�(ℎ(2))) ⋅ (ac(ℎ(1) ⋅ b)) =
= t ⋅ (ac(ℎ ⋅ b)) = ⟨a, (c#ℎ)⊳ b⟩.
Therefore the maps [⋅, ⋅] and ⟨⋅, ⋅⟩ are well defined.
Theorem 1. (A#H,AH ,A#HAAH ,AHAA#H , [⋅, ⋅], ⟨⋅, ⋅⟩) is a Morita con-
text.
Proof. We must check that both are bimodule maps. From Lemma 1 it
follows that ⟨, ⟩ is a bimodule map, because t̂ is an AH −AH− bimodule
mapping. For the other map, let a, b ∈ A and c#ℎ ∈ A#H. On the left
we have
[(c#ℎ)⊳ a, b] = c(ℎ ⋅ a)tb = (c(ℎ ⋅ a)#t)(b#1H) =
=
∑
(c(ℎ(1) ⋅ a)#�(ℎ(2))t)(b#1H) =
=
∑
(c(ℎ(1) ⋅ a)#ℎ(2)t)(b#1H) =
= (c#ℎ)(a#t)(b#1H) = (c#ℎ)[a, b].
For the right side,
[a, b⊲ (c#ℎ)] =
= [a,
∑
�(ℎ(2))S
−1(ℎ(1)) ⋅ (bc)] =
= (a#t)(
∑
�(ℎ(2))S
−1(ℎ(1)) ⋅ (bc)#1H) =
=
∑
(a#tℎ(2))(S
−1(ℎ(1)) ⋅ (bc)#1H) =
= (a#1H)(
∑
t(1)ℎ(2) ⋅ (S
−1(ℎ(1)) ⋅ (bc))#t(2)ℎ(3)) =
= (a#1H)(
∑
(t(1)ℎ(2)S
−1(ℎ(1)) ⋅ (bc))(t(2)ℎ(3) ⋅ 1A)#t(3)ℎ(4)) =
= (a#1H)(
∑
(t(1) ⋅ (bc))(t(2)ℎ(1) ⋅ 1A)#t(3)ℎ(2)) =
= (a#1H)(
∑
(t(1) ⋅ ((bc)(ℎ(1) ⋅ 1A)))#t(2)ℎ(2)) =
= (a#1H)(1A#t)(b#1H)(c#ℎ) = [a, b](c#ℎ).
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.M. M. S. Alves, E. Batista 15
Now we must check the “associativity” of the brackets, i.e, [a, b]⊳ c =
a⟨b, c⟩ and a ⊲ [b, c] = ⟨a, b⟩c, and we are done. The first is straightfor-
ward, and the second is
a⊲ [b, c] =
= a⊲ (
∑
b(t(1) ⋅ c)#t(2)) =
=
∑
�(t(3))S
−1(t(2)) ⋅ (ab(t(1) ⋅ c)) =
=
∑
�(t(5))(S
−1(t(4)) ⋅ ab)(S
−1(t(3)) ⋅ 1A)(S
−1(t(2))t(1) ⋅ c) =
=
∑
�(t(3))(S
−1(t(2)) ⋅ ab)(S
−1(t(1)) ⋅ 1A)c =
=
∑
�(t(2))(S
−1(t(1)) ⋅ ab)c =
= ((
∑
�(t(2))(S
−1(t(1))) ⋅ (ab))c =
= (t ⋅ (ab))c =
= ⟨a, b⟩c,
where we used
∑
(�(t(2))(S
−1(t(1))) = t, which follows from S(t) =∑
�(t(2))t(1) (See the reference [10] for a proof of this result).
It is worth to mention that in reference [2], the authors considered a
partial coaction of a Hopf algebra H on a unital algebra A and established
a Morita context between the invariant subalgebra ACoH and the dual
smash #(H,A) ∼= ∗(A⊗H) but in that case the modules were A as a
ACoH −#(H,A) bimodule and Q as a #(H,A)−ACoH bimodule, where
Q = {q ∈ ∗(A⊗H)∣c(1)q(c(2)) = q(c)�(1A), ∀c ∈ (A⊗H)}.
