1. Introduction
Rota–Baxter algebras were introduced in [
1] in the context of differential operators on commutative Banach algebras. At present, Rota–Baxter algebras have become a useful tool in many fields in mathematics and mathematical physics such as combinatorics, Loday type algebras, Pre-Lie algebras, Pre-Possion algebras, multiple zeta values, quantum field theory and so on (cf. [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]).
Hopf quasigroups were introduced in [
12] in order to capture the quasigroup feature of the 7-sphere and they are not required to be associative as a generalisation of Hopf algebras. Let
H be a Hopf quasigroup. In [
13], the authors obtained a braided monoidal category equivalence between the category of Yetter–Drinfeld quasimodules over
H and the category of two-sided two-cosided Hopf quasimodules over
H under some suitable assumption (cf. [
14,
15,
16,
17,
18]).
Rota–Baxter bialgebras were introduced by Ma and Liu in [
19]. As a continue of Ma and Liu’s paper, in this article, we introduce and discuss the notions of Rota–Baxter bialgebra equation systems and Rota–Baxter Hopf algebras. We give a lot of examples of Rota–Baxter Hopf algebras based on Hopf quasigroups. Our ideal may be regarded as a guide for further development.
This paper is organized as follows. In
Section 2, we recall and investigate some basic definitions and properties related to Hopf quasigroups and Rota–Baxter bialgebras.
In
Section 3, we introduce and study the notions of Rota–Baxter (co)algebra equation systems. We give a construction of creating examples based on a two-side
H-Hopf quasimodule bialgebra and Radford’s admissible mapping system for Hopf algebras, respectively.
The final
Section 4 is devoted to introduce and study the notion of a compatible Rota–Baxter bialgebra and construct a lot of examples.
Through the paper, we fix a ground field
of characteristic 0 and work on
. We use the Sweedler’s notation to express the coproduct of a coalgebra
C as
(cf. [
20]). For a left (resp. right)
C-comodule coaction
(resp.
) on
M, we write
(resp.
) for any
. Let
be vector spaces and
and
be two linear maps. Then we write
simply for the composite
from
U to
W. For any vector space
V, we use
to denote the identity map from
V to itself and always write
simply for
. We always write
T for the flipping map:
. Finally, the natural identification
is assumed.
3. Rota–Baxter (Co)algebra Equation Systems
In this section, we introduce and study the notions of Rota–Baxter (co)algebra systems and give an approach to these equation systems.Our ideal may be regarded as a guide for further development.
Definition 2. Let be an algebra and P a linear endomorphism of A. We say that P satisfiesRota–Baxter algebra equation systemwith regard to ∇, if the one of the following equation systems is satisfied: Dually, we have
Definition 3. Let be a coalgebra and Q a linear endomorphism of C. We say that Q satisfiesRota–Baxter coalgebra equation systemwith regard to Δ, if the one of the following equation systems is satisfied: For convenience’sake, we have
Lemma 2. Let be an algebra and P a linear endomorphism of A. Let be a coalgebra and Q a linear endomorphism of C. Let be a bialgebra and R and L linear endomorphisms of B.
(1) If P is a solution to Rota–Baxter algebra equation system (RBa1) or (RBa2), then is a Rota–Baxter algebra of weight .
(2) If Q is a solution to Rota–Baxter coalgebra equation system (RBc1) or (RBc2), then is a Rota–Baxter coalgebra of weight .
(3) If R is a solution to Rota–Baxter algebra equation system (RBa1) or (RBa2), and L a solution to Rota–Baxter coalgebra equation system (RBc1) or (RBc2), then the pentuple is a Rota–Baxter bialgebra of weight .
Example 4. Let H be a Hopf quasigroup. In Example 1, we have a linear map from to itself and from to itself. In particular, we have two linear maps: and from to itself. It is easy to check that and are both Rota–Baxter algebra of weight and Rota–Baxter coalgebra of weight .
Furthermore, and are Rota–Baxter bialgebras of weight .
Definition 4. Let be a Hopf quasigroup and let B be a bialgebra. We say that B is atwo-side H-Hopf quasimodule bialgebraif is a left-left H-Hopf quasimodule and is a right-right H-Hopf quasimodule as defined in Definition 1, such thatfor any and . Note that if H is a Hopf algebra, then B is called a two-side H-Hopf module bialgebra.
Theorem 1. Let H be a Hopf quasigroup and let M be a two-side H-Hopf quasimodule bialgebra. Set and for any . If Equation (6) holds, then and are Rota–Baxter bialgebras of weight . Proof. We show that
is a solution to Rota–Baxter algebra equation system (RBa2). For any
, we have
and
Therefore, we conclude that is a solution to Rota–Baxter algebra equation system (RBa2). A similar computation for gives rise to that is a solution to Rota–Baxter algebra equation system (RBa1).
It follows from Lemma 1(1) that and are idempotent Rota–Baxter algebras with the weight .
We also compute, for any
and similarly one can obtain:
. Therefore,
is a solution to Rota–Baxter coalgebra equation system (RBc2) and so
is an idempotent Rota–Baxter coalgebra with the weight
.
