3 -L-dendriform algebras and generalized derivations
aa r X i v : . [ m a t h . R A ] A p r Taoufik Chtioui ∗ and Sami Mabrouk †
1. Faculty of Sciences, University of Sfax, BP 1171, 3000 Sfax, Tunisia1. University of Gafsa, Faculty of Sciences Gafsa, 2112 Gafsa, Tunisia
Abstract
The main goal of this paper is to introduce the notion of 3-L-dendriform algebras which arethe dendriform version of 3-pre-Lie algebras. In fact they are the algebraic structures behindthe O -operator of 3-pre-Lie algebras. They can be also regarded as the ternary analogous ofL-dendriform algebras. Moreover, we study the generalized derivations of 3-L-dendriformalgebras. Finally, we explore the spaces of quasi-derivations, the centroids and the quasi-centroids and give some properties. Key words : 3-Lie algebras, 3-pre-Lie algebras, 3-L-dendriform algebras, Representations, O -operator, Generalized derivations. Mathematics Subject Classification : 17A40,17A42,17B15.
Introduction
A dendriform algebra is a module equipped with two binary products whose sum is associative.This concept was introduced by Loday in the late 1990s in the study of periodicity in algebraic K -theory [15]. Several years later, Loday and Ronco introduced the concept of a tridendriformalgebra from their study of algebraic topology [16]. It is a module with three binary operationswhose sum is associative. Afterward, quite a few similar algebraic structures were introduced,such as the quadi-algebra and ennea algebra [1]. The notion of splitting of associativity wasintroduced by Loday to describe this phenomena in general for the associative operation (see[17, 12, 10], for more details).A similar two-part and three-part splittings of the Lie-algebra are found to be the pre-Lie algebraand the post-Lie algebra repectively from operadic study with applications to integrable systems[5, 21]. Further, a two-part and three-part splittings of the associative commutative operation givethe Zinbiel algebra and commutative tridendriform algebra respectively [15, 22]. Analogues of thedendriform and tridendriform algebra for Jordan algebra, alternative algebra and Poisson algebrahave also been obtained [3, 8, 7].Some n -ary algebraic structures like n -Lie algebras and n -associative algebras, which have beenwidely studied in the last few years, were also decomposed into two and three operations (for n =
3) giving 3-pre-Lie algebras and partially dendriform 3-algebras [20], using the procedure thatsplits the operations in algebraic operads. ∗ Corresponding author, E-mail: chtioui.taoufi[email protected] † Corresponding author, E-mail: [email protected] O -operators which is generalisation ofRota-Baxter operators.The following is an outline of the paper. In section 1, we summarize some definitions andknown results about 3-Lie algebras and 3-pre-Lie algebras which will be useful in the sequel.In section 2, we introduce the notion of 3-L-dendriform algebra. We establish that it has towassociated 3-pre-Lie algebras (horizontal and vertical). They have the same sub-adjacent 3-Liealgebra. In addition, the left multiplication operator of the first operation (called north-west: տ ) and the right multiplication operator of the second operation (called north-east: ր ) consista bimodule of the associated horizontal 3-pre-Lie algebra. Section 3 is devoted to the study ofgeneralized derivations on 3-L-dendriform algebras, 3-pre-Lie algebras and 3-Lie algebras.Throughout this paper K is a field of characteristic 0 and all vector spaces are over K . In this section, we give some general results on 3-Lie algebras and 3-pre-Lie algebras which areuseful throughout this paper.
Definition 1.1. [11] A 3-Lie algebra consists of a vector space A equipped with a skew-symmetriclinear map called 3-Lie bracket [ · , · , · ] : ⊗ A → A such that the following Fundamental Identityholds (for x i ∈ A , ≤ i ≤ x , x , [ x , x , x ]] = [[ x , x , x ] , x , x ] + [ x , [ x , x , x ] , x ] + [ x , x , [ x , x , x ]] (1.1)In other words, for x , x ∈ A , the operator ad x , x : A → A , ad x , x x : = [ x , x , x ] , ∀ x ∈ A , (1.2)is a derivation in the sense that ad x , x [ x , x , x ] = [ ad x , x x , x , x ] + [ x , ad x , x x , x ] + [ x , x , ad x , x x ] , ∀ x , x , x ∈ A . A morphism between 3-Lie algebras is a linear map that preserves the 3-Lie brackets.
Example 1.1. [11] Consider -dimensional -Lie algebra A generated by ( e , e , e , e ) with the followingmultiplication [ e , e , e ] = e , [ e , e , e ] = e , [ e , e , e ] = e , [ e , e , e ] = e . The notion of a representation of an n -Lie algebra was introduced in [13]. See also [9]. Definition 1.2.
A representation of a 3-Lie algebra A on a vector space V is a skew-symmetriclinear map ρ : ⊗ A → gl ( V ) satisfying(i) ρ ( x , x ) ρ ( x , x ) − ρ ( x , x ) ρ ( x , x ) = ρ ([ x , x , x ] , x ) − ρ ([ x , x , x ] , x ),2ii) ρ ([ x , x , x ] , x ) = ρ ( x , x ) ρ ( x , x ) + ρ ( x , x ) ρ ( x , x ) + ρ ( x , x ) ρ ( x , x ),for x i ∈ A , ≤ i ≤ V , ρ ) is a representation of a 3-Lie algebra ( A , [ · , · , · ]) if and only if there is a 3-Lie algebrastructure on the direct sum A ⊕ V of the underlying vector spaces A and V given by[ x + v , x + v , x + v ] A ⊕ V = [ x , x , x ] + ρ ( x , x ) v + ρ ( x , x ) v + ρ ( x , x ) v , (1.3)for x i ∈ A , v i ∈ V , ≤ i ≤
3. We denote it by A ⋉ ρ V . Now we recall the definition of 3-pre-Lie algebra and exhibit construction results in terms of O -operators on 3-Lie algebras (for more details see [4], [20]) Definition 1.3.
