# Rota-Baxter Leibniz Algebras and Their Constructions

Authors: Liangyun Zhang; Linhan Li; Huihui Zheng
Journal: Advances in Mathematical Physics
 (2018)

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## 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 weight0 and -1 on solvable and nilpotent Leibniz algebras of dimension ≤3, respectively.


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## Body

## 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 A with a linear operator P on A that satisfies the Baxter identity (1)PxPy=PPxy+xPyfor all x,y∈A.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 A with a linear operator P on A that satisfies the Rota-Baxter identity (2)PxPy=PPxy+xPy+λxyfor all x,y∈A, where λ (called the weight) is a fixed element in the base ring of the algebra A.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 weight0 and -1 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 0 and -1 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 fieldC 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.
LetA be a vector space. Then, (A,[-,-]) is called a (left)Leibniz algebra defined as in [5] if there is a bilinear map [-,-]:A⊗A→A,a⊗b↦[a,b] satisfying(3)a,b,c=a,b,c+b,a,cfor any a,b,c∈A.In the following, our considered Leibniz algebras are left Leibniz algebras unless otherwise specified.Let(A,[-,-]) be a Leibniz algebra. Write A1=A,Aκ+1=[A,Aκ] and A(1)=[A,A],A(κ+1)=[A(κ),A(κ)], for any integer κ≥1. In another, we denote a1 by a and [a,aκ] by aκ+1, for any a∈A.Let(A,[-,-]A),(B,[-,-]B) be Leibniz algebras. A linear map f:A→B is called aLeibniz algebra homomorphism from A to B, if f([a,b]A)=[f(a),f(b)]B, for any a,b∈A.Definition 2.
Let(A,[-,-]) be a Leibniz algebra. If there exists a linear map P:A→A and an element λ∈C satisfying(4)Pa,Pb=Pa,Pb+Pa,b+λa,bfor any a,b∈A, then, (A,[-,-]) is called aRota-Baxter Leibniz algebra of weightλ, and P is called aRota-Baxter operator on A. In what follows, we simply denote it by (A,[-,-],P).
Let(A,[-,-],P) be a Rota-Baxter Leibniz algebra of weight λ. The subspace M of A is called asubalgebra, if M is a Leibniz algebra under the multiplication of A, and P is still a Rota-Baxter operator of weight λ on M.Example 3.
( 1 ) Let A be a 2-dimensional vector space with basis {x,y}. Define a multiplication on A: (5)x,x=0=x,y,y,x=x=y,y,and a linear map P:A→A given by (6)Px=x,Py=2x-y.
Then,(A,[-,-],P) is a Rota-Baxter Leibniz algebra of weight -1.
( 2 ) Let (A,[-,-],P) be a Rota-Baxter Leibniz algebra of weight 0. Then, (A,[-,-],lP) is also a Rota-Baxter Leibniz algebra of weight 0, for any given element l∈C.
( 3 ) Let (Ai,[-,-],Pi),i∈I, be a family of Rota-Baxter Leibniz algebras of weight λ. Denote ⨁i∈I⁡Ai by A. Now define a linear map P:A→A, such that P((ai))=(Pi(ai)), for all (ai)∈A. Then (A,[-,-],P) is also a Rota-Baxter Leibniz algebra of weight λ by defining [(ai),(bi)]=([ai,bi]), for all (ai),(bi)∈A.Proof.
( 1 ) According to Example 2.1 in [5], we know that (A,[-,-]) is a (left) Leibniz algebra, but it is not a (right) Leibniz algebra since [[y,y],y]≠[y,[y,y]]+[[y,y],y].
It is easy to check thatP is a Rota-Baxter operator of weight -1 on A.
( 2 ) For any a,b∈A, we have (7)lPa,lPb=lPlPa,b+a,lPb.
( 3 ) It is straightforward to check that (A,[-,-],P) is a Rota-Baxter Leibniz algebra of weight λ.In what follows, we will give some constructions of Rota-Baxter Leibniz algebras.Proposition 4.
LetA be an algebra and P an algebra map from A to A with P2=P. Then the following conclusions hold.
(1) Define a linear map[-,-]P:A⊗A→A by (8)a,bP=Pab-bPa.
