Research Article Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices

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1 Hindawi Advances in Mathematical Physics Volume 07, Article ID 680, 7 pages Research Article Rota-Baxter Operators on 3-Dimensional Lie Algebras and the Classical R-Matrices Linli Wu, Mengping Wang, and Yongsheng Cheng School of Mathematics and Statistics, Henan University, Kaifeng , China Correspondence should be addressed to Yongsheng Cheng; yscheng@henu.edu.cn Received 6 September 07; Accepted 7 November 07; Published 8 December 07 Academic Editor: Boris G. Konopelchenko Copyright 07 Linli Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Our aim is to classify the Rota-Baxter operators of weight 0 on the 3-dimensional Lie algebra whose derived algebra s dimension is. We explicitly determine all Rota-Baxter operators (of weight zero) on the 3-dimensional Lie algebras g. Furthermore, we give the corresponding solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebras g ad g and the induced left-symmetry algebra structures on g.. Introduction In physics, the Yang-Baxter equation is a consistency equation which was first introduced in the field of statistical mechanics. It depends on the idea that, in some scattering situations, particles may preserve their momentum while changing their quantum internal states. Rota-Baxter algebra started with the probability study and has since found applications in many areas of mathematics and physics, such as quasi-symmetric functions, number theory, dendriform algebras, and Yang-Baxter equations. A Rota-Baxter operator (of weight zero) on an associative algebra A isdefinedtobealinearmapp:g gsatisfying P (x) P(y)=P(P(x) y + xp (y)), x, y A. () Rota-Baxter operators (on associative algebras) were introduced by Baxter to solve an analytic formula in probability [ 4]. It has been related to other areas in mathematics and mathematical physics [5 9]. A Rota-Baxter operator (of weight zero) on a Lie algebra (g, [, ]) is a linear operator P:g gsuch that [P (x), P (y)] = P ([P (x),y]+[x,p(y)]), x, y g. In fact, a Rota-Baxter operator is also called the operator form of the classical Yang-Baxter equation [0 3]. Let g be a () Lie algebra and r= i a i b i g g. r is called a classical Rmatrix if it is a solution of the classical Yang-Baxter equation (CYBE) in g:thatis, [r,r 3 ] + [r,r 3 ] + [r 3,r 3 ] =0 (3) in U(g),whereU(g) is the universal enveloping algebra of g and r = a i b i, i r 3 = a i b i, i r 3 = a i b i. i Set r = i b i a i.itiseasytoobtainthatr is skew-symmetric if and only if r = r. Semenov-Tian-Shansky proved in [4] that r is skew-symmetric and there is a nondegenerate symmetric invariant bilinear form on Lie algebra g; relation () is equivalent to relation (3) when the weight is zero. Furthermore, Rota-Baxter operators of weights 0 and on a Lie algebra g give rise to solutions of CYBE on the double Lie algebra g ad g over the direct sum g g of the Lie algebra g and its dual space g [, 5, 6]. Moreover, we can get some solutions of CYBE in g ad g Lie algebras through Rota-Baxter operators of any weight on g. (4)

