Bimodules and matched pairs of noncommutative BiHom-(pre)-Poisson algebras
aa r X i v : . [ m a t h . R A ] F e b Bimodules and matched pairs of noncommutativeBiHom-(pre)-Poisson algebras
Ismail Laraiedh ∗ February 24, 2021
Abstract
The purpose of this paper is to introduce the notion of noncommutative BiHom-pre-Poisson algebra. Also we establish the bimodules and matched pairs of noncom-mutative BiHom-(pre)-Poisson algebras and related relevant properties are also given.Finally, we exploit the notion of O -operator to illustrate the relations existing betweennoncommutative BiHom-Poisson and noncommutative BiHom pre-Poisson algebras. Keywords:
Noncommutative BiHom-(pre)-Poisson algebras, Bimodules, Matched pairs, O -operator. Algebraic structure appeared in the Physics literature related to string theory, vertex modelsin conformal field theory, quantum mechanics and quantum field theory, such as the q-deformed Heisenberg algebras, q-deformed oscillator algebras, q-deformed Witt, q-deformedVirasoro algebras and related q-deformations of infinite-dimensional algebras [4, 14–20, 24,25, 33–35].Hom-type algebras satisfy a modified version of the Jacobi identity involving a homomor-phism, and were called Hom-Lie algebras by Hartwig, Larsson and Silvestrov in [22], [29].Afterwards, Hom-analogues of various classical algebraic structures have been introducedin the literature, such as Hom-associative algebras, Hom-dendriform algebras, Hom-pre-Liealgebras Hom-(pre)-Poisson algebras etc [1, 5–8, 10, 37–40, 43–48].The notion of a noncommutative Poisson algebra was first given by Xu in [42]. A non-commutative Poisson algebra consists of an associative algebra together with a Lie algebrastructure, satisfying the Leibniz identity. Noncommutative Poisson algebras are used in manyfields in mathematics and physics. Aguiar introduced the notion of a pre-Poisson algebra in [3] ∗ Departement of Mathematics, Faculty of Sciences, Sfax University, BP 1171, 3000 Sfax, Tunisia. E.mail:[email protected] and Departement of Mathematics, College of Sciences and Humanities - Kowaiyia,Shaqra University, Kingdom of Saudi Arabia. E.mail: [email protected] A, · , {· , ·} , α , α ) be a noncommutative BiHom-Poisson algebra. A noncommutativeBiHom-Poisson A -module V is simultaneously a BiHom-associative A -module ( λ, β , β , V )and a BiHom-Lie A -module ( ρ, β , β , V ) satisfying the BiHom-Leibniz conditions: ρ ( α α ( x ) , λ ( y, v )) = λ ( { α ( x ) , y } , β ( v )) + λ ( α ( y ) , ρ ( α ( x ) , v )) ,ρ ( µ ( α ( x ) , y ) , β ( v )) = λ ( α α ( x ) , ρ ( y, v )) + λ ( α ( y ) , ρ ( α ( x ) , v )) , for x, y ∈ A, v ∈ V , (for more details see Definition 5.3 in [9]). A noncommutative BiHom-pre-Poisson algebra gives rise to a noncommutative BiHom-Poisson algebra naturally throughthe sub-adjacent BiHom-associative algebra of the BiHom-dendriform algebra [31] and thesub-adjacent BiHom-Lie algebra of the BiHom-pre-Lie algebra [32]. We also introduce the no-tion of O -operators of noncommutative BiHom-Poisson algebra and we will prove that givena noncommutative BiHom-Poisson algebra and an O -operator give rise to a noncommutativeBiHom-pre-Poisson algebra. All that is illustrated by the following diagramBiHom-dendriform alg + BiHom-pre-Lie alg / / (cid:15) (cid:15) noncomm BiHom-pre-Poisson alg (cid:15) (cid:15) BiHom-associative alg + BiHom-Lie alg O O / / noncomm BiHom-Poisson alg O O The paper is organized as follows. In section 2, we introduce the notions of representa-tion and matched pair of noncommutative BiHom-Poisson algebra with a connection to arepresentations and matched pairs of BiHom-Lie algebra and BiHom-associative algebra. Insection 3, we establish definition of noncommutative BiHom-pre-Poisson algebra and we givesome key of constructions. Their bimodule and matched pair are defined and their relatedrelevant properties are also given. In section 4, we study the notion of O -operator and weillustrate the relations existing between noncommutative BiHom-Poisson and noncommuta-tive BiHom pre-Poisson algebras.Throughout this paper, all graded vector spaces are assumed to be over a field K ofcharacteristic different from 2. In this section we recall the definition of noncommutative BiHom-Poisson algebra [36] and westudy the representation and the matched pair of noncommutative BiHom-Poisson algebras2ith a connection to a representations and matched pairs of BiHom-associative algebra andBiHom-Lie algebra. Moreover we provide some key constructions.
Definition 2.1.
A BiHom-module is a triple (
V, α V , β V ) consisting of a K -vector space V andtwo linear maps α V , β V : V −→ V such that α V β V = β V α V . A morphism f : ( V, α V , β V ) → ( W, α W , β W ) of BiHom-modules is a linear map f : V −→ W such that f α V = α W f and f β V = β W f. Definition 2.2.
A BiHom-algebra is a quadruple (
A, µ, α , α ) in which ( A, α , α ) is aBiHom-module, µ : A ⊗ → A is a bilinear map. The BiHom-algebra ( A, µ, α , α ) is said tobe multiplicative if α ◦ µ = µ ◦ α ⊗ and α ◦ µ = µ ◦ α ⊗ (BiHom-multiplicativity).Let us recall now the definition and the notion of bimodule of a BiHom-associative givenin [21]. Definition 2.3.
A BiHom-associative algebra is a quadruple (
A, µ, α , α ) consisting of avector space A on which the operation µ : A ⊗ A → A and α , α : A → A are linear mapssatisfying α ◦ α = α ◦ α , (2.1) α ◦ µ ( x, y ) = µ ( α ( x ) , α ( y )) , (2.2) α ◦ µ ( x, y ) = µ ( α ( x ) , α ( y )) , (2.3) µ ( α ( x ) , µ ( y, z )) = µ ( µ ( x, y ) , α ( z )) , (2.4)for any x, y, z ∈ A . Remark . Clearly, a Hom-associative algebra (
A, µ, α ) can be regarded as a BiHom-associative algebra (
A, µ, α, α ). Definition 2.5. [21] Let (
A, µ, α , α ) be a BiHom-associative algebra. A left A -module isa triple ( M, β , β ), where M is a linear space, β , β : M → M are linear maps, with, inaddition, another linear map: A ⊗ M → M, a ⊗ m a · m, such that, for all a, a ′ ∈ A, m ∈ M : β ◦ β = β ◦ β , β ( a · m ) = α ( a ) · β ( m ) ,β ( a · m ) = α ( a ) · β ( m ) , α ( a ) · ( a ′ · m ) = ( aa ′ ) · β ( m ) . Definition 2.6.
Let ( A, · , α , α ) be a BiHom-associative algebra, and let ( V, β , β ) be aBiHom-module. Let l, r : A → gl ( V ) , be two linear maps. The quintuple ( l, r, β , β , V ) iscalled a bimodule of A if l ( x · y ) β ( v ) = l ( α ( x )) l ( y ) v, (2.5) r ( x · y ) β ( v ) = r ( α ( y )) r ( x ) v, (2.6) l ( α ( x )) r ( y ) v = r ( α ( y )) l ( x ) v, (2.7) β ( l ( x ) v ) = l ( α ( x )) β ( v ) , (2.8) β ( r ( x ) v ) = r ( α ( x )) β ( v ) , (2.9) β ( l ( x ) v ) = l ( α ( x )) β ( v ) , (2.10) β ( r ( x ) v ) = r ( α ( x )) β ( v ) , (2.11)for all x, y ∈ A, v ∈ V . 3 roposition 2.7. Let ( l, r, β , β , V ) be a bimodule of a BiHom-associative algebra ( A, · , α , α ) .Then, the direct sum A ⊕ V of vector spaces is turned into a BiHom-associative algebra bydefining multiplication in A ⊕ V by ( x + v ) · ′ ( x + v ) := x · x + ( l ( x ) v + r ( x ) v ) , ( α ⊕ β )( x + v ) := α ( x ) + β ( v ) , ( α ⊕ β )( x + v ) := α ( x ) + β ( v ) , for all x , x ∈ A, v , v ∈ V . We denote such a BiHom-associative algebra by ( A ⊕ V, · ′ , α + β , α + β ) , or A × l,r,α ,α ,β ,β V. Theorem 2.8. [23] Let ( A, · A , α , α ) and ( B, · B , β , β ) be two BiHom-associative algebras.Suppose that there are linear maps l A , r A : A → gl ( B ) and l B , r B : B → gl ( A ) such that ( l A , r A , β , β , B ) is a bimodule of A and ( l B , r B , α , α , A ) is a bimodule of B satisfy l A ( α ( x ))( a · B b ) = l A ( r B ( a ) x ) β ( b ) + ( l A ( x ) a ) · B β ( b ) (2.12) r A ( α ( x ))( a · B b ) = r A ( l B ( b ) x ) β ( a ) + β ( a ) · B ( r A ( x ) b ) (2.13) l A ( l B ( a ) x ) β ( b ) + ( r A ( x ) a ) · B β ( b ) − r A ( r B ( b ) x ) β ( a ) − β ( a ) · B ( l A ( x ) b ) = 0 (2.14) l B ( β ( a ))( x · A y ) = l B ( r A ( x ) a ) α ( y ) + ( l B ( a ) x ) · A α ( y ) (2.15) r B ( β ( a ))( x · A y ) = r B ( l A ( y ) a ) α ( x ) + α ( x ) · A ( r B ( a ) y ) (2.16) l B ( l A ( x ) a ) α ( y ) + ( r B ( a ) x ) · A α ( y ) − r B ( r A ( y ) a ) α ( x ) − α ( x ) · A ( l B ( a ) y ) = 0 (2.17) for any, x, y ∈ A, a, b ∈ B . Then ( A, B, l A , r A , β , β , l B , r B , α , α ) is called a matched pairof BiHom-associative algebras. In this case, there is a BiHom-associative algebra structureon the direct sum A ⊕ B of the underlying vector spaces of A and B given by ( x + a ) · ( y + b ) := x · A y + ( l A ( x ) b + r A ( y ) a ) + a · B b + ( l B ( a ) y + r B ( b ) x ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) . Proof.