Let ( A , [ · , · , · ]) be a 3-Lie algebra and ( V , ρ ) a representation. A linear operator T : V → A is called an O -operator associated to ( V , ρ ) if T satisfies[ Tu , Tv , Tw ] = T (cid:0) ρ ( Tu , Tv ) w + ρ ( Tv , Tw ) u + ρ ( Tw , Tu ) v (cid:1) , ∀ u , v , w ∈ V . (1.4) Definition 1.4.
A 3-pre-Lie algebra is a pair ( A , {· , · , ·} ) consisting of a a vector space A and a linearmap {· , · , ·} : A ⊗ A ⊗ A → A such that the following identities hold: { x , y , z } = −{ y , x , z } , (1.5) { x , x , { x , x , x }} = { [ x , x , x ] C , x , x } + { x , [ x , x , x ] C , x } + { x , x , { x , x , x }} , (1.6) { [ x , x , x ] C , x , x } = { x , x , { x , x , x }} + { x , x , { x , x , x }} + { x , x , { x , x , x }} , (1.7)where x , y , z , x i ∈ A , ≤ i ≤ · , · , · ] C is defined by[ x , y , z ] C = { x , y , z } + { y , z , x } + { z , x , y } , ∀ x , y , z ∈ A . (1.8) Example 1.2.
Let A be a -dimensional vector space generated by ( e , e , e , e ) and consider the bracket {· , · , ·} : A ⊗ A ⊗ A → A given by { e , e , e } = −{ e , e , e } = e + e and all the other brackets are zero. Then ( A , {· , · , ·} ) is a -pre-Lie algebra. Example 1.3.
Let A be a -dimensional vector space generated by ( e , e , e , e ) and let the bracket {· , · , ·} : A ⊗ A ⊗ A → A given by { e , e , e } = e , { e , e , e } = e , { e , e , e } = −{ e , e , e } = e , { e , e , e } = −{ e , e , e } = e , and all the other brackets are zero. Then ( A , {· , · , ·} ) is a -pre-Lie algebra. Proposition 1.4.
Let ( A , {· , · , ·} ) be a -pre-Lie algebra. Then the induced -commutator given by Eq. (1.8) defines a -Lie algebra. Definition 1.5.
Let ( A , {· , · , ·} ) be a 3-pre-Lie algebra. The 3-Lie algebra ( A , [ · , · , · ] C ) is called thesub-adjacent 3-Lie algebra of ( A , {· , · , ·} ) and ( A , {· , · , ·} ) is called a compatible 3-pre-Lie algebra ofthe 3-Lie algebra ( A , [ · , · , · ] C ). 3et ( A , {· , · , ·} ) be a 3-pre-Lie algebra. Define a skew-symmetric linear map L : ⊗ A → gl ( A ) by L ( x , y ) z = { x , y , z } , ∀ x , y , z ∈ A . (1.9)By the definitions of a 3-pre-Lie algebra and a representation of a 3-Lie algebra, we immediatelyobtain Proposition 1.5.
With the above notations, ( A , L ) is a representation of the -Lie algebra ( A , [ · , · , · ] C ) . Onthe other hand, let A be a vector space with a linear map {· , · , ·} : A ⊗ A ⊗ A → A satisfying Eq. (1.5) . Then ( A , {· , · , ·} ) is a -pre-Lie algebra if [ · , · , · ] C defined by Eq (1.8) is a -Lie algebra and the left multiplication Ldefined by Eq. (1.9) gives a representation of this -Lie algebra. Proposition 1.6.
Let ( A , [ · , · , · ]) be a -Lie algebra and ( V , ρ ) a representation. Suppose that the linear mapT : V → A is an O -operator associated to ( V , ρ ) . Then there exists a -pre-Lie algebra structure on V givenby { u , v , w } = ρ ( Tu , Tv ) w , ∀ u , v , w ∈ V . (1.10) Corollary 1.7.
With the above conditions, ( V , [ · , · , · ] C ) is a -Lie algebra as the sub-adjacent -Lie algebraof the -pre-Lie algebra given in Proposition 1.6, and T is a -Lie algebra morphism from ( V , [ · , · , · ] C ) to ( A , [ · , · , · ]) . Furthermore, T ( V ) = { Tv | v ∈ V } ⊂ A is a -Lie subalgebra of A and there is an induced -pre-Lie algebra structure {· , · , ·} T ( V ) on T ( V ) given by { Tu , Tv , Tw } T ( V ) : = T { u , v , w } , ∀ u , v , w ∈ V . (1.11) Proposition 1.8.
Let ( A , [ · , · , · ]) be a -Lie algebra. Then there exists a compatible -pre-Lie algebra if andonly if there exists an invertible O -operator on A. Definition 1.6. [19] A representation of a 3-pre-Lie algebra ( A , {· , · , ·} ) on a vector space V consistsof a pair ( l , r ), where l : ∧ A → gl ( V ) is a representation of the 3-Lie algebra A c on V and r : ⊗ A → gl ( V ) is a linear map such that for all x , x , x , x ∈ A , the following equalities hold: l ( x , x ) r ( x , x ) = r ( x , x ) µ ( x , x ) + r ([ x , x , x ] C , x ) + r ( x , { x , x , x } ) , (1.12) r ([ x , x , x ] C , x ) = l ( x , x ) r ( x , x ) + l ( x , x ) r ( x , x ) + l ( x , x ) r ( x , x ) , (1.13) r ( x , { x , x , x } ) = r ( x , x ) µ ( x , x ) − r ( x , x ) µ ( x , x ) + l ( x , x ) r ( x , x ) , (1.14) r ( x , x ) µ ( x , x ) = l ( x , x ) r ( x , x ) − r ( x , { x , x , x } ) + r ( x , { x , x , x } ) , (1.15)where µ ( x , y ) = l ( x , y ) + r ( x , y ) − r ( y , x ), for any x , y ∈ A .Define the left multiplication L : ∧ A −→ gl ( A ) by L ( x , y ) z = { x , y , z } for all x , y , z ∈ A . Then( A , L ) is a representation of the 3-Lie algebra A c . Moreover, we define the right multiplication R : ⊗ A → gl ( A ) by R ( x , y ) z = { z , x , y } . It is obvious that ( A , L , R ) is a representation of a 3-pre-Liealgebra on itself, which is called the adjoint representation. Proposition 1.9.