2 Advances in Mathematical Physics In [], the authors gave all Rota-Baxter operators (of weight zero) on 3-dimensional simple Lie algebra sl(, C).The aim of this paper is to determine the Rota-Baxter operators (of weight zero) on the 3-dimensional Lie algebra which is not simple, and the dimension of its derived algebra is. We will determine the Rota-Baxter operators on the Lie algebra g and give a family of solutions of CYBE in g ad g.thispaperis organized as follows. In Section, we give the classification theorem of Rota-Baxter operators (of weight zero) on g. In Section 3, we give the corresponding solutions of CYBE in g ad g. In Section 4, we give the corresponding leftsymmetry structure on g.. The Rota-Baxter Operators on g (of Weight Zero).. Notations and the Classification Theorem. Let g be a 3- dimensional linear Lie algebra whose standard (Cartan-Weyl) basis consists of e, e, e 3 over the field of complex numbers C with the following Lie brackets: [e,e ]=e, [e,e 3 ]=0, [e,e 3 ]=e +e 3. Thus, a linear operator P:g gis determined by (5) P(e ) b b b 3 e ( P(e ) )=( b b b 3 )( e ), (6) P (e 3 ) b 3 b 3 b 33 e 3 where b ij C, i,j 3. P is a Rota-Baxter operator on g if the above matrix (b ij ) 3 3 satisfies (). Here is our main theorem. Theorem. All Rota-Baxter operators of weight zero on g are listed in their matrices form with respect to the Cartan-Weyl basis below, where a, b, and c are nonzero complex numbers. P =( ), a 0 P =( 0 a a ), 0 a P 3 =( ), 0 0 P 4 =( ab a a ), b a P 5 =( a 0 ), P 6 =( 0 0), P 7 =( 0 0 ), P 8 =( ), P 9 =( a 0 b), 0 0 P 0 =( a 0 0), 0 0 P =( 0 0 a), 0 0 P =( ), 0 0 P 3 =( a b), P 4 =( a 0), P 5 =( 0 a), P 6 =( 0 0),

3 Advances in Mathematical Physics P 7 =( a 0 P 8 =( P 9 =( 0 0 b ), a b a a 0 a a 0 b 0 0 a a P 0 =( 0 0 a ), P =( a 0 0 ), 0 0 a ) (a = ) (a = a 0 P =( ) (a = a a 0 a 0 0 P 3 =( ), 0 0 a 0 P 4 =( b 0 c ) (a = a a 0 a 0 P 5 =( 0 0 a ), 0 a P 9 =( 0 a a ), 0 a a a P 30 =( ), a a 0 0 P 3 =( ), 0 0 a (a+b) P 3 =( ab b b ab ). a (a+b).. Reduction to Quadratic Equations. In order to show that P is a Rota-Baxter operator, we only check the following: [P (e ),P(e )]=P([P(e ),e ]+e,p(e )], [P (e ),P(e 3 )]=P([P(e ),e 3 ]+e,p(e 3 )], [P (e ),P(e 3 )]=P([P(e ),e 3 ]+e,p(e 3 )]. It follows from (5) and (6) that [P (e ),P(e 3 )]=(b b 3 b b 3 +b b 33 b 3 b 3 )e +(b b 33 b 3 b 3 )e 3, while P([P(e ),e 3 ]+[e,p(e 3 )]) =P((b b 3 +b 33 )e ) P ((b +b 33 )e 3 ) (7) (8) (9) P 6 =( a 0 0 ), 4 0 =(b b b 3 b +b 33 b +b b 3 +b 33 b 3 )e +(b b b 3 b +b 33 b +b b 3 +b 33 b 3 )e +(b b 3 b 3 b 3 +b 33 b 3 +b b 33 +b 33 b 33 )e 3. Comparing the coefficients in (9) and (0), we have (0) 0 P 7 =( ), P 8 =( a 0 b ), 4 0 b b 3 b b 3 +b b 33 b 3 b 3 b b +b 3 b () b 33 b b 33 b 3 =0, b b b 3 b +b 33 b +b b 3 +b 33 b 3 =0, () b 3 b 3 +b b 3 b 3 b 3 +b 33 b 3 +b 33 b 33 =0. (3) Similarly, from [P (e ),P(e )]=P([P(e ),e ]+e,p(e )], (4) [P (e ),P(e 3 )]=P([P(e ),e 3 ]+e,p(e 3 )],

