Globalizations for partial (co)actions on coalgebras
In this paper, we introduce the notion of globalization for partial module coalgebra and for partial comodule coalgebra. We show that every partial module coalgebra is globalizable exhibiting a standard globalization. We also show the existence of globalization for a partial comodule coalgebra, prov...
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| Цитувати: | Globalizations for partial (co)actions on coalgebras / F. Castro, G. Quadros // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 212–242. — Бібліогр.: 16 назв. — англ. |
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Castro, F. Quadros, G. 2023-03-01T15:38:10Z 2023-03-01T15:38:10Z 2019 Globalizations for partial (co)actions on coalgebras / F. Castro, G. Quadros // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 212–242. — Бібліогр.: 16 назв. — англ. 1726-3255 2010 MSC: 16T15; 16T99, 16W22 https://nasplib.isofts.kiev.ua/handle/123456789/188434 In this paper, we introduce the notion of globalization for partial module coalgebra and for partial comodule coalgebra. We show that every partial module coalgebra is globalizable exhibiting a standard globalization. We also show the existence of globalization for a partial comodule coalgebra, provided a certain rationality condition. Moreover, we show a relationship between the globalization for the (co)module coalgebra and the usual globalization for the (co)module algebra. The authors were partially supported by CNPq, Brazil. The authors would like to thank Antonio Paques, Alveri Sant’Ana and the referee, whose comments, corrections and suggestions were very useful to improve the manuscript. We would like to thank Lourdes Haase for her corrections and suggestions about the paper writing. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Globalizations for partial (co)actions on coalgebras Article published earlier |
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Globalizations for partial (co)actions on coalgebras Castro, F. Quadros, G. |
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In this paper, we introduce the notion of globalization for partial module coalgebra and for partial comodule coalgebra. We show that every partial module coalgebra is globalizable exhibiting a standard globalization. We also show the existence of globalization for a partial comodule coalgebra, provided a certain rationality condition. Moreover, we show a relationship between the globalization for the (co)module coalgebra and the usual globalization for the (co)module algebra.
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Globalizations for partial (co)actions on coalgebras / F. Castro, G. Quadros // Algebra and Discrete Mathematics. — 2019. — Vol. 27, № 2. — С. 212–242. — Бібліогр.: 16 назв. — англ. |
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“adm-n2” — 2019/7/14 — 21:27 — page 212 — #62
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 27 (2019). Number 2, pp. 212–242
c© Journal “Algebra and Discrete Mathematics”
Globalizations for partial (co)actions
on coalgebras∗
Felipe Castro and Glauber Quadros
Communicated by R. Wisbauer
Abstract. In this paper, we introduce the notion of glo-
balization for partial module coalgebra and for partial comodule
coalgebra. We show that every partial module coalgebra is glo-
balizable exhibiting a standard globalization. We also show the
existence of globalization for a partial comodule coalgebra, provided
a certain rationality condition. Moreover, we show a relationship
between the globalization for the (co)module coalgebra and the
usual globalization for the (co)module algebra.
1. Introduction
Partial actions of groups were first considered in the context of op-
erators algebra (cf. [12]). Dokuchaev and Exel in [10] introduced partial
actions of groups in a purely algebraic context, obtained several classical
results in the setting of partial actions of groups and covering the actions
of groups. The actions of Hopf algebras on algebras also generalize the
theory of actions of groups (cf. [13, Example 4.1.6]). The notion of action of
∗The authors were partially supported by CNPq, Brazil.
The authors would like to thank Antonio Paques, Alveri Sant’Ana and the referee,
whose comments, corrections and suggestions were very useful to improve the manuscript.
We would like to thank Lourdes Haase for her corrections and suggestions about the
paper writing.
2010 MSC: 16T15; 16T99, 16W22.
Key words and phrases: Hopf algebras, partial action, partial coaction, globali-
zation, partial module coalgebra, partial comodule coalgebra.
“adm-n2” — 2019/7/14 — 21:27 — page 213 — #63
F. Castro, G. Quadros 213
groups was extended in two directions, in both contexts extensive theories
were developed, where we highlight the Morita and Galois theories. As
a natural task, Caenepeel and Janssen in [7] introduced the concept of
partial actions of Hopf algebras on algebras, called partial module algebras.
This new theory arose unifying all the above theories.
The globalization process, which consists of constructing an action of
a group such that a given partial action can be seen inside it, was first
described by Abadie (cf. [1]) and it plays an important role in the context
of partial actions of groups. Alves and Batista successfully translated it
to the new context of partial module algebras (cf. [2, 3]).
Furthermore, the structure of Hopf algebra allows us to define new
objects in the theory, named the comodule algebra and the partial comod-
ule algebra, which are the duals of module algebra and partial module
algebra, respectively (cf. [7]). Alves and Batista have shown the relation
between these structures through a globalization (cf. [4]).
Since coalgebra is the dual notion of algebra, thus one can think
about these partial actions and coactions on a coalgebra, generating new
objects and studying its properties. Batista and Vercruysse defined these
structures, called partial module coalgebra and partial comodule coalgebra,
which are the dual notions of partial module algebra and partial comodule
algebra respectively. All these four partial structures are closely related.
These dual structures were studied, and it was shown some interesting
properties between them (cf. [5]).
The aim of this work is to study the existence of globalization in
the setting of partial module coalgebras and partial comodule coalgebras,
relating our results with the results obtained by Alves and Batista in [3].
This work is organized as follows.
In the second section we recall some preliminary results about partial
actions of Hopf algebras, which are necessary for a full understanding of
this work. An expert on this area can pass through this section, and go
directly to the next section.
The third section is devoted to the study of partial actions on coal-
gebras. We study the correspondence between this new structure and
the partial module algebra, some examples and useful properties. The
purpose of this section is to discuss the existence of a globalization for
partial action on coalgebras. For this, we start by introducing the notion
of induced partial action on coalgebras and showing how to obtain it via a
comultiplicative projection satisfying a special condition (see Proposition
3.18). After that, we define a globalization for partial module coalgebras
(see Definition 3.20) and we also discuss its affinity with the well known
“adm-n2” — 2019/7/14 — 21:27 — page 214 — #64
214 Globalization for partial (co)actions
notion of globalization for a partial module algebra, obtaining a direct
relation between them (see Theorem 3.23). Finally, we construct a glo-
balization for any given partial module coalgebra, called the standard
globalization (see Theorem 3.25).
In the fourth section, we study partial coactions on coalgebras. We
present some examples and important properties related to this structure.
We also show a correspondence among these four partial objects, asking
for special conditions, like density or finite dimension, getting a one to
one correspondence among all of them. In a similar way as made in Sec-
tion 3, we define the induced partial coaction assuming the existence of
a comultiplicative projection satisfying a special condition (see Propo-
sition 4.18). After that, we define globalization for a partial comodule
coalgebra (see Definition 4.20) and relate it with the notion of a partial
module coalgebra earlier defined in Section 3 (see Theorem 4.22). Suppos-
ing a rational module hypothesis, we show that every partial comodule
coalgebra is globalizable, constructing the standard globalization for it
(see Theorem 4.23).
Throughout this paper k denotes a field, all objects are k-vector spaces
(i.e. algebra, coalgebra, Hopf algebra, etc., mean respectively k-algebra,
k-coalgebra, k-Hopf algebra, etc.), linear maps mean k-linear maps and
unadorned tensor product means ⊗k. We use the well known Sweedler’s
Notation for comodules and coalgebras, in this way: given a coalgebra D,
with coproduct ∆, we will denote, for any c ∈ C
∆(c) = c1 ⊗ c2,
where the summation is understood; and given a left C-comodule M via
λ, we will denote, for any m ∈ M
λ(m) = m−1 ⊗m0,
where the summation is also understood.
Moreover, we will call π a projection if it is a linear map such that
π ◦ π = π.