Let ( A , {· , · , ·} ) be a -pre-Lie algebra, V a vector space and l , r : ⊗ A → gl ( V ) two linearmaps. Then ( V , l , r ) is a representation of A if and only if there is a -pre-Lie algebra structure ( called thesemi-direct product ) on the direct sum A ⊕ V of vector spaces, defined by [ x + u , x + u , x + u ] A ⊕ V = { x , x , x } + l ( x , x ) u − r ( x , x ) u + r ( x , x ) u , (1.16) for x i ∈ A , u i ∈ V , ≤ i ≤ . We denote this semi-direct product -Lie algebra by A ⋉ l , r V . V be a vector space. Define the switching operator τ : ⊗ V −→ ⊗ V by τ ( T ) = x ⊗ x , ∀ T = x ⊗ x ∈ ⊗ V . Proposition 1.10.
Let ( V , l , r ) be a representation of a -pre-Lie algebra ( A , {· , · , ·} ) . Then l − r τ + r is arepresentation of the sub-adjacent -Lie algebra ( A c , [ · , · , · ] C ) on the vector space V. Proposition 1.11.
Let ( l , r ) be a representation of a -pre-Lie algebra ( A , {· , · , ·} ) on a vector space V. Then ( l ∗ − r ∗ τ + r ∗ , − r ∗ ) is a representation of the -pre-Lie algebra ( A , {· , · , ·} ) on the vector space V ∗ , which iscalled the dual representation of the representation ( V , l , r ) . If ( l , r ) = ( L , R ) is the adjoint representation of a 3-pre-Lie algebra ( A , {· , · , ·} ), then we obtain( l ∗ − r ∗ τ + r ∗ , − r ∗ ) = ( ad ∗ , − R ∗ ). Definition 1.7.
Let ( A , {· , · , ·} ) be a 3-pre-Lie algebra and ( V , l , r ) be a representation. A linearoperator T : V → A is called and O -operator associated to ( V , l , r ) if T satisfies { Tu , Tv , Tw } = T ( l ( Tu , Tv ) w − r ( Tu , Tw ) v + r ( Tv , Tw ) u ) , ∀ u , v , w ∈ V . (1.17)If V = A , then T is called a Rota-Baxter operator on A of weight zero. That is[ R ( x ) , R ( y ) , R ( z )] = R (cid:16) [ R ( x ) , R ( y ) , z ] + [ R ( x ) , y , R ( z )] + [ x , R ( y ) , R ( z )] (cid:17) , for all x , y , z ∈ A . Example 1.12.
Let the -dimensional -pre-Lie algebra given in Example 1.2. Define R : A → A byR ( e ) = e + e , R ( e ) = e + e , R ( e ) = R ( e ) = . By a direct computation, we can verify that R is a Rota-Baxter operator. -L-dendriform algebras In this section, we introduce the notion of a 3-L-dendriform algebra which is exactly the ternaryversion of a L-dendriform algebra. We provide some construction results in terms of O -operatorand symplectic structure. Definition 2.1.
Let A be a vector space with two linear maps տ , ր : ⊗ A → A . The tuple ( A , տ , ր )is called a 3-L-dendriform algebra if the following identities hold տ ( x , x , x ) + տ ( x , x , x ) = , (2.1) տ ( x , x , տ ( x , x , x )) − տ ( x , x , տ ( x , x , x )) = տ ([ x , x , x ] C , x , x ) − տ ([ x , x , x ] C , x , x ) , (2.2) տ ( x , x , ր ( x , x , x )) − ր ( x , x , { x , x , x } h ) = ր ( x , [ x , x , x ] C , x ) + ր ( { x , x , x } v , x , x ) , (2.3) ր ( x , x , { x , x , x } h ) − տ ( x , x , ր ( x , x , x )) = ր ( { x , x , x } v , x , x ) − ր ( { x , x , x } v , x , x ) , (2.4) տ ([ x , x , x ] C , x , x ) = (cid:9) , , տ ( x , x , տ ( x , x , x )) , (2.5)5 ( x , [ x , x , x ] C , x ) = (cid:9) , , տ ( x , x , ր ( x , x , x )) , (2.6) տ ( x , x , ր ( x , x , x )) + ր ( x , x , { x , x , x } h ) = ր ( x , x , { x , x , x } h ) + ր ( { x , x , x } v , x , x ) , (2.7)for all x i ∈ A , 1 ≤ i ≤
5, where { x , y , z } h = տ ( x , y , z ) + ր ( x , y , z ) − ր ( y , x , z ) , (2.8) { x , y , z } v = տ ( x , y , z ) + ր ( z , x , y ) − ր ( z , y , x ) , (2.9)[ x , y , z ] C = (cid:9) x , y , z { x , y , z } h = (cid:9) x , y , z { x , y , z } v , (2.10)for any x , y , z ∈ A . Remark . Let ( A , տ , ր ) be a 3-L-dendriform algebra. if ր =
0, then ( A , տ ) is a 3-pre-Lie algebra. Proposition 2.1.