4 4 Advances in Mathematical Physics we obtain the following six equations: b b +b b b 3 b b 3 b 3 b b 3 +b 3 b =0, (5) b b +b b b 3 b b 3 b 3 =0, (6) b b 3 +b b 3 b 3 b 3 b 3 b 33 b b 3 =0, (7) b b 33 b b 3 b 3 b 3 b b =0, (8) b b +b 3 b =0, (9) b b 3 +b 3 b 3 =0. (0).3. Solving the Quadratic Equations. Equation (9) implies b (b +b 3 ) = 0. To solve the quadratic equations (), (), (3), (5), (6), (7), (8), (9), and (0), we distinguish the following cases depending on whether b =0or not. Case. b =0, b +b 3 =0.Thatis,b =0, b 3 =0,taking b 3 =.Equation(6)impliesb 3 =0. Equation (5) implies b =0. Equation () implies b =b b 3 +b +b 3.Equation () implies b 33 = b. Equation (3) implies b 3 = b 33 = b.weobtain P=( b b 3 b b b 3 b ). () Taking b =0, b 3 =a,weobtainp.takingb =a, b 3 =0, we obtain P.Takingb =0, b 3 =0,weobtainP 3.Taking b =a, b 3 =b,weobtainp 4. Case. Assume b =0, b +b 3 =0.Thatis,b =0, b 3 =0. We distinguish the two cases depending on whether b 3 =0 or not. Subcase.. If b 3 =0, then (3) implies b 33 =0.Equation (5) implies b =0. Equation () implies b b 3 =0. Subcase... If b =0, b 3 =0,weobtain P=( b 0 b 3 ). () Taking b =a, b 3 =,weobtainp 5.Takingb =, b 3 =0, we obtain P 6.Takingb =0, b 3 =,weobtainp 7.Taking b =0, b 3 =0,weobtainP 8. Subcase... If b =0, b 3 =0,takingb 3 =,weobtain P=( b 0 b 3 ). (3) 0 0 Taking b =a, b 3 =b,weobtainp 9.Takingb =a, b 3 =0, we obtain P 0.Takingb =0, b 3 =a,weobtainp.taking b =0, b 3 =0,weobtainP. Subcase..3. If b =0, b 3 =0,takingb =,weobtain P=( b b 3 ). (4) Taking b =a, b 3 =b,weobtainp 3.Takingb =a, b 3 =a, we obtain P 4.Takingb =0, b 3 =a,weobtainp 5.Taking b =0, b 3 =0,weobtainP 6. Subcase.. If b 3 = 0,takingb 3 =,(7)impliesb = (b 33 b + )/. Then (5) implies b 3 =(b 3b + b 33 + )/. Equation (3) implies b +b 33 =b b 33.Thatis,(b + b 33 )(b b 33 )=0. Subcase... If b +b 33 =0, b b 33 =0, (b =/), and then b =0, b 3 =b b,weobtain b 0 P=( b 0 b 3 ), (b = ). (5) b b 0 b Taking b =0, b =a, b 3 =b,weobtainp 7.Takingb =a, b =0, b 3 =b(a=/),weobtainp 8.Takingb =a, b =b, b 3 =0(a=/),weobtainP 9.Takingb =0, b =0, b 3 =a,weobtainp 0.Takingb =0, b =a, b 3 =0, we obtain P.Takingb =a, b =0, b 3 =0(a=/),we obtain P.Takingb =0, b =0, b 3 =0,weobtainP 3. Taking b =a, b =b, b 3 =c(a=/),weobtainp 4. Subcase... If b +b 33 =0, b b 33 =0(b =/),and then b =( b +)/, b 3 =(b 4b +)/, (3.4) implies 8b 3 b +6b =0.Thenwehave8(b /) 3 =0, b =/, giving a contradiction. Subcase..3. If b +b 33 =0, b b 33 = 0,thatis,b =/, b 33 = /,andthenb =0, b 3 = /4,weobtain 0 P=( b 0 b 3 ). (6) 4 0 Taking b =0, b 3 =a,weobtainp 5.Takingb =a, b 3 =0, we obtain P 6.Takingb =0, b 3 =0,weobtainP 7.Taking b =a, b 3 =b,weobtainp 8. Case 3. Assume b =0, (b +b 3 )=0.Takingb =,then b 3 = /.Equation(6)impliesb 3 =b +b.equations () and (8) imply b 3 =(b +b 33 )/. Equation(7)implies b 3 = b b b b b 33 b b 33. Equation (5) implies b = b b b b 33 b b 33 b b. Equations (), (5), and (7) imply b +b +b 33 +b b +b b 33 +b b 33 =0.

