Abstract
In this paper, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. And we construct a number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the end, we give all Rota-Baxter operators of weight and on solvable and nilpotent Leibniz algebras of dimension 3, respectively.
1. Introduction
The Leibniz algebra [1] was mentioned by Bloh at the first time, which was called a D-algebra in 1965. Later, Loday improved and named it as Leibniz algebra. In Loday’s work, he was mainly interested in the properties of the corresponding homology theory on “group level" (“Leibniz K-Theory").
Leibniz algebras are a well-established algebraic structure generalizing Lie algebras with their own structure and homology theory. Moreover, they have much more applications in homological algebra, noncommutative geometry, physics, and so on (see [1–9]).
The Baxter algebra was firstly found in the work [3] of Baxter in 1960, which was used to solve the problem of probability [10]. A Baxter algebra is an associative algebra with a linear operator on that satisfies the Baxter identity for all .
In the 1960s, Rota began a study of Rota-Baxter algebras from an algebraic and combinatorial perspective in connection with hypergeometric functions, incidence algebras, and symmetric functions and obtained some interesting results (see [11–13]). A Rota-Baxter algebra is an associative algebra with a linear operator on that satisfies the Rota-Baxter identity for all , where (called the weight) is a fixed element in the base ring of the algebra .
In recent years, many scholars such as Andrews, Guo, and Bai et al. found and established the relations between Rota-Baxter algebras and Hopf algebras, Lie algebras, shuffle products, and dendriform algebras. Rota-Baxter algebras have been more and more important and have attracted much attention nowadays (see [11, 14–22]).
In this paper, our main aims are to introduce the concept of Rota-Baxter Leibniz algebras and to obtain a large number of Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra, as well as construct all Rota-Baxter operators of weight and on solvable and nilpotent non-Lie Leibniz algebras of dimension 3.
The paper is organized as follows. In the second section, we introduce the concept of Rota-Baxter Leibniz algebras and explore two characterizations of Rota-Baxter Leibniz algebras. One is a generalization of the Atkinson factorization [23, 24]. One is new for a Rota-Baxter Leibniz algebra under the assumption of quasi-idempotency. And we construct a large number of Rota-Baxter Leibniz algebras from Leibniz algebras and associative algebras, respectively, and discover some Rota-Baxter Leibniz algebras from augmented algebra, bialgebra, and weak Hopf algebra. In the third section, we construct all Rota-Baxter operators of weight and on solvable and nilpotent non-Lie Leibniz algebras of dimension 3.
Throughout the paper, all algebras, linear maps, and tensor products are taken over the complex field unless otherwise specified.
2. Rota-Baxter Leibniz Algebras
In this section, we mainly give some characterizations of Rota-Baxter Leibniz algebras and construct a large number of Rota-Baxter Leibniz algebras from Leibniz algebras, augmented algebra, and weak Hopf algebra, respectively.
Definition 1. Let be a vector space. Then, is called a (left) Leibniz algebra defined as in [5] if there is a bilinear map satisfyingfor any
In the following, our considered Leibniz algebras are left Leibniz algebras unless otherwise specified.
Let be a Leibniz algebra. Write and , for any integer In another, we denote by and by , for any
Let be Leibniz algebras. A linear map is called a Leibniz algebra homomorphism from to , if , for any
Definition 2. Let be a Leibniz algebra. If there exists a linear map and an element satisfyingfor any , then, is called a Rota-Baxter Leibniz algebra of weight , and is called a Rota-Baxter operator on . In what follows, we simply denote it by .
Let be a Rota-Baxter Leibniz algebra of weight . The subspace of is called a subalgebra, if is a Leibniz algebra under the multiplication of , and is still a Rota-Baxter operator of weight on .
Example 3. Let be a 2-dimensional vector space with basis . Define a multiplication on : and a linear map given by Then, is a Rota-Baxter Leibniz algebra of weight .
Let be a Rota-Baxter Leibniz algebra of weight . Then, is also a Rota-Baxter Leibniz algebra of weight , for any given element .
Let , be a family of Rota-Baxter Leibniz algebras of weight . Denote by . Now define a linear map , such that , for all . Then is also a Rota-Baxter Leibniz algebra of weight by defining , for all .
Proof. According to Example 2.1 in [5], we know that is a (left) Leibniz algebra, but it is not a (right) Leibniz algebra since .