Let ( A , տ , ր ) be a -L-dendriform algebra.1. The bracket given in (2.8) defines a -pre-Lie algebra structure on A which is called the associatedhorizontal -pre-Lie algebra of ( A , տ , ր ) and ( A , տ , ր ) is also called a compatible -L-dendriformalgebra structure on the -pre-Lie algebra ( A , {· , · , ·} h ) .2. The bracket given in (2.9) defines a -pre-Lie algebra structure on A which is called the associatedvertical -pre-Lie algebra of ( A , տ , ր ) and ( A , տ , ր ) is also called a compatible -L-dendriformalgebra structure on the -pre-Lie algebra ( A , {· , · , ·} v ) .Proof. We will just prove item 1. Note, first that { x , y , z } h = −{ y , x , z } h and { x , y , z } v = −{ y , x , z } v , forany x , y , z ∈ A .Let x i ∈ A , ≤ i ≤
5. Then { x , x , { x , x , x } h } h − { x , x , { x , x , x } h } h − { [ x , x , x ] C , x , x } h + { [ x , x , x ] C , x , x } h = r + r + r + r + r , where r = տ ( x , x , տ ( x , x , x )) − տ ( x , x , տ ( x , x , x )) − տ ([ x , x , x ] C , x , x ) + տ ([ x , x , x ] C , x , x ) , r = ր ( x , [ x , x , x ] C , x ) + ր ( x , x , { x , x , x } h ) − տ ( x , x , ր ( x , x , x )) + ր ( { x , x , x } v , x , x ) , r = ր ( x , [ x , x , x ] C , x ) + ր ( x , x , { x , x , x } h ) − տ ( x , x , ր ( x , x , x )) + ր ( { x , x , x } v , x , x ) , r = ր ( x , x , { x , x , x } h ) − տ ( x , x , ր ( x , x , x )) − ր ( { x , x , x } v , x , x ) + ր ( { x , x , x } v , x , x ) , r = ր ( x , x , { x , x , x } h ) − տ ( x , x , ր ( x , x , x )) − ր ( { x , x , x } v , x , x ) + ր ( { x , x , x } v , x , x ) . From identities (2.2)-(2.4), we obtain immediately r i = , ∀ ≤ i ≤
5. This imply that (1.6) hold.On the other hand, we have { [ x , x , x ] C , x , x } h − { x , x , { x , x , x } h } h − { x , x , { x , x , x } h } h − { x , x , { x , x , x } h } h s + s + s + s + s , where s = տ ([ x , x , x ] C , x , x ) − (cid:9) , , տ ( x , x , տ ( x , x , x )) , s = ր ( x , [ x , x , x ] C , x ) − (cid:9) , , տ ( x , x , ր ( x , x , x )) , s = տ ( x , x , ր ( x , x , x )) + ր ( x , x , { x , x , x } h ) − ր ( x , x , { x , x , x } h ) − ր ( { x , x , x } v , x , x ) , s = տ ( x , x , ր ( x , x , x )) + ր ( x , x , { x , x , x } h ) − ր ( x , x , { x , x , x } h ) − ր ( { x , x , x } v , x , x ) , s = տ ( x , x , ր ( x , x , x )) + ր ( x , x , { x , x , x } h ) − ր ( x , x , { x , x , x } h ) − ր ( { x , x , x } v , x , x ) . From identities (2.5)-(2.7), we obtain immediately s i = , ∀ ≤ i ≤
5. This imply that (1.7) hold. (cid:3)
Corollary 2.2.
Let ( A , տ , ր ) be a -L-dendriform algebra. Then the bracket defined in (2.10) defines a -Lie algebra structure on A which is called the associated -Lie algebra of ( A , տ , ր ) . The following Proposition is obvious.
Proposition 2.3.
Let ( A , տ , ր ) be a -L-dendriform algebra. Define L տ , R ր : ⊗ A → gl ( A ) byL տ ( x , y ) z = տ ( x , y , z ) , R ր ( x , y ) z = ր ( z , x , y ) , ρ ( x , y ) z = տ ( x , y , z ) + ր ( z , x , y ) − ր ( z , y , x ) for all x , y , z ∈ A . Then(1) ( A , L տ , R ր ) is a representation of its horizontal associated -pre-Lie algebra ( A , {· , · , ·} h ) .(2) ( A , L տ ) is a representation of its associated -Lie algebra ( A , [ · , · , · ] C ) .(3) ( A , ρ ) is a representation of its associated -Lie algebra ( A , [ · , · , · ] C ) .Remark . In the sense of the above conclusion (1), a 3-L-dendriform algebra is understood asa ternary algebra structure whose left and right multiplications give a bimodule structure onthe underlying vector space of the 3-pre-Lie algebra defined by certain commutators. It can beregarded as the rule of introducing the notion of 3-L-dendriform algebra, which more generally, isthe rule of introducing the notions of 3-pre-Lie algebras, the Loday algebras and their Lie,Jordanand alternative algebraic analogues.
Theorem 2.4.
Let ( A , {· , · , ·} ) be a -pre-Lie algebra and ( V , l , r ) be a representation. Suppose that T : V → Ais an O -operator associated to ( V , l , r ) . Then there exists a -L-dendriform algebra structure on V given by տ ( u , v , w ) = l ( Tu , Tv ) w , ր ( u , v , w ) = r ( Tv , Tw ) u , ∀ u , v , w ∈ V . (2.11) Therefore, there exists two associated -pre-Lie algebra structures on V and T is a homomorphism of -pre-Lie algebras. Moreover, T ( V ) = { T ( v ) | v ∈ V } is -pre-Lie subalgebra of ( A , {· , · , ·} ) and there is an induced -L-dendriform algebra structure on T ( V ) given by տ ( Tu , Tv , Tw ) = T ( տ ( u , v , w )) , ր ( Tu , Tv , Tw ) = T ( ր ( u , v , w )) , ∀ u , v , w ∈ V . (2.12)7 roof. Let u , v , w ∈ V . Define {· , · , ·} hV , {· , · , ·} vV , [ · , · , · ] CV : ⊗ V → V by { u , v , w } hV = տ ( u , v , w ) + ր ( u , v , w ) − ր ( v , u , w ) , { u , v , w } vV = տ ( u , v , w ) + ր ( w , u , v ) − ր ( w , v , u ) , [ u , v , w ] CV = (cid:9) u , v , w { u , v , w } hV . Using identity (1.17), we have T { u , v , w } hV = T ( տ ( u , v , w ) + ր ( u , v , w ) − ր ( v , u , w )) = T ( l ( Tu , Tv ) w − r ( Tu , Tw ) v + r ( Tv , Tw ) u ) = { Tu , Tv , Tw } h and T [ u , v , w ] CV = (cid:9) u , v , w T { u , v , w } V = (cid:9) u , v , w { Tu , Tv , Tw } h = [ Tu , Tv , Tw ] C . It is straightforward that տ ( u , v , w ) + տ ( v , u , w ) = ( l ( Tu , Tv ) + l ( Tv , Tu )) w = . Furthermore, for any u i ∈ V , ≤ i ≤