For any x, y, z ∈ A and a, b, c ∈ B we have( α + β )( x + a ) · (( y + b ) · ( z + c ))= ( α ( x ) + β ( a ))[ y · A z + l B ( b ) z + r B ( c ) y + b · c + l A ( y ) c + r A ( z ) b )= α ( x ) · A ( y · A z ) + α ( x ) · A l B ( b ) z + α ( x ) · A r B ( c ) y + l B ( β ( a ))( y · A z )+ l B ( β ( a )) l B ( b ) z + l B ( β ( a )) r B ( c ) y + r B ( b · B c ) α ( x ) + r B ( l A ( y ) c ) α ( x )+ r B ( r A ( z ) b ) α ( x ) + β ( a ) · B ( b · B c ) + β ( a ) · B l A ( y ) c + β A ( α ( x )) l A ( y ) c + l A ( α ( x )) r A ( z ) b + r A ( y · A z ) β ( a ) + r A ( l A ( b ) z ) β ( a ) + r A ( r B ( c ) y ) β ( a ) .
4n the other hand, we have(( x + a ) · ( y + b )) · ( α + β )( z + c )= ( x · A y + l B ( a ) y + r B ( b ) x + a · B b + l A ( x ) b + r A ( y ) a ) · ( α ( z ) + β ( c ))= ( x · A y ) · A α ( z ) + l B ( a ) y · A α ( z ) + r B ( b ) x · A α ( z ) + l B ( a · B b ) α ( z )+ l B ( l A ( x ) b ) α ( z ) + l B ( r A ( y ) a ) α ( z ) + r B ( β ( c ))( x · A y ) + r A ( β ( c )) l B ( a ) y + r B ( β ( c )) r B ( b ) x + ( a · B b ) · B β ( c ) + ( l A ( x ) b ) · B β ( c ) + ( r A ( y ) a ) · B β ( c )+ r A ( α ( z ))( a · B b ) + r A ( α ( z ))( l A ( x ) b ) + r A ( α ( z ))( r A ( y ) a ) + l A ( x · A y ) β ( c )+ l A ( l B ( a ) y ) β ( c ) + ( r B ( b ) x ) β ( c ) . Then by (2.4) and (2.12)-(2.17), we deduce that ( α + β )( x + a ) · (( y + b ) · ( z + c )) =(( x + a ) · ( y + b )) · ( α + β )( z + c ). This finishes the proof.We denote this BiHom-associative algebra by A ⊲⊳ l A ,r A ,β ,β l B ,r B ,α ,α B .Let us recall now the definition and the notion of bimodule of a BiHom-Lie algebra givenin [21] Definition 2.9.
A BiHom-Lie algebra is a quadruple ( A, [ · , · ] , α , α ) consisting of a linearspace A , a bilinear map [ · , · ] : ∧ A → A and two linear maps α , α : A → A satisfying α ◦ α = α ◦ α and the following conditions, ∀ x, y, z ∈ A ,(i) α ([ x, y ]) = [ α ( x ) , α ( y )] and α ([ x, y ]) = [ α ( x ) , α ( y )] , (ii) [ α ( x ) , α ( y )] = − [ α ( y ) , α ( x )] . (iii) [ α ( x ) , [ α ( y ) , α ( z )]] + [ α ( z ) , [ α ( x ) , α ( y )]] + [ α ( y ) , [ α ( z ) , α ( x )]] = 0 . Definition 2.10.
Let ( A, [ · , · ] , α , α ) be a BiHom-Lie algebra and ( V, β , β ) be a BiHom-module. Let ρ : A → gl ( V ) be a linear map. The quadruple ( ρ, β , β , , V ) is called a repre-sentation of A if for all x, y ∈ A, v ∈ V , we have ρ ([ α ( x ) , y ]) β ( v ) = ρ ( α α ( x )) ◦ ρ ( y ) v − ρ ( α ( y )) ◦ ρ ( α ( x )) v, (2.18) β ( ρ ( x ) v ) = ρ ( α ( x )) β ( v ) , (2.19) β ( ρ ( x ) v ) = ρ ( α ( x )) β ( v ) . (2.20) Proposition 2.11.
Let ( ρ, β , β , V ) be a representation of a BiHom-Lie algebra ( A, [ · , · ] , α , α ) such that α , β are bijectives. Then, the direct sum A ⊕ V of vector spaces is turned into aBiHom-Lie algebra by defining the multiplication in A ⊕ V by [ x + v , x + v ] ρ = [ x , x ] + ρ ( x ) v − ρ ( α − α ( x )) β β − ( v ) , ( α ⊕ β )( x + v ) = α ( x ) + β ( v ) , ( α ⊕ β )( x + v ) = α ( x ) + β ( v ) , for all x , x ∈ A, v , v ∈ V . We denote such a BiHom-Lie algebra by ( A ⊕ V, [ · , · ] ρ , α + β , α + β ) , or A × ρ,α ,α ,β ,β V. Now, we introduce the notion of matched pair of BiHom-Lie algebra5 heorem 2.12.
Let ( A, {· , ·} A , α , α ) and ( B, {· , ·} B , β , β ) be two BiHom-Lie algebras.Suppose that there are linear maps ρ A : A → gl ( B ) and ρ B : B → gl ( A ) such that ( ρ A , β , β , B ) is a bimodule of A and ( ρ B , α , α , A ) is a bimodule of B satisfies ρ B ( β β ( a )) { α ( x ) , α ( y ) } A = { ρ B ( β ( a )) α ( x ) , α α ( y ) } A + { α α ( x ) , ρ B ( β ( a )) α ( y ) } A + ρ B ( ρ A ( α ( y )) β ( a )) α ( x ) − ρ B ( ρ A ( α ( x )) β ( a )) α α ( y )(2.21) ρ A ( α α ( x )) { β ( a ) , β ( b ) } B = { ρ A ( α ( x )) β ( a ) , β β ( b ) } B + { β β ( a ) , ρ A ( α ( x )) β ( b ) } B + ρ A ( ρ B ( β ( b )) α ( x )) β ( a ) − ρ A ( ρ B ( β ( a )) α ( x )) β β ( b )(2.22) for any, x, y ∈ A, a, b ∈ B . Then ( A, B, ρ A , β , β , ρ B , α , α ) is called a matched pair ofBiHom-Lie algebras. Moreover, assume that ( A, {· , ·} A , α , α ) and ( B, {· , ·} B , β , β ) be tworegular BiHom-Lie algebras, then there exists a BiHom-Lie algebra structure on the vectorspace A ⊕ B of the underlying vector spaces of A and B given by [ x + a, y + b ] := { x, y } A + ρ A ( x ) b − ρ A ( α − α ( y )) β β − ( a )+ { a, b } B + ρ B ( a ) y − ρ B ( β − β ( b )) α α − ( x ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) . Proof.
First, we prove the BiHom-multiplicativity. For all x, y ∈ A, a, b ∈ B , we have( α + β )[ x + a, y + b ]=( α + β )( { x, y } A + ρ A ( x ) b − ρ A ( α − α ( y )) β β − ( a )+ { a, b } B + ρ B ( a ) y − ρ B ( β − β ( b )) α α − ( x ))= α ( { x, y } A ) + α ρ B ( a ) y − α ρ B ( β − β ( b )) α α − ( x )+ β ρ A ( x ) b − β ρ A ( α − α ( y )) β β − ( a ) + β { a, b } B = { a ( x ) , α ( y ) } A + ρ B ( β ( a )) α ( y ) − ρ B ( β − β ( β ( b )) α α − ( α ( x ))+ ρ A ( α ( x )) β ( b ) − ρ A ( α − α ( α ( y )) β β − ( β ( a )) + { b ( a ) , β ( b ) } B =[ α ( x ) + β ( a ) , α ( y ) + β ( b )]=[( α + β )( x + a ) , ( α + β )( y + b )] . In the same way, ( α + β )[ x + a, y + b ] = [( α + β )( x + a ) , α + β )( y + b )] . Now, we prove the Bihom-skewsymmetry. For all x, y ∈ A, a, b ∈ B , we have[( α + β )( x + a ) , ( α + β )( y + b )]=[ α ( x ) + β ( a ) , α ( y ) + β ( b )]= { α ( x ) , α ( y ) } A + ρ A ( α ( x )) β ( b ) − ρ A ( α − α ( α ( y )) β β − ( β ( a ))+ { β ( a ) , β ( b ) } B + ρ B ( β ( a )) α ( y ) − ρ B ( β − β ( β ( b ))) α α − ( α ( x ))= − { a ( y ) , α ( x ) } A − ρ A ( α ( y )) β ( a ) + ρ A ( α − α ( α ( x )) β β − ( β ( b )) − { β ( b ) , β ( a ) } B − ρ B ( β ( b )) α ( x ) + ρ B ( β − β ( β ( a ))) α α − ( α ( y ))6 − [( α + β )( y + b ) , ( α + β )( x + a )] . Finally, we prove the Bihom-Jacobi identity. For any x, y, z ∈ A, a, b, c ∈ B , we have[( α + β ) ( x + a ) , [( α + β )( y + b ) , ( α + β )( z + c )]]=[ α ( x ) + β ( a ) , [ α ( y ) + β ( b ) , α ( z ) + β ( c )]]=[ α ( x ) + β ( a ) , { α ( y ) , α ( z ) } + ρ A ( α ( y )) β ( c ) − ρ A ( α − α ( α ( z )))) β β − ( β ( b ))+ { β ( b ) , β ( c ) } B + ρ B ( β ( b )) α ( z ) − ρ B ( β − β ( β ( c ))) α α − ( α ( y ))]=[ α ( x ) + β ( a ) , { α ( y ) , α ( z ) } + ρ A ( α ( y )) β ( c ) − ρ A ( α ( z )) β ( b )+ { β ( b ) , β ( c ) } B + ρ B ( β ( b )) α ( z ) − ρ B ( β ( c )) α ( y )]= { α ( x ) , { α ( y ) , α ( z ) } A } A + { α ( x ) , ρ B ( β ( b )) α ( z ) } A − { α ( x ) , ρ B ( β ( c )) α ( y ) } A + ρ A ( α ( x )) ρ A ( α ( y )) β ( c ) − ρ A ( α ( x )) ρ A ( α ( z )) β ( b ) + ρ A ( α ( x )) { β ( b ) , β ( c ) } B − ρ A ( { α − α ( y ) , α ( z ) } A ) β β ( a ) − ρ A ( ρ B ( β − β ( b )) α ( z )) β β ( a )+ ρ A ( ρ B ( β − β ( c )) α ( y )) β β ( a ) + { β ( a ) , ρ A ( α ( y )) β ( c ) } B − { β ( a ) , ρ A ( α ( z )) β ( b ) } B + { β ( a ) , { β ( b ) , β ( c ) } B } B + ρ B ( β ( a ) { α ( y ) , α ( z ) } A + ρ B ( β ( a )) ρ B ( β ( b )) α ( z ) − ρ B ( β ( a )) ρ B ( β ( c )) α ( y ) − ρ B ( ρ A ( α − α ( y )) β ( c )) α α ( x )+ ρ B ( ρ A ( α − α ( z )) β ( b )) α α ( x ) − ρ B ( { β − β ( b ) , β ( c ) } B ) α α ( x ) . By a direct computation we verify that (cid:9) x,y,z [( α + β ) ( x + a ) , [( α + β )( y + b ) , ( α + β )( z + c )]] = 0. This ends the proof.We denote this BiHom-Lie algebra by A ⊲⊳ ρ A ,β ,β ρ B ,α ,α B . Definition 2.13.
A noncommutative BiHom-Poisson algebra is a 5-tuple ( A, · , {· , ·} , α , α ),where ( A, · , α , α ) is a BiHom-associative algebra and ( A, {· , ·} , α , α ) is a BiHom-Lie al-gebra, such that the BiHom-Leibniz identity: { α α ( x ) , y · z } = { α ( x ) , y } · α ( z ) + α ( y ) · { α ( x ) , z } . (2.23) Proposition 2.14. [2] Let ( A, · , α , α ) be a regular BiHom-associative algebra. Then A − = ( A, {· , ·} , · , α, β ) is a regular noncommutative BiHom-Poisson algebra, where for all x, y ∈ A , { x, y } = x · y − α − α ( y ) · α α − ( x ) . In the following we introduce the notions of representation and matched pair of noncom-mutative BiHom-Poisson algebras.
Definition 2.15.
Let ( A, · , {· , ·} , α , α ) be a noncommutative BiHom-Poisson algebra. Arepresentation of A is a 6-tuple ( l, r, ρ, β , β , V ) such that ( l, r, β , β , V ) is a bimodule of theBiHom-associative algebra ( A, · , α , α ) and ( ρ, β , β , V ) is a representation of the BiHom-Lie algebra ( A, {· , ·} , α , α ) satisfying, for all x, y ∈ A, v ∈ V.l ( { α ( x ) , y } ) β ( v ) = ρ ( α α ( x )) l ( y ) v − l ( α ( y )) ρ ( α ( x )) v (2.24) r ( { α ( x ) , y } ) β ( v ) = ρ ( α α ( x )) r ( y ) v − r ( α ( y )) ρ ( α ( x )) v (2.25) ρ ( x · y ) β β ( v ) = l ( α ( x )) ρ ( y ) β ( v ) + r ( α ( y )) ρ ( x ) β ( v ) . (2.26)7 roposition 2.16. Let ( l, r, ρ, β , β , V ) be a representation of noncommutative BiHom-Poisson algebra ( A, · , {· , ·} , α , α ) such that α , β are bijectives. Then ( A ⊕ V, · ′ , {· , ·} ′ , α + β , α + β ) is a noncommutative BiHom-Poisson algebra, where ( A ⊕ V, · ′ , α + β , α + β ) isthe semi-direct product BiHom-associative algebra A × l,r,α ,α ,β ,β V and ( A ⊕ V, {· , ·} ′ , α + β , α + β ) is the semi-direct product BiHom-Lie algebra A × ρ,α ,α ,β ,β V Proof.
We prove only the BiHom-Leibniz identity. For all x , x , x ∈ A, v , v , v ∈ V. { ( α α + β β )( x + v ) , ( x + v ) · ′ ( x + v ) } ′ − { ( α + β )( x + v ) · ′ { ( α + β )( x + v ) , x + v } ′ − ( α + β )( x + v ) · ′ { ( α + β )( x + v ) , x + v } ′ = { α α ( x ) + β β ( v ) , x · x + l ( x ) v + r ( x ) v } ′ − (cid:16) { α ( x ) , x } + ρ ( α ( x )) v − ρ ( α − α ( x )) β ( v ) (cid:17) · ′ ( α ( x ) + β ( v )) − ( α ( x ) + β ( v )) · ′ (cid:16) { α ( x ) , x } + ρ ( α ( x )) v − ρ ( α − α ( x )) β β − ( v ) (cid:17) = { α α ( x ) , x · x } + ρ ( α α ( x )) l ( x ) v + ρ ( α α ( x )) r ( x ) v − ρ ( α − α ( x ( x )) β ( v ) − { α ( x ) , x } · α ( x ) − l ( { a ( x ) , x } ) β ( v )) − r ( α ( x )) ρ ( α ( x )) v + r ( α ( x )) ρ ( α − α ( x )) β ( v ) − α ( x ) · { α ( x ) , x } − l ( α ( x )) ρ ( α ( x )) v + l ( α ( x )) ρ ( α − α ( x )) β β ( v ) − r ( { α ( x ) , x } ) β = (cid:16) { α α ( x ) , x · x } − { ‘ α ( x ) , x } · α ( x ) − α ( x ) { α ( x ) , x } (cid:17) + (cid:16) ρ ( α α ( x )) l ( x ) v − l ( { α ( x ) , x } ) β ( v ) − l ( α ( x )) ρ ( α ( x )) v (cid:17) + (cid:16) ρ ( α α ( x )) r ( x ) v − r ( α ( x )) ρ ( α ( x )) v − r ( { α ( x )) , x } ) β ( v ) (cid:17) − (cid:16) ρ ( α − α ( x · x )) β ( v ) + r ( α ( x )) ρ ( α − α ( x )) β ( v )+ l ( α ( x )) ρ ( α − α ( x )) β β − ( v ) (cid:17) = 0 + 0 + 0 + 0 = 0 . Then ( A ⊕ V, · ′ , {· , ·} ′ , α + β , α + β ) is a noncommutative BiHom-Poisson algebra and wedenote by A × l,r,ρ,α ,α ,β ,β V. Example . Let ( A, · , {· , ·} , α , α ) be a noncommutative BiHom-Poisson algebra. Then( L · , R · , ad, α , α , A ) is a regular representation of A , where L · ( a ) b = a · b, R · ( a ) b = b · a and ad ( a ) b = [ a, b ], for all a, b ∈ A . Theorem 2.18.
Let ( A, · A , {· , ·} A , α , α ) and ( B, · B , {· , ·} B , β , β ) be two noncommutativeBiHom-Poisson algebras. Suppose that there are linear maps l A , r A , ρ A : A → gl ( B ) and l B , r B , ρ B : B → gl ( A ) such that A ⊲⊳ ρ A ,β ,β ρ B ,α ,α B is a matched pair of BiHom-Lie algebras and A ⊲⊳ l A ,r A ,β ,β l B ,r B ,α ,α B is a matched pair of BiHom-associative algebras and for all x, y ∈ A, a, b ∈ B , the following equalities hold: ρ A ( α α ( x ))( β ( a ) · B β ( b )) = ( ρ A ( α ( x )) β ( a )) · B β β ( b ) + β β ( a ) · B ρ A ( α α ( x )) β ( b )8 l A ( ρ B ( β ( a )) α ( x )) β β ( b ) − r A ( ρ B ( β ( b )) α ( x )) β β ( a ) , (2.27) l A ( α α ( x )) { β ( a ) , β ( b ) } B = { β β ( a ) , l A ( α ( x )) β ( b ) } B − ρ A ( r B ( β ( b )) α ( x )) β ( a ) − l A ( ρ B ( β ( a )) α ( x )) β β ( b ) + ( ρ A ( α ( x )) β ( a )) · B β β ( b ) , (2.28) ρ B ( β β ( a ))( α ( x ) · A α ( y )) = ( ρ B ( β ( a )) α ( x )) · A α α ( y ) + α α ( x ) · A ρ B ( β β ( a )) α ( y ) − l B ( ρ A ( α ( x )) β ( a )) α α ( y ) − r B ( ρ A ( α ( y )) β ( a )) α α ( x ) , (2.29) l B ( β β ( a )) { α ( x ) , α ( y ) } A = { α α ( x ) , l B ( β ( a )) α ( y ) } A − ρ B ( r A ( α ( y )) β ( a )) α ( x ) − l B ( ρ A ( β ( x )) β ( a )) α α ( y ) + ( ρ B ( β ( a )) α ( x )) · A α α ( y ) . (2.30) Then ( A, B, l A , r A , ρ A , β , β , l B , r B , ρ B , α , α ) is called a matched pair of noncommutativeBiHom-Poisson algebras. Moreover, assume that ( A, {· , ·} A , α , α ) and ( B, {· , ·} B , β , β ) betwo regular noncommutative BiHom-Poisson algebras, then, there exists a noncommutativeBiHom-Poisson algebra structure on the direct sum A ⊕ B of the underlying vector spaces of A and B given by ( x + a ) · ( y + b ) := x · A y + ( l A ( x ) b + r A ( y ) a ) + a · B b + ( l B ( a ) y + r B ( b ) x ) , [ x + a, y + b ] := { x, y } A + ρ A ( x ) b − ρ A ( α − α ( y )) β β − ( a )+ { a, b } B + ρ B ( a ) y − ρ B ( β − β ( b )) α α − ( x ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) . for any x, y ∈ A, a, b ∈ B. Proof.
