1. Introduction
The notion of Hopf algebra was given by Borel in 1953 (see [
1]), honoring the foundational work of Heinz Hopf in algebraic topology. Nowadays, progress obtained in understanding the structure of Hopf algebras and their representations have been entwined with the development of different areas of mathematics such as knot theory [
2], topology, conformal field theory, algebraic geometry, ring theory [
3], category theory (see, [
4]), combinatorics, etc. More generally, a Hopf quasigroup (non-associative algebra) was introduced by Klim and Majid [
5], whose dual notion is a Hopf coquasigroup, a particular case of the notion of unital coassociative bialgebra introduced in [
6]. For more research on these aspects, see the papers [
7,
8,
9]. For some basic and recent papers related to non-associative BCC-algebras and B-filters in this field, we refer to [
10,
11,
12].
In the paper [
13], the authors constructed the relevance between an algebra
A and its subalgebra of invariants
under the action of a finite-dimensional Hopf algebra
H by using the smash product
of
A by
H. Furthermore, they studied the conception of Hopf Galois extension
and gave some equivalent conditions.
It is now natural to ask whether the results in [
13] hold in the framework of Hopf coquasigroups. This becomes our motivation of writing this paper. We will overcome non-associativity in Hopf quasigroups and non-coassociativity in Hopf coquasigroups by introducing some new notions.
Actually, in this paper, we introduce some several actions of Hopf coquasigroup H on an algebra A and consider the existence of a Morita context relating and . Then, we introduce the definition of a Hopf quasigroup Galois extension and apply the Morita setting to prove that for finite dimensional H, there are equivalent conditions for to be Galois. Finally, we consider a special case of a quasigroup.
This paper is organized as follows.
In
Section 2 of this paper, the notions of a Hopf (co)quasigroups, a smash product and an integral are recalled. Meanwhile, we introduce and study the notion of quasigroup-graded algebras.
In
Section 3 of this paper, we mainly construct a Morita context linking
and
by the connecting bimodules being both
A (see Theorem 1).
In
Section 4, we introduce the notion of a Hopf quasigroup Galois extension and study the surjectivity of the Morita context map (see Theorems 5 and 6), and in the final
Section 5, we treat a special case of a quasigroup (see Theorem 7).
Throughout this paper, unless stated otherwise,
k is a field, and all vector spaces are over
k. By linear maps, we means
k-linear maps. Unadorned ⊗ means
. We make use of the Sweedler’s notation for the comultiplication,
, where the summation sign is suppressed (see [
1]).
3. The Morita Context
We will establish a Morita context linking and for A our left quasi-H-module algebra.
Lemma 1. Let H be a Hopf coquasigroup and A a left quasi-H-module algebra. Then, A is a left or right -module via left or right multiplication.
Proposition 7. Let H be a Hopf coquasigroup and A a left quasi-H-module algebra. Then,
- (i)
A is a left -module, via , for any and .
- (ii)
If H has a bijective antipode S, then A is a right -module, via , for any and .
Proof. - (i)
By calculations, we have
and
for all
.
- (ii)
The process for the part (ii) is generally similar, but it is slightly more complicated because of the non-coassociativity. When the antipode
S of
H is bijective, we have for all
and
,
In addition,
and
where Equation (
4) was used to obtain the last equality. □
From now until the end of this section, we will assume that
H is a finite dimensional Hopf coquasigroup. Let us here make some comments on integrals for the Hopf coquasigroup, which will be helpful for our later discussion. By a duality to ([
17], Corollary 3.6), we conclude that if
H is a finite dimensional Hopf coquasigroup, then the antipode
S is always bijective.
We will use the following lemmas later.
Lemma 2. Let V be a n-dimensional vector space with a base of , and be any n scalars. Then, there exists a unique linear function λ on V such that for all .
Lemma 3 ([
18]).
Let A be an algebra. Then,- (i)
, where denotes the set of algebra homomorphisms from A to k.
- (ii)
is a linearly independent subset of .
Now, choose a non-zero , also for any . Since , it follows from Lemma 3 that , for some . Moreover, clearly, is an algebra homomorphism from A to k, and so by Lemma 4, it is a group-like element of . We note that is determined by the above equation and that does not depend on the choice of t. Similar to the classical one, the element of constructed above is called an H-distinguished grouplike element of .
By means of the above
, we can define a map from
H to itself as follows:
for all
.
Proposition 8. With the above f.
- (i)
We have that f is an algebraic automorphism on H.
- (ii)
We can extend f to an automorphism of by the following formula:
Proof. (i) The algebraic endomorphism is easy. Define a map
for all
. By computing, one has
where Equation (
5) was used in the last equation. Similarly,
.