In the context of (partial) actions on algebras, it is usual to assume
that all modules are left modules. In order to respect the correspondences
among the four partial structures (see Sections 3 and 4), we assume the
following convention: (partial) module algebras are left (partial) module
algebras; (partial) comodule algebras are right (partial) comodule algebras;
(partial) module coalgebras are right (partial) module coalgebras; and
(partial) comodule coalgebras are left (partial) comodule coalgebras.
“adm-n2” — 2019/7/14 — 21:27 — page 215 — #65
F. Castro, G. Quadros 215
2. Partial Actions
Definition 2.1 (Module algebra). A left H-module algebra is a pair
(A, ⊲), where A is an algebra and ⊲ : H ⊗ A → A is a linear map, such
that the following conditions hold, for any h, k ∈ H and a, b ∈ A:
(MA1) 1H ⊲ a = a;
(MA2) h ⊲ (ab) = (h1 ⊲ a)(h2 ⊲ b);
(MA3) h ⊲ (k ⊲ a) = (hk) ⊲ a.
Remark 2.2. The standard definition of a module algebra (cf. [9]) con-
tains the additional condition, for any h ∈ H:
h ⊲ 1A = ε(h)1A.
This condition should be required for modules algebra over bialgebras.
Since H is a Hopf algebra, so it is a consequence of the others.
Definition 2.3 (Partial module algebra [7]). A left partial H-module
algebra is a pair (A,→), where A is an algebra and → : H ⊗A → A is a
linear map, such that the following conditions are satisfied, for all h, k ∈ H
and a, b ∈ A:
(PMA1) 1H → a = a;
(PMA2) h → (ab) = (h1 → a)(h2 → b);
(PMA3) h → (k → a) = (h1 → 1A)(h2k → a).
The partial action → is said symmetric if the following additional condition
is satisfied:
(PMA4) h → (k → a) = (h1k → a)(h2 → 1A).
The conditions (PMA2) and (PMA3) can be replaced by
h → (a(g → b)) = (h1 → a)(h2g → b),
for all h, g ∈ H and a, b ∈ A. It can be useful if working with non-unital
algebras.
A natural way to get a partial module algebra is inducing from a global
action, as follows:
Proposition 2.4 ([3, Proposition 1]). Given a (global) module algebra
B and a right ideal A of B with unity 1A. Then A is a partial H-module
algebra via
h · a = 1A(h ⊲ a)
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216 Globalization for partial (co)actions
In this context, an enveloping action (or globalization) of a partial
module algebra A is a pair (B, θ), where B is a module algebra, θ : A → B
is an algebra monomorphism, such that
(i) θ(A) is a right ideal of B;
(ii) the partial action on A is equivalent to the induced partial action
on θ(A);
(iii) B = H ⊲ θ(A).
Alves and Batista showed that any partial action has an enveloping
action.
Theorem 2.5 ([3, Theorem 1]). Let A be a left partial H-module algebra,
ϕ : A → Hom(H,A) the map given by ϕ(a)(h) = h · a and B = H ⊲ ϕ(A).
Then (B,ϕ) is an enveloping action of A.
Remark 2.6. In the construction made above, we can highlight some
points that are important for this paper.
1) The image of A should be a right ideal of an enveloping action B,
but it does not need to be a left ideal, neither the partial action needs to
be symmetric (cf. [3, Proposition 4]).
2) The construction of an enveloping action supposes that the partial
module algebra is unital. However, this restriction may be overcome by
appropriate projections, as shown below.
Let B be an H-module algebra (not necessarily unital) and A a
subalgebra of B. Given a multiplicative projection
π : B −→ A
such that, for all h, k ∈ H and x, y ∈ A, the condition
π(h ⊲ (x(k ⊲ y))) = π(h ⊲ (xπ(k ⊲ y))) (1)
holds, so we can define a structure of partial module algebra in A by
h · x = π(h ⊲ x).
Note that the converse is also true. In fact, supposing that the projec-
tion π induces a structure of partial module algebra in A, then Equation
(1) holds.
Now, since we have the notion of induced partial action, we can define
a globalization (or enveloping action) of A.
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F. Castro, G. Quadros 217
Definition 2.7 (Globalization for a partial module algebra). Given a right
partial H-module algebra A with partial action →, a globalization of A is
a triple (B, θ, π), where B is a right H-module algebra via ⊲, θ : A → B
is an algebra monomorphism and π is a multiplicative projection from B
onto θ(A), satisfying the following conditions:
(GMA1) the partial action on A is equivalent to the partial action in-
duced by ⊲ on θ(A), that is, θ(h → a) = h → θ(a) = π(h ⊲ θ(a));
(GMA2) B is the H-module algebra generated by θ(A), that is, B =
H ⊲ θ(A),
for all h ∈ H, a ∈ A and b ∈ B.
It is a simple task to check that any partial module algebra has a
globalization, in this sense.
Since the induced partial action defined by Alves and Batista is a
particular case of the construction above, where the projection is given
by left multiplication by the idempotent 1A, then this construction of
globalization generalizes the construction made in [3].
3) Since the notion of induction by a central idempotent is a particular
case of induction by projection, then it inspires us to define the induced
partial (co)module coalgebra using projections (see Definitions 3.18 and
4.18).
3. Partial actions on coalgebras
3.1. Partial module coalgebras
Definition 3.1 (Module coalgebra). A right H-module coalgebra is a
pair (D,◭), where D is a coalgebra and ◭ : D ⊗H → D is a linear map
such that for any g, h ∈ H and d ∈ D, the following properties hold:
(MC1) d ◭ 1H = d;
(MC2) ∆(d ◭ h) = d1 ◭ h1 ⊗ d2 ◭ h2;
(MC3) (d ◭ h) ◭ g = d ◭ hg.
In this case we say that H acts on D via ◭, or that ◭ is an action
of H on D. We sometimes use the terminology global module algebra to
differ the above structure from the partial one.
Remark 3.2. The above definition can be seen in a categorical sense,
as follows (cf. [14, Definition 11.2.8]): a k-vector space D is said a right
H-module coalgebra if it is a coalgebra object in the category of right
H-modules.
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218 Globalization for partial (co)actions
Note that from this categorical approach, it is required a fourth property
for an H-module coalgebra. More precisely, the following equality needs
to hold:
εD(d ◭ h) = εD(d)εH(h),
for all h ∈ H and d ∈ D.
Since H is a Hopf algebra, so it has an antipode, then this additional
condition is a consequence from the others axioms of Definition 3.1, as
stated below.
Proposition 3.3. Let H be a Hopf algebra and D a right H-module
coalgebra. Then εD(d ◭ h) = εD(d)εH(h), for any d ∈ D,h ∈ H.
Remark 3.4. From Definition 3.1 and Proposition 3.3 it follows that one
can define, without loss of generality, module coalgebra for non-counital
coalgebras, and both definitions coincide when the coalgebra is counital.
Now we present some classical examples of module coalgebras (cf.
[9, 13,14].
Example 3.5. Any Hopf algebra is a right module coalgebra over itself
via right multiplication.
Example 3.6. Let D be a right H-module coalgebra and C a coalgebra,
then C ⊗D is a right H-module coalgebra with action given by
(c⊗ d) ◭ h = c⊗ (d ◭ h)
for any c⊗ d ∈ C ⊗D and h ∈ H.
Definition 3.7 (Partial module coalgebra). [5, Definition 5.1.] A right
partial H-module coalgebra is a pair (C,↼), where C is a coalgebra C
and ↼ : C ⊗H → C is a linear map, such that the following conditions
are satisfied, for any g, h ∈ H and c ∈ C:
(PMC1) c ↼ 1H = c;
(PMC2) ∆(c ↼ h) = c1 ↼ h1 ⊗ c2 ↼ h2;
(PMC3) (c ↼ h) ↼ g = ε(c1 ↼ h1)(c2 ↼ h2g).