For
, one has
for any
. Thus,
satisfies the first identity in (RBc1). Similarly for the second one in (RBc1) and so
is a solution to Rota–Baxter coalgebra equation system (RBc1).
Therefore, it follows from Lemma 2(3) and Lemma 1(1) that and are idempotent Rota–Baxter bialgebra with the weight .
This completes the proof. □
In what follows, we use the notion of Radford’s admissible mapping system for Hopf algberas ([
25]) to construct Rota–Baxter bialgebra.
Proposition 2. Let B be a unital counital associative coassociative bialgebra and H a Hopf algebra with an antipode S. Suppose there are two bialgebra homomorphism and such that . Define two linear maps by and by for any . Then and are Rota–Baxter bialgebras with the weight . Furthermore, and .
Proof. For this, we want show that is a solution to the equation systems (RBa1) and (RBc1), and is a solution to the equation systems (RBa2) and (RBc2).
In fact, we have, for any
and
This shows that is a solution to Rota–Baxter algebra equation system (RBa1).
Similarly for satisfying (RBa2) and (RBc2).
Therefore, it follows from Lemma 2.3(3) that and are Rota–Baxter bialgebras with the weight .
Finally, we have
and similarly
. □
Proposition 3. Let A be a bialgebra. If P and Q satisfy Rota–Baxter Equation (co)algebra systems (RBa1)and (RBc1) (resp. (RBa2) and (RBc2)) such that , then the composite of P and Q satisfies Rota–Baxter (co)algebra systems (RBa1) and (RBc1) (resp. (RBa2) and (RBc2)).
Proof. For (RBa1), computing we have, for any
and
Similar for (RBa2).
For (RBc1), one computes, for any
and
Similar for (RBc2).
This completes the proof. □
We finish this section with the following result.
Proposition 4. Let be a direct sum of two bi-ideals and of a bialgebra B. Then for any on (resp. on ) satisfying Rota–Baxter (co)algebra equation systems (RBa1) and (RBc1) (resp. (RBa2) and (RBc2)), the linear map given by (resp. given by ) for any , solves Rota–Baxter (co)algebra equation systems (RBa1) and (RBc1) (resp. (RBa2) and (RBc2)).
4. Compatible Rota–Baxter Bialgebras and Rota–Baxter Hopf Algebras
We begin with the notion of a compatible Rota–Baxter bialgebra by modifying the definition of a Rota–Baxter bialgebra given in [
19] as follows.
Definition 5. A septuple is called acompatible Rota–Baxter bialgebraof weight if is a bialgebra, is a Rota–Baxter algebra of weight λ, and is a Rota–Baxter coalgebra of weight γ such thator As a matter of convenience, we have
Lemma 3. Let P and Q be two linear endomorphisms of a bialgebra . Let P be a solution to Rota–Baxter algebra equation system (RBa1) and (RBa2), and let Q be a solution to Rota–Baxter coalgebra equation system (RBc1) and (RBc2). Then and are Rota–Baxter bialgebra of weight .
Proposition 5. With the notations as in Proposition 2. Then and are compatible Rota–Baxter bialgebras of weight .
Definition 6. An octuple is called aRota–Baxter Hopf algebraof weight if the quadruple is a Hopf algebra with antipode S and the septuple is a compatible Rota–Baxter bialgebra, such that the antipode S of H is compatible with P and Q in the following sense that Example 5. Let H be a Hopf algebra and A a braided Hopf algebra in the Yetter–Drinfeld module category . Recall from [25] that the Radford’s biproduct have the following Hopf algebra structures:for all and . Then there are two bialgebra homomorphisms and such that . By Proposition 2, we can define two linear maps byand . Therefore, is a solution to Rota–Baxter algebra equation system (RBa1) and coalgebra equation system (RBc1), and is a solution to Rota–Baxter algebra equation system (RBa2) and coalgebra equation system (RBc2). Furthermore, one can directly show that as follows:Furthermore, similar for . Thus, and satisfy the Condition (14). Therefore, is a Rota–Baxter Hopf algebra of weight .
Proposition 6. Let be a Hopf algebra with antipode S. Let A be a Hopf algebra with antipode S. If A is a two-side H-Hopf module bialgebra such that, for then is a Rota–Baxter Hopf algebra of weight . Proof. By Theorem 1, we only show that
and
. In fact, we have
and similarly for
.
This completes the proof. □
Theorem 2. With the notations as in Proposition 2. If B is a Hopf algebra with antipode S, then is a Rota–Baxter Hopf algebra of weight .
Proof. By Proposition 5, we only show that
and
. Actually, we have
and similarly for
. □
Example 6. With the notations as in Proposition 2. Then B can be made into a two-side H-Hopf module bialgebra with the following H-module and H-comodule structures, for any and It is easy to check that the conditions: (10)–() are satisfied. Equation (6) also holds. In order to check Equation (15), we haveand similarly for . Doing calculation one hasand . This proves Equation (). Then we have two linear maps byandfor any . It follows from Proposition 6 that we also can obtain Theorem 2.