5, we have տ ( u , u , տ ( u , u , u )) − տ ( u , u , տ ( u , u , u )) − տ ([ u , u , u ] CV , u , u ) + տ ([ u , u , u ] CV , u , u ) = l ( T ( u ) , T ( u )) l ( T ( u ) , T ( u )) u − l ( T ( u ) , T ( u )) l ( T ( u ) , T ( u )) u − l ( T [ u , u , u ] CV , T ( u )) u + l ( T [ u , u , u ] CV , T ( u )) u = (cid:16) [ l ( T ( u ) , T ( u )) , l ( T ( u ) , T ( u ))] − l ([ T ( u ) , T ( u ) , T ( u )] C , T ( u )) + l ([ T ( u ) , T ( u ) , T ( u )] C , T ( u )) (cid:17) u = . This implies that (2.2) holds. Moreover, (2.3) holds. Indeed, ր ( u , [ u , u , u ] CV , u ) − ր ( u , u , { u , u , u } hV ) − տ ( u , u , ր ( u , u , u )) + ր ( { u , u , u } vV , u , u ) = r ( T [ u , u , u ] CV , T ( u )) u − r ( T ( u ) , T { u , u , u } hV ) u − l ( T ( u ) , T ( u )) r ( T ( u ) , T ( u )) u + r ( T ( u ) , T ( u )) { u , u , u } vV = r ([ Tu , Tu , Tu ] C , T ( u )) u − r ( T ( u ) , { T ( u ) , T ( u ) , T ( u ) } h ) u − l ( T ( u ) , T ( u )) r ( T ( u ) , T ( u )) u + r ( T ( u ) , T ( u )) µ ( T ( u ) , T ( u )) u = տ ( u , u , ր ( u , u , u )) + ր ( u , u , { u , u , u } hV ) − ր ( u , u , { u , u , u } hV ) − ր ( { u , u , u } vV , u , u ) = l ( Tu , Tu ) r ( Tu , Tu ) u + r ( Tu , { Tu , Tu , Tu } h ) u − r ( Tu , { Tu , Tu , Tu } h ) u − r ( Tu , Tu ) µ ( Tu , Tu ) u = . The other conclusions follow immediately. (cid:3) orollary 2.5. Let ( A , {· , · , ·} ) be a -pre-Lie algebra and R : A → A is a Rota-Baxter operator of weight .Then there exists a -L-dendriform algebra structure on A given by տ ( x , y , z ) = { R ( x ) , R ( y ) , z } , ր ( x , y , z ) = { x , R ( y ) , R ( z ) } , (2.13) for all x , y , z ∈ A. Example 2.6.
Let the -dimensional -pre-Lie algebra given in Example 1.3. Define R : A → A byR ( e ) = e , R ( e ) = − e , R ( e ) = e , R ( e ) = e . Then R is a Rota-Baxter operator. Using Corollary 2.5, we can construct a -Lie dendriform algebra givenby the structures տ , ր : A ⊗ A ⊗ A → A defined in the basis ( e , e , e , e ) , by տ ( e , e , e ) = e , տ ( e , e , e ) = e , ր ( e , e , e ) = ր ( e , e , e ) = e , ր ( e , e , e ) = ր ( e , e , e ) = e , ր ( e , e , e ) = − ր ( e , e , e ) = − e , ր ( e , e , e ) = − ր ( e , e , e ) = − e and all the other products are zero. Proposition 2.7.
Let ( A , {· , · , ·} ) be a -pre-Lie algebra. Then there exists a compatible 3-L-dendriformalgebra if and only if there exists an invertible O -operator on A.Proof. Let T be an invertible O -operator of A associated to a representation ( V , l , r ). Then thereexists a 3-L-dendriform algebra structure on V defined by տ ( u , v , w ) = l ( Tu , Tv ) w , ր ( u , v , w ) = r ( Tv , Tw ) u , ∀ u , v , w ∈ V . (2.14)In addition there exists a 3-L-dendriform algebra structure on T ( V ) = A given by տ ( Tu , Tv , Tw ) = T ( l ( Tu , Tv ) w ) , ր ( Tu , Tv , Tw ) = T ( r ( Tv , Tw ) u ) , ∀ u , v , w ∈ V . (2.15)If we put x = Tu , y = Tv and z = Tw , we get տ ( x , y , z ) = T ( l ( x , y ) T − ( z )) , ր ( x , y , z ) = T ( r ( y , z ) T − ( x )) , ∀ w , y , z ∈ A . (2.16)It is a compatible 3-L-dendriform algebra structure on A . Indeed, տ ( x , y , z ) + ր ( x , y , z ) − ր ( y , x , z ) = T ( l ( x , y ) T − ( z )) + T ( r ( y , z ) T − ( x )) − T ( r ( x , z ) T − ( y )) = { TT − ( x ) , TT − ( y ) , TT − ( z ) } = { x , y , z } . Conversely, let ( A , {· , · , ·} ) be a 3-pre-Lie algebra and ( A , տ , ր ) its compatible 3-L-dendriformalgebra. Then the identity map id : A → A is an O -operator of ( A , {· , · , ·} ) associated to ( A , L , R ). (cid:3) Definition 2.2.
Let ( A , {· , · , ·} ) be a 3-pre-Lie algebra and B be a skew-symmetric bilinear form on A . We say that B is closed if it satisfies B ( { x , y , z } , w ) − B ( z , [ x , y , w ] C ) − B ( y , { w , x , z } ) + B ( x , { w , y , z } ) = , (2.17)for any x , y , z , w ∈ A . If in addition B is nondegenerate , then B is called symplectic.A 3-pre-Lie algebra ( A , {· , · , ·} ) equipped with a symplectic form is called a symplectic 3-pre-Liealgebra and denoted by ( A , {· , · , ·} , B ). 9 roposition 2.8. Let ( A , {· , · , ·} , B ) be a symplectic -pre-Lie algebra. Then there exists a compatible -L-dendriform algebra structure on A given byB ( տ ( x , y , z ) , w ) = B ( z , [ x , y , w ] C ) , B ( ր ( x , y , z ) , w ) = − B ( x , { w , y , z } ) , ∀ x , y , z , w ∈ A . (2.18) Proof.