It is easy to check that is a Rota-Baxter operator of weight on .
For any , we have It is straightforward to check that is a Rota-Baxter Leibniz algebra of weight .
In what follows, we will give some constructions of Rota-Baxter Leibniz algebras.
Proposition 4. Let be an algebra and an algebra map from to with . Then the following conclusions hold.
(1) Define a linear map by Then is a Rota-Baxter Leibniz algebra of weight .
(2) Define a linear map by Then, is a Rota-Baxter Leibniz algebra of weight .
(3) Suppose that is a commutative algebra. Define a linear map by Then, is a Rota-Baxter Leibniz algebra of any weight .
Proof. According to Example 2.2 in [5], we know that the conclusion holds.
For any , we can prove that It is obvious that is a Leibniz algebra by .
Remark 5. Let be an algebra. If is a augmented algebra as in [25] in the sense that there exists an algebra homomorphism , then, by Proposition 4, is a Rota-Baxter Leibniz algebra of weight , where the operator on is defined by for any , and the operator on is given by for any .
Let be a bialgebra or a Hopf algebra as in [26]. Then, the counit map is an algebra map. So, by , is a Rota-Baxter Leibniz algebra of weight .
Let be a weak Hopf algebra with an antipode given in [27]. Define a linear map (called the target map) by , where is denoted by .
Then, according to Corollary 2.2 in [28], we know that is idempotent. Furthermore, if is commutative, then, by Corollary 2.2 in [28], is also an algebra map. So, by Proposition 4, is a Rota-Baxter Leibniz algebra of any weight , with the product
Proposition 6. Let be a Rota-Baxter Leibniz algebra of nonzero weight . Then, is a Rota-Baxter Leibniz algebra of weight , where , for any given element .
Proof. It is straightforward to check that is a Rota-Baxter Leibniz algebra of weight .
Example 7. Let be an algebra and an algebra map from to with . Then, according to Proposition 4 and Proposition 6, we know that is a Rota-Baxter Leibniz algebra of weight , for any given element .
Proposition 8. Let be a Rota-Baxter Leibniz algebra of weight . Define a new binary product withThen we have the following conclusions.
(1) .
(2) is a Rota-Baxter Leibniz algebra of weight . So is a Leibniz algebra map from to .
Proof. It is just the Rota-Baxter Leibniz algebra equation.
By the definition of and the equality of Rota-Baxter Leibniz algebra, we easily prove so is a Leibniz algebra. It is easy to see that is a Rota-Baxter operator of weight .
In the following, we give two differentiated conditions for a Leibniz algebra to be a Rota-Baxter Leibniz algebra.
Theorem 9. Let be a nondegenerate Leibniz algebra and be a linear map.
(1) Suppose that satisfies , for any . Then, is a Rota-Baxter Leibniz algebra of weight , if and only if is quasi-idempotent of weight .
(2) DenoteThen, is a subalgebra of such that , for all .
(3) Suppose that is a Rota-Baxter operator of weight on and is idempotent (i.e., ). Then, is quasi-idempotent of weight . Conversely, if is quasi-idempotent of weight , then is a Rota-Baxter Leibniz algebra of weight .
Proof. For any , if is a Rota-Baxter algebra of weight , then, we easily prove that So we know that . This implies that
Conversely, if , then, for any , we have as desired.
In order to prove that is a subalgebra of , we only need to prove that , for all .
In fact, we have so , that is, . Hence is a subalgebra of .
Suppose that is a Rota-Baxter operator of weight on . Then, for any , , that is, , and So we get that is, . Hence by .
Conversely, if is quasi-idempotent of weight , then, for any , we have that is, , so . Hence, according to items and , we easily see that is a Rota-Baxter operator of weight on .
Theorem 10. Let be a Leibniz algebra. If is a Rota-Baxter Leibniz algebra of nonzero weight , then, for any given , there is an element , such that where as in Proposition 6.
Conversely, if there exists an element satisfying the above equalities and the annihilator of in has only zero, then, is a Rota-Baxter Leibniz algebra of nonzero weight .
Proof. For any , and , we have Taking , then, we can obtain Conversely, if there is an element such that for any given Then, when , we have that so we have Since has not trivial annihilator in , we have So This means that as desired.
In the following, we describe some properties of Rota-Baxter Leibniz algebras.
Proposition 11. Let be a Rota-Baxter Leibniz algebra of weight and idempotent. Then, for any ,
Proof. This proof is straightforward by Proposition 8.
Proposition 12. Let be a Rota-Baxter Leibniz algebra of weight . If is quasi-idempotent of weight , that is, Then, for any , In particular, if is idempotent, then, is a Leibniz algebra homomorphism from to .
Proof. The proof is left for the readers.
Proposition 13. Let be a Rota-Baxter Leibniz algebra of nonzero weight . Then the following conclusions are satisfied.
(1) .
(2) .
(3) [29] For any integer , and , (4) ,
where and so does .
In particular, taking , we haveHere is defined in Proposition 6 and defined in Proposition 8.
Proof. For any and , we easily prove According to and the equality for any , we can prove .
We can prove this conclusion by using induction on .
This follows from by taking .
3. Rota-Baxter Operators on Low-Dimensional Leibniz Algebras
In this section, we mainly focus on the Rota-Baxter operators of weight and , and give all Rota-Baxter operators on solvable and nilpotent Leibniz algebras of dimension 3.
Suppose is a Lebniz algebra with basis . Then, for any given Rota-Baxter operator of weight on , it can be presented by a matrix , that is, there are elements , such thatsatisfiesfor any .
By [5], we know that any non-Lie Leibniz algebra in dimension and nilpotent Leibniz algebra are solvable.
In what follows, we choose the low-dimensional nilpotent Leibniz algebras and the solvable ones to construct Rota-Baxter operators. And we denote the set of all the Rota-Baxter operators on by RBO(A).
Firstly, we recall the conception of nilpotent Leibniz algebras and solvable ones.
Definition 14. A Leibniz algebra is solvable if for some integers .
Definition 15. A Leibniz algebra is nilpotent of class if but for some integer .
Lemma 16. Let be a non-Lie Leibniz algebra and . Then, by Theorem 6.1 in [5], is isomorphic to a cyclic Leibniz algebra generated by a single element with (hence is nilpotent), or (hence is solvable).
In the following, for the proofs of our given results, the readers can see Appendix.
Theorem 17. The Rota-Baxter operators on 2-dimensional non-Lie Leibniz algebra are given in Table 1.
In what follows, Lemmas 18 and 20 follow from Theorems 6.4 and 6.5 in [5], respectively.
Lemma 18. Let be a non-Lie nilpotent Leibniz algebra and . Then, is isomorphic to a Leibniz algebra spanned by with the nonzero product given by one of the following:(1);(2);(3);(4);(5)
Theorem 19(A). The Rota-Baxter operators of weight on 3-dimensional non-Lie nilpotent Leibniz algebra are given in Table 2.
Theorem 19(B). The Rota-Baxter operators of weight on 3-dimensional non-Lie nilpotent Leibniz algebra are given in Table 3.
Lemma 20. Let be a non-Lie nonnilpotent solvable Leibniz algebra and . Then, is isomorphic to a Leibniz algebra spanned by with the nonzero product given by one of the following:(1);(2);(3);(4);(5);(6);(7)
Theorem 21(A). The Rota-Baxter operators of weight on 3-dimensional non-Lie and nonnilpotent solvable Leibniz algebra are given in in Table 4.
Theorem 21(B). The Rota-Baxter operators of weight on 3-dimensional non-Lie and nonnilpotent solvable Leibniz algebra are given in Table 5.
Appendix
In this section, we mainly give the proof of some results in Section 3.
Proof of Theorem 17. We firstly construct Rota-Baxter operators of weight in the case . Assume thatwith . Then, we have that is,By computing, we easily get that is, the following equations hold: Again by computing, we can obtain the following solutions: In a similar way, we can show the other cases.
Proof of Theorem 19(A). In the following, we firstly prove the case . Suppose that is Rota-Baxter operator of weight , with the representationApplying the above equalities to Rota-Baxter identity, we obtain the following: That is, we have Since , we haveTransposing and amalgamating, we get that It is easy to see that by the above equations. So and . Hence we get In a light of the above first equality: , we consider two cases: or
Suppose that Then, according to the above equalities, it is easy to see that and . So or .
Suppose that Then, by the above equalities, we have Since , we know that . So . It means that
Hence we obtain all solutions as follows: The other cases can be similarly proved.
Similar to Theorem 19(A), we can prove Theorems 19(B), 21(A), and 21(B).
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by the National Natural Science Foundation of China (11571173).