A partial module coalgebra is said symmetric if the following additional
condition is satisfied:
(PMC4) (c ↼ h) ↼ g = (c1 ↼ h1g)ε(c2 ↼ h2).
“adm-n2” — 2019/7/14 — 21:27 — page 219 — #69
F. Castro, G. Quadros 219
One can define left partial module coalgebra in an analogous way.
The definition of right partial module coalgebra can be extended to
non-counital coalgebras, as follows.
Definition 3.8. A right partial H-module coalgebra is a pair (C,↼),
where C is a (non necessarily counital) coalgebra and ↼ : C ⊗ H → C
is a linear map, such that, for any h, k ∈ H and c ∈ C, the following
conditions hold:
(PMC′1) c ↼ 1H = c;
(PMC′2) (c ↼ h)1 ⊗ ((c ↼ h)2 ↼ k) = (c1 ↼ h1)⊗ (c2 ↼ h2k).
Moreover, it is symmetric if the following additional condition holds:
(PMC′3) ((c ↼ h)1 ↼ k)⊗ (c ↼ h)2 = (c1 ↼ h1k)⊗ (c2 ↼ h2).
It is straightforward to check the following statement.
Proposition 3.9. If C is a counital coalgebra, then Definition 3.8 is
equivalent to Definition 3.7.
Batista and Vercruysse [5] related right partial module coalgebras and
left partial module algebra, using non-degenerated dual pairing between
an algebra A and a coalgebra C. Here we consider a special case, where
the algebra is the dual of the coalgebra.
Proposition 3.10 ([5, Theorem 5.12]). Let C be a coalgebra, and suppose
that → : H ⊗ C∗ → C∗ and ↼ : C ⊗H → C are linear maps satisfying
the following compatibility:
(h → α)(c) = α(c ↼ h) (2)
for any h ∈ H and c ∈ C. Then C is a partial H-module coalgebra via ↼
if and only if C∗ is a partial H-module algebra via →.
Remark 3.11. The existence of a linear map
↼ : C ⊗H −→ C
implies the existence of a linear map
→ : H ⊗ C∗ −→ C∗
h⊗ α 7−→ (h → α) : c 7−→ α(c ↼ h),
for all h ∈ H,α ∈ C∗ and c ∈ C. Clearly these two maps satisfy the
Equation (2).
“adm-n2” — 2019/7/14 — 21:27 — page 220 — #70
220 Globalization for partial (co)actions
Remark 3.12. It is not clear to the authors if the converse of Remark
3.11 is true in general. However, it holds if C ≃ C∗∗ via
∧ : C −→ C∗∗
c 7−→ ĉ : α 7−→ α(c).
In fact, given h ∈ H and c ∈ C we consider ξh,c ∈ C∗∗ given by
ξh,c(α) = (h → α)(c).
Hence, we can define the linear map
↼ : C ⊗H −→ C
c⊗ h 7−→ c ↼ h = ∧−1(ξh,c),
that clearly satisfies the Equation (2).
The next result follows from Remarks 3.11 and 3.12.
Proposition 3.13. Let C be a coalgebra. Then the following statements
hold:
(i) If C is a partial H-module coalgebra, then C∗ is a partial H-module
algebra.
(ii) If C∗ is a partial H-module algebra and C ≃ C∗∗ via ∧, then C is
a partial H-module coalgebra.
Examples and the induced partial action
Example 3.14. Any global right H-module coalgebra is a partial one.
Example 3.15. Let α be a linear functional on H , then the ground field
k is a right partial H-module coalgebra via x ↼ h = xα(h) if and only if
the following conditions hold, for any x ∈ k and h, k ∈ H:
(i) α(1H) = 1k;
(ii) α(h)α(k) = α(h1)α(h2k).
Example 3.16. Let G be a group and H = kG the group algebra.
Consider α ∈ kG∗ and let N = {g ∈ G | α(g) 6= 0}. Then k is a partial
kG-module coalgebra if and only if N is a subgroup of G. In this case, we
have that
α(g) =
{
1, if g lies in N
0, otherwise.
“adm-n2” — 2019/7/14 — 21:27 — page 221 — #71
F. Castro, G. Quadros 221
Example 3.17 ([5, Theorem 5.7]). A group G acts partially on a coalgebra
C if and only if C is a symmetric partial left kG-module coalgebra. In
this case, g · c = θg(Pg−1(c)) and Pg(c) = ε(g−1 · c1)c2 = c1ε(g
−1 · c2).
Now, we construct a partial action on a coalgebra from a global one.
Let D be a right H-module coalgebra and C ⊆ D a subcoalgebra.
Since D is a right H-module, we could try to induce a partial action
restricting the action of D on C. But one can note that the range of this
restriction does not need to be contained in C, thus we need to project
it on C. By this way, let π : D → C be a projection from D over C (as
vector spaces) and consider the following map
ı↼ : C ⊗H
◭
−→ D
π
−→ C,
where ◭ denotes the right action of H on D.
In the sequel, we exhibit a necessary and sufficient condition to the
above map to be a partial action on C. In the next proposition, we denote
by C ◭ H the vector space spanned by the elements c ◭ h, for c ∈ C and
h ∈ H.
Proposition 3.18 (Induced partial module coalgebra). Let D be a right
H-module coalgebra, C ⊆ D a subcoalgebra and π : D → C a comultiplica-
tive projection satisfying
π[π(x) ◭ h] = π[ε(π(x1))x2 ◭ h], (3)
for any x ∈ C ◭ H.
Consider ı↼ : C ⊗H → C the linear map given by
c ı↼ h := π(c ◭ h), (4)
then C becomes a right partial H-module coalgebra via ı↼ .
Proof. (PMC1): Let c ∈ C, so c ı↼ 1H = π(c ◭ 1H) = π(c) = c, where
the last equality holds because π is a projection.
(PMC2): If π is a comultiplicative map (i.e., ∆ ◦ π = (π ⊗ π) ◦∆),
then for any c ∈ C and h ∈ H, we have
∆(c ı↼ h) = ∆(π(c ◭ h))
= (π ⊗ π)(∆(c ◭ h))
(MC2)
= (π ⊗ π)(c1 ◭ h1 ⊗ c2 ◭ h2)
“adm-n2” — 2019/7/14 — 21:27 — page 222 — #72
222 Globalization for partial (co)actions
= π(c1 ◭ h1)⊗ π(c2 ◭ h2)
= c1 ı↼ h1 ⊗ c2 ı↼ h2.
(PMC3): For h, k ∈ H and c ∈ C, we have
ε(c1 ı↼ h1)(c2 ı↼ h2k) = ε[π(c1 ◭ h1)]π(c2 ◭ h2k)
(MC3)
= ε[π(c1 ◭ h1)]π[(c2 ◭ h2) ◭ k]
(MC2)
= ε[π((c ◭ h)1)]π[((c ◭ h)2) ◭ k]
= π[ε[π((c ◭ h)1)](c ◭ h)2 ◭ k]
(3)
= π[π(c ◭ h) ◭ k]
= (c ı↼ h) ı↼ k.
Hence, C is a partial module coalgebra.
Remark 3.19. One can note that the converse of Proposition 3.18 is also
true. Indeed, supposing C a subcoalgebra of a module coalgebra D and
π : D → C a comultiplicative projection such that C is a partial module
coalgebra by c ı↼ h = π(c ◭ h), hence π[π(x) ◭ h] = π[ε(π(x1))x2 ◭ h]
for any x ∈ C ◭ H.
The proof of the above statement follows straight from the calculations
made in Proposition 3.18.
With the construction of induced partial action we have the necessary
tools to define a globalization for a partial module coalgebra. This is our
next goal.