(ii) Straightforward. □
In order to construct a Morita context better, we now define a new right action of
on
A using Equation (
17). For any
and
, set
Notice that if
, we have the following linear isomorphism
Then, if
, it follows from ([
17], Lemma 4.2) that
, and consequently,
is distinct of zero. Thus, it always possible to choose some
and
satisfying
.
Consider the right action of
H on
defined by
It is easy to verify that
is a right
H-module under ≺. It should be emphasized that in the previous discussion,
H is not a right
-module, because when
H is a Hopf coquasigroup,
is a Hopf quasigroup, and it is non-associative as algebra. Dually, if
, we also have the following linear isomorphism
which is called a
Fourier transformation in
H defined by Klim in [
17].
Combining the above two situations, we have
Proposition 9. For with and , we have .
Proof. On the one hand, since
t is a left integral in
H, we have
Notice that
is a right integral in
H; on the other hand, we have
This completes the proof. □
Proposition 10. Let φ, ψ, t and T be as above. If , then In particular, and .
Proof. For all
, we have
and
This finishes the proof. □
Proposition 11. Let t, λ and f be as above. Then, .
Proof. First of all, we remind that and are right integrals in H. In fact, when there exists a unique such that because of the bijectivity of f; then, , and it follows that is a right integral in H.
It follows from ([
17], Lemma 3.3(1)) that
is a right integral in
H as well. Therefore,
and
are linearly dependent.
Without loss of generality, we may assume that
for some
, and we have the following calculations:
which shows that
. Therefore,
. □
Proposition 12. If , then for all .
Proof. Suppose that
and
; then, we have
from which the proposition follows. □
In the remainder of this section, we will show the existence of a Morita context relating and when H is a finite dimensional Hopf coquasigroup. Before we do that, let us give a few results that we will use later.
Proposition 13. Let A be a left quasi-H-module algebra. Then, , where the element form is given byfor all . Proof. For any
and
, we have
where Equation (
4) was used to obtain the last equality. □
Proposition 14. Let A be a left quasi-H-module algebra. Suppose that t is a left integral in H and λ is an H-distinguished grouplike element in . Then, for all , consider A as a right -module as in Lemma 2; then, for any and ,
- (i)
.
- (ii)
.
Proof. (i) For any
and
, we compute.
(ii) For any
, one also has
This completes the proof. □
At last, after some preparations, we give the main theorem of this section, which generalizes the well-known results in [
13] to the case of finite dimensional Hopf coquasigroups.
Theorem 1. Let A be a left quasi-H-module algebra. Then, we can form a Morita context with the following module actions and maps:
- (1)
Consider A as a left (respectively, right) module via left (respectively, right) multiplication;
- (2)
Consider A as a left (respectively, right) -module via Proposition 7(i) (respectively, Equation (18)); - (3)
- (4)
Proof. First, we show that
A is an
-
-bimodule. Based on Lemma 1 and Proposition 7, the associativity conditions need to be supplemented. In fact, for all
and
, we have
Similarly, A is an --bimodule.
Next, we will show the linearity of these two maps and , respectively.
For the Morita map
, it is expected to be left and right
-linear and to be middle
-linear at the same time. Explicitly, for all
and
, we have
This complete the proof of left -linear.
As for right
-linear, we still need detailed verification, since the definition of left and right
-action on A is not symmetrical. For all
, and
, we have
On the other hand, we have
where Proposition 14(i) was used for the third equality. Hence, right
-linear.
As the middle
-linear, we have for all
and
,
and
Analogously, for the Morita map
, we need prove three linearities as well, respectively, left
-linear, right
-linear, and middle
-linear.
and
, we have
This shows that is a left -linear. Likewise, it is a right -linear.
Furthermore, for all
and
, we have
and so
is middle
-linearity.
At last, we are left with only the proof of compatibility conditions of
with
. For all
, we have
and
So far, we have constructed the Morita context. □
Then, it follows immediately that:
Corollary 1. Let and be as above. Then, we have
- (i)
is an ideal of .
- (ii)
is an ideal of .
For a right ideal
I of
and a left ideal
J of
, we put
respectively.
For a right
-submodule
N of
and a left
-submodule
P of
, one puts
respectively.
We now obtain the following results (see [
19]).
Theorem 2. With the above notation as in Theorem 1. Let be surjective. Then,
- (1)
and are inverses and are lattice isomorphisms between the lattice of right ideals of and the lattice of submodules of . Moreover, these induce lattice isomorphisms between the lattice of ideals of and the lattice of submodules of .
- (2)
and are inverses and are lattice isomorphisms between the lattice of left ideals of and the lattice of submodules of . Moreover, these induce lattice isomorphisms between the lattice of ideals of and the lattice of submodules of .
