Relative Rota-Baxter systems on Leibniz algebras
aa r X i v : . [ m a t h . R A ] J a n Relative Rota-Baxter systems on Leibniz algebras
Apurba Das , Shuangjian Guo ∗
1. Department of Mathematics and Statistics, Indian Institute of TechnologyKanpur 208016, Uttar Pradesh, IndiaEmail: [email protected]. School of Mathematics and Statistics, Guizhou University of Finance and EconomicsGuiyang 550025, P. R. of ChinaEmail: [email protected]
ABSTRACT
In this paper, we introduce relative Rota-Baxter systems on Leibniz algebrasand give some characterizations and new constructions. Then we constructa graded Lie algebra whose Maurer-Cartan elements are relative Rota-Baxtersystems. This allows us to define a cohomology theory associated with arelative Rota-Baxter system. Finally, we study formal deformations and ex-tendibility of finite order deformations of a relative Rota-Baxter system interms of the cohomology theory.
Key words : Relative Rota-Baxter system, Leibniz algebra, Cohomology, de-formation.
Introduction
The notion of Rota-Baxter operators on associative algebras was introduced in 1960by Baxter [2] in his study of fluctuation theory in probability. Recently, it has been foundmany applications, including in Connes-Kreimer’s algebraic approach to the renormaliza-tion in perturbative quantum field theory [7]. In the Lie algebra context, a Rota-Baxteroperator of weight zero was introduced independently in the 1980’s as the operator formof the classical Yang-Baxter equation, whereas the classical Yang-Baxter equation playsimportant roles in many fields in mathematics and mathematical physics such as quantumgroups and integrable systems [8, 25]. ∗ Corresponding author K -theory. Leibniz algebras werestudied from different aspects due to applications in both mathematics and physics. Inparticular, integration of Leibniz algebras was studied in [5, 9] and deformation quantiza-tion of Leibniz algebras was considered in [12]. As the underlying structure of embeddingtensor, Leibniz algebras also have application in higher gauge theories, see [20,27] for moredetails. Recently, relative Rota-Baxter operators on Leibniz algebras were studied in [26],which is the main ingredient in the study of the twisting theory and the bialgebra theoryfor Leibniz algebras. Moreover, relative Rota-Baxter operators on a Leibniz algebra canbe seen as the Leibniz algebraic analogue of Poisson structures. Generally, Rota-Baxteroperators can be defined on operads, which give rise to the splitting of operads [3, 24]. Forfurther details on Rota-Baxter operators, see [16].The deformation of algebraic structures began with the seminal work of Gersten-haber [14, 15] for associative algebras and followed by its extension to Lie algebras byNijenhuis and Richardson [21, 22]. In general, the deformation theory of algebras overbinary quadratic operads was developed by Balavoine [1]. Deformations of morphismsand O -operators (also called relative Rota-Baxter operators) were developed in [10, 13]and [28, 29]. Rota-Baxter systems as a generalization of a Rota-Baxter operator wereintroduced by Brzezi´nski [6]. In a Rota-Baxter system, two operators are acting on thealgebra and satisfying some Rota-Baxter type identities. Rota-Baxter systems in the pres-ence of bimodule were introduced and their deformation theory was studied by Das [11].They are called generalized Rota-Baxter systems. Our main objective in this paper isgeneralized Rota-Baxter systems in the context of Leibniz algebra. We call them relativeRota-Baxter systems, motivated by the terminology of relative Rota-Baxter operatorsof [26]. Our aim in this paper is to study the cohomology and deformation theory ofrelative Rota-Baxter systems in the context of Leibniz algebras.The paper is organized as follows. In Section 2, we introduce relative Rota-Baxtersystems with respect to a representation of a Leibniz algebra and give some characteriza-tions and new constructions. In Section 3, we emphasis on relative Rota-Baxter systemswith respect to the regular representation. In Section 4, we construct a graded Lie algebrawhose Maurer-Cartan elements are relative Rota-Baxter systems, which leads us to definecohomology for a relative Rota-Baxter system. Finally, in Section 5, we consider formaldeformations of relative Rota-Baxter systems.Throughout this paper, K is a field of characteristic zero and Z denotes the set of allintegers. 2 Leibniz algebras
In this preliminary section, we recall Leibniz algebras and their representations [17,18].
Definition 1.1.
A Leibniz algebra is a vector space g together with a bilinear operation [ · , · ] g : g ⊗ g → g satisfying [ x, [ y, z ] g ] g = [[ x, y ] g , z ] g + [ y, [ x, z ] g ] g , for x, y, z ∈ g. Definition 1.2.
A representation of a Leibniz algebra ( g, [ · , · ] g ) is a triple ( V, ρ L , ρ R ) ,where V is a vector space, ρ L , ρ R : g → gl ( V ) are linear maps such that the followingequalities hold : for all x, y ∈ g , (1) ρ L ([ x, y ] g ) = [ ρ L ( x ) , ρ L ( y )] , (2) ρ R ([ x, y ] g ) = [ ρ L ( x ) , ρ R ( y )] , (3) ρ R ( y ) ◦ ρ L ( x ) = − ρ R ( y ) ◦ ρ R ( x ) . Let ( g, [ · , · ]) be a Leibniz algebra. Define the left multiplication L : g → gl ( g ) and theright multiplication R : g → gl ( g ) by L x y = [ x, y ] g and R x y = [ y, x ] g , for all x, y ∈ g .Then ( g, L, R ) is a representation of ( g, [ · , · ] g ), called the regular representation. Definetwo linear maps L ∗ , R ∗ : g → gl ( g ∗ ) with x L ∗ x and x R ∗ x respectively by h L ∗ x ξ, y i = −h ξ, [ x, y ] g i , h R ∗ x ξ, y i = −h ξ, [ y, x ] g i , for x, y ∈ g, ξ ∈ g ∗ . Then it has been shown in [26] that ( g ∗ , L ∗ , − L ∗ − R ∗ ) is a representation. This iscalled the dual of the regular representation. Definition 1.3.
A quadratic Leibniz algebra is a Leibniz algebra ( g, [ · , · ] g ) equipped witha nondegenerate skew-symmetric bilinear form ω ∈ ∧ g ∗ such that the following invariantcondition holds: ω ( x, [ y, z ] g ) = ω ([ x, z ] g + [ z, x ] g , y ) , for x, y, z ∈ g. Proposition 1.4. ( [26]) Let ( g, [ · , · ] g , ω ) be a quadratic Leibniz algebra. Then the map ω ♮ : g → g ∗ , ω ♯ ( x )( y ) = ω ( x, y ) , for x, y ∈ g is an isomorphism from the regular representation ( g, L, R ) to its dual representation ( g ∗ , L ∗ , − L ∗ − R ∗ ) . Relative Rota-Baxter systems with respect to a represen-tation
In this section, we introduce relative Rota-Baxter systems with respect to a represen-tation of a Leibniz algebra.Let (
V, ρ L , ρ R ) be a representation of a Leibniz algebra ( g, [ · , · ] g ). Definition 2.1. (1) A relative Rota-Baxter system on ( g, [ · , · ] g ) with respect to the repre-sentation ( V, ρ L , ρ R ) consists of a pair ( R, S ) of linear maps R, S : V → g satisfying [ Ru, Rv ] g = R ( ρ L ( Ru ) v + ρ R ( Sv ) u ) , [ Su, Sv ] g = S ( ρ L ( Ru ) v + ρ R ( Sv ) u ) , for u, v ∈ V .(2) A Rota-Baxter system on ( g, [ · , · ] g ) is a relative Rota-Baxter system on ( g, [ · , · ] g ) with respect to the regular representation. Example 2.2.
A relative Rota-Baxter operator [26] on ( g, [ · , · ] g ) with respect to the rep-resentation ( V, ρ L , ρ R ) is a linear map R : V → g satisfying [ Ru, Rv ] g = R ( ρ L ( Ru ) v + ρ R ( Rv ) u ) , for u, v ∈ V. Thus R is a relative Rota-Baxter operator if and only if the pair ( R, R ) is a relative Rota-Baxter system. Example 2.3.
Consider the 2-dimensional Leibniz algebra ( g, [ · , · ]) given with respect toa basis { e , e } by [ e , e ] = 0 , [ e , e ] = 0 , [ e , e ] = e , [ e , e ] = e . Let { e ∗ , e ∗ } be the dual basis. Then R = a a a a ! , S = b b b b ! is a relativeRota-Baxter system on ( g, [ · , · ]) with respect to the representation ( g ∗ , L ∗ , − L ∗ − R ∗ ) if andonly if [ Re ∗ i , Re ∗ j ] = R ( L ∗ Re ∗ i e ∗ j − L ∗ Se ∗ j e ∗ i − R ∗ Se ∗ j e ∗ i ) , [ Se ∗ i , Se ∗ j ] = S ( L ∗ Re ∗ i e ∗ j − L ∗ Se ∗ j e ∗ i − R ∗ Se ∗ j e ∗ i ) , i, j = 1 , . It is straightforward to deduce that L e ( e , e ) = ( e , e ) ! , L e ( e , e ) = ( e , e ) ! ,R e ( e , e ) = ( e , e ) ! , R e ( e , e ) = ( e , e ) ! , nd L ∗ e ( e ∗ , e ∗ ) = ( e ∗ , e ∗ ) ! , L ∗ e ( e ∗ , e ∗ ) = ( e ∗ , e ∗ ) − − ! ,R ∗ e ( e ∗ , e ∗ ) = ( e ∗ , e ∗ ) − ! , R ∗ e ( e ∗ , e ∗ ) = ( e ∗ , e ∗ ) − ! . We have [ Re ∗ , Re ∗ ] = [ a e + a e , a e + a e ] = a ( a + a ) e , and R ( L ∗ Re ∗ e ∗ − L ∗ Se ∗ e ∗ − R ∗ Se ∗ e ∗ )= − a ( R ( e ∗ ) + R ( e ∗ )) + b ( R ( e ∗ ) + R ( e ∗ )) + ( b + b ) R ( e ∗ )= − a ( a e + a e + a e + a e ) + b ( a e + a e + a e + a e )+( b + b )( a e + a e )= (( b + b ) a + ( a + a )( b − a )) e + (( b + b ) a + ( a + a )( b − a )) e , [ Se ∗ , Se ∗ ] = [ b e + b e , b e + b e ] = b ( b + b ) e , and S ( L ∗ Re ∗ e ∗ − L ∗ Se ∗ e ∗ − R ∗ Se ∗ e ∗ )= − a ( R ( e ∗ ) + R ( e ∗ )) + b ( S ( e ∗ ) + S ( e ∗ )) + ( b + b ) S ( e ∗ )= − a ( a e + a e + a e + a e ) + b ( b e + b e + b e + b e )+( b + b )( b e + b e )= (( b + b ) b + ( b + b ) b − ( a + a ) a ) e +(( b + b ) b + ( b + b ) b − ( a + a ) a ) e . Thus, we obtain a ( a + a ) = ( b + b ) a + ( a + a )( b − a ) , ( b + b ) a + ( a + a )( b − a ) = 0 ,b ( b + b ) = ( b + b ) b + ( b + b ) b − ( a + a ) a , ( b + b ) b + ( b + b ) b − ( a + a ) a = 0 . imilarly, we obtain a ( a + a ) = b ( a + a ) + ( b + b ) a ,b ( a + a ) + ( b + b ) a = 0 ,b ( b + b ) = b ( b + b ) + ( b + b ) b ,b ( b + b ) + ( b + b ) b = 0 , a ( a + a ) = b ( b + b ) = 0 ,a ( a + a ) = a ( a + a ) , − a ( a + a ) = 0 ,b ( b + b ) = a ( b + b ) , − a ( b + b ) = 0 . Summarize the above discussion, we have(1) If a = b = 0 and a = b , then R = a a a ! , S = b b b ! is arelative Rota-Baxter system on ( g, [ · , · ]) with respect to the representation ( g ∗ , L ∗ , − L ∗ − R ∗ ) if and only if ( b − a ) a = ( b − a ) b = 0 ,a ( a + a ) = ( b + b ) a ,b ( b + b ) = ( b + b ) b + ( b + b ) b − ( a + a ) a , (2) If a = b = 0 and a = b , then R = a a a a ! , S = b b b b ! is arelative Rota-Baxter system on ( g, [ · , · ]) with respect to the representation ( g ∗ , L ∗ , − L ∗ − R ∗ ) if and only if a = − a = − a = a , b = − b = − b = b . We will give some more examples of Rota-Baxter systems on Leibniz algebras in thenext section.In the following, we give some characterizations of relative Rota-Baxter systems. Let(
V, ρ L , ρ R ) be a representation of a Leibniz algebra ( g, [ · , · ] g ). Then there is a Leibnizalgebra structure on g ⊕ g ⊕ V given by[ x + x + u, y + y + v ] = [ x , y ] g + [ x , y ] g + ρ L ( x ) v + ρ R ( y ) u. This is exactly the semidirect product if we consider the Leibniz algebra structure on g ⊕ g and define its representation on V by ρ L ( x + x ) v = ρ L ( x ) v and ρ R ( x + x ) v = ρ R ( x ) v . Proposition 2.4.
A pair ( R, S ) of linear maps from V to g is a relative Rota-Baxtersystem with respect to the representation ( V, ρ L , ρ R ) if and only if the pair ( e R, e S ) of maps e R : g ⊕ g ⊕ V → g ⊕ g ⊕ V, x + x + u R ( u ) + 0 + 0 , e S : g ⊕ g ⊕ V → g ⊕ g ⊕ V, x + x + u S ( u ) + 0 , is a Rota-Baxter system on the Leibniz algebra g ⊕ g ⊕ V . roof. For any x , x , y , y ∈ g and u, v ∈ V , we have[ e R ( x + x + u ) , e R ( y + y + v )] = [ R ( u ) , R ( v )] g + 0 + 0and e R ([ e R ( x + x + u ) , y + y + v ] + [ x + x + u, e S ( y + y + v )])= R ( ρ L ( Ru ) v + ρ R ( Sv ) u ) + 0 + 0 . Similarly, we have[ e S ( x + x + u ) , e S ( y + y + v )] = 0 + [ S ( u ) , S ( v )] g + 0and e S ([ e R ( x + x + u ) , y + y + v ] + [ x + x + u, e S ( y + y + v )]= 0 + S ( ρ L ( Ru ) v + ρ R ( Sv ) u ) + 0 . Hence (
R, S ) is a relative Rota-Baxter system if and only if ( e R, e S ) is a Rota-Baxter system. (cid:3) Recall that a Nijenhuis operator on a Leibniz algebra ( g, [ · , · ] g ) is a linear map N : g → g satisfying [ N x, N y ] g = N ([ N ( x ) , y ] g + [ x, N ( y )] g − N [ x, y ] g ) , for x, y ∈ g. The following result relates to relative Rota-Baxter systems and Nijenhuis operators.
Proposition 2.5.
A pair ( R, S ) of linear maps from V to g is a relative Rota-Baxtersystem if and only if N ( R,S ) = R S : g ⊕ g ⊕ V → g ⊕ g ⊕ V is a Nijenhuis operator on the Leibniz algebra g ⊕ g ⊕ V . Proof.
For any x , x , y , y ∈ g and u, v ∈ V , by a simple calculation, we have[ N ( R,S ) ( x + y + u ) , N ( R,S ) ( x + y + v )] = [ R ( u ) , R ( v )] g + [ S ( u ) , S ( v )] g + 0and N ( R,S ) ([ N ( R,S ) ( x + y + u ) , x + y + v ] + [ x + y + u, N ( R,S ) ( x + y + v )] − N ( R,S ) [ x + y + u, x + y + v ])= R ( ρ L ( Ru ) v + ρ R ( Sv ) u ) + S ( ρ L ( Ru ) v + ρ R ( Sv ) u ) + 0 . It follows that N ( R,S ) is a Nijenhuis operator if and only if ( R, S ) is a relative Rota-Baxtersystem. (cid:3) efinition 2.6. Let ( V, ρ L , ρ R ) be a representation of a Leibniz algebra ( g, [ · , · ] g ) . Supposethat dim ( g ) = dim ( V ) . A pair (Φ , Ψ) of invertible linear maps from g to V is said to bean invertible 1-cocycle system if they satisfy Φ([ x, y ] g ) = ρ L ( x )Φ( y ) + ρ R (Ψ − ◦ Φ( y ))Φ( x ) , Ψ([ x, y ] g ) = ρ L (Φ − ◦ Ψ( x ))Ψ( y ) + ρ R ( y )Ψ( x ) , for x, y ∈ g . It follows from the above definition that (Φ , Φ) is an invertible 1-cocycle system if andonly if Φ : g → V is an invertible derivation. Proposition 2.7.
Let ( V, ρ L , ρ R ) be a representation of a Leibniz algebra ( g, [ · , · ] g ) . Sup-pose that dim ( g ) = dim ( V ) . A pair ( R, S ) of invertible linear maps from V to g is arelative Rota-Baxter system if and only if ( R − , S − ) is an invertible 1-cocycle system. Proof.
For any u, v ∈ V and x, y ∈ g , by taking R ( u ) = x, R ( v ) = y , the first identityof Definition 2.1 is equivalent to R − [ x, y ] g = ρ L ( x ) R − y + ρ R (( S − ) − ◦ R − ( y )) R − x. Similarly, for any u, v ∈ V and x, y ∈ g , by taking S ( u ) = x, S ( v ) = y , the second identityof Definition 2.1 is equivalent to S − [ x, y ] g = ρ L (( R − ) − ◦ S − ( x )) S − y + ρ R ( y ) S − x. It follows that (
R, S ) of invertible linear maps from V to g is a relative Rota-Baxter systemif and only if ( R − , S − ) is an invertible 1-cocycle system. (cid:3) Proposition 2.8.
Let ( g, [ · , · ] g , ω ) be a quadratic Leibniz algebra and R, S : g ∗ → g be twolinear maps. Then ( R, S ) is a relative Rota-Baxter system on ( g, [ · , · ] g ) with respect to therepresentation ( g ∗ , L ∗ , − L ∗ − R ∗ ) if and only if ( R ◦ ω ♮ , S ◦ ω ♮ ) is a Rota-Baxter systemon ( g, [ · , · ] g ) . Proof.
For any x, y ∈ g , we have R ◦ ω ♮ ([ R ◦ ω ♮ ( x ) , y ] g + [ x, S ◦ ω ♮ ( y )] g )= R ( ω ♮ ( L R ◦ ω ♮ ( x ) y ) + ω ♮ ( R S ◦ ω ♮ ( y ) x ))= R ( L ∗ R ◦ ω ♮ ( x ) ω ♮ ( y ) − L ∗ S ◦ ω ♮ ( y ) ω ♮ ( x ) − R ∗ S ◦ ω ♮ ( y ) ω ♮ ( x )) . Similarly, we have S ◦ ω ♮ ([ R ◦ ω ♮ ( x ) , y ] g + [ x, S ◦ ω ♮ ( y )] g )= S ( ω ♮ ( L R ◦ ω ♮ ( x ) y ) + ω ♮ ( R S ◦ ω ♮ ( y ) x ))= S ( L ∗ R ◦ ω ♮ ( x ) ω ♮ ( y ) − L ∗ S ◦ ω ♮ ( y ) ω ♮ ( x ) − R ∗ S ◦ ω ♮ ( y ) ω ♮ ( x )) . R ◦ ω ♮ , S ◦ ω ♮ ) is a Rota-Baxter system on ( g, [ · , · ] g ) if and only if[ R ◦ ω ♮ ( x ) , R ◦ ω ♮ ( y )] g = R ( L ∗ R ◦ ω ♮ ( x ) ω ♮ ( y ) − L ∗ S ◦ ω ♮ ( y ) ω ♮ ( x ) − R ∗ S ◦ ω ♮ ( y ) ω ♮ ( x )) , [ S ◦ ω ♮ ( x ) , S ◦ ω ♮ ( y )] g = S ( L ∗ R ◦ ω ♮ ( x ) ω ♮ ( y ) − L ∗ S ◦ ω ♮ ( y ) ω ♮ ( x ) − R ∗ S ◦ ω ♮ ( y ) ω ♮ ( x )) . Since ω ♮ is an isomorphism, these identities hold if and only if ( R, S ) is a relative Rota-Baxter system on ( g, [ · , · ] g ) with respect to the representation ( g ∗ , L ∗ , − L ∗ − R ∗ ). (cid:3) In this section, we mainly provide examples of Rota-Baxter systems on Leibniz alge-bras. As mentioned earlier, they are relative Rota-Baxter operators with respect to theregular representation.
Example 3.1.
Consider the three-dimensional Leibniz algebra ( g, [ · , · ] g ) given with respectto a basis { e , e , e } by [ e , e ] g = e . Then R = a a a a a a a a a , S= b b b b b b b b b is a Rota-Baxter system on ( g, [ · , · ] g ) if and only if [ Re i , Re j ] g = R ([ Re i , e j ] g + [ e i , Se j ] g ) , [ Se i , Se j ] g = S ([ Re i , e j ] g + [ e i , Se j ] g ) , for i, j = 1 , , . We have [ Re , Re ] g = [ a e + a e + a e , a e + a e + a e ] = a e , and R ([ Re , e ] g + [ e , Se ] g )= R ([ a e + a e + a e , e ] g + [ e , b e + b e + b e ] g )= ( a + b ) Re = ( a + b ) a e + ( a + b ) a e + ( a + b ) a e . Thus, by [ Re , Re ] g = R ([ Re , e ] g + [ e , Se ] g ) , we have ( a + b ) a = 0 , ( a + b ) a = 0 , a = ( a + b ) a . Similarly, by [ Se , Se ] g = S ([ Re , e ] g + [ e , Se ] g ) , we have ( a + b ) b = 0 , ( a + b ) b = 0 , b = ( a + b ) b . y considering other choices of e i and e j , we obtain a a = b a , b a = 0 , b a = 0 ,b b = b b , b b = 0 , b b = 0 ,a a = b a , b a = 0 , b a = 0 ,b b = b b , b b = 0 , b b = 0 ,a a = a a , a a = 0 , a a = 0 ,b b = a b , a b = 0 , a b = 0 ,a a = a a , a a = 0 , a a = 0 ,b b = a b , a b = 0 , a b = 0 ,a = 0 , a = 0 , a a = 0 , b = 0 , b = 0 , b b = 0 . Summarize the above discussion, we have(1) If a = b = a = b = a = b = 0 , then any R = a a a a a a ,S= b b b b b b is a Rota-Baxter system on ( g, [ · , · ] g ) with respect to the regularrepresentation.(2) If a = b = a = b = 0 and a = b = 0 , a = b = 0 , then any any R = a a a a a a a , S= b b b b b b b is a Rota-Baxter system on ( g, [ · , · ] g ) withrespect to the regular representation. We have seen that relative Rota-Baxter systems generalize relative Rota-Baxter op-erators. In the following, we show that they also generalize Rota-Baxter operators ofarbitrary weight.
Definition 3.2.
Let ( g, [ · , · ] g ) be a Leibniz algebra. A linear map R : g → g is said to bea Rota-Baxter operator of weight λ if R satisfies [ Rx, Ry ] g = R ([ Rx, y ] g + [ x, Ry ] g + λ [ x, y ] g ) , for x, y ∈ g. Proposition 3.3.
Let ( g, [ · , · ] g ) be a Leibniz algebra and R : g → g be a Rota-Baxteroperator of weight λ . Then ( R, R + λId ) and ( R + λId, R ) are Rota-Baxter systems on ( g, [ · , · ] g ) . roof. For any x, y ∈ g , we have[ Rx, Ry ] g = R ([ Rx, y ] g + [ x, Ry ] g + λ [ x, y ] g )= R ([ Rx, y ] g + [ x, ( R + λId ) y ] g )= R ([( R + λId ) x, y ] g + [ x, Ry ] g ) , and [( R + λId ) x, ( R + λId ) y ] g = [ Rx, Ry ] g + λ [ Rx, y ] g + λ [ x, Ry ] g + [ λx, λy ] g = R ([ Rx, y ] g + [ x, Ry ] g + λ [ x, y ] g ) + λ [ Rx, y ] g + λ [ x, Ry ] g + [ λx, λy ] g = ( R + λId )([ Rx, y ] g + [ x, ( R + λId ) y ] g )= ( R + λId )([( R + λId ) x, y ] g + [ x, Ry ] g ) . This shows that (
R, R + λId ) and ( R + λId, R ) are Rota-Baxter systems on ( g, [ · , · ] g ). (cid:3) Let ( g, [ · , · ] g ) be a Leibniz algebra. A linear map T : g → g is said to be a left g -linear map (resp. right g -linear map) if T [ x, y ] = [ x, T y ] (resp. T [ x, y ] = [ T x, y ]), for any x, y ∈ g . Lemma 3.4.
Let ( g, [ · , · ] g ) be a Leibniz algebra. Suppose that R : g → g is a left g -linearmap and S : g → g is a right g -linear map. Then ( R, S ) is a Rota-Baxter system on ( g, [ · , · ] g ) if and only if [ x, R ◦ S ( y )] g = 0 = [ S ◦ R ( x ) , y ] g , for x, y ∈ g. Proof.
For any x, y ∈ g , we observe that R ([ Rx, y ] g + [ x, Sy ] g ) = [ Rx, Ry ] g + [ x, R ◦ S ( y )] g , and S ([ Rx, y ] g + [ x, Sy ] g ) = [ R ◦ S ( x ) , Ry ] g + [ Sx, Sy ] g . It follows from the above two identities that (
R, S ) is a Rota-Baxter system on ( g, [ · , · ] g )if and only if [ x, R ◦ S ( y )] g = 0 = [ S ◦ R ( x ) , y ] g , for x, y ∈ g. (cid:3) A Leibniz algebra ( g, [ · , · ] g ) is said to be nondegenerate if the bracket [ · , · ] g satisfy thefollowings [ x, y ] g = 0 , for all y implies that x = 0 , [ x, y ] g = 0 , for all x implies that y = 0 . orollary 3.5. Let ( g, [ · , · ] g ) be a nondegenerate Leibniz algebra. Let R : g → g be a left g -linear map and S : g → g be a right g -linear map. Then ( R, S ) is a Rota-Baxter systemon ( g, [ · , · ] g ) if and only if R ◦ S = S ◦ R = 0 . Another class of Rota-Baxter systems arise from twisted Rota-Baxter operators. Let( g, [ · , · ] g ) be a Leibniz algebra and σ : g → g be a Leibniz algebra morphism. Definition 3.6.
A linear map R : g → g is said to be a σ -twisted Rota-Baxter operator if R satisfies [ Rx, Ry ] g = R ([ Rx, y ] g + [ x, ( σ ◦ R ) y ] g ) , for all x, y ∈ g. (3. 1)When σ = Id , a σ -twisted Rota-Baxter operator is nothing but a Rota-Baxter operator. Example 3.7.
A differential Rota-Baxter Leibniz algebra of weight λ is a Leibniz algebra ( g, [ · , · ] g ) together with linear maps R, ∂ : g → g satisfying the following set of identities ( dR
1) [
Rx, Ry ] g = R ([ Rx, y ] g + [ x, Ry ] g + λ [ x, y ] g ) , ( dR ∂ [ x, y ] g = [ ∂x, y ] g + [ x, ∂y ] g + λ [ ∂x, ∂y ] g , ( dR ∂ ◦ R = Id.
Let ( g, R, ∂ ) be a differential Rota-Baxter Leibniz algebra of weight λ . It follows from ( dR that the map σ : g → g, σ ( x ) = x + λ∂ ( x ) , for x ∈ g is a Leibniz algebra morphism. On the other hand, ( dR implies that ( σ ◦ R )( x ) = R ( x ) + λx, for x ∈ g. Hence, by ( dR , we get [ Rx, Ry ] g = R ([ Rx, y ] g + [ x, ( σ ◦ R ) y ] g ) , for x, y ∈ g. This shows that R is a σ -twisted Rota-Baxter operator. Proposition 3.8.
Let R be a σ -twisted Rota-Baxter operator on a Leibniz algebra ( g, [ · , · ] g ) .Then ( R, σ ◦ R ) is a Rota-Baxter system on ( g, [ · , · ] g ) . Proof.
Note that the condition Eq. (3.1) is same as the first condition of a Rota-Baxter system. To prove the second one, we observe that[( σ ◦ R ) x, ( σ ◦ R ) y ] g = σ [ Rx, Ry ] g = ( σ ◦ R )([ Rx, y ] g + [ x, ( σ ◦ R ) y ] g ) . This shows that (
R, σ ◦ R ) is a Rota-Baxter system on ( g, [ · , · ] g ). (cid:3) xample 3.9. Let ( W, [ · , · ] W ) be the Witt Lie algebra generated by basis elements { l n } n ∈ Z and the Lie bracket given by [ l m , l n ] W = ( m − n ) l m + n , for m, n ∈ Z . View this Lie algebra as a Leibniz algebra. Let q ∈ K be a nonzero scalar that is not a rootof unity. We define linear maps σ, R : W → W by σ ( l n ) = q n l n , R ( l n ) = 1 − q − q n l n , for n ∈ Z . Then σ is clearly a Leibniz algebra morphism. Moreover, it is easy to verify that R satisfies [ R ( l m ) , R ( l n )] W = R ([ R ( l m ) , l n ] W + [ l m , ( σ ◦ R )( l n )] W ) , for m, n ∈ Z . Therefore, R is a σ -twisted Rota-Baxter operator. Hence, ( R, σ ◦ R ) is a Rota-Baxtersystem on W . In [23] the authors introduced a notion of weak pseudotwistor on an associative algebraand showed that a weak pseudotwistor induce a new associative algebra structure. A Rota-Baxter system on an associative algebra gives rise to a weak pseudotwistor, hence a newassociative algebra structure. This is not true for Rota-Baxter systems on Leibniz algebras.However, if we concentrate on Rota-Baxter operators, they induce a new Leibniz algebrastructure via a Leibniz analogue of weak pseudotwistor. Let us first recall the new Leibnizalgebra associated to a Rota-Baxter operator on a Leibniz algebra.Let ( g, [ · , · ] g ) be a Leibniz algebra, and R : g → g be a Rota-Baxter operator, i.e., R satisfies [ Rx, Ry ] g = R ([ Rx, y ] + [ x, Ry ]) , for x, y ∈ g. Then the vector space g carries a new Leibniz algebra structure with bracket[ x, y ] R = [ Rx, y ] + [ x, Ry ] , for x, y ∈ g. Here we give a new example of Rota-Baxter operator on a Leibniz algebra inducedfrom dialgebra [19].
Definition 3.10.
A dialgebra is a vector space D together with two bilinear operations ⊣ , ⊢ : D ⊗ D → D satisfying the following identities a ⊣ ( b ⊣ c ) = ( a ⊣ b ) ⊣ c = a ⊣ ( b ⊢ c ) , ( a ⊢ b ) ⊣ c = a ⊢ ( b ⊣ c ) , ( a ⊣ b ) ⊢ c = ( a ⊢ b ) ⊢ c = a ⊢ ( b ⊢ c ) , for a, b, c ∈ D.
13 dialgebra as above may be denoted by the triple ( D, ⊣ , ⊢ ). Any associative algebrais obviously a dialgebra with both the bilinear maps coincide with the associative product.See Loday [19] for more examples of dialgebras. Proposition 3.11.
Let ( D, ⊣ , ⊢ ) be a dialgebra. Then ( D, [ · , · ] D ) is a Leibniz algebra,where [ a, b ] D := a ⊢ b − b ⊣ a, for a, b ∈ D. Proof.
For any a, b, c ∈ D , we have[[ a, b ] D , c ] D + [ b, [ a, c ] D ] D = [ a ⊢ b − b ⊣ a, c ] D + [ b, a ⊢ c − c ⊣ a ] D = ( a ⊢ b − b ⊣ a ) ⊢ c − c ⊣ ( a ⊢ b − b ⊣ a ) + b ⊢ ( a ⊢ c − c ⊣ a ) − ( a ⊢ c − c ⊣ a ) ⊣ b = a ⊢ ( b ⊢ c − c ⊣ b ) − ( b ⊢ c − c ⊣ b ) ⊣ a = [ a, [ b, c ] D ] D . Hence ( D, [ · , · ] D ) is a Leibniz algebra. (cid:3) The Leibniz algebra in the above proposition is called the Leibniz algebra induced fromthe dialgebra ( D, ⊣ , ⊢ ). Definition 3.12.
Let ( D, ⊣ , ⊢ ) be a dialgebra. A Rota-Baxter operator on D consists ofa linear map R : D → D satisfying R ( a ) ∗ R ( b ) = R ( R ( a ) ∗ b + a ∗ R ( b )) , for all a, b ∈ D and ∗ = ⊣ , ⊢ . Proposition 3.13.
Let ( D, ⊣ , ⊢ ) be a dialgebra and R be a Rota-Baxter operator on it.Then R is a Rota-Baxter operator on the induced Leibniz algebra ( D, [ · , · ] D ) . Proof.
For any a, b ∈ D , we have[ Ra, Rb ] D = Ra ⊢ Rb − Rb ⊣ Ra = R ( R ( a ) ⊢ b + a ⊢ R ( b )) − R ( R ( b ) ⊣ a + b ⊣ R ( a ))= R ([ Ra, b ] D + [ a, Rb ] D ) . Hence the result follows. (cid:3)
The Leibniz bracket [ · , · ] R induced from a Rota-Baxter operator R can be understoodin terms of the weak pseudotwistor on a Leibniz algebra.14 efinition 3.14. Let ( g, [ · , · ] g ) be a Leibniz algebra with the Leibniz bracket denoted bythe product µ . A linear map T : g ⊗ g → g ⊗ g is said to be a weak pseudotwistor if thereexists a linear map τ : g ⊗ g ⊗ g → g ⊗ g ⊗ g with ( η ⊗ Id ) ◦ τ = τ ◦ ( η ⊗ Id ) andcommuting the following diagram g ⊗ g ⊗ g Id ⊗ µ / / g ⊗ g T (cid:15) (cid:15) g ⊗ g ⊗ g µ ⊗ Id o o g ⊗ g ⊗ g Id ⊗ T ♣♣♣♣♣♣♣♣♣♣♣ τ ' ' ◆◆◆◆◆◆◆◆◆◆◆ g ⊗ g ⊗ g τ w w ♣♣♣♣♣♣♣♣♣♣♣ T ⊗ Id g g ◆◆◆◆◆◆◆◆◆◆◆ g ⊗ g ⊗ g Id ⊗ µ / / g ⊗ g g ⊗ g ⊗ g µ ⊗ Id o o Here η : g ⊗ g → g ⊗ g is the flip map η ( x ⊗ y ) = y ⊗ x . The map τ is called a weakcompanion of T . Proposition 3.15.
Let ( g, [ · , · ] g ) be a Leibniz algebra and T : g ⊗ g → g ⊗ g be a weakpseudotwistor. Then ( g, µ ◦ T ) is a new Leibniz algebra structure on g . Proof.
We have( µ ◦ T ) ◦ ( Id ⊗ ( µ ◦ T ))= µ ◦ ( Id ⊗ µ ) ◦ τ = µ ◦ ( µ ⊗ Id ) ◦ τ + µ ◦ ( Id ⊗ µ ) ◦ ( η ⊗ Id ) ◦ τ = ( µ ◦ T ) ◦ (( µ ◦ T ) ⊗ Id ) + µ ◦ ( Id ⊗ µ ) ◦ τ ◦ ( η ⊗ Id )= ( µ ◦ T ) ◦ (( µ ◦ T ) ⊗ Id ) + ( µ ◦ T ) ◦ ( Id ⊗ ( µ ◦ T )) ◦ ( η ⊗ Id ) . This shows that µ ◦ T defines a Leibniz bracket on g . (cid:3) Proposition 3.16.
Let ( g, [ · , · ] g ) be a Leibniz algebra and R : g → g be a Rota-Baxteroperator on it. Then the map T : g ⊗ g → g ⊗ g defined by T ( x ⊗ y ) = R ( x ) ⊗ y + x ⊗ R ( y ) is a weak pseudotwistor on g . Consequently, g carries a new Leibniz algebra structure withbracket [ x, y ] R = [ Rx, y ] g + [ x, Ry ] g , for x, y ∈ g . Proof.
We define τ : g ⊗ g ⊗ g → g ⊗ g ⊗ g by τ ( x ⊗ y ⊗ z ) = R ( x ) ⊗ R ( y ) ⊗ z + R ( x ) ⊗ y ⊗ R ( z ) + x ⊗ R ( y ) ⊗ R ( z ) , for x, y, z ∈ g. We will show that T is a weak pseudotwistor with a weak companion τ . First observe that(( η ⊗ Id ) ◦ τ )( x ⊗ y ⊗ z )= R ( y ) ⊗ R ( x ) ⊗ z + y ⊗ R ( x ) ⊗ R ( z ) + R ( y ) ⊗ x ⊗ R ( z )= τ ( y ⊗ x ⊗ z ) = ( τ ◦ ( η ⊗ Id ))( x ⊗ y ⊗ z ) . T ◦ ( Id ⊗ µ ◦ T ))( x ⊗ y ⊗ z )= R ( x ) ⊗ µ ( R ( y ) ⊗ z + y ⊗ R ( z )) + x ⊗ µ ( R ( y ) ⊗ R ( z ))= (( Id ⊗ µ ) ◦ τ )( x ⊗ y ⊗ z ) . Similarly, we have T ◦ (( µ ◦ T ) ⊗ Id ) = ( µ ⊗ Id ) ◦ τ. Hence, the result follows. (cid:3)
Remark 3.17.
The notion of weak pseudotwistor on a Leibniz algebra is a generalizationof weak pseudotwistor on an associative algebra introduced by Panaite and Oystaeyen [23].In the associative context, a Rota-Baxter system induces a weak pseudotwistor on theunderlying associative algebra. It is remarked that given a Leibniz algebra ( g, [ · , · ] g ) and aRota-Baxter system ( R, S ) on g , the map T : g ⊗ g → g ⊗ g, T ( x ⊗ y ) = R ( x ) ⊗ y + x ⊗ S ( y ) is not a weak pseudotwistor on g with weak companion τ ( x ⊗ y ⊗ z ) = R ( x ) ⊗ R ( y ) ⊗ z + R ( x ) ⊗ y ⊗ S ( z ) + x ⊗ S ( y ) ⊗ S ( z ) as ( η ⊗ Id ) ◦ τ = τ ◦ ( η ⊗ Id ) . In the section, we construct a graded Lie algebra that characterize relative Rota-Baxtersystems as Maurer-Cartan elements. Using this characterization, we define the cohomologyassociated to a relative Rota-Baxter system. We first recall some results from [1].A permutation σ ∈ S n is called an ( i, n − i )-shuffle if σ (1) < · · · < σ ( i ) and σ ( i + 1) < · · · < σ ( n ). If i = 0 or n we assume σ = Id . The set of all ( i, n − i )-shuffles will be denotedby S ( i,n − i ) .Let M be a vector space. We consider the graded vector space C ∗ ( M, M ) = ⊕ n ≥ C n ( M, M ) = ⊕ n ≥ Hom ( ⊗ n M, M )of multilinear maps on M . The Balavoine bracket is a degree − C ∗ ( M, M ) given by[ f, g ] B := f ◦ g − ( − pq g ◦ f, f ∈ C p +1 ( M, M ) , g ∈ C q +1 ( M, M ) . Here f ◦ g ∈ C p + q +1 ( M, M ) is defined by f ◦ g = p +1 X k =1 ( − ( k − q f ◦ k g, with ( f ◦ k g )( x , · · · , x p + q +1 )= X σ ∈ S ( k − ,q ) ( − σ f ( x σ (1) , · · · , x σ ( k − , g ( x σ ( k ) , · · · , x σ ( k + q − , x k + q ) , x k + q +1 , · · · , x p + q +1 ) . Theorem 4.1. ( [1]) With the above notations, ( C ∗ ( M, M ) , [ · , · ] B ) is a degree − gradedLie algebra. In other words ( C ∗ +1 ( M, M ) , [ · , · ] B ) is a graded Lie algebra. Its Maurer-Cartan elements are precisely the Leibniz algebra structures on M . Let (
V, ρ L , ρ R ) be a representation of a Leibniz algebra ( g, [ · , · ] g ). Consider the semidi-rect product Leibniz algebra structure on g ⊕ g ⊕ V . We denote the correspondingLeibniz product by b µ . Then b µ is a Maurer-Cartan element in the graded Lie algebra( C ∗ +1 ( g ⊕ g ⊕ V, g ⊕ g ⊕ V ) , [ · , · ] B ).Consider the graded vector subspace C ∗ ( V, g ) ⊂ C ∗ ( g ⊕ g ⊕ V, g ⊕ g ⊕ V ) given by C ∗ ( V, g ) := ⊕ n ≥ C n ( V, g ) := ⊕ n ≥ Hom ( V ⊗ n , g ⊕ g ) . Theorem 4.2.
With the above notations, ( C ∗ ( V, g ) , [[ · , · ]]) is a graded Lie algebra, wherethe graded Lie bracket [[ · , · ]] : C m ( V, g ) × C n ( V, g ) → C m + n ( V, g ) is defined by [[( P, Q ) , ( P ′ , Q ′ )]] := ( − m [[ b µ, ( P, Q )] B , ( P ′ , Q ′ )] B , for any ( P, Q ) ∈ C m ( V, g ) , ( P ′ , Q ′ ) ∈ C n ( V, g ) . Moreover, its Maurer-Cartan elements arerelative Rota-Baxter systems on the Leibniz algebra ( g, [ · , · ] g ) with respect to the represen-tation ( V, ρ L , ρ R ) . Let
P r , P r : g ⊕ g → g denote the projection maps onto the first and second factor,17espectively. Then the explicit description of the above graded Lie bracket is given by P r ([[( P, Q ) , ( P ′ , Q ′ )]]( v , · · · , v m + n ))= m X k =1 X σ ∈ S ( k − ,n ) ( − ( k − n ( − σ P ( v σ (1) , · · · , v σ ( k − , ρ L ( P ′ ( v σ ( k ) , · · · , v σ ( k + n − )) v k + n , · · · , v m + n )+ m X k =2 X σ ∈ S ( k − ,n, ( − kn ( − σ P ( v σ (1) , · · · , v σ ( k − , ρ R ( Q ′ ( v σ ( k ) , · · · , v σ ( k + n − )) v σ ( k + n − , v k + n , · · · , v m + n )+ n X k =1 X σ ∈ S ( k − ,m ) ( − ( k + n − m ( − σ P ′ ( v σ (1) , · · · , v σ ( k − , ρ L ( P ( v σ ( k ) , · · · , v σ ( k + m − )) v σ ( k + m ) , · · · , v m + n ) n X k =1 X σ ∈ S ( k − ,m, ,σ ( k + m − k + m ( − ( k + n − m +1 ( − σ P ′ ( v σ (1) , · · · , v σ ( k − , ρ R ( Q ( v σ ( k ) , · · · , v σ ( k − m ) )) v σ ( k + m ) , v k + m +1 , · · · , v m + n )+ X σ ∈ S ( m,n − ( − mn +1 ( − σ [ P ( v σ (1) , · · · , v σ ( m ) ) , P ′ ( v σ ( m +1) , · · · , v σ ( m + n − , v m + n ] g + m X k =1 X σ ∈ S ( k − ,n − ( − ( k − n ( − σ [ P ′ ( v σ ( k ) , · · · , v σ ( k + n − ) , P ( v σ (1) , · · · , v σ ( k − , v k + n , · · · , v m + n )] g , and P r ([[( P, Q ) , ( P ′ , Q ′ )]]( v , · · · , v m + n ))= m X k =1 X σ ∈ S ( k − ,n ) ( − ( k − n ( − σ Q ( v σ (1) , · · · , v σ ( k − , ρ L ( P ′ ( v σ ( k ) , · · · , v σ ( k + n − )) v k + n , · · · , v m + n )+ m X k =2 X σ ∈ S ( k − ,n, ( − kn ( − σ Q ( v σ (1) , · · · , v σ ( k − , ρ R ( Q ′ ( v σ ( k ) , · · · , v σ ( k + n − )) v σ ( k + n − , v k + n , · · · , v m + n )+ n X k =1 X σ ∈ S ( k − ,m ) ( − ( k + n − m ( − σ Q ′ ( v σ (1) , · · · , v σ ( k − , ρ L ( P ( v σ ( k ) , · · · , v σ ( k + m − )) v σ ( k + m ) , · · · , v m + n ) n X k =1 X σ ∈ S ( k − ,m, ,σ ( k + m − k + m ( − ( k + n − m +1 ( − σ Q ′ ( v σ (1) , · · · , v σ ( k − , ρ R ( Q ( v σ ( k ) , · · · , v σ ( k − m ) )) v σ ( k + m ) , v k + m +1 , · · · , v m + n )+ X σ ∈ S ( m,n − ( − mn +1 ( − σ [ Q ( v σ (1) , · · · , v σ ( m ) ) , Q ′ ( v σ ( m +1) , · · · , v σ ( m + n − , v m + n ] g + m X k =1 X σ ∈ S ( k − ,n − ( − ( k − n ( − σ [ Q ′ ( v σ ( k ) , · · · , v σ ( k + n − ) , Q ( v σ (1) , · · · , v σ ( k − , v k + n , · · · , v m + n )] g , for any ( P, Q ) ∈ C m ( V, g ) , ( P ′ , Q ′ ) ∈ C n ( V, g ).18 roof.
The graded Lie algebra ( C ∗ ( V, g ) , [[ · , · ]]) is obtained via the derived bracket [26].First consider the graded Lie algebra ( C ∗ +1 ( g ⊕ g ⊕ V, g ⊕ g ⊕ V ) , [ · , · ] B ). Since b µ is thesemidirect product Leibniz algebra structure on the vector space g ⊕ g ⊕ V , we deduce that( C ∗ +1 ( g ⊕ g ⊕ V, g ⊕ g ⊕ V ) , [ · , · ] B , d = [ b µ, · ] B ) is a differential graded Lie algebra. Obviously C ∗ +1 ( V, g ) is an abelian subalgebra. Therefore, by the derived bracket construction, wedefine a bracket on the shifted graded vector space C ∗ ( V, g ) by[[(
P, Q ) , ( P ′ , Q ′ ]] := ( − m [ d (( P, Q )) , ( P ′ , Q ′ )] B = ( − m [[ b µ, ( P, Q )] , ( P ′ , Q ′ )] , for any ( P, Q ) ∈ C m ( V, g ) , ( P ′ , Q ′ ) ∈ C n ( V, g ). The derived bracket [[ · , · ]] is closed on C ∗ ( V, g ), which implies that ( C ∗ ( V, g ) , [[ · , · ]]) is a graded Lie algebra.For ( R, S ) ∈ C ( V, g ), we have
P r ([[( R, S ) , ( R, S )]]( u, v )) = 2([
Ru, Rv ] g − R ( ρ L ( Ru ) v ) − R ( ρ R ( Sv ) u )) ,P r ([[( R, S ) , ( R, S )]]( u, v )) = 2([
Su, Sv ] g − S ( ρ L ( Ru ) v ) − S ( ρ R ( Sv ) u )) . Thus, (
R, S ) is a Maurer-Cartan element (i.e. [[(
R, S ) , ( R, S )]] = 0) if and only if (
R, S )is a relative Rota-Baxter systems on g with respect to the representation ( V, ρ L , ρ R ). Theproof is finished. (cid:3) Thus, relative Rota-Baxter systems can be characterized as Maurer-Cartan elementsin a graded Lie algebra. It follows from the above theorem that if (
R, S ) is a relativeRota-Baxter system, then d ( R,S ) := [[( R, S ) , · ]] is a differential on C ∗ ( V, g ) and makes thegLa ( C • ( V, g ) , [[ · , · ]]) into a differential graded Lie algebra.The cohomology of the cochain complex ( C • ( V, g ) , d ( R,S ) ) is called the cohomology ofthe relative Rota-Baxter system ( R, S ). We denote the corresponding cohomology groupssimply by H • ( V, g ).The following theorem describes the Maurer-Cartan deformation of a relative Rota-Baxter system.
Theorem 4.3.
Let ( R, S ) be a relative Rota-Baxter system on a Leibniz algebra ( g, [ · , · ] g ) with respect to a representation ( V, ρ L , ρ R ) . For any pair ( R ′ , S ′ ) of linear maps from V to g , the pair of sums ( R + R ′ , S + S ′ ) is a relative Rota-Baxter system if and only if ( R ′ , S ′ ) is a Maurer-Cartan element in the differential graded Lie algebra ( C ∗ ( V, g ) , [[ · , · ]] , d ( R,S ) ) ,i.e. [[( R + R ′ , S + S ′ ) , ( R + R ′ , S + S ′ )]] = 0 ⇔ d ( R,S ) ( R ′ , S ′ ) + 12 [[( R ′ , S ′ ) , ( R ′ , S ′ )]] = 0 . Deformations of relative Rota-Baxter systems
Let K [[ t ]] be the ring of power series in one variable t . For any K -linear space V , let V [[ t ]] denotes the vector space of formal power series in t with coefficients from V . Ifin addition, ( g, [ · , · ] g ) is a Leibniz algebra over K , then there is a K [[ t ]]-Leibniz algebrastructure on g [[ t ]] given by[ + ∞ X i =0 x i t i , + ∞ X j =0 y j t j ] g = + ∞ X k =0 X i + j = k [ x i , y j ] t k , for all x i , y j ∈ g. Let (
V, ρ L , ρ R ) be a representation of the Leibniz algebra ( g, [ · , · ] g ). Then there is a rep-resentation ( V [[ t ]] , ρ L , ρ R ) of the K [[ t ]]-Leibniz algebra g [[ t ]]. Here ρ L and ρ R are givenby ρ L ( + ∞ X i =0 x i t i )( + ∞ X j =0 v j t j ) = + ∞ X k =0 X i + j = k ρ L ( x i )( v j ) t k ,ρ R ( + ∞ X i =0 x i t i )( + ∞ X j =0 v j t j ) = + ∞ X k =0 X i + j = k ρ R ( x i )( v j ) t k , for all x i ∈ g, v j ∈ V. Let (
R, S ) be a relative Rota-Baxter system on the Leibniz algebra ( g, [ · , · ] g ) with respectto the representation ( V, ρ L , ρ R ). We consider two power series R t = + ∞ X i =0 R i t i and S t = + ∞ X j =0 S j t j , where R i , S j ∈ Hom K ( V, g ) . That is, both R t and S t are in Hom K ( V, g )[[ t ]]. Extend them to K [[ t ]]-linear maps from V [[ t ]] to g [[ t ]]. We still denote them by same symbols. Definition 5.1. If R t = P + ∞ i =0 R i t i and S t = P + ∞ j =0 S j t j with R = R , S = S satisfy [ R t u, R t v ] g = R t ( ρ L ( R t u ) v + ρ R ( S t v ) u ) , [ S t u, S t v ] g = S t ( ρ L ( R t u ) v + ρ R ( S t v ) u )) , we say that ( R t , S t ) is a formal deformation of the relative Rota-Baxter system ( R, S ) . By expanding these equations and comparing coefficients of various powers of t , weobtain for k ≥ + ∞ X k =0 X i + j = k [ R i u, R j v ] g = + ∞ X k =0 X i + j = k R i ( ρ L ( R j u ) v + ρ R ( S j v ) u ) , + ∞ X k =0 X i + j = k [ S i u, S j v ] g = + ∞ X k =0 X i + j = k S i ( ρ L ( R j u ) v + ρ R ( S j v ) u )) . k = 0 as ( R, S ) is a relative Rota-Baxter system. For k = 1, we get[ Ru, R v ] g + [ R u, Rv ] g = R ( ρ L ( Ru ) v + ρ R ( Sv ) u ) + R ( ρ L ( R u ) v + ρ R ( S v ) u ) , [ Su, S v ] g + [ S u, Sv ] g = S ( ρ L ( Ru ) v + ρ R ( Sv ) u ) + S ( ρ L ( R u ) v + ρ R ( S v ) u, for u, v ∈ V . These identities are equivalent to the single condition[[( R, S ) , ( R , S )]] = 0 . As a consequence, we get the following.
Proposition 5.2.
Let ( R t = P + ∞ i =0 R i t i , S t = P + ∞ j =0 S j t j ) be a formal deformation ofa relative Rota-Baxter system ( R, S ) on the Leibniz algebra ( g, [ · , · ] g ) with respect to arepresentation ( V, ρ L , ρ R ) . Then ( R , S ) is a -cocycle in the cohomology of the relativeRota-Baxter system ( R, S ) , that is, d ( R,S ) ( R , S ) = 0 . Definition 5.3.
Let ( R, S ) be a relative Rota-Baxter system on the Leibniz algebra ( g, [ · , · ] g ) with respect to a representation ( V, ρ L , ρ R ) . The 1-cocycle ( R , S ) is called the infinites-imal of the formal deformation ( R t = P + ∞ i =0 R i t i , S t = P + ∞ j =0 S j t j ) of the relative Rota-Baxter system ( R, S ) . Definition 5.4.
Two formal deformations ( R t , S t ) and ( R ′ t , S ′ t ) of a relative Rota-Baxtersystem ( R, S ) on the Leibniz algebra ( g, [ · , · ] g ) with respect to a representation ( V, ρ L , ρ R ) are said to be equivalent if there exist two elements x, y ∈ g and linear maps φ i , ϕ i ∈ gl ( g ) and ψ i ∈ gl ( V ) for i ≥ such that for φ t = Id g + t ( L x − R x ) + + ∞ X i =2 φ i t i , ϕ t = Id g + t ( L y − R y ) + + ∞ X i =2 ϕ i t i and ψ t = Id V + t ( ρ L ( x ) − ρ R ( y )) + + ∞ X i =2 ψ i t i , the following conditions hold: ( i ) [ φ t ( z ) , φ t ( w )] g = φ t ([ z, w ] g ) , [ ϕ t ( z ) , ϕ t ( w )] g = ϕ t ([ z, w ] g );( ii ) ψ t ( ρ L ( z ) u ) = ρ L ( φ t ( z )) ψ t ( u );( iii ) ψ t ( ρ R ( z ) u ) = ρ R ( ϕ t ( z )) ψ t ( u );( iv ) R ′ t ◦ ψ t ( u ) = φ t ◦ R t ( u ) , S ′ t ◦ ψ t ( u ) = ϕ t ◦ S t ( u ) , for all z, w ∈ g and u ∈ V .
21y expanding the identities in (iv) and equating coefficients of t from both sides, weobtain( R , S )( u ) − ( R ′ , S ′ )( u ) = [ R ( u ) , x ] g − R ( ρ R ( y ) u ) − [ x, R ( u )] g + R ( ρ L ( x ) u )+[ S ( u ) , y ] g − S ( ρ R ( y ) u ) − [ y, S ( u )] g + S ( ρ L ( x ) u )= ( d ( R,S ) ( x, y ))( u ) . Thus, we have the following.
Theorem 5.5.
The cohomology class of the infinitesimal of a deformation of a relativeRota-Baxter system depends only on the equivalence class of the deformation.
In this subsection, we introduce a cohomology class associated to any order n deforma-tion of a relative Rota-Baxter system, and show that an order n deformation is extensibleif and only if this cohomology class is trivial. Thus, we call this cohomology class theobstruction class of the order n deformation being extensible. Definition 5.6.
Let ( R, S ) be a relative Rota-Baxter system on a Leibniz algebra ( g, [ · , · ] g ) with respect to a representation ( V, ρ L , ρ R ) . If the finite sums R t = n X i =0 R i t i and S t = n X j =0 S j t j with R = R, S = S as K [[ t ]] / ( t n +1 ) -module maps from V [[ t ]] / ( t n +1 ) to the Leibniz algebra g [[ t ]] / ( t n +1 ) satisfy [ R t u, R t v ] g = R t ( ρ L ( R t u ) v + ρ R ( S t v ) u ) , [ S t u, S t v ] g = S t ( ρ L ( R t u ) v + ρ R ( S t v ) u )) , for u, v ∈ V, we say that ( R t , S t ) is an order n deformation of the relative Rota-Baxter system ( R, S ) . Definition 5.7.
Let ( R t , S t ) be an order n deformation of the relative Rota-Baxter system ( R, S ) on a Leibniz algebra ( g, [ · , · ] g ) with respect to a representation ( V, ρ L , ρ R ) . If thereexists a pair ( R n +1 , S n +1 ) of linear maps from V to g such that ( b R t = R t + t n +1 R n +1 , b S t = S t + t n +1 S n +1 ) is a deformation of order n + 1 , we say that ( R t , S t ) is extensible. Let ( R t , S t ) be an order n deformation of the relative Rota-Baxter system ( R, S ) on aLeibniz algebra ( g, [ · , · ] g ) with respect to a representation ( V, ρ L , ρ R ). Define an element Ob ( R t ,S t ) ∈ C ( V, g ) by Ob ( R t ,S t ) = − X i + j = n +1 ,i,j ≥ [[( R i , S i ) , ( R j , S j )]] . (5. 1)22 roposition 5.8. The -cochain Ob ( R t ,S t ) is a 2-cocycle, that is, d ( R,S ) ( Ob ( R t ,S t ) ) = 0 . Proof.
We have d ( R,S ) ( Ob ( R t ,S t ) )= − X i + j = n +1 ,i,j ≥ [[( R, S ) , [[( R i , S i ) , ( R j , S j )]]]]= − X i + j = n +1 ,i,j ≥ ([[[[( R, S ) , ( R i , S i )]] , ( R j , S j )]] − [[( R i , S i ) , [[( R, S ) , ( R j , S j )]]]])= 14 X i + i + j = n,i ,i ,j ≥ [[[[( R i , S i ) , ( R i , S i )]] , ( R j , S j )]] − X i + j + j = n,i,j ,j ≥ [[( R i , S i ) , [[( R j , S j ) , ( R j , S j )]]]]= 12 X i + j + k = n +1 ,i,j,k ≥ [[[[( R i , S i ) , ( R j , S j )]] , ( R k , S k )]]= 0 . The proof is finished. (cid:3)
Definition 5.9.
Let ( R t , S t ) be an order n deformation of the relative Rota-Baxter system ( R, S ) on a Leibniz algebra ( g, [ · , · ] g ) with respect to a representation ( V, ρ L , ρ R ) . Thecohomology class [ Ob ( R t ,S t ) ] ∈ H ( V, g ) is called the obstruction class for ( R t , S t ) beingextensible. As a consequence of Eq. (5.1) and Proposition 5.8, we obtain the following.
Theorem 5.10.
Let ( R t , S t ) be an order n deformation of the relative Rota-Baxter system ( R, S ) on a Leibniz algebra ( g, [ · , · ] g ) with respect to a representation ( V, ρ L , ρ R ) . Then ( R t , S t ) is extensible if and only if the obstruction class [ Ob ( R t ,S t ) ] is trivial. Corollary 5.11. If H ( V, g ) = 0 then every -cocycle in the cohomology of a relativeRota-Baxter system ( R, S ) is the infinitesimal of some formal deformation of ( R, S ) . ACKNOWLEDGEMENT
The work of A. Das is supported by the fellowship of Indian Institute of Technology(IIT) Kanpur. The work of S. Guo is supported by the NSF of China (No. 11761017) andGuizhou Provincial Science and Technology Foundation (No. [2020]1Y005).
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