3.2. Globalization for partial module coalgebra
From now on, given a left (resp., right) H-module M with the action
denoted by ⊲ (resp., ◭), we consider the k-vector space H ⊲ M (resp.,
M ◭ H) as the k-vector space generated by the elements h ⊲ m (resp.,
m ◭ h) for all h ∈ H and m ∈ M . Clearly, it is an H-submodule of M .
Definition 3.20 (Globalization for partial module coalgebra). Given a
right partial H-module coalgebra C with a partial action ↼, a globalization
of C is a triple (D, θ, π), where D is a right H-module coalgebra via ◭,
θ : C → D is a coalgebra monomorphism and π is a comultiplicative
projection from D onto θ(C), satisfying the following conditions for all
h ∈ H, c ∈ C and d ∈ D:
(GMC1) π[π(d) ◭ h] = π[ε(π(d1))d2 ◭ h];
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F. Castro, G. Quadros 223
(GMC2) θ(c ↼ h) = θ(c) ı↼ h;
(GMC3) D is the H-module generated by θ(C), i.e., D = θ(C) ◭ H.
Remark 3.21. The first condition of Definition 3.20 says that we can
induce a structure of a partial module coalgebra on θ(C). The second
one says that this induced partial action on θ(C) is equivalent to the
partial action on C. The last one says that does not exists any submodule
coalgebra of D containing θ(C).
Correspondence between globalizations. Our next aim is to estab-
lish relations between the globalization for partial module coalgebras, as
defined in Definition 3.20, and for partial module algebras (cf. [3]). For
this we use the fact that a partial module coalgebra C naturally induces
a structure of a partial module algebra on C∗.
Given a right partial H-module coalgebra C, it follows from Proposition
3.13 that the dual C∗ is a left partial H-module algebra. The same is true
for (global) module coalgebras (cf. [14]).
Remark 3.22. Let C be a right partial H-module coalgebra, D a right
H-module coalgebra, θ : C → D a coalgebra monomorphism and π : D →
θ(C) a comultiplicative projection. Since θ is injective, hence it has an
inverse θ−1, defined in θ(C) = π(D).
Consider the linear map ϕ : C∗ → D∗, given by the transpose of θ−1◦π,
i.e., for any α ∈ C∗, we have ϕ(α) := (θ−1 ◦ π)∗(α) = α ◦ θ−1 ◦ π. This is
clearly a multiplicative monomorphism.
So, one can also define the following linear maps:
ı↼ : π(D)⊗H −→ π(D)
π(d)⊗ h 7−→ π(π(d) ◭ h)
and
→ı : H ⊗ ϕ(C∗) −→ D∗
h⊗ ϕ(α) 7−→ ϕ(εC) ∗ (h ⊲ ϕ(α)).
Note that, there is a correspondence between these maps, given by
(h →ı ϕ(α))(π(d)) = ϕ(α)(π(d) ı↼ h), (5)
for any h ∈ H, d ∈ D and α ∈ C∗. In fact,
(h →ı ϕ(α))(π(d)) = (ϕ(ε) ∗ (h ⊲ ϕ(α)))(π(d))
= ϕ(ε)(π(d)1) ((h ⊲ ϕ(α)))(π(d)2)
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224 Globalization for partial (co)actions
= ϕ(ε)(π(d)1) ϕ(α)(π(d)2 ◭ h)
= ϕ(ε)(π(d1)) ϕ(α)(π(d2) ◭ h)
= ε(θ−1(π(π(d1)))) α(θ
−1(π(π(d2) ◭ h)))
= εθ(C)(π(d1)) α(θ
−1(π(π(π(d2) ◭ h))))
= εθ(C)(π(d)1) ϕ(α)(π(π(d)2 ◭ h))
= ϕ(α)(π(π(d) ◭ h))
= ϕ(α)(π(d) ı↼ h),
for any h ∈ H, d ∈ D and α ∈ C∗.
Notice that the maps ı↼ and →ı are the induced partial actions on
π(D) and ϕ(C∗), respectively.
In fact, the next statement shows that the map ı↼ is the induced
right partial action in π(D) if and only if the map →ı is the induced left
partial action in ϕ(C∗) and, moreover, it relates the globalization of the
partial module coalgebra to the globalization of the dual partial module
algebra. Therefore, for simplicity we will write ↼ instead of ı↼ and →
instead of →ı (even for induced partial actions).
Theorem 3.23. Let C be a partial module coalgebra. With the above
notations, we have that (θ(C) ◭ H, θ, π) is a globalization for C if and
only if (H ⊲ ϕ(C∗), ϕ) is a globalization for C∗.
Proof. If (θ(C) ◭ H, θ, π) is a globalization for C, it follows that
(ϕ(εC) ∗ (h ⊲ ϕ(α)))(d) = (ϕ(εC)(d1))((h ⊲ ϕ(α))(d2))
= εC(θ
−1(π(d1)))ϕ(α)(d2 ◭ h)
= εθ(C)(π(d1))α(θ
−1(π(d2 ◭ h)))
= α(θ−1(π(εθ(C)(π(d1))d2 ◭ h)))
(GMC1)
= α(θ−1(π(π(d) ◭ h)))
(4)
= α(θ−1(π(d) ↼ h))
(GMC2)
= α((θ−1(π(d))) ↼ h)
(3.13)
= (h → α)(θ−1(π(d)))
= ϕ(h → α)(d),
for every h ∈ H,α ∈ C∗ and d ∈ D. Thus, ϕ(C∗) is a right ideal of
H⊲ϕ(C∗) and, moreover, h → ϕ(α) = ϕ(h → α). Therefore, (H⊲ϕ(C∗), ϕ)
is a globalization for C∗, as desired.
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F. Castro, G. Quadros 225
Conversely, if (H ⊲ ϕ(C∗), ϕ) is a globalization for C∗, then
(GMC1): Given α ∈ C∗, h ∈ H and d ∈ D we have
α(θ−1(π(π(d) ◭ h))) = α(θ−1(π(π(d) ↼ h)))
= ϕ(α)(π(d) ↼ h)
(5)
= (h → ϕ(α))(π(d))
(GMA1)
= ϕ(h → α)(π(d))
= (h → α)(θ−1(π(π(d))))
= (h → α)(θ−1(π(d)))
= ϕ(h → α)(d)
= (ϕ(εC) ∗ (h ⊲ ϕ(α)))(d)
= ϕ(εC)(d1)(h ⊲ ϕ(α))(d2)
= ϕ(εC)(d1)ϕ(α)(d2 ◭ h)
= ϕ(εC)(d1)α(θ
−1(π(d2 ◭ h)))
= εC(θ
−1(π(d1)))α(θ
−1(π(d2 ◭ h)))
= ε(π(d1))α(θ
−1(π(d2 ◭ h)))
= α(θ−1(π(ε(π(d1))d2 ◭ h))).
Since α is an arbitrary linear functional on C and θ ◦ θ−1 = IC , then
π(π(d) ◭ h) = π(ε(π(d1))d2 ◭ h).
(GMC2): Given α ∈ C∗, h ∈ H and c ∈ C, then
(α ◦ θ−1)(θ(c) ↼ h) = (α ◦ θ−1)(π(θ(c) ◭ h))
= (α ◦ θ−1 ◦ π)(θ(c) ◭ h)
= ϕ(α)(θ(c) ◭ h)
= [h ⊲ ϕ(α)](θ(c))
= [h ⊲ ϕ(α)](θ(εC(c1) c2))
= εC(c1) [h ⊲ ϕ(α)](θ(c2))
= (εC ◦ θ−1 ◦ θ)(c1) [h ⊲ ϕ(α)](θ(c2))
= (εC ◦ θ−1 ◦ π ◦ θ)(c1) [h ⊲ ϕ(α)](θ(c2))
= ϕ(εC)(θ(c1)) [h ⊲ ϕ(α)](θ(c2))
= ϕ(εC)(θ(c)1) [h ⊲ ϕ(α)](θ(c)2)
= [ϕ(εC) ∗ (h ⊲ ϕ(α))] (θ(c))
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226 Globalization for partial (co)actions
= (h → ϕ(α)) (θ(c))
(GMA1)
= ϕ(h → α) (θ(c))
= (h → α) (θ−1(π(θ(c))))
= (h → α) (θ−1(θ(c)))
= (h → α)(c)
(3.13)
= α (c ↼ h)
= (α ◦ θ−1 ◦ θ)(c ↼ h)
= (α ◦ θ−1)(θ(c ↼ h)).