5 Advances in Mathematical Physics 5 Thenwehave(b +b +b 33 ) =0.Sob 33 = (b +b ), b =b b, b 3 = b b b, b 3 = b /.Weobtain b b +b P=( b b b b b b ). (7) b (b +b ) Taking b =0, b =a,weobtainp 9.Takingb =a, b =0, we obtain P 30.Takingb =0, b =0,weobtainP 3.Taking b =a, b =b,weobtainp Solutions of the CYBE in g ad g In this section, we will give some solutions of CYBE in g ad g.let(g, [, ]) be a Lie algebra and β : g gl(v) a representation of g. On the vector space g V,thereisnatural Lie algebra structure (denoted by g β V) given by r 3 =e e 3 e 3 e, r 4 =(abe +ae a e 3 ) e + (be +e ae 3 ) e 3 e (abe +ae a e 3 ) e 3 (be +e ae 3 ), r 5 =(ae +e 3 ) e e (ae +e 3 ), r 6 =e e e e, r 7 =e 3 e e e 3, r 8 =0, r 9 =(ae +be 3 ) e +e e 3 e (ae +be 3 ) e 3 e, r 0 =ae e +e e 3 e ae e 3 e, [x + V,x + V ]=[x,x ]+β(x ) V β(x ) V, x,x g, V, V V. (8) r =ae 3 e +e e 3 e ae 3 e 3 e, r =e e 3 e 3 e, Let β :g gl(v ) be the dual representation of β.alinear map P : V g canbeidentifiedasanelement P in g V (g β V ) (g β V ) as follows. Let {V, V,...,V m } be abasisofv and {V, V,...,V m } the dual basis in V :thatis, V i (V j) = δ ij.let{e,e,...,e n } be a basis of g. SetP(V i ) = n i= a ije j, i n. Since, as a vector space, Hom(V, g) g V,then P = n i= m P(V i ) V i = a ij e j V i n i=j= (g β V ) (g β V ). (9) Lemma (see [5]). Let g be a Lie algebra; let (V, β) be a gmodule. A linear map P:g gis a Rota-Baxter operator if and only if r=p P is a skew-symmetric solution of CYBE in g ad g. Now consider the adjoint representation of g, (g, ad) which is a g-module. Let e, e, e 3 be the Cartan-Weyl basis. Using Lemma and relation (9), we can obtain a family of solutions of CYBE in g ad g through the Rota-Baxter operators on g given in Theorem. Theorem 3. The following tensors are solutions of the classical Yang-Baxter equation in g ad g,wherea, b, andc are nonzero complex numbers r =(ae +e ) e 3 e 3 (ae +e ), r =(ae a e 3 ) e +(e ae 3 ) e 3 e (ae a e 3 ) e 3 (e ae 3 ), r 3 =(ae +e +be 3 ) e e (ae +e +be 3 ), r 4 =(ae +e ) e e (ae +e ), r 5 =(e +ae 3 ) e e (e +ae 3 ), r 6 =e e e e, r 7 =e 3 e +(ae +be 3 ) e e 3 e 3 e e 3 e (ae +be 3 )+e 3 e 3, r 8 =(ae +e 3 ) e +be 3 e +((a a)e + (a ) e 3 ) e 3 e (ae +e 3 ) e be 3 e 3 ((a a)e + (a ) e 3 ), r 9 =(ae +e 3 ) e +be e (a = +((a a)e + (a ) e 3 ) e 3 e (ae +e 3 ) e be e 3 ((a a)e + (a ) e 3 ), (a r 0 =e 3 e +ae 3 e e 3 e 3 e e 3 e ae 3 +e 3 e 3, r =e 3 e +ae e e 3 e 3 e e 3 e ae +e 3 e 3, =

6 6 Advances in Mathematical Physics r =(ae +e 3 ) e +((a a)e + (a ) e 3 ) e 3 e (ae +e 3 ) e 3 ((a a)e + (a ) e 3 ) r 3 =e 3 e e 3 e 3 e e 3 +e 3 e 3, (a = r 3 =e e e e 3 e e +e 3 e, r 3 =(ae +e +(a+b) e 3 ) e +(abe +be (b +ab)e 3 ) e ( a e + e + (a+b) e 3 ) e 3 e r 4 =(ae +e 3 ) e +(be +ce 3 ) e +((a a)e + (a ) e 3 ) e 3 e (ae +e 3 ) e (be +ce 3 ) e 3 (ae +e +(a+b) e 3 ) e (abe +be (b +ab)e 3 ) e +e 3 ( a e + e + (a+b) e 3 ). ((a a)e + (a ) e 3 ), (a= r 5 =( e +e 3 ) e +ae 3 e ( 4 e + e 3) e 3 e ( e +e 3 ) e ae 3 +e 3 ( 4 e + e 3), r 6 =( e +e 3 ) e +ae e ( 4 e + e 3) e 3 e ( e +e 3 ) e ae +e 3 ( 4 e + e 3), r 7 =( e +e 3 ) e ( 4 e + e 3) e 3 e ( e +e 3 )+e 3 ( 4 e + e 3), r 8 =( e +e 3 ) e +(ae +be 3 ) e ( 4 e + e 3) e 3 e ( e +e 3 ) e (ae +be 3 )+e 3 ( 4 e + e 3), r 9 =(e +ae 3 ) e +(ae +a e 3 ) e ( e +ae 3 ) e 3 e (e +ae 3 ) e (ae +a e 3 )+e 3 ( e +ae 3 ), r 30 =(ae +e +ae 3 ) e (a e + e +ae 3 ) e 3 e (ae +e +ae 3 )+e 3 ( a e + e +ae 3 ), (30) Onecancheckthatallofthetensorsabovearesolutions of the classical Yang-Baxter equation in g ad g. 4. Induced Left-Symmetric Algebras from Rota-Baxter Operators of Weight 0 on g A left-symmetric algebra structure on g is a bilinear product :g g gsatisfying the condition x (y z) (x y) z=y (x z) (y x) z (3) for all x, y, z g. There are many examples of Lie algebras which do not admit a left-symmetric product. For example, it is easy to see that there are no left-symmetric algebras with semisimple Lie algebra. Equation (3) implies that the commutators [x, y] = x y y x satisfy the Jacobi identity; thatistosayeachleft-symmetricproducthasanassociated commutation Lie algebra, which is called the subadjacent Lie algebra. If R is a Rota-Baxter operator on a left-symmetric algebra, then R is a solution of CYBE on its subadjacent Lie algebra [7]. Clearly, each associative algebra product is a leftsymmetric product. Given a Lie algebra g, it is a fundamental problem to decide whether g admits a left-symmetric product andtogiveaclassificationofsuchproducts[8].asan application of Yang-Baxter operators, we can use them to construct left-symmetric algebras with respect to g 6. Lemma 4 (see [3]). Let g be a Lie algebra; P is called a solution of the classical Yang-Baxter equation. Define a new operation on g by x y=[p(x),y], x,y g. (3) Then (g, ) is a left-symmetric algebra. According to Theorem 3 and Lemma 4, we can get some left-symmetric algebras of g. Theorem 5. Some left-symmetric algebras of g (of weight zero) are determined: () e 3 e = e, e 3 e =ae, e 3 e 3 =e +e 3 ; () e e = ae, e e =a (e +e 3 ), e e 3 =a(e +e 3 ), e 3 e = e, e 3 e =a(e +e 3 ), e 3 e 3 =e +e 3 ;

7 Advances in Mathematical Physics 7 (3) e 3 e = e, e 3 e 3 =e +e 3 ; (4) e e = ae, e e =(a + ab)e +a e 3, e e 3 = a(e +e 3 ), e 3 e = e, e 3 e =(a+b)e +ae 3, e 3 e 3 =e +e 3 ; (5) e e =(a )e e 3 ; (6) e e =e ; (7) e e = (e +e 3 ); (8) e e =(a b)e be 3, e 3 e =e ; (9) e e =ae, e 3 e =e ; (0) e e = a(e +e 3 ), e 3 e =e ; () e 3 e =e ; () e e = e, e e =(a b)e be 3, e e 3 =e +e 3 ; (3) e e = e, e e =ae, e e 3 =e +e 3 ; (4) e e = e, e e = a(e +e 3 ), e e 3 =e +e 3 ; (5) e e = e, e e 3 =e +e 3 ; (6) e e = (e +e 3 ), e e =(a b)e be 3, e 3 e = e +e 3 ; (7) e e =(a )e e 3, e e = b(e +e 3 ), e 3 e = (a ) e (a )e 3 ; (8) e e = (a )e e 3, e e = be, e 3 e = (a ) e (a )e 3 ; (9) e e = (e +e 3 ), e e = a(e +e 3 ), e 3 e =e +e 3 ; (0) e e =e +e 3, e e =ae, e 3 e =e +e 3 ; () e e =(a )e e 3, e 3 e =(a ) e (a )e 3 ; () e e = (e +e 3 ), e 3 e =e +e 3 ; (3) e e =(a )e e 3, e e =(b c)e ce 3, e 3 e =(a ) e (a )e 3 ; (4) e e = (/)e e 3, e e = a(e +e 3 ), e 3 e = (/4)e + (/)e 3 ; (5) e e = (/)e e 3, e e =ae, e 3 e = (/4)e + (/)e 3 ; (6) e e = (/)e e 3, e 3 e = (/4)e + (/)e 3 ; (7) e e = (/)e e 3, e e =(a b)e be 3, e 3 e = (/4)e + (/)e 3 ; (8) e e = e, e e = a(e +e 3 ), e e 3 =e +e 3, e e = ae, e e = a (e +e 3 ), e e 3 = a(e +e 3 ), e 3 e = (/)e, e 3 e = a(e +e 3 ), e 3 e 3 = (/)(e +e 3 ); (9) e e = e, e e = ae ae 3, e e 3 =e +e 3, e 3 e = (/)e, e 3 e = (a/)e +ae 3, e 3 e 3 = (/)(e +e 3 ); (30) e e 3 =e +e 3, e 3 e 3 = (/)(e +e 3 ); (3) e e = e, e e = (a + b)e (a+b)e 3, e e 3 =e +e 3, e e = be, e e =(b +3ab)e + (b + ab)e 3, e e 3 =b(e +e 3 ), e 3 e = (/)e, e 3 e = (a/+b)e +(a+b)e 3, e 3 e 3 = (/)(e +e 3 ). Conflicts of Interest The authors declare no conflicts of interest. Acknowledgments This work is supported by the National Science Foundation of China (Grants nos and 77) and the Science and Technology Program of Henan Province (Grant no ). References [] G. Baxter, An analytic problem whose solution follows from a simple algebraic identity, Pacific Mathematics,vol.0, pp , 960. [] K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation, Letters in Mathematical Physics,vol.6, no.,pp , 00. [3] L. Guo, An Introdction to Roto-Baxter Algebra, International Press, Beijing, China, 0. [4] L. Guo and W. Keigher, Baxter algebras and shuffle products, Advances in Mathematics,vol.50,no.,pp.7 49,000. [5] R.Bai,L.Guo,J.Li,andY.Wu, Rota-Baxter3-Liealgebras, Mathematical Physics, vol.54,no.6,articleid , 03. [6] A. A. Belavin and V. G. Drinfel d, Solutions of the classical Yang - Baxter equation for simple Lie algebras, Functional Analysis and its Applications, vol. 6, no. 3, pp , 98. [7] Y. Cheng and Y. Su, Quantum deformations of the Heisenberg- Virasoro algebra, Algebra Colloquium, vol. 0, no., pp , 03. [8] A. Makhlouf and D. Yau, Rota-Baxter Hom-Lie-admissible algebras, Communications in Algebra, vol.4,no.3,pp.3 57, 04. [9] X. Tang, Y. Zhang, and Q. Sun, Rota-Baxter operators on 4-dimensional complex simple associative algebras, Applied Mathematics and Computation,vol.9,pp.73 86,04. [0] Y. Cheng and Y. Shi, Lie bialgebra structures on the q-analog Virasoro-like algebras, Communications in Algebra, vol. 37, no. 4, pp , 009. [] P. Etingof, T. Schedler, and A. Soloviev, Set-theoretical solutions to the quantum Yang-Baxter equation, Duke Mathematical Journal,vol.00,no.,pp.69 09,999. [] J. Pei, C. Bai, and L. Guo, Rota-Baxter operators on sl(, C) and solutions of the classical Yang-Baxter equation, Mathematical Physics,vol.55,no.,ArticleID070,04. [3] X. Li, D. Hou, and C. Bai, Rota-Baxter operators on pre-lie algebras, Nonlinear Mathematical Physics,vol.4,no.,pp.69 89,007. [4] M. A. Semenov-Tyan-Shanskii, What is a classical r-matrix? Functional Analysis and its Applications, vol.7,no.4,pp.59 7, 983. [5] C. Bai, A unified algebraic approach to the classical Yang- Baxter equation, Physics A: Mathematical and General,vol.40,no.36,pp ,007. [6] C. Bai, L. Guo, and X. Ni, Generalizations of the classical Yang- Baxter equation and O-operators, Mathematical Physics, vol. 5, no. 6, Article ID 06355, 0. [7] H. An and C. Bai, From Rota-Baxter algebras to pre-lie algebras, Physics A: Mathematical and Theoretical, vol. 4, no., Article ID 050, 008. [8] O. Baues, Left-symmetric algebras for gl(n), Transactions of the American Mathematical Society, vol.35,no.7,pp , 999.

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