Define the linear map T : A ∗ → A by h T − x , y i = B ( x , y ). Using Eq. (2.17), we obtain that T isan invertible O -operator on A associated to the dual representation ( A ∗ , ad ∗ , − R ∗ ). By Proposition2.7, there exists a compatible 3-L-dendriform algebra structure given by տ ( x , y , z ) = T ( ad ∗ ( x , y ) T − ( z )) , ր ( x , y , z ) = − T ( R ∗ ( y , z ) T − ( x )) , ∀ w , y , z ∈ A . (2.19)Then we have B ( տ ( x , y , z ) , w ) = B ( T ( ad ∗ ( x , y ) T − ( z )) , w ) = h ad ∗ ( x , y ) T − ( z ) , w i = h T − ( z ) , [ x , y , w ] C i = B ( z , [ x , y , w ] C )and B ( ր ( x , y , z ) , w ) = − B ( T ( R ∗ ( y , z ) T − ( x ))) , w ) = −h R ∗ ( y , z ) T − ( x ) , w i = − h T − ( x ) , { w , y , z }i = − B ( x , { w , y , z } ) . The proof is finished. (cid:3)
Corollary 2.9.
Let ( A , {· , · , ·} , B ) be a symplectic -pre-Lie algebra and let ( A , [ · , · , · ] C ) be its associated -Liealgebra. Then there exists a -pre-Lie algebraic structure ( A , {· , · , ·} ′ ) on A given byB ( { x , y , z } ′ , w ) = B ( z , [ x , y , w ] C ) − B ( z , { w , x , y } ) + B ( z , { w , y , x } ) = . (2.20) Lemma 2.10.
Let { R , R } be a pair of of commuting Rota-Baxter operators (of weight zero) on a -Liealgebra ( A , [ · , · , · ]) . Then R is a Rota-Baxter operator (of weight zero) on the associated -pre-Lie algebradefined by { x , y , z } = [ R ( x ) , R ( y ) , z ] .Proof. For any x , y , z ∈ A , we have { R ( x ) , R ( y ) , R ( z ) } = [ R R ( x ) , R R ( y ) , R ( z )] = [ R R ( x ) , R R ( y ) , R ( z )] = R ([ R R ( x ) , R R ( y ) , z ] + [ R ( x ) , R R ( y ) , R ( z )] + [ R R ( x ) , R ( y ) , R ( z )]) = R ( { R ( x ) , R ( y ) , z } + { x , R ( y ) , R ( z ) } + { R ( x ) , y , R ( z ) } ) . Hence R is a Rota-Baxter operator (of weight zero) on the 3-pre-Lie algebra ( A , {· , · , ·} ). (cid:3) Proposition 2.11.
Let { R , R } be a pair of of commuting Rota-Baxter operators (of weight zero) on a -Liealgebra ( A , [ · , · , · ]) . Then there exists a -L-dendriform algebra structure on A defined by տ ( x , y , z ) = [ R R ( x ) , R R ( y ) , z ] , ր ( x , y , z ) = [ R ( x ) , R R ( y ) , R ( z )] , ∀ x , y , z ∈ A . Proof.
It follows immediately from Lemma 2.10 and Corollary 2.5. (cid:3) emark . Let ( A , [ · , · ]) be a Lie-algebra. Recall that a trace function τ : A → K is a linear mapsuch that τ ([ x , y ]) = , ∀ x , y ∈ A . When τ is a trace function, it is well known [2] that ( A , [ · , · , · ] τ ) isa 3-Lie algebra, where [ x , y , z ] τ : = (cid:9) x , y , z ∈ A τ ( x )[ y , z ] , ∀ x , y , z ∈ A . Now let { R , R } be a pair of of commuting Rota-Baxter operators (of weight zero) on the 3-Liealgebra ( A , [ · , · , · ] τ ). Then we can construct a 3-L-dendriform structure on A , given by տ ( x , y , z ) = τ ( R R ( x ))[ R R ( y ) , z ] + τ ( R R ( y ))[ z , R R ( x )] + + τ ( z )[ R R ( x ) , R R ( y )] , ր ( x , y , z ) = τ ( R ( x ))[ R R ( y ) , R ( z )] + τ ( R R ( y ))[ R ( z ) , R ( x )] + τ ( R ( z ))[ R ( x ) , R R ( y )] , for any x , y , z ∈ A . -L-dendriform algebras This section is devoted to investigate generalized derivations of 3-Lie algebras, 3-pre-Lie algebrasand 3-L-dendriform algebras. Throughout the sequel ( A , < · , · , · > ) denotes a 3-(pre)-Lie algebra. Definition 3.1.
A linear map D : A → A is said to be a derivation on A , if it satisfies the followingcondition (for x , y , z ∈ A ) D ( < x , y , z > ) = < D ( x ) , y , z > + < x , D ( y ) , z > + < x , y , D ( z ) > . (3.1) Definition 3.2.
Let ( A , տ , ր ) be a 3-L-dendriform algebra. A linear map D : A → A is said to be aderivation on A , if it satisfies the following condition ( for x , y , z ∈ A ) D ( տ ( x , y , z )) = տ ( D ( x ) , y , z ) + տ ( x , D ( y ) , z ) + տ ( x , y , D ( z )) , (3.2) D ( ր ( x , y , z )) = ր ( D ( x ) , y , z ) + ր ( x , D ( y ) , z ) + ր ( x , y , D ( z )) . (3.3)We denote the set of all derivations on A by Der ( A ) . We can easily show that Der ( A ) is equippedwith a Lie algebra structure. In fact, for D ∈ Der ( A ) and D ′ ∈ Der ( A ), we have [ D , D ′ ] ∈ Der ( A ),where [ D , D ′ ] is the standard commutator defined by [ D , D ′ ] = DD ′ − D ′ D . Definition 3.3.
A linear mapping D ∈ End ( A ) is said to be a quasi-derivation of A if there existlinear mapping D ′ ∈ End ( A ) such that D ′ ( < x , y , z > ) = < D ( x ) , y , z > + < x , D ( y ) , z > + < x , y , D ( z ) >, (3.4)for all x , y , z ∈ A . We say that D associates with D ′ . Definition 3.4.