Then(A,[-,-]P,P) is a Rota-Baxter Leibniz algebra of weight -1.
(2) Define a linear map[-,-]P:A⊗A→A by (9)a,bP=Pab-bPa.
Then,(A,[-,-]P,P) is a Rota-Baxter Leibniz algebra of weight -1.
(3) Suppose thatA is a commutative algebra. Define a linear map [-,-]P:A⊗A→A by (10)a,bP=Pab-Pab.
Then,(A,[-,-]P,P) is a Rota-Baxter Leibniz algebra of any weight l∈C.Proof.
( 1 ) According to Example 2.2 in [5], we know that the conclusion (1) holds.
( 2 ) For any a,b,c∈A, we can prove that (11)a,b,cPP=a,bP,cP+b,a,cPP.Pa,PbP+PPa,bP-Pa,bP=Pa,PbP.(3) It is obvious that (A,[-,-]) is a Leibniz algebra by (2).Remark 5.
( 1 ) Let A be an algebra. If A is a augmented algebra as in [25] in the sense that there exists an algebra homomorphism f:A→C, then, by Proposition 4(2), (A,[-,-]Pf,Pf) is a Rota-Baxter Leibniz algebra of weight -1, where the operator Pf on A is defined by Pf(a)=f(a)1A for any a∈A, and the operator [-,-]Pf on A⊗A is given by (12)a,bPf=fab1A-fabfor any a,b∈A.
( 2 ) Let A be a bialgebra or a Hopf algebra as in [26]. Then, the counit map ε:A→C is an algebra map. So, by (1), (A,[-,-]Pε,Pε) is a Rota-Baxter Leibniz algebra of weight -1.
( 3 ) Let A be a weak Hopf algebra with an antipode S given in [27]. Define a linear map ΠL:A→A (called the target map) by ΠL(a)=ε(11a)12, where Δ(1A) is denoted by 11⊗12∈A⊗A.
Then, according to Corollary 2.2(1) in [28], we know that ΠL is idempotent. Furthermore, if A is commutative, then, by Corollary 2.2 (4) in [28], ΠL is also an algebra map. So, by Proposition 4(3), (A,[-,-]ΠL,ΠL) is a Rota-Baxter Leibniz algebra of any weight l, with the product [a,b]ΠL=ΠL(ab)-ΠL(a)b.Proposition 6.
Let(A,[-,-],P) be a Rota-Baxter Leibniz algebra of nonzero weight λ. Then, (A,[-,-],QP) is a Rota-Baxter Leibniz algebra of weight -l, where QP=l/λP+lid, for any given element l∈C.Proof.
It is straightforward to check that(A,[-,-],QP) is a Rota-Baxter Leibniz algebra of weight -l.Example 7.
LetA be an algebra and P an algebra map from A to A with P2=P. Then, according to Proposition 4(1) and Proposition 6, we know that (A,[-,-]P,QP=-lP+lid) is a Rota-Baxter Leibniz algebra of weight -l, for any given element l∈C.Proposition 8.
Let(A,[-,-],P) be a Rota-Baxter Leibniz algebra of weight λ. Define a new binary product [-,-]P:A⊗A→A with(13)a,bP=a,Pb+Pa,b+λa,b.
Then we have the following conclusions.
(1)[P(a),P(b)]=P([a,b]P).
(2)(A,[-,-]P,P) is a Rota-Baxter Leibniz algebra of weight λ. So P is a Leibniz algebra map from (A,[-,-]P) to (A,[-,-]).Proof.
( 1 ) It is just the Rota-Baxter Leibniz algebra equation.
( 2 ) By the definition of [-,-]P and the equality of Rota-Baxter Leibniz algebra, we easily prove (14)a,b,cPP=a,bP,cP+b,a,cPP,so (A,[-,-]P) is a Leibniz algebra. It is easy to see that P is a Rota-Baxter operator of weight λ.In the following, we give two differentiated conditions for a Leibniz algebra to be a Rota-Baxter Leibniz algebra.Theorem 9.
Let(A,[-,-]) be a nondegenerate Leibniz algebra and P:A→A be a linear map.
(1) Suppose thatP satisfies P([a,b])=[a,P(b)], for any a,b∈A. Then, (A,[-,-],P) is a Rota-Baxter Leibniz algebra of weight λ, if and only if P is quasi-idempotent of weight λ.