Since α ∈ C∗ is arbitrary, we obtain that θ(c) ↼ h = θ(c ↼ h).
Therefore, (θ(C) ◭ H, θ, π) is a globalization for C.
The standard globalization. Now our next aim is to show that every
partial module coalgebra has a globalization, constructing the standard
globalization.
Remark 3.24. Let C be a right partial H-module coalgebra. Consider
the coalgebra C ⊗H with the natural structure of the tensor coalgebra,
the coalgebra monomorphism from C into C ⊗H, given by the natural
embedding
ϕ : C −→ C ⊗H
c 7−→ c⊗ 1H ,
and the projection from C ⊗H onto ϕ(C), given by
π : C ⊗H −→ θ(C)
c⊗ h 7−→ (c ↼ h)⊗ 1H .
We claim that π is comultiplicative. Indeed, for c ∈ C and h ∈ H we have
∆(π(c⊗ h)) = ∆((c ↼ h)⊗ 1H)
= (c ↼ h)1 ⊗ 1H ⊗ (c ↼ h)2 ⊗ 1H
(PMC2)
= c1 ↼ h1 ⊗ 1H ⊗ c2 ↼ h2 ⊗ 1H
= π(c1 ⊗ h1)⊗ π(c2 ⊗ h2)
= (π ⊗ π)∆(c⊗ h).
With the above noticed we are able to construct a globalization for a
partial module coalgebra.
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F. Castro, G. Quadros 227
Theorem 3.25. Every right partial H-module coalgebra has a globaliza-
tion.
Proof. Let C be a left partial H-module coalgebra, so from Examples
3.5 and 3.6, we know that C ⊗H is an H-module coalgebra, with action
given by right multiplication in H. By the above noticed, we have the
maps ϕ : C → C ⊗H and π : C ⊗H → ϕ(C), as required in Definition
3.20. Then we only need to show that the conditions (GMC1) and (GMC2)
hold.
(GMC1): For every h, k ∈ H and c ∈ C, we have
π[ε(π((c⊗ h)1))(c⊗ h)2 ◭ k]
(PMC2)
= π[ε(π(c1 ⊗ h1))(c2 ⊗ h2) ◭ k]
= ε(c1 ↼ h1)π[(c2 ⊗ h2) ◭ k]
= ε(c1 ↼ h1)π[c2 ⊗ h2k]
= ε(c1 ↼ h1)(c2 ↼ h2k)⊗ 1H
(PMC3)
= (c ↼ h) ↼ k ⊗ 1H
= π[(c ↼ h)⊗ k]
= π[((c ↼ h)⊗ 1H) ◭ k]
= π[π(c⊗ h) ◭ k].
(GMC2): Let h ∈ H and c ∈ C, then
ϕ(c) ↼ h = π[ϕ(c) ◭ h]
= π[(c⊗ 1H) ◭ h]
= π[c⊗ h]
= c ↼ h⊗ 1H
= ϕ(c ↼ h).
Moreover, by the definitions of π, ϕ and ◭ it follows that ϕ(C) ◭ H =
C ⊗H. Therefore C ⊗H is a globalization for C.
The globalization above constructed is called the standard globalization
and it is close related with the standard globalization for partial module
algebras, as follows.
Theorem 3.26. Let C be a right partial H-module coalgebra, then the
standard globalization for C generates the standard globalization for C∗ as
left partial H-module algebra.
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228 Globalization for partial (co)actions
Proof. From Theorem 3.25, we have that (C ⊗H,ϕ, π) is the standard
globalization for C. Consider the multiplicative map
φ : C∗ −→ (C ⊗H)∗
α 7−→ α ◦ ϕ−1 ◦ π.
Thus, by the Theorem 3.23, we have that (H ⊲φ(C∗), φ) is a globalization
for C∗, where the action on (C ⊗H)∗ is given by
⊲ : H ⊗ (C ⊗H)∗ −→ (C ⊗H)∗
h⊗ ξ 7−→ (h ⊲ ξ)(c⊗ k) = ξ(c⊗ k h),
for every ξ ∈ (C ⊗H)∗, c ∈ C and h, k ∈ H.
Now, consider the following algebra isomorphism given by the adjoint
isomorphism
Ψ: (C ⊗H)∗ −→ Hom(H, C∗)
ξ 7−→ [Ψ(ξ)(h)](c) = ξ(c⊗ h),
which is an H-module morphism. In fact, let h, k ∈ H, c ∈ C and
ξ ∈ (C ⊗H)∗, so
{[Ψ(h ⊲ ξ)](k)}(c) = [(h ⊲ ξ)](c⊗ k)
= ξ(c⊗ k h)
= {[Ψ(ξ)](k h)}(c)
= {[h ⊲Ψ(ξ)](k)}(c)
and, therefore, Ψ is an H-module map. Moreover, composing Ψ with φ
we obtain
{[Ψ ◦ φ(α)](h)}(c) = φ(α)(c⊗ h)
= α(ϕ−1(π(c⊗ h)))
= α(ϕ−1(c ↼ h⊗ 1H))
= α(c ↼ h)
= (h → α)(c)
= [Φ(α)(h)](c),
where Φ: C∗ → Hom(H, C∗), given by Φ(α)(h) = h → α, for all h ∈ H
and α ∈ C∗ is the multiplicative map that appears in the construction of
the standard globalization, replacing A by C∗ (cf. [3, Theorem 1]).
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F. Castro, G. Quadros 229
4. Partial coaction on coalgebras
4.1. Partial comodule coalgebra
Given two vector spaces V and W , we write τV,W to denote the standard
isomorphism between V ⊗W and W ⊗ V .
Definition 4.1 (Comodule coalgebra). A left H-comodule coalgebra is
a pair (D,λ), where D is a coalgebra and λ : D → H ⊗D is a linear map,
such that, for all d ∈ D, the following conditions hold:
(CC1) (εH ⊗ I)λ(d) = d;
(CC2) (I ⊗∆D)λ(d) = (mH ⊗ I ⊗ I)(I ⊗ τD,H ⊗ I)(λ⊗ λ)∆D(d);
(CC3) (I ⊗ λ)λ(d) = (∆H ⊗ I)λ(d).
In this case, we say that H coacts on D via λ, or that λ is a coaction of
H on D. We will also call it a global comodule coalgebra to differ explicitly
from the partial one.
We can also see the above definition in a categorical approach, in
the following sense (cf. [14, Definition 11.3.7]): a k-vector space D is a
left H-comodule coalgebra if it is a coalgebra object in the category of left
H-comodules.
From this categorical point of view, one additional condition is required
in Definition 4.1, that is,
(I ⊗ εD)λ(d) = εD(d)1H , (6)
for all d ∈ D.
Since H is a Hopf algebra (so it has an antipode), thus this additional
condition may be obtained from the another ones, as stated below.
Proposition 4.2. Let D be a left H-comodule coalgebra in the sense of
Definition 4.1. Then (I ⊗ εD)λ(d) = εD(d)1H , for any d ∈ D.
Remark 4.3. From the above proposition one can extend the notion of
comodule coalgebra for non-counital coalgebras.
Now we exhibit some classical examples of comodule coalgebras (cf. [14,
Section 11.3]).
Example 4.4. A Hopf algebra H is an H-comodule coalgebra with
coaction λ : H → H ⊗H given by
λ(h) = h1S(h3)⊗ h2,
for any h ∈ H.
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230 Globalization for partial (co)actions
Example 4.5. Any coalgebra D is an H-comodule coalgebra with coac-
tion λ : D → H ⊗D defined by λ(d) = 1H ⊗ d, for every d ∈ D.
Example 4.6. If the Hopf algebra H is finite dimensional, then H∗ is
an H-comodule coalgebra with structure given by λ : H∗ → H ⊗H∗ with
λ(f) =
n∑
i=1
hi ⊗ f ∗ h∗i , where {hi}
n
i=1 and {h∗i }
n
i=1 are dual basis for H
and H∗, respectively.
Example 4.7. Let C be a left H-comodule coalgebra with coaction λ
and D a coalgebra, then C ⊗D is a left H-comodule coalgebra via λ⊗ ID.
Definitions and correspondences
Definition 4.8 (Partial comodule coalgebra). [5, Definition 6.1] A left
partial H-comodule coalgebra is a pair (C, λ′), where C is a coalgebra and
λ′ : C → H ⊗ C is a linear map, such that, for any c ∈ C, the following
conditions hold:
(PCC1) (εH ⊗ I)λ′(c) = c;
(PCC2) (I ⊗∆C)λ
′(c) = (mH ⊗ I ⊗ I)(I ⊗ τC,H ⊗ I)(λ′ ⊗ λ′)∆C(c);
(PCC3) (I ⊗ λ′)λ′(c) = (mH ⊗ I ⊗ I){∇ ⊗ [(∆H ⊗ I)λ′]}∆C(c),
where ∇ : C → H is defined by ∇(c) = (I ⊗ εC)λ
′(c).
The partial comodule coalgebra is said symmetric if the following
additional condition holds, for any c ∈ C:
(PCC4) (I⊗λ′)λ′(c) = (mH⊗I⊗I)(I⊗τ
H⊗C,H
){[(∆H⊗I)λ′]⊗∇}∆C(c).
Remark 4.9. For a partial comodule coalgebra C via λ′ we use the
Sweedler’s notation λ′(c) = c−1 ⊗ c0, where the summation is under-
stood. The bar over the upper index is useful to distinguish partial from
global comodule coalgebras when working with both structures in a single
computation.
Proposition 4.10. [5, Lemma 6.3 and Corollary 6.4] Let C be left partial
H-comodule coalgebra, then, for all c ∈ C, the following equalities hold:
c−1 ⊗ c0 = ∇(c1)c2
−1 ⊗ c2
0 = c1
−1∇(c2)⊗ c1
0 (7)
and
∇(c1)∇(c2) = ∇(c). (8)
Example 4.11. Every left H-comodule coalgebra is a left partial H-
comodule coalgebra.
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F. Castro, G. Quadros 231
The next result gives us a simple method to construct new examples
of partial comodule coalgebras. The proof is straightforward and it will
be omitted.
Proposition 4.12. Let λ′ : k → H ⊗ k be a linear map and h ∈ H such
that λ′(1k) = h⊗ 1k. Then the ground field k is a left partial H-comodule
coalgebra if and only if the following conditions hold:
1) εH(h) = 1k;
2) h⊗ h = (h⊗ 1H)∆(h).
As an application of the above result, we present the next example.
Example 4.13. Let G be a group, λ′ : k → kG⊗ k a linear map and x =∑
g∈G
αgg in kG such that λ′(1) = x⊗ 1. Consider N = {g ∈ G | αg 6= 0}
and suppose that the characteristic of k does not divides |N |. Then k is
a left partial kG-comodule coalgebra via λ′ if and only if N is a finite
subgroup of G. In this case, we have that
αg =
1
|N |
,
for all g ∈ N
Proposition 4.14. [5] Let C be a left partial H-comodule coalgebra via
λ′, then it is a (global) H-comodule coalgebra if and only if
c−1εC(c
0) = εC(c)1H ,
for all c ∈ C.
Given a Hopf algebra H , we say that the finite dual H0 separate points
if it is dense on H∗ in the finite topology, i.e., if h ∈ H is such that
f(h) = 0, for all f ∈ H0, then h = 0.
For a coalgebra C and a linear map λ′ : C → H ⊗ C (denoting by
λ′(c) = c−1 ⊗ c0) we have two induced linear maps λ′↼ : C ⊗H∗ → C and
λ′→ : H∗ ⊗ C∗ → C∗, given respectively by
c λ′↼f = f(c−1)c0 (9)
(f λ′→α)(c) = f(c−1)α(c0), (10)
for all c ∈ C,α ∈ C∗ and f ∈ H∗. Since, in general, H∗ is not a Hopf
algebra, thus we can restrict λ′↼ and λ′→ to the subspaces C ⊗H0 and
H0 ⊗ C∗, respectively. Therefore, under the assumption that C is a left
partial H-comodule coalgebra, we can show the following.
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232 Globalization for partial (co)actions
Theorem 4.15. With the above notations, if C is a left partial H-comodule
coalgebra via λ′, then the following statements hold:
(1) C is a right partial H0-module coalgebra via λ′↼ ;
(2) C∗ is a left partial H0-module algebra via λ′→ .
Proof. (1) : Taking the dual pairing between H andH0 given by evaluation,
then by Theorem 6.7 of [5] we have the desired.
(2) : In this case, taking the dual pairing between C and C∗ given by
evaluation, then the desired follows from Theorem 6.8 of [5].
It is not clear if the converse of the above theorem is true in general,
but whenever H0 separate points (so the dual pairing between H and H0
is non-degenerate) it holds, as stated in the next theorem.
Theorem 4.16. With the above notations, if H0 separate points, then
the following conditions are equivalent:
(1) C is a right partial H0-module coalgebra via λ′↼ ;
(2) C is a left partial H0-module algebra via λ′→ ;
(3) C is a left partial H-comodule coalgebra via λ′.
The above theorem can be translated in the following commutative
diagram:
(C, λ′, H) //
oo H0 sep points
��
OO
H0 sep points
(C∗, λ′→ , H0)
(C, λ′↼,H0)
uu
55
(11)
Theorems 4.15 and 4.16 show relations between a partial comodule
coalgebra, a partial module coalgebra and a partial module algebra, when-
ever we start from a partial coaction λ′. In general, we can not start from
an action and induce a coaction. To do this we require a more strong
hypothesis on H, more precisely, we assume that H is finite dimensional.
In fact, if H is finite dimensional, then H0 = H∗ (and so H0 separate
points). Moreover, given a linear map ↼ : C ⊗H∗ → C and a dual basis
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F. Castro, G. Quadros 233
{hi, h
∗
i }
n
i=1 for H and H∗, then one can induce a linear map λ′
↼ : C →
H ⊗ C, by
λ′
↼(c) =
n∑
i=1
hi ⊗ c ↼ h∗i
and it is clear that
f(c−1)c0 = c ↼ f,
for all c ∈ C and f ∈ H∗.
Thus, for a finite dimensional Hopf algebra H , we can induce a coaction
of H on a coalgebra C from a given action of H∗ on C.
In [2], there is a similar construction for (right) partial H-comodule
algebras and (left) partial H∗-module algebras. Hence we have that these
four partial structures are close related, in the following sense:
Theorem 4.17. Let C be a coalgebra and suppose that the Hopf algebra
H is finite dimensional. Then the following statements are equivalent:
(1) C is a left partial H-comodule coalgebra;
(2) C∗ is a right partial H-comodule algebra;
(3) C is a right partial H∗-module coalgebra;
(4) C∗ is a left partial H∗-module algebra.