4. Partial Hopf Galois theory
In this section, we give some necessary and sufficient conditions for the
Morita context (AH , A#H,A, ⟨⋅, ⋅⟩, [⋅, ⋅]) to be strict. This envolves a gen-
eralization for the partial case of the concept of a Hopf-Galois extension
of algebras. First, given a partial coaction � : A→ A⊗H, we can define
an A − A bimodule structure on A ⊗H: the left A module structure is
given by the multiplication and the right A module structure is given by
(a ⊗ ℎ)b =
∑
ab[0] ⊗ ℎb[1]. In what follows, let us consider a sub A − A
bimodule of A⊗H defined by
A⊗H = (A⊗H)1A = {
∑
a1
[0]
A ⊗ ℎ1
[1]
A ∣ a ∈ A, ℎ ∈ H}.
Definition 9. (Partial Hopf Galois Extension, [2]) Let (A, �) be a partial
right H-comodule algebra. The extension AcoH ⊂ A is partial H-Hopf
Galois if the canonical map � : A⊗AcoH A→ A⊗H, given by �(a⊗ b) =
(a⊗ 1)�(b) =
∑
ab[0] ⊗ b[1], is a bijective A−A bimodule morphism.
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.16 Partial Hopf actions and partial invariants
Lemma 3. If H is a finite dimensional Hopf algebra, A is a partial H-
module algebra and � : A → A ⊗H∗ is the induced partial H∗-comodule
structure on A, then AH = AcoH∗
.
Proof. Let {ℎi}
n
i=1 be a basis of H and let {ℎ∗i }
n
i=1 be the dual basis. If
a ∈ AH then
�(a) =
∑
(ℎi ⋅ a)⊗ ℎ∗i =
∑
a(ℎi ⋅ 1A)⊗ ℎ∗i = a�(1A)
(for �(1A) =
∑
ℎi ⋅ 1A ⊗ ℎ∗i )
Conversely, if a ∈ AcoH∗
then �(a) =
∑
a1
[0]
A ⊗ 1
[1]
A , and therefore
ℎ ⋅ a = (Id⊗ evℎ)�(a) =
∑
a1
[0]
A 1
[1]
A (ℎ) = a(ℎ ⋅ 1A).
This last lemma says that when H is finite dimensional, we may
consider on a partial H-module algebra A the induced structure of partial
H∗-comodule algebra and, since AH = AcoH∗
, we get the map
� : A⊗AH A→ A⊗H∗.
Theorem 2. Let H be a finite-dimensional Hopf algebra, 0 ∕= t ∈
∫ l
H
,
and let A be a partial H-module algebra such that the canonical map
� : A⊗AH A→ A⊗H∗ is surjective. Then
(i) There exist a1, . . . , an and b1, . . . , bn in A such that �i : A → AH
given by �i(a) = t ⋅ bia is an AH module map, and a =
∑
ai�i(a)
for each a ∈ A; hence {ai}
n
i=1 is a projective basis over AH and A
is a finitely generated projective AH right module.
(ii) � is bijective.
Proof. (i) Consider the canonical isomorphism [9] induced by the nonzero
left integral t ∈
∫ l
H
given by
� : H∗ → H
f 7→ �(f) = t ↼ f
where t ↼ f =
∑
f(t(1))t(2). Then, as � is surjective, there exists T ∈ H∗
such that 1H = t ↼ T . Using also the surjectivity of the canonical map
�, there exist a1, . . ., an and b1, . . ., bn, elements of A such that
1
[0]
A ⊗ T1
[1]
A = �(
n∑
i=1
ai ⊗AH bi).
Now, consider any element a ∈ A, then
a = 1H ⋅ a = (t ↼ T ) ⋅ a = (t ↼ T )(1Aa) =
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.M. M. S. Alves, E. Batista 17
=
∑
1
[0]
A a
[0](1
[1]
A a
[1](t ↼ T )) =
=
∑
1
[0]
A a
[0]T (t(1))(1
[1]
A a
[1](t(2))) =
=
∑
1
[0]
A a
[0]T (t(1))1
[1]
A (t(2))a
[1](t(3)) =
=
∑
1
[0]
A a
[0](T1
[1]
A (t(1)))a
[1](t(2)) =
=
n∑
i=1
∑
aib
[0]
i a
[0]b
[1]
i (t(1))a
[1](t(2)) =
=
n∑
i=1
∑
aib
[0]
i a
[0](b
[1]
i a
[1](t)) =
=
n∑
i=1
ai(t ⋅ bia).