(2) Denote(15)CA≔a∈A∣Pa,b=a,Pb,∀b∈A
Then,CA is a subalgebra of A such that P([a,b])=[a,P(b)], for all a∈CA,b∈A.
(3) Suppose thatPCA is a Rota-Baxter operator of weight λ on CA and CA is idempotent (i.e., [CA,CA]=CA). Then, PCA is quasi-idempotent of weight λ. Conversely, if P is quasi-idempotent of weight λ, then (CA,[-,-],PCA) is a Rota-Baxter Leibniz algebra of weight λ.Proof.
( 1 ) For any a,b∈A, if (A,[-,-],P) is a Rota-Baxter algebra of weight λ, then, we easily prove that (16)Pa,Pb=a,P2b+Pa,Pb+λa,Pb.
So we know that[a,P2(b)]+λ[a,P(b)]=[a,P2(b)+λP(b)]=0. This implies that P2+λP=0.
Conversely, ifP2+λP=0, then, for any a,b∈A, we have (17)Pa,Pb+Pa,b+λa,b=Pa,Pbas desired.
( 2 ) In order to prove that CA is a subalgebra of A, we only need to prove that [a,b]∈CA, for all a,b∈CA.
In fact, we have(18)a,b,Pc=a,b,Pc-b,a,Pc=a,Pb,c-b,Pa,c=Pa,b,c-Pb,a,c=Pa,b,c,so P([[a,b],c])=[[a,b],P(c)], that is, [a,b]∈CA. Hence CA is a subalgebra of A.
( 3 ) Suppose that PCA is a Rota-Baxter operator of weight λ on CA. Then, for any a,b∈CA, P(a)∈CA, that is, [PCA(a),PCA(b)]=PCA([PCA(a),b]), and [PCA(a),PCA(b)]-PCA([a,P∣CA(b)])-PCA([PCA(a),b])-λPCA([a,b])=0. So we get (19)0=PCAa,PCAb-PCAa,PCAb-PCAPCAa,b-λPCAa,b=PCAa,PCAb-PCA2a,b-PCAPCAa,b-λPCAa,b=-PCA2a,b-λPCAa,b,that is, -PCA2([a,b])-λPCA([a,b])=0. Hence PCA2=-λPCA by [CA,CA]=CA.
Conversely, ifP is quasi-idempotent of weight λ, then, for any a∈CA,b∈A, we have (20)0=Pa,Pb-Pa,Pb-PPa,b-λPa,b=Pa,Pb-P2a,b-PPa,b-λPa,b=Pa,Pb-PPa,b,that is, [P(a),P(b)]=P[P(a),b], so P(a)∈CA. Hence, according to items (1) and (2), we easily see that PCA is a Rota-Baxter operator of weight λ on CA.Theorem 10.
Let(A,[-,-]) be a Leibniz algebra. If (A,[-,-],P) is a Rota-Baxter Leibniz algebra of nonzero weight λ, then, for any given a,b∈A, there is an element c∈A, such that (21)lλPa,Pb=Pc,Qa,Qb=Qc,where Q=l/λP+lid as in Proposition 6.
Conversely, if there exists an elementc∈A satisfying the above equalities and the annihilator of l∈C in A has only zero, then, (A,[-,-],P) is a Rota-Baxter Leibniz algebra of nonzero weight λ.Proof.
For anya,b∈A, and λ∈C, we have (22)Pa,Pb=Pa,Pb+Pa,b+λa,b.
Takingc=l/λ([a,P(b)]+[P(a),b]+λ[a,b]), then, we can obtain (23)Pc=lλPa,Pb+Pa,b+λa,b=lλPa,Pb,Qc=Qa,Qb.
Conversely, if there is an elementc∈A such that (24)lλPa,Pb=Pc,Qa,Qb=Qc,for any given a,b∈A. Then, when Q=l/λP+lid, we have that (25)Qa,Qb=lλPa+la,lλPb+lb=lλPa,lλPb+lb+la,lλPb+lb=lλPa,lλPb+lλPa,lb+la,lλPb+la,lb=l2λ2Pa,Pb+l2λPa,b+l2λa,Pb+l2a,b=Qc=lλPc+lc=l2λ2Pa,Pb+lc.so we have (26)lc=l2λPa,b+a,Pb+λa,b.