Let ( A , տ , ր ) be a 3-L-dendriform algebra. A linear map D : A → A is said to be aquasi-derivation on A , if it satisfies the following condition D ′ ( տ ( x , y , z )) = տ ( D ( x ) , y , z ) + տ ( x , D ( y ) , z ) + տ ( x , y , D ( z )) , (3.5) D ′ ( ր ( x , y , z )) = ր ( D ( x ) , y , z ) + ր ( x , D ( y ) , z ) + ր ( x , y , D ( z )) , (3.6)for all x , y , z ∈ A . We say that D associates with D ′ . Definition 3.5.
A linear mapping D ∈ End ( A ) is said to be a generalized derivation of A if thereexist linear mappings D ′ , D ′′ , D ′′′ ∈ End ( A ) such that D ′′′ ( < x , y , z > ) = < D ( x ) , y , z > + < x , D ′ ( y ) , z > + < x , y , D ′′ ( z ) >, (3.7)for all x , y , z ∈ A . We also say that D associates with ( D ′ , D ′′ , D ′′′ ).11 efinition 3.6. A linear map D : A → A is said to be a generalized derivation of a 3-L-dendriformalgebra ( A , տ , ր ), if it satisfies the following condition D ′′′ ( տ ( x , y , z )) = տ ( D ( x ) , y , z ) + տ ( x , D ′ ( y ) , z ) + տ ( x , y , D ′′ ( z )) , (3.8) D ′′′ ( ր ( x , y , z )) = ր ( D ( x ) , y , z ) + ր ( x , D ′ ( y ) , z ) + ր ( x , y , D ′′ ( z )) , (3.9)for all x , y , z ∈ A . We say that D associates with ( D ′ , D ′′ , D ′′′ ).The sets of quasi-derivations and generalized derivations will be denoted by QDer ( A ) and GDer ( A ) , respectively. It is easy to see that Der ( A ) ⊂ QDer ( A ) ⊂ GDer ( A ) . Definition 3.7.
A linear map Θ ∈ End ( A ) is said to be a centroid of A if Θ ( < x , y , z > ) = < Θ ( x ) , y , z > = < x , Θ ( y ) , z > = < x , y , Θ ( z ) >, ∀ x , y , z ∈ A . (3.10) Definition 3.8.
A linear map Θ : A → A is said to be a Centroid of a 3-L-dendriform algebra( A , տ , ր ), if it satisfies the following conditions: Θ ( տ ( x , y , z )) = տ ( Θ ( x ) , y , z ) = տ ( x , Θ ( y ) , z ) = տ ( x , y , Θ ( z )) , (3.11) Θ ( ր ( x , y , z )) = ր ( Θ ( x ) , y , z ) = ր ( x , Θ ( y ) , z ) = ր ( x , y , Θ ( z )) , ∀ x , y , z ∈ A . (3.12)The set of centroids of A is denoted by C ( A ). Definition 3.9.
A linear map Θ ∈ End ( A ) is said to be a quasi-centroid of A if < Θ ( x ) , y , z > = < x , Θ ( y ) , z > = < x , y , Θ ( z ) >, ∀ x , y , z ∈ A . (3.13) Definition 3.10.
A linear map Θ : A → A is said to be a quasi-Centroid of a 3-L-dendriform algebra( A , տ , ր ), if it satisfies the following conditions: տ ( Θ ( x ) , y , z ) = տ ( x , Θ ( y ) , z ) = տ ( x , y , Θ ( z )) , (3.14) ր ( Θ ( x ) , y , z ) = ր ( x , Θ ( y ) , z ) = ր ( x , y , Θ ( z )) , ∀ x , y , z ∈ A . (3.15)The set of quasi-centroids of A is denoted by QC ( A ). It is obvious that C ( A ) ⊂ QC ( A ). Proposition 3.1.
Let ( A , տ , ր ) be a -L-dendriform algebra, D ∈ Der ( A ) and Θ ∈ C ( A ) . Then [ D , Θ ] ∈ C ( A ) . Proof.
Assume that D ∈ Der ( A ) , Θ ∈ C ( A ). For arbitrary x , y ∈ A , we have D Θ ( տ ( x , y , z )) = D ( տ ( Θ ( x ) , y , z )) = տ ( D Θ ( x ) , y , z ) + տ ( Θ ( x ) , D ( y ) , z ) + տ ( Θ ( x ) , y , D ( z )) (3.16)and Θ D ( տ ( x , y , z )) = Θ ( տ ( D ( x ) , y , z ) + տ ( Θ ( x ) , D ( y ) , z ) + տ ( Θ ( x ) , y , D ( z ))) = տ ( Θ D ( x ) , y , z ) + տ ( Θ ( x ) , D ( y ) , z ) + տ ( Θ ( x ) , y , D ( z )) . (3.17)12y making the di ff erence of equations (3.16) and (3.17), we get[ D , Θ ]( տ ( x , y , z )) = տ ([ D , Θ ]( x ) , y , z ) . Similarly we can proof that, for all x , y , z ∈ A , [ D , Θ ]( տ ( x , y , z )) = տ ( x , [ D , Θ ]( y ) , z ) , [ D , Θ ]( տ ( x , y , z > )) = տ ( x , y , [ D , Θ ]( z ))and [ D , Θ ]( ր ( x , y , z )) = ր ([ D , Θ ]( x ) , y , z ) = ր ( x , [ D , Θ ]( y ) , z ) = ր ( x , y , [ D , Θ ]( z )) . (cid:3) Proposition 3.2. C ( A ) ⊆ QDer ( A ) .Proof. straightforward. (cid:3) Proposition 3.3.