(ii) the proof of this item follows the same steps as in [9] Theorem
8.3.1.
Proposition 6. Let H be a finite dimensional Hopf algebra and A a
partial H-module algebra. Then AH ∼= End(A#HA)
op as algebras.
Proof. Let � : AH → End(A#HA)
op be the map that takes a to the
endomorphism �(a) : b 7→ ba. Then � is a algebra map and, if f ∈
End(A#HA) and a ∈ A, then
f(a) = f(a1A) = f((a#1) ⋅ 1A) = (a#1) ⋅ f(1A) = af(1A).
Hence f = �f(1A); besides, f(1A) is indeed in AH , since
ℎ ⋅ f(1A) =
∑
((ℎ(1) ⋅ 1A)#ℎ(2)) ⋅ f(1A) = f((ℎ ⋅ 1A)1A) = (ℎ ⋅ 1A)f(1A).
Finally, the next result provides a relation between the surjectivity of
the canonical map, the extension being Hopf-Galois, and the surjectivity
of the map [, ] of the Morita context.
Theorem 3. Let H be a finite dimensional Hopf algebra with a nonzero
integral t, and let A be a (left) partial H-module algebra. Then, the
following affirmations are equivalent:
(i) The canonical map � : A⊗AHA→ A⊗H∗, given by �(
∑
xi⊗yi) =∑
xiy
[0]
i ⊗ y
[1]
i is surjective.
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.18 Partial Hopf actions and partial invariants
(ii) The algebra A is a finitely generated projective right AH module and
AH ⊂ A is a partial H∗-Galois extension.
(iii) The algebra A is a finitely generated projective right AH module,
and the map [⋅, ⋅] : A⊗AH A→ A#H is surjective
Proof. (i) ⇒ (ii) This is basically the content of the Theorem 2
(ii) ⇒ (iii)
Let � : H∗ → H be the H-module isomorphism ' 7→ t ↼ ' =∑
'(t1)t2, and let � be the canonical map (�(a ⊗ b) =
∑
ab[0] ⊗ b[1]).
Then [a, b] = (I ⊗ �)�(a⊗ b). In fact,
(I ⊗ �)�(a⊗ b) = ab[0] ⊗ �(b[1])
=
∑
ab[0] ⊗ t ↼ (b[1])
=
∑
ab[0] ⊗ (b[1])(t1)t2
=
∑
a(t1 ⋅ b)⊗ t2
=
∑
(a(t1 ⋅ 1A)⊗ t2)(b⊗ 1H)
= (a#t)(b#1H)
= (a#1H)(1A#t)(b#1H)
= [a, b]
This formula assures that [⋅, ⋅] is surjective if, and only if � is surjective.
Hence, if (ii) holds then � is bijective, in particular, �, therefore [, ] is
onto.
(iii)⇒ (i) Note that, the formula [⋅, ⋅] = (I ⊗ �)� implies that the
surjectivity of the map [⋅, ⋅] assures the surjectivity of �.
Note that the previous theorem gives us a necessary and sufficient
condition for the Morita context (AH , A#H,A, ⟨⋅, ⋅⟩, [⋅, ⋅]) to be strict.
In fact, assuming that H is a finite dimensional Hopf algebra with a
nonzero integral t and that the map t̂ : A → AH , defined as t̂(a) = t ⋅ a,
is surjective, we have that the Morita context is strict if, and only if, the
extension AH ⊂ A is partial H∗-Hopf Galois.
Aknowledgements
This work is dedicated to Prof. Miguel Ferrero, whose contribution to
the development of algebra in Brazil is remarkable.
The authors would like to thank to Piotr Hajac and Joost Vercruysse
for fuitfull discussions and suggestions.
References
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Contact information
M. M. S. Alves Departamento de Matemática
Universidade Federal do Paraná
Centro Politécnico
C.P. 19081, Jardim das Américas
81531-990 Curitiba - PR
Brazil
E-Mail: marcelomsa@ufpr.br
URL: http://www.mat.ufpr.br
E. Batista Departamento de Matemática
Universidade Federal de Santa Catarina
88.040-900 Florianópolis-SC
Brasil
E-Mail: ebatista@mtm.ufsc.br
URL: http://www.mtm.ufsc.br/
Received by the editors: 31.08.2009
and in final form 21.09.2009.
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