Sincel has not trivial annihilator in A, we have (27)c=lλa,Pb+Pa,b+λa,b.
So(28)lλPa,Pb=Pc=Plλa,Pb+Pa,b+λa,b.
This means that[P(a),P(b)]=P([a,P(b)]+[P(a),b]+λ[a,b]) as desired.In the following, we describe some properties of Rota-Baxter Leibniz algebras.Proposition 11.
Let(A,[-,-],P) be a Rota-Baxter Leibniz algebra of weight λ and P idempotent. Then, for any a,b∈A,(29)1+λPa,Pb=0,1+λPPa,b=0,1+λPa,Pb-λPa,b=0.Proof.
This proof is straightforward by Proposition8.Proposition 12.
Let(A,[-,-],P) be a Rota-Baxter Leibniz algebra of weight λ. If P is quasi-idempotent of weight λ, that is, P2=-λP. Then, for any x,y∈Im⁡P, (30)Px,Py=-λPx,y.
In particular, ifP is idempotent, then, P is a Leibniz algebra homomorphism from Im⁡P to Im⁡P.Proof.
The proof is left for the readers.Proposition 13.
Let(A,[-,-],P) be a Rota-Baxter Leibniz algebra of nonzero weight λ. Then the following conclusions are satisfied.
(  1) - l 2 / λ [ a , b ] P = l 2 / λ 2 [ P ( a ) , P ( b ) ] - [ Q ( a ) , Q ( b ) ].
(  2) - l 2 / λ [ P ( a ) , P ( b ) ] = P ( l 2 / λ 2 [ P ( a ) , P ( b ) ] - [ Q ( a ) , Q ( b ) ] ).
(  3) [29] For any integer n≥2, and ai∈A,i=1,2,…,n, (31)-lnλn-1Pa1,Pa2,⋯Pan-1,Pan⋯=PlnλnPa1,Pa2,⋯Pan-1,Pan⋯-Qa1,Qa2,⋯Qan-1,Qan⋯.
(  4) - l n / λ n - 1 P ( a ) n = P ( l n / λ n P ( a ) n - Q ( a ) n ),
whereP(a)n=Pa,Pa,⋯Pa,Pa⋯ and so does Q(a)n.
In particular, takingl=λ, we have(32)-λPan=PPan-Qan.
HereQ=l/λP+lid is defined in Proposition 6 and [-,-]P defined in Proposition 8.Proof.
( 1 ) For any a,b∈A and l∈C, we easily prove (33)l2λ2Pa,Pb-Qa,Qb=-l2λa,bP.
( 2 ) According to (1) and the equality P([a,b]P)=[P(a),P(b)] for any a,b∈A, we can prove (2).
( 3 ) We can prove this conclusion by using induction on n.
( 4 ) This follows from (3) by taking ai=a,i=1,2,…,n.


## 3. Rota-Baxter Operators on Low-Dimensional Leibniz Algebras

In this section, we mainly focus on the Rota-Baxter operators of weightλ=0 and λ=-1, and give all Rota-Baxter operators on solvable and nilpotent Leibniz algebras of dimension ≤3.SupposeA is a Lebniz algebra with basis {a1,a2,…,an}. Then, for any given Rota-Baxter operator P of weight λ on A, it can be presented by a matrix R=(rij)n×n, that is, there are n2 elements rij∈C(i=1,2,…n;j=1,2,…n), such that(34)Pa1a2⋮an=r11r12⋯r1nr21r22⋯r2n⋮⋮⋱⋮rn1rn2⋯rnna1a2⋮ansatisfies(35)Px,Py=Px,Py+Px,y+λx,yfor any x,y∈A.By [5], we know that any non-Lie Leibniz algebra in dimension ≤3 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 onA by RBO(A).Firstly, we recall the conception of nilpotent Leibniz algebras and solvable ones.Definition 14.
A Leibniz algebraA issolvable if A(κ)=0 for some integers κ≥0.Definition 15.
A Leibniz algebraA isnilpotent of classκ if Aκ+1=0 but Aκ≠0 for some integer κ≥0.Lemma 16.