The relations between the correspondent actions and coactions are given
in the following way, for any c ∈ C, α ∈ C∗ and f ∈ H∗:
α(c−1)c0 = α0(c)α+1 (12)
(f → α)(c) = α(c ↼ f) (13)
c ↼ f = f(c−1)c0 (14)
f → α = α0f(α+1), (15)
where λ′ : c 7→ c−1 ⊗ c0 and ρ′ : α 7→ α0 ⊗ α+1 are the partial coactions on
C and C∗, respectively.
Using Theorem 4.17, we can extend the Diagram (11) to the following
commutative diagram, under the hypothesis that the Hopf algebra is finite
dimensional:
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234 Globalization for partial (co)actions
(C, λ′, H) oo //
OO
��
(C∗, λ′→ , H0)
(C, λ′↼,H0) (C∗, ρ′, H)
��
OO
//oo
(16)
4.2. Globalization for partial comodule coalgebras
In this section, our goal is to introduce the concept of globalization
for partial comodules coalgebras.
Let D be a left H-comodule coalgebra via λ : d 7→ d−1 ⊗ d0 ∈ H ⊗D
and C a subcoalgebra of D. In order to induce a coaction on C we can
restrict the coaction λ to C, but in general λ(C) 6⊆ H ⊗ C. However, if
there is a linear map π : D → C we consider the composite map
λ′: C −→ H ⊗ C
c 7−→ c−1 ⊗ π(c0). (17)
The following result gives us conditions on the map π for the above
map becomes a partial coaction on C.
Proposition 4.18 (Induced partial comodule coalgebra). Let (D,λ) be
a left H-comodule coalgebra, C a subcoalgebra of D and π : D → C a
comultiplicative projection such that
(I ⊗ I ⊗ π)(I ⊗ λπ)λ(c) = (I ⊗ I ⊗ π)(I ⊗ λ⊗ επ)(I ⊗ τ∆)λ(c), (18)
for any c ∈ C. Then C is a left partial H-comodule coalgebra, with structure
given by Equation (17).
Proof. First of all, since π is a projection from D onto C, then λ′ satisfies
the condition (PCC1). In fact, given c ∈ C, we have
(ε⊗ I)λ′(c) = ε(c−1)π(c0)
= π(ε(c−1)c0)
(CC1)
= π(c)
= c,
where the last equality holds since π is a projection (and so π(c) = c, for
all c ∈ C).
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F. Castro, G. Quadros 235
Now, since π is a comultiplicative map (i.e. ∆ ◦ π = (π ⊗ π) ◦∆) then
it follows that λ′ satisfies the condition (PCC2). In fact, let c ∈ C, so
(I ⊗∆C)λ
′(c) = c−1 ⊗∆(π(c0))
= c−1 ⊗ (π ⊗ π)(∆(c0))
= c−1 ⊗ π(c01)⊗ π(c02)
(CC2)
= c1
−1c2
−1 ⊗ π(c1
0)⊗ π(c2
0)
= (mH ⊗ I ⊗ I)(I ⊗ τC,H ⊗ I)(λ′ ⊗ λ′)∆C(c).
Finally, since π satisfies Equation (18) we have that, for all c ∈ C
(I ⊗ λ′)λ′(c) = c−1 ⊗ c0−1 ⊗ c00
= c−1 ⊗ π(c0)−1 ⊗ π(π(c0)0)
(18)
= c−1 ⊗ c02
−1 ⊗ ε(π(c01))π(c
0
2
0)
(CC2)
= c1
−1c2
−1 ⊗ c2
0−1 ⊗ ε(π(c1
0))π(c2
00)
(CC3)
= c1
−1ε(π(c1
0))c2
−1
1 ⊗ c2
−1
2 ⊗ π(c2
0)
= c1
−1ε(c1
0)c2
−1
1 ⊗ c2
−1
2 ⊗ c2
0
= (mH ⊗ I ⊗ I){∇ ⊗ [(∆H ⊗ I)λ′]}∆C(c).
Therefore, C is a left partial H-comodule coalgebra.
Remark 4.19. One can note that the converse of Proposition 4.18 is
also true. Indeed, supposing C a subcoalgebra of a comodule coalgebra
D and π : D → C a comultiplicative projection such that C is a partial
comodule coalgebra by λ′(c) = c−1 ⊗ π(c0), hence Equation (18) holds.
The proof of the above statement follows straight from the calculations
made in Proposition 4.18.
We are now able to define a globalization for a partial comodule
coalgebra, as follows.
Definition 4.20 (Globalization for a partial comodule coalgebra). Let
C be a left partial H-comodule coalgebra. A globalization for C is a
triple (D, θ, π), where D is an H-comodule coalgebra, θ is a coalgebra
monomorphism from C into D and π is a comultiplicative projection from
D onto θ(C), such that the following conditions hold:
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236 Globalization for partial (co)actions
(GCC1) x−1 ⊗ π(x0)−1 ⊗ π(π(x0)0) =
= x−1 ⊗ x02
−1 ⊗ ε(π(x01))π(x
0
2
0),
for all x ∈ θ(C);
(GCC2) θ is an equivalence of partial H-comodule coalgebra;
(GCC3) D is the H-comodule coalgebra generated by θ(C).
Remark 4.21. The first item in Definition 4.20 tells us that it is possible
to define the induced partial comodule coalgebra on θ(C).
The second one tells us that this induced partial coaction coincides
with the original, and this fact is translated in the commutative diagram
bellow:
C
λ′
//
θ
��
H ⊗ C
I⊗θ
��
θ(C)
λπ // H ⊗ θ(C)
� (19)
Moreover, the second condition can be seen as
θ(c)−1 ⊗ π(θ(c)0) = c−1 ⊗ θ(c0), (20)
for all c ∈ C.
Finally, the last condition of Definition 4.20 tells us that there is no
proper subcomodule coalgebra of D containing θ(C).
Correspondence between globalizations. Given a left partial
H-comodule coalgebra C we can induce a structure of a right partial
H0-module coalgebra on C (see Theorem 4.15). It is also true that
a (global) H-comodule coalgebra induces a (global) H0-module coalgebra.
Therefore, given a globalization (D, θ, π) for C, one can ask: Is there some
relation between D and C when viewed as H0-module coalgebras (global
and partial, respectively)? Here we study a little bit more these structures
in order to answer this question. The notations previously used are kept.
Let C be a left partial H-comodule coalgebra and suppose that (D, θ, π)
is a globalization for C. From Theorem 4.15, we have that C is a right
partial H0-module coalgebra with partial action given by
c ↼ f = f(c−1)c0,
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F. Castro, G. Quadros 237
for all c ∈ C and f ∈ H0. Clearly, the same is true for D, i.e., we have a
structure of H0-module coalgebra on D given by
d ◭ f = f(d−1)d0,
for all d ∈ D and f ∈ H0.
Theorem 4.22. Let C be a left partial H-comodule coalgebra and suppose
that (D, θ, π) is a globalization for C. If H0 separate points, then (D, θ, π)
is also a globalization for C, as right partial H0-module coalgebra.
Proof. Since θ is a coalgebra monomorphism from C into D and π is
a comultiplicative projection from D onto θ(C), in order to induce a
structure of partial H0-module coalgebra on θ(C) we just need to check
that the Equation (3) holds (see Proposition 3.18). For this, let x = θ(c) ◭
g ∈ θ(C) ◭ H0 and f ∈ H0, so
π(π(x) ◭ f) = π(π(θ(c)0g(θ(c)−1)) ◭ f)
= g(θ(c)−1)π(π(θ(c)0)0f(π(θ(c)0)−1))
= g(θ(c)−1)f(π(θ(c)0)−1)π(π(θ(c)0)0)
(18)
= g(θ(c)−1)f(θ(c)02
−1)π(θ(c)02
0)επ(θ(c)01)
= f((θ(c) ◭ g)2
−1)π((θ(c) ◭ g)2
0)επ((θ(c) ◭ g)1)
= f(x2
−1)π(x2
0)επ(x1)
= π(x2 ◭ f)επ(x1)
= π(ε(π(x1))x2 ◭ f).
Thus, θ(C) has a structure of a partial H0-module coalgebra induced from
the structure of module coalgebra of D.
Now we show that θ is a morphism of partial actions. In fact, let c ∈ C,
so
θ(c) ↼ f = π(θ(c) ◭ f)
= f(θ(c)−1)π(θ(c)0)
(20)
= f(c−1)θ(c0)
= θ(f(c−1)c0)
= θ(c ↼ f).
Therefore, we just need to show that the (GMC3) in Definition 3.20
holds.
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238 Globalization for partial (co)actions
Let M be any H0-submodule coalgebra of D containing θ(C). We
need to show that M = D, and for this it is enough to show that M is an
H-subcomodule coalgebra of D.
Take f ∈ H0, m ∈ M , and consider {hi} a basis of H. Let {h∗i } be
the set contained in H∗ whose elements are all the dual maps of the hi’s.
Then write λ(m) ∈ H ⊗D in terms of the basis of H, i.e.,
λ(m) =
n∑
i=0
hi ⊗mi,
where the mi’s are non-zero elements, at least, in D.
Since D is an H-comodule, so it is an H∗-module via the same action
of H0. Moreover, the action of H0 on D is a restriction of the action of H∗.
Since H0 separate points, it follows by Jacobson Density Theorem that if
m ∈ M then there exists
{
h0(m)i ∈ H0
}
such that m ◭ h0(m)i = m ◭ h∗i ,
for each i. Thus we have that
m ◭ h0(m)j = m ◭ h∗j =
n∑
i=0
h∗j (hi)mi = mj
and so each mi lies in M . Then M is an H-subcomodule of D, so M is an
H-subcomodule coalgebra of D containing θ(C), which implies M = D,
and the proof is complete.
Constructing a globalization. Now we construct a globalization for
a left partial comodule coalgebra C in a special situation. First of all,
remember that if M is a right H0-module and H0 separate points, then
we have a linear map
ϕ : M −→ Hom(H0,M)
m 7−→ ϕ(m)(f) = m · f
and an injective linear map
γ : H ⊗M −→ Hom(H0,M)
h⊗m 7−→ γ(h⊗m)(f) = f(h)m.
In the above situation, we say that M is a rational H0-module if ϕ(M) ⊆
γ(H ⊗M) (cf. [9, Definition 2.2.2]).
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F. Castro, G. Quadros 239
This definition can be seen in the following commutative diagram:
Hom(H0,M)
M
ϕ
@@
λ // H ⊗M
0 P
γ
aa
(21)
Notice that, given a rational H0-module M we have on it a structure
of H-comodule via λ : M → H ⊗M satisfying, for any m ∈ M
λ(m) =
∑
hi ⊗mi ⇐⇒ m · f =
∑
f(hi)mi, for all f ∈ H0. (22)
From Theorem 4.22, it follows that exists a natural way to look for a
globalization for a partial coaction of H on C, i.e., to get a globalization
for C we should see it as a partial module coalgebra and then consider its
standard globalization as module, under the hypothesis that H0 separate
points.
Thus, given a left partial H-comodule coalgebra (C, λ′) we have from
Theorem 3.25 that (C ⊗H0, θ, π) is a globalization for C as right partial
H0-module coalgebra.
We desire C ⊗ H0 to be a globalization for C as partial comodule
coalgebra, but, in general, it is not even an H-comodule coalgebra.
In order to overcome this problem we will suppose that C ⊗H0 is a
rational H0-module and, therefore, we have that C⊗H0 is an H-comodule
with coaction satisfying Equation (22), i.e., the following holds
λ(c⊗ f) =
∑
hi ⊗ ci ⊗ fi ⇐⇒ c⊗ (f ∗ g) =
∑
g(hi)ci ⊗ fi, (23)
for any c⊗ f ∈ C ⊗H0 and g ∈ H0
Therefore, by Theorem 4.16, C⊗H0 is an H-comodule coalgebra. Now
we are in position to show that C ⊗H0 is a globalization for C as partial
comodule colagebra, as follows.
Theorem 4.23. Let C be a left partial H-comodule coalgebra. With the
above notations, if C⊗H0 is a rational H0-module and H0 separate points,
then (C ⊗H0, θ, π) is a globalization for C.
Proof. By the above discussed, C ⊗ H0 is an H-comodule coalgebra,
θ : C → C ⊗H0 is a coalgebra map and π : C ⊗H0 ։ θ(C) is a comulti-
plicative projection.
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240 Globalization for partial (co)actions
Now, we can show directly that the conditions (GCC1)-(GCC3) hold.
Let x ∈ C ⊗H0, and f, g ∈ H0, so
(f ⊗ g ⊗ I)[x−1 ⊗ π(x0)−1 ⊗ π(π(x0)0)] =
= f(x−1)g(π(x0)−1)π(π(x0)0)
= f(x−1)π(g(π(x0)−1)π(x0)0)
= f(x−1)π[π(x0) ◭ g]
= π[π(x0f(x−1)) ◭ g]
= π[π(x ◭ f) ◭ g]
(GMC2)
= π[επ((x ◭ f)1)(x ◭ f)2 ◭ g]
= π[επ(x1 ◭ f1)(x2 ◭ f2) ◭ g]
= π[επ(x1
0f1(x1
−1))(x2
0f2(x2
−1) ◭ g)]
= f1(x1
−1)f2(x2
−1)επ(x1
0)π(x2
0
◭ g)
= f(x1
−1x2
−1)επ(x1
0)π(x2
0
◭ g)
(PCC2)
= f(x−1)επ(x01)π(x
0
2 ◭ g)
= f(x−1)επ(x01)π(x
0
2
0g(x02
−1))
= f(x−1)g(x02
−1)επ(x01)π(x
0
2
0)
= (f ⊗ g ⊗ I)(x−1 ⊗ x02
−1 ⊗ επ(x01)π(x
0
2
0)).
Since H0 separate points, the condition (GCC1) is satisfied.
To prove the condition (GCC2) take c ∈ C and note that
(g ⊗ I)[θ(c)−1 ⊗ π(θ(c)0)] = g(θ(c)−1)π(θ(c)0)
= π(g(θ(c)−1)θ(c)0)
= π(θ(c) ◭ g)
= θ(c ↼ g)
= g(c−1)θ(c0)
= (g ⊗ I)[c−1 ⊗ θ(c0)].
Since H0 separate points, thus θ is an equivalence of partial coactions.
Finally, to show that C ⊗H0 is generated by θ(C), consider a subco-
module coalgebra M of C ⊗H0 containing θ(C). By Theorem 4.15, M is
an H0-submodule coalgebra of C ⊗H0 containing θ(C). Thus, it follows
from condition (GMC3) that M = C ⊗H0.
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F. Castro, G. Quadros 241
Therefore C ⊗H0 is a globalization for C as a partial H-comodule
coalgebra.
The globalization above constructed is called the standard globalization
for a partial comodule coalgebra.
Remark 4.24. When the Hopf algebra is finite dimensional, H0 = H∗
and so it separate points. Furthermore, in this case C ⊗H∗ is a rational
H∗-module with coaction given by
λ : c⊗ f 7−→
n∑
i=1
hi ⊗ c⊗ f ∗ h∗i ,
where {hi, h
∗
i }
n
i=1 is a dual basis for H and H∗.
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Contact information
F. Castro Departamento de Matemática
Universidade Federal de Santa Catarina
88040-900 Brazil
E-Mail(s): f.castro@ufsc.br
Web-page(s): http://mtm.ufsc.br/~fcastro
G. Quadros Coordenadoria Acadêmica
Universidade Federal de Santa Maria
96506-322 Brazil
E-Mail(s): glauber.quadros@ufsm.br
Received by the editors: 15.07.2016
and in final form 08.05.2019.
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