Similar treats for : For a right ideal X of and a left ideal Y of , we putrespectively. For a right -submodule U of and a left -submodule V of , one putsrespectively. Theorem 3. With the above notation as in Theorem 1. Let be surjective. Then,
- (1)
and are inverses and are lattice isomorphisms between the lattice of right ideals of and the lattice of submodules of . Moreover, these induce lattice isomorphisms between the lattice of ideals of and the lattice of submodules of .
- (2)
and are inverses and are lattice isomorphisms between the lattice of left ideals of and the lattice of submodules of . Moreover, these induce lattice isomorphisms between the lattice of ideals of and the lattice of submodules of .
Corollary 2. With the above notation as in Theorem 1. Let the Morita maps and be surjective. Then, and have isomorphic lattices of ideals.
5. A Special Case
We will treat a special case in order to illustrate results in
Section 3 about our Morita theory.
It was known that a G-grading of an algebra A is the same as an -module algebra action of , the Hopf algebra dual to , on A. Similarly, we have
Proposition 15. Let G be a quasigroup. Then, a G-grading of an algebra A is the same as a left quasi--module algebra action of on A.
Proof. Suppose that A is graded by G; then, we have that . Any may be written uniquely as , where . We define the action of on A by , where forms the dual basis for . That is, is the projection onto the gth part of any element of A. Using , it is clear that A is a -module. In addition, for any , we have , and is trivial. Therefore, A is a quasi--module algebra.
Conversely, we say that A is a quasi--module algebra, and we denote the action of on A by for any . Let ; since , and , it is clear that . Since the given action satisfies , and , it follows that . Thus, , for any so that A is graded by G. □
Proposition 16. A is a G-graded algebra if and only if A is a (right) -comodule algebra.
Proof. We already know from Proposition 15 that , a G-graded vector space, and that for . Thus, ; that is, , for all , and also . Thus, A is a G-graded algebra.
If A is a G-graded algebra, perform the same process: write , and it is not difficult to check that Equations (13) and (14) hold, and so A is a (right) -comodule algebra.
This completes the proof. □
If
A is graded, then we have the smash product
. For
, and basis elements
, the product is given by
using
and given the fact that
represents orthogonal idempotents and that
.
In addition, A may be identified with , and with in . We have some results about below.
Proposition 17. Let A be graded by the finite quasigroup G. Then, is the free right and left A-module with basis , a set of orthogonal idempotents whose sum is 1, and with the product given as above. Furthermore, the following statements hold:
- (1)
, for all .
- (2)
, for all .
- (3)
Each centralizes .
- (4)
, for all and for all graded ideal I.
- (5)
, which is isomorphic as a ring to .
Proof. The values are a free k-basis for , and so the values are a free left A-basis for . Using (1), A may be identified with , and it is clear they are also a free right A-basis. Since are orthogonal idempotents in , , and the values are also orthogonal idempotents whose sum is 1 in .
(1) , for all .
(2) By (1), , for all . However, = 0 unless , in which case and . Hence, .
(3) By (2), take g=1, .
(4) and (5) are straightforward. □
As a corollary of Theorem 1, we have
Theorem 7. Let G be a finite quasigroup and A be a G-graded algebra. Then, we can form a Morita context with the following module actions and maps:
- (1)
Consider A as a left (respectively, right) -module via left (respectively, right) multiplication;
- (2)
Consider A as a left (respectively, right)-module via Proposition 7(i) (respectively, Equation (18)); - (3)
;
- (4)
.
Proof. We here give a sketch of the proof: given a G-graded algebra A with finite quasigroup G, then A is also a quasi--module algebra by Proposition 15.
(1) The invariants subalgebra of A is . Notice that and ; this means for all , , where , that is, . It is obvious that A is a left (respectively, right) -module via left (respectively, right) multiplication.
(2) By Proposition 7(i), A is a left -module via , for all .
The right modular action is slightly more complicated. First, we compute the left integral space of
,
, and we assume that
, so we have
, for which we have used the fact that
are orthogonal idempotents in
and
. So, by comparing degrees, it can be concluded that the coefficients
of
are all 0 unless
. Therefore, t=
, that is,
. Second, we compute the distinguished group-like element
(as mentioned in
Section 2), and choosing a left integral
, we have
, which means
, the coefficient of the
term. The fact that
shows that
, the identity element in G, where we identified
with k[G]. Finally,
.
By Equation (
18), we have that
A is a right
module via
, for all
.
(3) Applying Theorem 1(3) here, we have .
(4) Applying Theorem 1(4) here, we have .
This finishes the proof. □