Let ( A , տ , ր ) be a -L-Dendriform algebra and D ∈ GDer ( A ) associates with ( D ′ , D ′′ , D ′′′ ) .Then D is a generalized derivation of associated horizontal -pre-Lie algebras ( A , {· , · , ·} h ) and vertical -pre-Lie algebras ( A , {· , · , ·} v ) defined in Proposition 2.1 associates with ( D ′ , D ′′ , D ′′′ ) .Proof. Let x , y , z ∈ A and using definition of bracket {· , · , ·} h in (2.8), we have D ′′′ ( { x , y , z } h ) = D ′′′ ( տ ( x , y , z ) + ր ( x , y , z ) − ր ( y , x , z )) = տ ( D ( x ) , y , z ) + տ ( x , D ′ ( y ) , z ) + տ ( x , y , D ′′ ( z )) + ր ( D ( x ) , y , z ) + ր ( x , D ′ ( y ) , z ) + ր ( x , y , D ′′ ( z )) − ր ( D ( y ) , x , z ) − ր ( y , D ′ ( x ) , z ) − ր ( y , x , D ′′ ( z )) = { D ( y ) , x , z } h + { y , D ′ ( x ) , z } h + { y , x , D ′′ ( z ) } h . Similarly, we can proof that D ′′′ ( { x , y , z } v ) = { D ( y ) , x , z } v + { y , D ′ ( x ) , z } v + { y , x , D ′′ ( z ) } v . (cid:3) Proposition 3.4.
Let ( A , {· , · , ·} ) be a -pre-Lie algebra and D : A → A be a generalized derivation on Aassociates with ( D ′ , D ′′ , D ′′′ ) and let R : A → A be a Rota-Baxter operator of weight commuting withD , D ′ , D ” and D ′ ” . Then D is a generalized derivation of the compatible -L-dendriform algebra defined inCorollary 2.5 associates with ( D ′ , D ′′ , D ′′′ ) .Proof. Let x , y , z ∈ A . Using the definition of structures տ , ր given in (2.13), we have D ′′′ ( տ ( x , y , z )) = D ′′′ ( { Rx , Ry , z } ) = { D ( Rx ) , Ry , z } + { Rx , D ′ R ( y ) , z } + { Rx , Ry , D ′′ ( z ) } = տ ( D ( x ) , y , z ) + տ ( x , D ′ ( y ) , z ) + տ ( x , y , D ′′ ( z ))and D ′′′ ( ր ( x , y , z )) = D ′′′ ( { x , Ry , Rz } ) = { D ( x ) , Ry , Rz } + { x , D ′ R ( y ) , Rz } + { x , Ry , D ′′ ( Rz ) } = ր ( D ( x ) , y , z ) + ր ( x , D ′ ( y ) , z ) + ր ( x , y , D ′′ ( z )) . (cid:3) roposition 3.5. Let ( A , տ , ր ) be a -L-dendriform algebra. Thena) D ( A ) ⊕ C ( A ) ⊂ QD ( A ) .b) QD ( A ) + QC ( A ) ⊂ GD ( A ) .Proof. Straightforward. (cid:3)
References [1] M. Aguiar and J.-L. Loday,
Quadri-algebras , J. Pure Applied Algebra, 191, (2004), 205-221.[2] J. Arnlind, A.Makhlouf and S. Silvestrov,
Ternary Hom-Nambu-Lie algebras induced by Hom-Liealgebras , J. Math. Phys., 51(4) (2010), 043515, 11 pp[3] C.M. Bai, X. Ni,
Pre-alternative algebras and pre-alternative bialgebras , Pacific J. Math. (2010)355-390.[4] C. Bai, G. Li and Y. Sheng.
Bialgebras, the classical Yang-Baxter equation and Manin triples for3-Lie algebras . Advances in Theoretical and Mathematical Physics 23.1 (2019): 27-74.[5] D. Burde,
Left-symmetric algebras, or pre-Lie algebras in geometry and physics . Cent. Eur. J. Math.4, 323-357 (2006).[6] 6 G. Baxter, An analytic problem whose solution follows from a simple algebraic identity,Pac. J. Math. 10, 731742 (1960).[7] H. Dongping and C. Bai,
J-dendriform algebras, Frontiers of Mathematics in China
Pre-Jordan algebras , Mathematica Scandinavica (2013) 19-48.[9] A. S. Dzhumadil ′ daev, Representations of vector product n-Lie algebras , Comm. Alg.
32 (2004),3315-3326.[10] K. Ebrahimi-Fard, L. Guo,
Unit actions on operads and Hopf algebras . Theory Appl. Categ. 18,348-371 (2007).[11] V. T. Filippov, n-Lie algebras , Sib. Mat. Zh.
26 (1985), 126-140.[12] R. Holtkamp,
On Hopf algebra structures over operads . Adv. Math. 207, 544-565 (2006)[13] S. M. Kasymov,
Theory of n-Lie algebras . Algebra i Logika 26 (1987), no. 3, 277-297. (Englishtranslation:
Algebra and Logic
26 (1988), 155-166).[14] B. A. Kupershmidt,
What a classical r-matrix really is , J. Nonlinear Math. Phys. 6, 448-488(1999).[15] J.-L. Loday,
Dialgebras, in “Dialgebras and related operads” , Lecture Notes in Math., ,Springer, Berlin, (2001) 7-66.[16] J.L. Loday, M.Ronco,
Trialgebras and families of polytopes, Homotopy Theory: Relations withAlgebraic Geometry, Group Cohomology, and Algebraic K-theory . Comtep. Math. 346, 369-398(2004) 1417] J.-L. Loday,
Scindement d’associativit´e et alg`ebres de Hopf , Proceeding of the Conference inhonor of Jean Leray, Nantes (2002), S´eminaire et Congr`es (SMF), (2004) 155-172.[18] J. Pei, C. Bai and G. Li. Splitting of operads and Rota-Baxter operators on operads . AppliedCategorical Structures 25.4 (2017): 505-538.[19] Y. Sheng and T. Rong .
Symplectic, product and complex structures on 3-Lie algebras . Journal ofAlgebra 508 (2018): 256-300.[20] J. Pei, C. Bai and G. Li.
Splitting of operads and Rota-Baxter operators on operads . AppliedCategorical Structures 25.4 (2017): 505-538.[21] B. Vallette,
Manin products, Koszul duality, Loday algebras and Deligne conjecture . J. Reine Angew.Math. 620, 105-164 (2008).[22] Zinbiel, G.W.: