Twisted relative Rota-Baxter operators on Leibniz algebras and NS-Leibniz algebras
aa r X i v : . [ m a t h . R A ] F e b Twisted relative Rota-Baxter operators on Leibnizalgebras and NS-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 twisted relative Rota-Baxter operators on a Leibnizalgebra as a generalization of twisted Poisson structures. We define the coho-mology of a twisted relative Rota-Baxter operator K as the Loday-Pirashvilicohomology of a certain Leibniz algebra induced by K with coefficients ina suitable representation. Then we consider formal deformations of twistedrelative Rota-Baxter operators from cohomological points of view. Finally, weintroduce and study NS-Leibniz algebras as the underlying structure of twistedrelative Rota-Baxter operators. Key words : Twisted Rota-Baxter operator, Cohomology, Formal deforma-tion, NS-Leibniz algebra. : 17A32, 17B38, 17B62.
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 beenfound many connections with dendriform algebras, pre-Lie algebras, and have applicationsincluding in Connes-Kreimer’s algebraic approach to the renormalization in perturbativequantum field theory [5]. Relative Rota-Baxter operators on Leibniz algebras were studiedin [17] which is the main ingredient in the study of the twisting theory and the bialgebra ∗ Corresponding author (Shuangjian Guo), Email: [email protected] ( g , [ · , · ]) be a Leibniz algebra and ( V, ρ L , ρ R ) be a representation of it. A linear map K : V → g is a relative Rota-Baxter operator on g with respect to the representation V if it satisfies [ Ku, Kv ] = K ( ρ L ( Ku ) v + ρ R ( Kv ) u ) , for u, v ∈ V. (0. 1)Such operators can be seen as the Leibniz algebraic analogue of Poisson structures. Gen-erally, Rota-Baxter operators can be defined on algebraic operads, which give rise to thesplitting of operads [3, 15]. For further details on Rota-Baxter operators, see [9].Deformation theory of algebraic structures began with the seminal work of Gersten-haber [8] for associative algebras and followed by its extension to Lie algebras by Nijenhuisand Richardson [13, 14]. In general, deformation theory was developed for algebras overbinary quadratic operads by Balavoine [1]. Recently, deformations of relative Rota-Baxteroperators (also called O -operators) are developed in [18, 6, 19].In [16] ˘Severa and Weinstein introduced a notion of twisted Poisson structure as a Diracstructure in a certain twisted Courant algebroid. Twisted Poisson structures also studiedby Klim˘cík and Strobl from geometric points of view [10]. The corresponding algebraicnotion, called twisted Rota-Baxter operators was introduced by Uchino [20] in the contextof associative algebras and find relations with NS-algebras of Leroux [11]. Recently, one ofthe present authors introduces twisted Rota-Baxter operators on Lie algebras and considersNS-Lie algebras that are related to twisted Rota-Baxter operators in the same way pre-Liealgebras are related to Rota-Baxter operators [7].Our aim in this paper is to consider twisted (relative) Rota-Baxter operators on Leib-niz algebras. We show that a twisted relative Rota-Baxter operator K induces a newLeibniz algebra structure and there is a suitable representation of it. The correspondingLoday-Pirashvili cohomology is called the cohomology of the twisted relative Rota-Baxteroperator. As an application of the cohomology, we study deformations of a twisted relativeRota-Baxter operator K . We show that the infinitesimal in a formal deformation of K isa -cocycle in the cohomology of K . Moreover, we define a notion of equivalence betweentwo formal deformations of K . The infinitesimals corresponding to equivalent deformationsare shown to be cohomologous. We introduce Nijenhuis elements associated with a twistedrelative Rota-Baxter operator that are obtained from trivial linear deformations. We alsofind a sufficient condition for the rigidity of a twisted relative Rota-Baxter operator interms of Nijenhuis elements.In the last, we introduce a new algebraic structure, called NS-Leibniz algebras. We showthat Ns-Leibniz algebras split Leibniz algebras and the underlying structure of a twistedrelative