LetA be a non-Lie Leibniz algebra and dim⁡(A)=2. Then, by Theorem 6.1 in [5], A is isomorphic to a cyclic Leibniz algebra generated by a single element a with [a,a2]=0 (hence A is nilpotent), or [a,a2]=a2 (hence A 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 algebraA are given in Table 1.Table 1 Non-Lie Leibniz algebra A RBO(A) of weightλ=0 RBO(A) of weightλ=-1 1[a,a2]=0 0 r 12 0 r 22,2r22r120r22. r 11 r 12 0 r 22 r 11 2 = 2 r 11 - 1 r 22 .  2[a,a2]=a2 0 r 12 0 0,r11-r1100. 0 r 12 0 0,r11-r1100,1r1201, r111-r1101.In what follows, Lemmas18 and 20 follow from Theorems 6.4 and 6.5 in [5], respectively.Lemma 18.
LetA be a non-Lie nilpotent Leibniz algebra and dim⁡(A)=3. Then, A is isomorphic to a Leibniz algebra spanned by {a,b,c} with the nonzero product given by one of the following:(1)
[a,a]=b,[a,b]=c;(2)
[a,a]=c;(3)
[a,b]=c,[b,a]=c;(4)
[a,b]=c,[b,a]=-c,[b,b]=c;(5)
[a,b]=c,[b,a]=αc,α∈C\{1,-1}.Theorem 19 (A).The Rota-Baxter operators of weight0on 3-dimensional non-Lie nilpotent Leibniz algebraAare givenin Table2.Table 2 Non-Lie nilpotent Leibniz algebraA RBO(A) of weightλ=0 (1)[a,a]=b;[a,b]=c 0 0 r 13 0 0 r 23 0 0 r 33,0r12r130r22r23000, 2r223r23r130r22r230023r22.  (2)[a,a]=c 0 r 12 r 13 0 r 22 r 23 0 r 32 r 33 ,r11r12r130r22r230012r11.  (3)[a,b]=c;[b,a]=c r 11 0 r 13 r 21 0 r 23 0 0 0,0r12r130r22r23000,r11r112r21r13r21r11r2300r11r21≠0, r 11 0 r 13 0 r 22 r 23 0 0 r 33 r 11 + r 22 r 33 = r 11 r 22.  (4)[a,b]=c;[b,a]=-c;[b,b]=c r 11 0 r 13 r 21 0 r 23 0 0 0,00r13r210r2300r33,2r330r13r212r33r2300r33.  (5)[a,b]=c;[b,a]=αcα∈C\1,-1 0 0 r 13 r 21 0 r 23 0 0 0,r110r13r210r23000,0r12r130r22r23000, r 11 0 r 13 0 r 22 r 23 0 0 r 33 r 22 r 33 + r 11 r 33 - r 11 r 22 = 0.Theorem 19 (B).The Rota-Baxter operators of weight-1on 3-dimensional non-Lie nilpotent Leibniz algebraAare givenin Table3.Table 3 Non-Lie nilpotent Leibniz algebraA RBO(A) of weightλ=-1 (1)[a,a]=b;[a,b]=c r 11 r 12 r 13 0 r 11 2 2 r 11 - 1 r 12 r 11 - r 33 2 r 11 - 1 0 0 r 33 r 11 ≠ 1 2 r 11 2 + r 11 - 1 2 r 11 - 1 r 33 = r 11 3.  (2)[a,a]=c r 11 r 12 r 13 0 r 22 r 23 0 0 r 11 2 2 r 11 - 1 r 11 ≠ 1 2.  (3)[a,b]=c;[b,a]=c r 11 0 r 13 r 21 0 r 23 0 0 0,r110r13r211r23001,0r12r130r22r23000,1r12r130r22r23001, r 11 0 r 13 0 r 22 r 23 0 0 r 33 r 11 + r 22 - 1 r 33 = r 11 r 22, r 11 r 11 2 - r 11 r 21 r 13 r 21 r 11 r 23 0 0 r 11 r 21 ≠ 0.  (4)[a,b]=c;[b,a]=-c;[b,b]=c r 11 0 r 13 r 21 r 22 r 23 0 0 r 22 2 2 r 22 - 1 r 22 ≠ 1 2 r 11 + r 22 - 1 r 22 2 = r 11 r 22 2 r 22 - 1.  (5)[a,b]=c;[b,a]=αc(α∈C\{1,-1}) 0 r 12 r 13 0 r 22 r 23 0 0 0,1r12r130r22r23001,r110r13r210r23000,r110r13r211r23001, r 11 0 r 13 0 r 22 r 23 0 0 r 33 r 11 + r 22 - 1 r 33 = r 11 r 22 .Lemma 20.
LetA be a non-Lie nonnilpotent solvable Leibniz algebra and dim⁡(A)=3. Then, A is isomorphic to a Leibniz algebra spanned by {a,b,c} with the nonzero product given by one of the following:(1)
[c,a]=a;(2)
[c,a]=αa(α∈C\{0}),[c,b]=b,[b,c]=-b;(3)
[c,b]=b,[b,c]=-b,[c,c]=a;(4)
[c,a]=2a,[b,b]=a,[c,b]=b,[b,c]=-b,[c,c]=a;(5)
[c,a]=a+b,[c,b]=b;(6)
[c,a]=b,[c,b]=b,[c,c]=a;(7)
[c,b]=b,[c,a]=αa(α∈C\{0}).Theorem 21 (A).The Rota-Baxter operators of weight0on 3-dimensional non-Lie and nonnilpotent solvable Leibniz algebraAare given inin Table4.Table 4 Non-Lie solvable Leibniz algebraA RBO(A) of weightλ=0 (1)[c,a]=a 0 0 0 0 r 22 r 23 0 r 32 r 33,0r1200r2200r320,000r21r220r31r320.  (2)[c,a]=αa(α∈C\{0});[c,b]=b;[b,c]=-b 0 0 0 r 21 0 0 r 31 r 32 0,0r1200000r320,0000r22r230r32-r22r222+r23r32=0.  (3)[c,b]=b;[b,c]=-b;[c,c]=a r 11 0 0 r 21 0 0 r 31 r 32 0,r1100000r31r322r11,r11r120000r31r320.  (4)[c,a]=2a;[b,b]=a;[c,b]=b;[b,c]=-b;[c,c]=a 0 0 0 r 21 0 0 r 31 0 0,000r210-2r21000, 0 0 0 0 0 0 r 31 r 32 r 33 r 32 2 + 2 r 33 r 31 + r 33 2 = 0, 0 0 0 r 21 r 22 r 23 r 31 r 32 r 33 r 22 2 + 2 r 23 r 21 + r 23 2 = 0 r 31 = r 22 3 + r 22 r 23 2 2 r 23 2 , r 32 = - r 22 2 r 23 r 22 + r 33 = 0 , r 23 ≠ 0.  (5)[c,a]=a+b;[c,b]=b 0 r 12 0 0 0 0 0 0 r 33,000000r31r320,000r21-r210r31-r310,0r1200000r320, r 11 r 12 0 r 21 r 22 0 r 31 r 32 0 r 12 = - r 11 2 - r 11 r 21 r 21 r 32 = r 31 r 22 r 21 , r 21 ≠ 0 r 22 = - r 11 - r 21.  (6)[c,a]=b;[c,b]=b;[c,c]=a r 11 r 12 0 0 0 0 r 31 r 32 0,r11-r110r21-r210r31-r310,r11-r110000r31-2r11-r312r11.  (7)[c,b]=b;[c,a]=αa(α∈C\{0}) 0 0 0 r 21 0 0 r 31 0 0,0r1200000r320,000000r31r320,00000000r33, r 11 r 12 0 r 21 - α r 11 0 r 31 r 32 0 r 31 = r 11 r 32 r 12 r 21 = - α r 11 2 r 12 , r 12 ≠ 0.Theorem 21 (B).The Rota-Baxter operators of weight-1on 3-dimensional non-Lie and nonnilpotent solvable Leibniz algebraAare givenin Table5.Table 5 Non-Lie solvable Leibniz algebraA RBO(A) of weightλ=-1 (1)[c,a]=a 0 0 0 r 21 r 22 0 r 31 r 32 0,0000r2200r32r33,0r1200r2200r321,1000r22r230r32r33, 1 r 12 0 0 r 22 0 0 r 32 0,100r21r220r31r321,0000r22r230r32r33.  (2)[c,a]=αa(α∈C\{0});[c,b]=b;[b,c]=-b 0 0 0 r 21 1 0 r 31 r 32 0,100r2100r31r321,1000100r32r33,1r1200000r320, 0 0 0 0 0 0 0 r 32 r 33,000000r31r320, 1000000r32r33,100000r31r321, 0 0 0 0 1 0 r 31 r 32 0,100010r31r321, 0r1200100r321,0000100r32r33, 0 0 0 0 r 22 r 23 0 1 - r 22 r 22 r 23 1 - r 22 r 23 ≠ 0, 1 0 0 0 r 22 r 23 0 1 - r 22 r 22 r 23 1 - r 22 r 23 ≠ 0.  ( 3 )   [ c , b ] = b ; [ b , c ] = - b ; [ c , c ] = a 1 0 0 r 21 0 0 r 31 r 32 1,r1100000r31r32r33r112r33-1=r332, 0 0 0 r 21 1 0 r 31 r 32 0,r1100010r31r32r33r112r33-1=r332.  ( 4 )   [ c , a ] = 2 a ; [ b , b ] = a ; [ c , b ] = b ; [ b , c ]  = - b ; [ c , c ] = a 0 0 0 0 0 0 r 31 r 32 r 33 r 32 2 + 2 r 33 r 31 + r 33 2 = 0, 0 0 0 - r 22 2 + r 23 2 2 r 23 r 22 r 23 - r 22 2 r 33 + r 23 2 r 33 2 r 23 2 r 33 r 22 r 23 r 33 r 22 + r 33 = 1 r 23 ≠ 0 .  (5)[c,a]=a+b;[c,b]=b 0 0 0 0 0 0 r 31 r 32 0,00000000r33,0r1200100r321,100100000, 1 r 12 0 0 0 0 0 r 32 0,10001000r33,100010r31r321, r 11 r 12 0 r 21 r 22 0 0 0 r 33 r 12 = r 33 r 21 + r 11 r 22 r 21 , r 21 ≠ 0 r 11 r 11 - 1 + r 11 + r 12 + r 33 - 1 r 21 = 0 r 12 r 11 - 1 + r 11 + r 12 + r 33 - 1 r 22 = r 33 r 11 r 11 + r 21 + r 22 - 1 = 0.  (6)[c,a]=b;[c,b]=b;[c,c]=a 0 0 0 0 0 0 r 31 r 32 0,r111-r110r111-r110r311-r310,r11-r110r11-11-r110r31-1-r311, r 11 1 - r 11 0 0 1 0 r 31 r 32 r 33 r 11 = r 33 2 2 r 33 - 1 , r 33 ≠ 1 2 2 r 33 - 1 1 - r 11 = ( r 33 - 1 ) ( r 31 + r 32 ), 1 0 0 0 1 0 r 31 r 32 1.   ( 7 )   [ c , b ] = b ; [ c , a ] = α a  ( α ∈ C \ { 0 } ) 1 0 0 0 0 0 1 0 1,100010000,000000r31r320,00000000r33,000010r3100, 0 0 0 0 1 0 0 0 r 33,0000100r321,1000000r320,-100010101,1000101r321, 0 0 0 r 21 1 0 r 31 0 0,000r210000-1α-1α≠1,000r211000r33α=1, 1 0 0 r 21 0 0 r 31 - r 31 r 21 0 α = 1 r 21 ≠ 0,100r2100r3101,100r211000αα-1α≠1, 0 r 12 0 0 1 0 - r 32 r 12 r 32 0 α = 1 r 12 ≠ 0,0r12000000αα-1α≠1,0r1200100r321, 0 r 12 0 0 1 0 0 0 r 33 α = 1,1r1200000r320,1r12001000-1α-1α≠1, 1 0 0 0 0 0 0 0 r 33,10001000r33,100010r31r321,1r12000000r33α=1, 1 r 12 0 0 0 0 r 31 r 31 r 12 1 α = 1,0r1200100r321α=1, 0 0 0 r 21 1 0 0 0 r 33 α = 1,r11-r11r11-1r210r211-r110r11-r11r11-1r211α=1r21≠0.



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*Source: 8540674-2018-12-02.xml*