Rota-Baxter operator. NS-Leibniz algebras also arise from Nijenhuis operators2n Leibniz algebras. Further study on NS-Leibniz algebras is postponed to a forthcomingarticle.The paper is organized as follows. In Section 2, we introduce twisted relative Rota-Baxter operators in the context of Leibniz algebras and give a characterization and somenew constructions. In Section 3, we define cohomology of a twisted relative Rota-Baxteroperator. This cohomology has been used in Section 4 to study deformations of a twistedrelative Rota-Baxter operator. Finally, in Section 5, we introduce NS-Leibniz algebras andfind its relation with twisted relative Rota-Baxter operators.Throughout this paper, all vector spaces, (multi)linear maps are over the field C ofcomplex numbers and all the vector spaces are finite-dimensional. In this section, we recall some basic definitions about Leibniz algebras and their coho-mology [12].
Definition 1.1.
A Leibniz algebra is a vector space g together with a bilinear operation(called bracket) [ · , · ] : g ⊗ g → g satisfying [ x, [ y, z ]] = [[ x, y ] , z ] + [ y, [ x, z ]] , for x, y, z ∈ g . A Leibniz algebra as above may be denoted by the pair ( g , [ · , · ]) or simply by g when noconfusion arises. A Leibniz algebra whose bilinear bracket is skewsymmetric is nothing buta Lie algebra. Thus, Leibniz algebras are the non-skewsymmetric analogue of Lie algebras. Definition 1.2.
A representation of a Leibniz algebra ( g , [ · , · ]) consists of a triple ( V, ρ L , ρ R ) of a vector space V and two linear maps ρ L , ρ R : g → gl ( V ) satisfying for x, y ∈ g , ρ L ([ x, y ]) = ρ L ( x ) ◦ ρ L ( y ) − ρ L ( y ) ◦ ρ L ( x ) ,ρ R ([ x, y ]) = ρ L ( x ) ◦ ρ R ( y ) − ρ R ( y ) ◦ ρ L ( x ) ,ρ R ([ x, y ]) = ρ L ( x ) ◦ ρ R ( y ) + ρ R ( y ) ◦ ρ R ( x ) . It follows that any Leibniz algebra g is a representation of itself with ρ L ( x ) = L x = [ x, · ] and ρ R ( x ) = R x = [ · , x ] , for x ∈ g . Here L x and R x denotes the left and right multiplications by x , respectively. This is calledthe regular representation.Let ( g , [ · , · ]) be a Leibniz algebra and ( V, ρ L , ρ R ) be a representation of it. The Loday-Pirashvili cohomology of g with coefficients in V is the cohomology of the cochain complex3 C ∗ ( g , V ) , ∂ } , where C n ( g , V ) = Hom ( g ⊗ n , V ) , ( n ≥ and the coboundary operator ∂ : C n ( g , V ) → C n +1 ( g , V ) given by ( ∂f )( x , · · · , x n +1 )= n X i =1 ( − i +1 ρ L ( x i ) f ( x , · · · , ˆ x i , · · · , x n +1 ) + ( − n +1 ρ R ( x n +1 ) f ( x , · · · , x n )+ X ≤ i A linear map K : V → g is said to a H -twisted relative Rota-Baxteroperator if K satisfies [ Ku, Kv ] = K ( ρ L ( Ku ) v + ρ R ( Kv ) u + H ( Ku, Kv )) , for u, v ∈ V. Example 2.2. Any relative Rota-Baxter operator (0. 1) is a H -twisted relative Rota-Baxter operator with H = 0 . Example 2.3. Let ( V, ρ L , ρ R ) be a representation of a Leibniz algebra ( g , [ · , · ]) . Suppose h ∈ C ( g , V ) is an invertible -cochain in the Loday-Pirashvili cochain complex of g withcoefficients in V . Take H = − ∂h . Then H ( Ku, Kv ) = ( − ∂h )( Ku, Kv ) = − ρ L ( K ( u )) v − ρ R ( K ( v )) u + h ([ Ku, Kv ]) , for u, v ∈ V. This shows that K = h − : V → g is a H -twisted relative Rota-Baxter operator. Example 2.4. Let ( g , [ · , · ]) be a Leibniz algebra and N : g → g be a Nijenhuis operator it,i.e., N satisfies [ N x, N y ] = N ([ N x, y ] + [ x, N y ] − N [ x, y ]) , for x, y ∈ g . n this case the vector space g carries a new Leibniz algebra structure with deformed bracket [ x, y ] N = [ N x, y ] + [ x, N y ] − N [ x, y ] , for x, y ∈ g . (2. 1) This deformed Leibniz algebra g N = ( g , [ · , · ] N ) has a representation on g by ρ L ( x ) y :=[ N x, y ] and ρ R ( x ) y := [ y, N x ] , for x ∈ g N , y ∈ g . With this representation, the map H : ( g N ) ⊗ → g , H ( x, y ) = − N [ x, y ] is a -cocycle in the Loday-Pirashvili cohomology of g N with coefficients in g . Moreover the identity map Id : g → g N is a H -twisted relativeRota-Baxter operator. Let K : V → g be a H -twisted relative Rota-Baxter operator. Suppose ( V ′ , ρ ′ L , ρ ′ R ) is a representation of another Leibniz algebra ( g ′ , [ · , · ] ′ ) and H ′ ∈ C ( g ′ , V ′ ) is a -cocycle.Let K ′ : V ′ → g ′ be a H ′ -twisted relative Rota-Baxter operator. Definition 2.5. A morphism of twisted relative Rota-Baxter operators from K to K ′ con-sists of a pair ( φ, ψ ) of a Leibniz algebra morphism φ : g → g ′ and a linear map ψ : V → V ′ satisfying φ ◦ K = K ′ ◦ ψ,ψ ( ρ L ( x ) u ) = ρ ′ L ( φ ( x )) ψ ( u ) , ψ ( ρ R ( x ) u ) = ρ ′ R ( φ ( x )) ψ ( u ) ,ψ ◦ H = H ′ ◦ ( φ ⊗ φ ) , for x ∈ g , u ∈ V. Given a -cocycle H in the Loday-Pirashvili cochain complex of g with coefficients in V , one can construct the twisted semidirect product algebra. More precisely, the directsum g ⊕ V carries a Leibniz algebra structure with the bilinear bracket given by [( x, u ) , ( y, v )] H = ([ x, y ] , ρ L ( x ) v + ρ R ( y ) u + H ( x, y )) , for x, y ∈ g , u, v ∈ V. We denote this H -twisted semidirect product Leibniz algebra by g ⋉ H V . Using this twistedsemidirect product, one can characterize twisted relative Rota-Baxter operators by theirgraph. Proposition 2.6. A linear map K : V → g is a H -twisted relative Rota-Baxter operator ifand only if its graph Gr ( K ) = { ( Ku, u ) | u ∈ V } is a subalgebra of the H -twisted semidirectproduct g ⋉ H V . Since Gr ( K ) is isomorphic to V as a vector space, as a consequence, we get the follow-ing. Proposition 2.7. Let K : V → g be an H -twisted relative Rota-Baxter operator. Thenthe vector space V carries a Leibniz algebra structure with the bracket [ u, v ] K := ρ L ( Ku ) v + ρ R ( Kv ) u + H ( Ku, Kv ) , for u, v ∈ V. .1 Some new constructions In this subsection, we construct new twisted relative Rota-Baxter operators out of anold one by suitable modifications. We start with the following. Proposition 2.8. Let ( V, ρ L , ρ R ) be a representation of a Leibniz algebra ( g , [ · , · ]) . For any -cocycle H ∈ C ( g , V ) and -cochain h ∈ C ( g , V ) , the twisted Leibniz algebras g ⋉ H V and g ⋉ H + ∂h V are isomorphic. Proof. We define an isomorphism Ψ h : g ⋉ H V → g ⋉ H + ∂h V of the underlying vectorspaces by Ψ h ( x, u ) := ( x, u − h ( x )) , for ( x, u ) ∈ g ⋉ H V . Moreover, we have Ψ h ([( x, u ) , ( y, v )] H )= Ψ h ([ x, y ] , ρ L ( x ) v + ρ R ( y ) u + H ( x, y ))= ([ x, y ] , ρ L ( x ) v + ρ R ( y ) u + H ( x, y ) − h [ x, y ])= ([ x, y ] , ρ L ( x ) v + ρ R ( y ) u + H ( x, y ) − ρ L ( x ) h ( y ) − ρ R ( y ) h ( x ) + ∂h ( x, y ))= [( x, u − h ( x )) , ( y, v − h ( y ))] H + ∂h = [Ψ h ( x, u ) , Ψ h ( y, v )] H + ∂h . This shows that Ψ h is in fact an isomorphism of Leibniz algebras. (cid:3) Proposition 2.9. Let K : V → g be a H -twisted Rota-Baxter operator. For any -cochain h ∈ C ( g , V ) , if the linear map ( Id V − h ◦ K ) : V → V is invertible then the map K ◦ ( Id V − h ◦ K ) − : V → g is a ( H + ∂h ) -twisted relative Rota-Baxter operator. Proof. Consider the subalgebra Gr ( K ) ⊂ g ⋉ H V of the H -twisted semidirect product.Thus by Proposition 2.8, we get that Ψ h ( Gr ( K )) = { ( Ku, u − hK ( u )) | u ∈ V } ⊂ g ⋉ H + ∂h V is a subalgebra. Since the map ( Id V − h ◦ K ) : V → V is invertible, we have Ψ h ( Gr ( K )) is the graph of the linear map K ◦ ( Id V − h ◦ K ) − . Hence by Proposition 2.6, the map K ◦ ( Id V − h ◦ K ) − is a ( H + ∂h ) -twisted relative Rota-Baxter operator. (cid:3) Let K : V → g be a H -twisted Rota-Baxter operator. Suppose B ∈ C ( g , V ) is a -cocycle in the Loday-Pirashvili cochain complex of g with coefficients in V . Then B issaid to be K -admissible if the linear map ( Id V + B ◦ K ) : V → V is invertible. With thisnotation, we have the following. Proposition 2.10. Let B ∈ C ( g , V ) be a K -admissible -cocycle. Then the map K ◦ ( Id V + B ◦ K ) − : V → g is a H -twisted Rota-Baxter operator. roof. Consider the deformed subspace τ B ( Gr ( K )) = { ( Ku, u + B ◦ K ( u )) | u ∈ V } ⊂ g ⋉ H V. Since B is a -cocycle, τ B ( Gr ( K )) ⊂ g ⋉ H V turns out to be a subalgebra. Further,the map ( Id V + B ◦ K ) is invertible implies that τ B ( Gr ( K )) is the graph of the map K ◦ ( Id V + B ◦ K ) − . Hence the result follows from Proposition 2.6. (cid:3) The H -twisted Rota-Baxter operator in the above proposition is called the gauge trans-formation of K associated with B . We denote this H -twisted relative Rota-Baxter operatorsimply by K B . Proposition 2.11. Let K be a H -twisted relative Rota-Baxter operator and B be a K -admissible -cocycle. Then the Leibniz algebra structures on V induced from the H -twistedRota-Baxter operators K and K B are isomorphic. Proof. Consider the linear isomorphism ( Id V + B ◦ K ) : V → V . Moreover, for any u, v ∈ V , we have [( Id V + B ◦ K )( u ) , ( Id V + B ◦ K )( v )] K B = ρ L ( K ( u ))( Id V + B ◦ K )( v ) + ρ R ( K ( v ))( Id V + B ◦ K )( u ) + H ( Ku, Kv )= ρ L ( K ( u )) v + ρ R ( K ( v )) u + ρ L ( K ( u ))( B ◦ K ( v )) + ρ R ( K ( v ))( B ◦ K ( u )) + H ( Ku, Kv )= ρ L ( K ( u )) v + ρ R ( K ( v )) u + B [ Ku, Kv ] + H ( Ku, Kv )= [ u, v ] K + B ◦ K ([ u, v ] K )= ( Id V + B ◦ K )([ u, v ] K ) . This shows that ( Id V + B ◦ K ) : ( V, [ · , · ] K ) → ( V, [ · , · ] K B ) is a Leibniz algebra isomorphism. (cid:3) In this section, we define cohomology of a H -twisted relative Rota-Baxter operator K as the Loday-Pirashvili cohomology of the Leibniz algebra ( V, [ · , · ] K ) constructed inProposition 2.7 with coefficients in a suitable representation on g . In the next section, wewill use this cohomology to study deformations of K . Proposition 3.1. Let K : V → g be a H -twisted relative Rota-Baxter operator. Definemaps ρ L , ρ R : V → gl ( g ) by ρ L ( u ) x = [ Ku, x ] − K ( ρ R ( x ) u ) − KH ( Ku, x ) , ρ R ( u ) x = [ x, Ku ] − K ( ρ L ( x ) u ) − KH ( x, Ku ) , for u ∈ V and x ∈ g . Then ( g , ρ L , ρ R ) is a representation of the Leibniz algebra ( V, [ · , · ] K ) . roof. For u, v ∈ V and x ∈ g , we have ρ L ( u ) ρ L ( v ) x − ρ L ( v ) ρ L ( u ) x = ρ L ( u )([ Kv, x ] − K ( ρ R ( x ) v ) − KH ( Kv, x )) − ρ L ( v )([ Ku, x ] − K ( ρ R ( x ) u ) − KH ( Ku, x ))= [ Ku, [ Kv, x ]] − [ Ku, K ( ρ R ( x ) v )] − [ Ku, KH ( Kv, x )] − K ( ρ R ([ Kv, x ]) u ) + K ( ρ R ( Kρ R ( x ) v ) u )+ K ( ρ R ( KH ( Kv, x )) u ) − KH ( Ku, [ Kv, x ]) + KH ( Ku, K ( ρ R ( x ) v )) + KH ( Ku, KH ( Kv, x )) − [ Kv, [ Ku, x ]] + [ Kv, K ( ρ R ( x ) u )] + [ Kv, KH ( Ku, x )] + K ( ρ R ([ Ku, x ]) v ) − K ( ρ R ( Kρ R ( x ) u ) v ) − K ( ρ R ( KH ( Ku, x )) v ) + KH ( Kv, [ Ku, x ]) − KH ( Kv, K ( ρ R ( x ) u )) − KH ( Kv, KH ( Ku, x ))= [[ Ku, Kv ] , x ] − K ( ρ R ( x ) ρ L ( Ku ) v ) − K ( ρ R ( x ) ρ R ( Kv ) u ) − K ( ρ R ( x ) H ( Ku, Kv )) − KH ( K [ u, v ] K , x )= [ K [ u, v ] K , x ] − K ( ρ R ( x )[ u, v ] K ) − KH ( K [ u, v ] K , x )= ρ L ([ u, v ] K ) x. The third equality is obtained by some cancellations and using the fact that H is a -cocycle. Thus, we deduce that ρ L ([ u, v ] K ) = ρ L ( u ) ρ L ( v ) − ρ L ( v ) ρ L ( u ) . We also have ρ L ( u ) ρ R ( v ) x − ρ R ( v ) ρ L ( u ) x = ρ L ( u )([ x, Kv ] − K ( ρ L ( x ) v ) − KH ( x, Kv )) − ρ R ( v )([ Ku, x ] − K ( ρ R ( x ) u ) − KH ( Ku, x ))= [ Ku, [ x, Kv ]] − [ Ku, K ( ρ L ( x ) v )] − [ Ku, KH ( x, Kv )] − K ( ρ R ([ x, Kv ]) u ) + K ( ρ R ( Kρ L ( x ) v ) u )+ K ( ρ R ( KH ( x, Kv )) u ) − KH ( Ku, [ x, Kv ]) + KH ( Ku, K ( ρ L ( x ) v )) + KH ( Ku, KH ( x, Kv )) − [[ Ku, x ] , Kv ] + [ K ( ρ R ( x ) u ) , Kv ] + [ KH ( Ku, x ) , Kv ] + K ( ρ L ([ Ku, x ]) v ) − K ( ρ L ( Kρ R ( x ) u ) v ) − K ( ρ L ( KH ( Ku, x )) v ) + KH ([ Ku, x ] , Kv ) − KH ( K ( ρ R ( x ) u ) , Kv ) − KH ( KH ( Ku, x ) , Kv )= [ x, [ Ku, Kv ]] − K ( ρ L ( x ) ρ L ( Ku ) v ) − K ( ρ L ( x ) ρ R ( Kv ) u ) − Kρ L ( x ) H ( Ku, Kv )) − KH ( x, K [ u, v ] K )= [ x, K [ u, v ] K ] − K ( ρ L ( x )[ u, v ] K ) − KH ( x, K [ u, v ] K )= ρ R ([ u, v ] K ) x which shows that ρ R ([ u, v ] K ) = ρ L ( u ) ρ R ( v ) − ρ R ( v ) ρ L ( u ) . Similarly, we can show that ρ R ([ u, v ] K ) = ρ R ( v ) ◦ ρ R ( u ) + ρ L ( u ) ◦ ρ R ( v ) . Therefore, ( g , ρ L , ρ R ) is a representation of the Leibniz algebra ( V, [ · , · ] K ) . (cid:3) We will now consider the Loday-Pirashvili cohomology of the Leibniz algebra ( V, [ · , · ] K ) with coefficients in the representation ( g , ρ L , ρ R ) . More precisely, we define C n ( V, g ) := Hom ( V ⊗ n , g ) , for n ≥ . ∂ K : C n ( V, g ) → C n +1 ( V, g ) by ( ∂ K f )( u , · · · , u n +1 )= n X i =1 ( − i +1 [ Ku i , f ( u , · · · , ˆ u i , · · · , u n +1 )] − n X i =1 ( − i +1 K ( ρ R ( f ( u , · · · , ˆ u i , · · · , u n +1 )) u i ) − n X i =1 ( − i +1 KH ( Ku i , f ( u , · · · , ˆ u i , · · · , u n +1 )) + ( − n +1 [ f ( u , · · · , u n ) , Ku n +1 ]+ ( − n K ( ρ L ( f ( u , · · · , u n )) u n +1 ) + ( − n KH ( f ( u , · · · , u n ) , Ku n +1 )+ X ≤ i A linear map K : V → g is said to generate a linear deformation of the H -twisted relative Rota-Baxter operator K if for all t ∈ C , the sum K t = K + tK is still a H -twisted relative Rota-Baxter operator. In this case, K t = K + tK is said to be a lineardeformation of K . K generates a linear deformation of K . Then we have [ K t u, K t v ] = K t (cid:0) ρ L ( K t u ) v + ρ R ( K t v ) u + H ( K t u, K t v ) (cid:1) , for u, v ∈ V. This is equivalent to the following conditions [ Ku, K v ] + [ K u, Kv ] = K ( ρ L ( Ku ) v + ρ R ( Kv ) u + H ( Ku, Kv ))+ K ( ρ L ( K u ) v + ρ R ( K v ) u + H ( K u, Kv ) + H ( Ku, K v )) , (4. 1) [ K u, K v ] = K ( ρ L ( K u ) v + ρ R ( K v ) u + H ( Ku, K v ) + H ( K u, Kv )) + KH ( K u, K v ) , (4. 2) K ( H ( K ( u ) , K ( v ))) = 0 . (4. 3)Note that Eq. (4.1) means that K is a -cocycle in the cohomology of K . Hence K induces an element in H K ( V, g ) . Definition 4.2. Two linear deformations K t = K + tK and K ′ t = K + tK ′ of K are saidto be equivalent if there exists an element x ∈ g such that ( φ t = Id g + tL x , ψ t = Id V + t ( ρ L ( x ) + H ( x, K − )) is a morphism of twisted Rota-Baxter operators from K t to K ′ t . The condition that φ t = Id g + tL x is a Leibniz algebra morphism of ( g , [ · , · ]) is equivalentto [[ x, y ] , [ x, z ]] = 0 , for y, z ∈ g . (4. 4)Further, The conditions ψ t ( ρ L ( y ) u ) = ρ L ( φ t ( y )) ψ t ( u ) and ψ t ( ρ R ( y ) u ) = ρ R ( φ t ( y )) ψ t ( u ) ,for y ∈ g , u ∈ V are respectively equivalent to ( H ( x, K ( ρ L ( y ) u )) = ρ L ( y ) H ( x, Ku ) ,ρ L ([ x, y ])( ρ L ( x ) u + H ( x, Ku )) = 0 , (4. 5) ( H ( x, K ( ρ R ( y ) u )) = ρ R ( y ) H ( x, Ku ) ,ρ R ([ x, y ])( ρ L ( x ) u + H ( x, Ku )) = 0 . (4. 6)Similarly, the conditions ψ t ◦ H = H ◦ ( φ t ⊗ φ t ) and φ t ◦ K t = K ′ t ◦ ψ t are respectivelyequivalent to ( ρ L ( x ) H ( y, z ) + H ( x, KH ( y, z )) = H ( x, [ y, z ]) + H ( y, [ x, z ]) ,H ([ x, y ] , [ x, z ]) = 0 , (4. 7)10 K ( u ) + [ x, Ku ] = K ( ρ L ( x ) u + H ( x, Ku )) + K ′ ( u ) , [ x, K u ] = K ′ ( ρ L ( x ) u + H ( x, Ku )) . (4. 8)It follows from the first identity in (4. 8) that K ( u ) − K ′ ( u ) = d K ( x )( u ) . Hence weobtain the following. Theorem 4.3. If two linear deformations K t = K + tK and K ′ t = K + tK ′ of a H -twisted relative Rota-Baxter operator K are equivalent, then K and K ′ are in the samecohomology class of H K ( V, g ) . Definition 4.4. A linear deformation K t = K + tK of a H -twisted relative Rota-Baxteroperator K is said to be trivial if K t is equivalent to the undeformed deformation K ′ t = K . We will now define Nijenhuis elements associated with a H -twisted relative Rota-Baxteroperator K in a way that a trivial deformation of K induces a Nijenhuis element. Definition 4.5. Let K be a H -twisted relative Rota-Baxter operator. An element x ∈ g iscalled a Nijenhuis element associated with K if x satisfies [ x, ρ R ( u )( x )] = 0 , for u ∈ V and Equations (4. 4), (4. 5), (4. 6), (4. 7) hold. The set of all Nijenhuis elements associated with K is denoted by Nij ( K ) . As mentionedearlier that a trivial deformation induces a Nijenhuis element. In the next subsection, wegive a sufficient condition for the rigidity of a twisted Rota-Baxter operator in terms ofNijenhuis elements. Let C [[ t ]] be the ring of power series in one variable t . For any C -linear space V , we let V [[ t ]] denotes the vector space of formal power series in t with coefficients in V . Moreover,if ( g , [ · , · ]) is a Leibniz algebra over C , then one can extend the Leibniz bracket on g [[ t ]] by C [[ t ]] -bilinearity. Furthermore, if ( V, ρ L , ρ R ) is a representation of the Leibniz algebra ( g , [ · , · ]) , then there is a representation ( V [[ t ]] , ρ L , ρ R ) of the Leibniz algebra g [[ t ]] . Here, ρ L and ρ R are also extended by C [[ t ]] -bilinearity. Similarly, the -cocycle H can be extendedto a -cocycle (which we denote by the same notation H ) on the Leibniz algebra g [[ t ]] withcoefficients in V [[ t ]] .Let K : V → g be a H -twisted relative Rota-Baxter operator on the Leibniz algebra ( g , [ · , · ]) with respect to the representation ( V, ρ L , ρ R ) and -cocycle H . We consider thepower series K t = + ∞ X i =0 K i t i , for K i ∈ Hom ( V, g ) with K = K. K t to a linear map from V [[ t ]] to g [[ t ]] by C [[ t ]] -linearity which we still denote by K t . Definition 4.6. A formal deformation of K is given by a formal sum K t = P + ∞ i =0 K i t i with K = K satisfying [ K t u, K t v ] = K t (cid:0) ρ L ( K t u ) v + ρ R ( K t v ) u + H ( K t ( u ) , K t ( v )) (cid:1) , for u, v ∈ V. (4. 9)It follows that K t is a H -twisted relative Rota-Baxter operator on the Leibniz algebra g [[ t ]] with respect to the representation V [[ t ]] and -cocycle H . Remark 4.7. If K t = P + ∞ i =0 K i t i is a formal deformation of a H -twisted relative Rota-Baxter operator K on a Leibniz algebra ( g , [ · , · ]) with respect to a representation ( V, ρ L , ρ R ) and -cocycle H , then [ · , · ] K t defined by [ u, v ] K t := + ∞ X i =0 (cid:0) ρ L ( K i u ) v + ρ R ( K i v ) u + X j + k = i H ( K j u, K k v ) (cid:1) t i , for u, v ∈ V, is a formal deformation of the associated Leibniz algebra ( V, [ · , · ] K ) . By expanding the identity (4. 9) and comparing coefficients of various powers of t , weobtain for n ≥ , X i + j = n [ K i u, K j v ] = X i + j = n K i ( ρ L ( K j u ) v + ρ R ( K j v ) u ) + X i + j + k = n K i H ( K j ( u ) , K k ( v )) , for u, v ∈ V . It holds for n = 0 as K is a H -twisted relative Rota-Baxter operator. For n = 1 , we obtain [ Ku, K v ] + [ K u, Kv ] = K ( ρ L ( Ku ) v + ρ R ( Kv ) u + H ( Ku, Kv ))+ K ( ρ L ( K u ) v + ρ R ( K v ) u + H ( K ( u ) , Kv ) + H ( K ( u ) , K v )) . This condition is equivalent to ( ∂ K ( K ))( u, v ) = 0 , for u, v ∈ V . Proposition 4.8. Let K t = P + ∞ i =0 K i t i be a formal deformation of a H -twisted relativeRota-Baxter operator K . Then K is a -cocycle in the cohomology of the H -twisted relativeRota-Baxter operator K , that is, ∂ K ( K ) = 0 . Definition 4.9. The -cocycle K is called the infinitesimal of the formal deformation K t = P + ∞ i =0 K i t i . Next, we define an equivalence between two formal deformations of a H -twisted relativeRota-Baxter operator. 12 efinition 4.10. Two formal deformations K t == P + ∞ i =0 K i t i and K ′ t == P + ∞ i =0 K ′ i t i ofa H -twisted relative Rota-Baxter operator K are said to be equivalent if there exists anelement x ∈ g , linear maps φ i ∈ gl ( g ) and ψ i ∈ gl ( V ) for i ≥ such that the pair (cid:0) φ t = Id g + tL x + + ∞ X i =2 φ i t i , ψ t = Id V + t ( ρ L ( x ) + H ( x, K − )) + + ∞ X i =2 ψ i t i (cid:1) is a morphism of H -twisted relative Rota-Baxter operators from K t to K ′ t . By equating coefficients of t from both sides of the identity φ t ◦ K t = K ′ t ◦ ψ t , we obtain K ( u ) − K ′ ( u ) = K ( ρ L ( x ) u + H ( x, Ku )) − [ x, Ku ] = d K ( x )( u ) , for u ∈ V. As a summary, we get the following. Theorem 4.11. The infinitesimal of a formal deformation of a H -twisted relative Rota-Baxter operator K is a -cocycle in the cohomology of K , and the corresponding cohomologyclass depends only on the equivalence class of the deformation of K . Definition 4.12. A H -twisted relative Rota-Baxter operator K is said to be rigid if anyformal deformation of K is equivalent to the undeformed deformation K ′ t = K . In the next theorem, we give a sufficient condition for the rigidity of a H -twisted relativeRota-Baxter operator in terms of Nijenhuis elements. Theorem 4.13. Let K be a H -twisted relative Rota-Baxter operator. If Z K ( V, g ) = ∂ K (Nij( K )) then K is rigid. Proof. Let K t = P + ∞ i =0 K i t i be any formal deformation of K . Then it follows fromProposition 4.8 that the linear term K is a -cocycle in the cohomology of K , i.e., K ∈ Z K ( V, g ) . Thus, by the hypothesis, there is a Nijenhuis element x ∈ Nij( K ) such that K = − ∂ K ( x ) . We take φ t = Id g + tL x and ψ t = Id V + t ( ρ L ( x ) + H ( x, K − )) , and define K ′ t = φ t ◦ K t ◦ ψ − t . Then K ′ t is a formal deformation equivalent to K t . For u ∈ V , we observe that K ′ t ( u ) = ( Id g + tL x )( K t ( u − tρ L ( x ) u − tH ( x, Ku ) + power of t ≥ ))= K ( u ) + t ( K u − Kρ L ( x ) u − KH ( x, Ku ) + [ x, Ku ]) + power of t ≥ . = K ( u ) + t K ′ ( u ) + · · · ( as K = − ∂ K ( x )) . Hence the coefficient of t in the expression of K ′ t is trivial. Applying the same processrepeatedly, we get that K t is equivalent to K . Therefore, K is rigid. (cid:3) NS-Leibniz algebras In this section, we introduce NS-Leibniz algebras as the underlying structure of twistedRota-Baxter operators. Here we study some properties of NS-Leibniz algebras and givesome examples. Further study on NS-Leibniz algebras is postponed to a forthcoming paper. Definition 5.1. An NS-Leibniz algebra is a quadruple ( A, ⊲, ⊳, ⋄ ) consisting of a vectorspace A together with three bilinear operations ⊲, ⊳, ⋄ : A ⊗ A → A satisfying for all x, y, z ∈ A , ( A x ⊲ ( y ∗ z ) = ( x ⊲ y ) ⊲ z + y ⊳ ( x ⊲ z ) , ( A x ⊳ ( y ⊲ z ) = ( x ⊳ y ) ⊲ z + y ⊲ ( x ∗ z ) , ( A x ⊳ ( y ⊳ z ) = ( x ∗ y ) ⊳ z + y ⊳ ( x ⊳ z ) , ( A x ⊳ ( y ⋄ z ) + x ⋄ ( y ∗ z ) = ( x ⋄ y ) ⊲ z + ( x ∗ y ) ⋄ z + y ⊳ ( x ⋄ z ) + y ⋄ ( x ∗ z ) , where x ∗ y = x ⊲ y + x ⊳ y + x ⋄ y . NS-Leibniz algebras are more general than Leibniz-dendriform algebras introduced in[17]. More precisely, an NS-Leibniz algebra ( A, ⊲, ⊳, ⋄ ) in which the bilinear operation ⋄ istrivial is a Leibniz-dendriform algebra.In the following, we show that NS-Leibniz algebras split Leibniz algebras. Proposition 5.2. Let ( A, ⊲, ⊳, ⋄ ) be an NS-Leibniz algebra. Then the vector space A withthe bilinear operation [ · , · ] ∗ : A ⊗ A → A, [ x, y ] ∗ := x ∗ y is a Leibniz algebra. Proof. By summing up the left hand sides of the identities (A1)-(A4), we simplyget [ x, [ y, z ] ∗ ] ∗ . On the other hand, by summing up the right hand sides of the identities(A1)-(A4), we get [[ x, y ] ∗ , z ] ∗ + [ y, [ x, z ] ∗ ] ∗ . Hence the result follows. (cid:3) The Leibniz algebra ( A, [ · , · ] ∗ ) of the above proposition is called the subadjacent Leibnizalgebra of ( A, ⊲, ⊳, ⋄ ) and ( A, ⊲, ⊳, ⋄ ) is called a compatible NS-Leibniz algebra structureon ( A, [ · , · ] ∗ ) . Proposition 5.3. Let ( g , [ · , · ]) be a Leibniz algebra and N : g → g be a Nijenhuis operatoron it. Then the bilinear operations x ⊲ y = [ x, N y ] , x ⊳ y = [ N x, y ] and x ⋄ y = − N [ x, y ] , for x, y ∈ g defines an NS-Leibniz algebra structure on g . roof. For any x, y, z ∈ g , we have x ⊲ ( y ∗ z ) = [ x, N ( y ∗ z )] = [ x, [ N y, N z ]]= [[ x, N y ] , N z ] + [ N y, [ x, N z ]]= ( x ⊲ y ) ⊲ z + y ⊳ ( x ⊲ z ) . Hence the identity (A1) of Definition 5.1 holds. Similarly, x ⊳ ( y ⊲ z ) = [ N x, [ y, N z ]] = [[ N x, y ] , N z ] + [ y, [ N x, N z ]]= ( x ⊳ y ) ⊲ z + y ⊲ ( x ∗ z ) , and x ⊳ ( y ⊳ z ) = [ N x, [ N y, z ]] = [[ N x, N y ] , z ] + [ N y, [ N x, z ]]= ( x ∗ y ) ⊳ z + y ⊳ ( x ⊳ z ) . Therefore, the identities (A2) and (A3) also hold. To prove the identity (A4), we firstrecall from [4] that the given Leibniz bracket [ · , · ] and the deformed Leibniz bracket [ · , · ] N given in (2. 1) are compatible in sense that their sum also defines a Leibniz bracket on g .This is equivalent to the fact that [ x, [ y, z ]] N + [ x, [ y, z ] N ] = [[ x, y ] , z ] N + [[ x, y ] N , z ] + [ y, [ x, z ]] N + [ y, [ x, z ] N ] , (5. 1)for x, y, z ∈ g . The identity (A4) of Definition 5.1 simply follows from (5. 1). Hence ( g , ⊲, ⊳, ⋄ ) is an NS-Leibniz algebra. (cid:3) Let ( A, ⊲, ⊳, ⋄ ) be an NS-Leibniz algebra. Define two linear maps L ⊳ : A → gl ( A ) , R ⊲ : A → gl ( A ) and a bilinear map H : A ⊗ A → A by L ⊳ ( x ) y = x ⊳ y, R ⊲ ( x ) y = y ⊲ x, H ( x, y ) = x ⋄ y, for x, y ∈ A. With these notations, we have the following. Proposition 5.4. Let ( A, ⊲, ⊳, ⋄ ) be an NS-Leibniz algebra. Then ( A, L ⊳ , R ⊲ ) is a repre-sentation of the subadjacent Leibniz algebra ( A, [ · , · ] ∗ ) , and H defined above is a -cocycle.Moreover, the identity map Id : A → A is a H -twisted relative Rota-Baxter operator onthe Leibniz algebra ( A, [ · , · ] ∗ ) with respect to the representation ( A, L ⊳ , R ⊲ ) . Proof. For any x, y, z ∈ A , we have L ⊳ ([ x, y ] ∗ ) z = [ x, y ] ∗ ⊳ z ( A = x ⊳ ( y ⊳ z ) − y ⊳ ( x ⊳ z )= (cid:0) L ⊳ ( x ) ◦ L ⊳ ( y ) − L ⊳ ( y ) ◦ L ⊳ ( x ) (cid:1) z. R ⊲ ([ x, y ] ∗ ) z = z ⊲ [ x, y ] ∗ ( A = x ⊳ ( z ⊲ y ) − ( x ⊳ z ) ⊲ y = L ⊳ ( x ) ◦ R ⊲ ( y )) z − R ⊲ ( y ) ◦ L ⊳ ( x ) z, and R ⊲ ([ x, y ] ∗ ) z = z ⊲ [ x, y ] ∗ ( A = ( z ⊲ x ) ⊲ y + x ⊳ ( z ⊲ y )= (cid:0) R ⊲ ( y ) ◦ R ⊲ ( x ) + L ⊳ ( x ) R ⊲ ( y ) (cid:1) z. Therefore, ( A, L ⊳ , R ⊲ ) is a representation of the subadjacent Leibniz algebra ( A, [ · , · ] ∗ ) .Moreover, the condition (A4) is equivalent that H is a -cocycle in the Loday-Pirashvilicochain complex of the Leibniz algebra ( A, [ · , · ] ∗ ) with coefficients in the representation ( A, L ⊳ , R ⊲ ) . Finally, we have Id ( L ⊳ ( Id x ) y + R ⊲ ( Id y ) x + H ( Id x, Id y )) = x ⊳ y + x ⊲ y + x ◦ y = [ Id x, Id y ] ∗ , which shows that Id : A → A is a H -twisted relative Rota-Baxter operator on the Leibnizalgebra ( A, [ · , · ] ∗ ) with respect to the representation ( A, L ⊳ , R ⊲ ) . (cid:3) Proposition 5.5. Let ( g , [ · , · ]) be a Leibniz algebra, ( V, ρ L , ρ R ) be a representation and H ∈ C ( g , V ) be a -cocycle. Let K : V → g be a H -twisted relative Rota-Baxter operator.Then there is an NS-Leibniz algebra structure on V with bilinear operations given by u ⊲ v := ρ R ( Kv ) u, u ⊳ v := ρ L ( Ku ) v, u ⋄ v := H ( Ku, Kv ) , for u, v ∈ V. Proof. For any u, v, w ∈ V , we have u ⊲ ( v ∗ w ) = ρ R ( K ( v ∗ w )) u = ρ R ([ Kv, Kw ]) u = ρ L ( Kv ) ρ R ( Kw ) u + ρ R ( Kw ) ρ R ( Kv ) u = v ⊳ ( u ⊲ w ) + ( u ⊲ v ) ⊲ w. Similarly, u ⊳ ( v ⊲ w ) = ρ L ( Ku ) ρ R ( Kw ) v = ρ R ([ Ku, Kz ]) v + ρ R ( Kw ) ρ L ( Ku ) v = v ⊲ ( u ∗ w ) + ( u ⊳ v ) ⊲ w, and u ⊳ ( v ⊳ w ) = ρ L ( Ku ) ρ L ( Kv )( w ) = ρ L ([ Ku, Kv ]) w + ρ L ( Kv ) ρ L ( Ku ) w = ( u ∗ v ) ⊳ w + v ⊳ ( u ⊳ w ) . H is a -cocycle, we have ( ∂H )( Ku, Kv, Kz ) = 0 , i.e., ρ L ( Ku ) H ( Kv, Kw ) − ρ L ( Kv ) H ( Ku, Kw ) − ρ R ( Kw ) H ( Ku, Kv ) − H ([ Ku, Kv ] , Kw ) − H ( Kv, [ Ku, Kw ]) + H ( Ku, [ Kv, Kw ]) = 0 . This is equivalent to the condition (A4) of Definition 5.1. Hence the proof. (cid:3) Remark 5.6. The subadjacent Leibniz algebra of the NS-Leibniz algebra constructed inProposition 5.5 is given by [ u, v ] ∗ = ρ L ( Ku ) v + ρ R ( Kv ) u + H ( Ku, Kv ) , for u, v ∈ V. This Leibniz algebra on V coincides with the one given in Proposition 2.7. In the following, we give a necessary and sufficient condition for the existence of acompatible NS-Leibniz algebra structure on a Leibniz algebra. Proposition 5.7. Let ( g , [ · , · ]) be a Leibniz algebra. Then there is a compatible NS-Leibnizalgebra structure on g if and only if there exists an invertible H -twisted relative Rota-Baxteroperator K : V → g on g with respect to a representation ( V, ρ L , ρ R ) and a -cocycle H .Furthermore, the compatible NS-Leibniz algebra structure on g is given by x ⊲ y := K ( ρ R ( y ) K − x ) , x ⊳ y := K ( ρ L ( x ) K − y ) , x ⋄ y = KH ( x, y ) , for x, y ∈ g . Proof. Let K : V → g be an invertible H -twisted relative Rota-Baxter operator on g with respect to a representation ( V, ρ L , ρ R ) and a -cocycle H . By Proposition 5.5, thereis an NS-Leibniz algebra structure on V given by u ¯ ⊲ v := ρ R ( Kv ) u, u ¯ ⊳ v := ρ L ( Ku ) v, u ¯ ⋄ v := H ( Ku, Kv ) , for u, v ∈ V. Since K is an invertible map, the bilinear operations x ⊲ y := K ( K − x ¯ ⊲ K − y ) = K ( ρ R ( y ) K − x ) ,x ⊳ y := K ( K − x ¯ ⊳ K − y ) = K ( ρ L ( x ) K − y ) ,x ⋄ y := K ( K − x ¯ ⋄ K − y ) = KH ( x, y ) , for x, y ∈ g defines an NS-Leibniz algebra on g . Moreover, we have x ⊲ y + x ⊳ y + x ⋄ y = K ( ρ R ( y ) K − x ) + K ( ρ L ( x ) K − y ) + KH ( x, y )= K ( ρ R ( K ◦ K − y ) K − x ) + K ( ρ L ( K ◦ K − x ) K − y ) + KH ( K ◦ K − x, K ◦ K − y )= [ K ◦ K − x, K ◦ K − y ] = [ x, y ] . ( g , ⊲, ⊳, ⋄ ) be a compatible NS-Leibniz algebra structure on g . By Propo-sition 5.4, ( g , L ⊳ , R ⊲ ) is a representation of the Leibniz algebra ( g , [ · , · ]) , and the identitymap Id : g → g is a H -twisted relative Rota-Baxter operator on the Leibniz algebra ( g , [ · , · ]) with respect to the representation ( g , L ⊳ , R ⊲ ) . Hence the proof. (cid:3) 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).