By Theorem 2.8 and Theorem 2.12, we deduce that ( A ⊕ B, · , α + β , α + β ) is aBiHom-associative algebra and ( A ⊕ B, [ · , · ] , α + β , α + β ) is a BiHom-Lie algebra. Now,the rest, it is easy ( in a similar way as for Proposition 2.12 and Proposition 2.8) to verifythe BiHom-Leibniz identity satisfied.We denote this noncommutative BiHom-Poisson algebra by A ⊲⊳ l A ,r A ,ρ A ,β ,β l B ,r B ,ρ B ,α ,α B . In this section we introduce the definition and bimodule of a noncommutative BiHom-pre-Poisson algebra. We also establish the matched pair of noncommutative BiHom-pre-Poissonalgebra and equivalently link them to a matched pair of their underlying noncommutativeBiHom-Poisson algebras. 9 efinition 3.1. [31] A BiHom-pre-Lie algebra ( A, ∗ , α , α ) is a vector space A equippedwith a bilinear product ∗ : A ⊗ A → A, and two linear maps α , α ∈ End ( A ), such that forall x, y, z ∈ A, α ( x ∗ y ) = α ( x ) ∗ α ( y ) , α ( x ∗ y ) = α ( x ) ∗ α ( y ) and the following equalityis satisfied:( α ( x ) ∗ α ( y )) ∗ α ( z ) − α α ( x ) ∗ ( α ( y ) ∗ z ) = ( α ( y ) ∗ α ( x )) ∗ α ( z ) − α α ( y ) ∗ ( α ( x ) ∗ z ) . (3.1)The equation (3.1) is called BiHom-pre-Lie identity. Lemma 3.2. [31] Let ( A, ∗ , α , α ) be a regular BiHom-pre-Lie algebra. Then ( A, [ · , · ] , α , α ) is a BiHom-Lie algebra with [ x, y ] = x ∗ y − α − α ( y ) ∗ α α − ( x ) , for any x, y ∈ A . We say that ( A, [ · , · ] , α , α ) is the sub-adjacent BiHom-Lie algebra of ( A, ∗ , α , α ) and denoted by A c . Let us recall now the notion of bimodule of a BiHom-pre-Lie algebra given in [13].
Definition 3.3.
Let ( A, ∗ , α , α ) be a BiHom-pre-Lie algebra, and let ( V, β , β ) be aBiHom-module. Let l ∗ , r ∗ : A → gl ( V ) be two linear maps. The quintuple ( l ∗ , r ∗ , β , β , V ) iscalled a bimodule of A if for all x, y ∈ A, v ∈ Vl ∗ ( { α ( x ) , α ( y ) } ) β ( v ) = l ∗ ( α α ( x )) l ∗ ( α ( y )) v − l ∗ ( α α ( y )) l ∗ ( α ( x )) v (3.2) r ∗ ( α ( y )) ρ ( α ( x )) β ( v ) = l ∗ ( α α ( x )) r ∗ ( y ) β ( v ) − r ∗ ( α ( x ) ∗ y ) β β ( v ) (3.3) β ( l ∗ ( x ) v ) = l ∗ ( α ( x )) β ( v ) , (3.4) β ( r ∗ ( x ) v ) = r ∗ ( α ( x )) β ( v ) , (3.5) β ( l ∗ ( x ) v ) = l ∗ ( α ( x )) β ( v ) , (3.6) β ( r ∗ ( x ) v ) = r ∗ ( α ( x )) β ( v ) (3.7).where { α ( x ) , α ( y ) } = α ( x ) ∗ α ( y ) − α ( y ) ∗ α ( x ) and ( ρ ◦ α ) β = ( l ∗ ◦ α ) β − ( r ∗ ◦ α ) β . Proposition 3.4.
Let ( l ∗ , r ∗ , β , β , V ) be a bimodule of a BiHom-pre-Lie algebra ( A, ∗ , α , α ) .Then, the direct sum A ⊕ V of vector spaces is turned into a BiHom-pre-Lie algebra by defin-ing multiplication in A ⊕ V by ( x + v ) ∗ ′ ( x + v ) := x ∗ x + ( l ∗ ( x ) v + r ∗ ( x ) v ) , ( α ⊕ β )( x + v ) := α ( x ) + β ( v ) , ( α ⊕ β )( x + v ) := α ( x ) + β ( v ) , for all x , x ∈ A, v , v ∈ V . We denote such a BiHom-pre-Lie algebra by ( A ⊕ V, ∗ ′ , α + β , α + β ) , or A × l ∗ ,r ∗ ,α ,α ,β ,β V. roposition 3.5. Let ( l ∗ , r ∗ , β , β , V ) be a bimodule of a regular BiHom-pre-Lie algebra ( A, ∗ , α , α ) such that β is bijective. Let ( A, {· , ·} , α , α ) be the subadjacent BiHom-Liealgebra of ( A, ∗ , α , α ) . Then ( l ∗ − ( r ∗ ◦ α α − ) β − β , β , β , V ) is a representation of Liealgebra ( A, {· , ·} , α , α ) .Proof. For all x, y ∈ Aβ ◦ ( l ∗ ( x ) − r ∗ ( α α − ( x )) β − β )( v )= β ◦ l ∗ ( x ) v − β ◦ ( r ∗ ( α α − ( x )) β − β ( v )= l ∗ ( α ( x )) β ( v ) − ( r ∗ ◦ α α − ( α ( x ))) β − β ( β ( v ))=( l ∗ − ( r ∗ ◦ α α − ) β − β )( α ( x )) β ( v ) . In the same way, we have β ◦ ( l ∗ − ( r ∗ ◦ α α − ))( x ) β − β )( v ) = ( l ∗ − ( r ∗ ◦ α α − ) β − β )( α ( x )) β ( v ).Finally, for all x, y ∈ A , we have( l ∗ − ( r ∗ ◦ α α − ) β − β )( α α ( x )) ◦ ( l ∗ − ( r ∗ ◦ α α − ) β − β )( y ) v − ( l ∗ − ( r ∗ ◦ α α − ) β − β )( α ( y ) ◦ ( l ∗ − ( r ∗ ◦ α α − ) β − β )( α ( x )) v = l ∗ ( α α ( x )) ◦ l ∗ ( y ) v − ( r ∗ ◦ α ( x )) β − β ◦ l ∗ ( y ) v − l ∗ ( α α ( x )) ◦ ( r ∗ ◦ α α − ( y )) β − β ( v ) + ( r ∗ ◦ α ( x )) β − β ◦ ( r ∗ ◦ α α − ( y )) β − β ( v ) − l ∗ ( α ( y )) ◦ l ∗ ( α ( x )) v + l ∗ ( α ( y ))( r ∗ ◦ α α − ( x )) β − β ( v )+ ( r ∗ ◦ α ( y )) β − β ◦ l ∗ ( α ( x )) v − ( r ∗ ◦ α ( y )) β − β ◦ ( r ∗ ◦ α α − ( x )) β − β ( v )= l ∗ ( α α ( x )) ◦ l ∗ ( y ) v − ( r ∗ ◦ α ( x )) ◦ l ∗ ( α − α ( y )) β − β ( v ) − l ∗ ( α α ( x )) ◦ ( r ∗ ◦ α α − ( y )) β − β ( v ) + ( r ∗ ◦ α ( x )) ◦ ( r ∗ ( y )) β − β ( v ) − l ∗ ( α ( y )) ◦ l ∗ ( α ( x )) v + l ∗ ( α ( y ))( r ∗ ◦ α α − ( x )) β − β ( v )+ ( r ∗ ◦ α ( y )) ◦ l ∗ ( α ( x )) β − β ( v ) − ( r ∗ ◦ α ( y )) ◦ ( r ∗ ◦ α ( x )) β − β ( v )= (cid:16) l ∗ ( α α ( x )) ◦ l ∗ ( y ) v − l ∗ ( α ( y )) ◦ l ∗ ( α ( x )) v (cid:17) + (cid:16) − l ∗ ( α α ( x )) ◦ ( r ∗ ◦ α α − ( y )) β − β ( v ) + ( r ∗ ◦ α ( y )) ◦ l ∗ ( α ( x )) β − β ( v ) − ( r ∗ ◦ α ( y )) ◦ ( r ∗ ◦ α ( x )) β − β ( v ) (cid:17) − (cid:16) ( r ∗ ◦ α ( x )) ◦ l ∗ ( α − α ( y )) β − β ( v ) − ( r ∗ ◦ α ( x )) ◦ ( r ∗ ( y )) β − β ( v ) − l ∗ ( α ( y ))( r ∗ ◦ α α − ( x )) β − β ( v ) (cid:17) = l ∗ ( { α ( x ) , y } ) β ( v ) − r ∗ ( α ( x ) ∗ α α − ( y )) β − β ( v ) + r ∗ ( y ∗ α α − ( x )) β − β ( v )= l ∗ ( { α ( x ) , y } ) β ( v ) − r ∗ ( α ( x ) ∗ α α − ( y ) − y ∗ α α − ( x )) β − β ( v )= l ∗ ( { α ( x ) , y } ) β ( v ) − r ∗ ( { α ( x ) , α α − ( y ) } ) β − β ( v )= ρ ( { α ( x ) , y } ) β ( v ) . Therefore, (2.18)-(2.20) are satisfied.Now, we introduce the notion of matched pair of BiHom-pre-Lie algebra:
Theorem 3.6.
Let ( A, ∗ A , α , α ) and ( B, ∗ B , β , β ) be two BiHom-pre-Lie algebras. Sup-pose that there are linear maps l ∗ A , r ∗ A : A → gl ( B ) and l ∗ B , r ∗ B : B → gl ( A ) such that l ∗ A , r ∗ A , β , β , B ) is a bimodule of A and ( l ∗ B , r ∗ B , α , α , A ) is a bimodule of B satisfy forany, x, y ∈ A, a, b ∈ B : r ∗ A ( α ( x )) { β ( a ) , β ( b ) } B = r ∗ A ( l ∗ B ( β ( b )) x ) β β ( a ) − r ∗ A ( l ∗ B ( β ( a ) x ) β β ( b ) + β β ( a ) ∗ B r ∗ A ( x ) β ( b ) − β β ( b ) ∗ B r ∗ A ( x ) β ( a ) , (3.8) l ∗ A ( α α ( x ))( β ( a ) ∗ B b ) = ( ρ A ( α ( x )) β ( a )) ∗ B β ( b ) − l ∗ A ( ρ B ( β ( a )) α ( x )) β ( b ) + β β ( a ) ∗ B ( l ∗ A ( α ( x )) b )+ r ∗ A ( r ∗ B ( b ) α ( x )) β β ( a ) , (3.9) r ∗ B ( β ( a )) { α ( x ) , α ( y ) } A = r ∗ B ( l ∗ A ( α ( y )) a ) α α ( x ) − r ∗ B ( l ∗ A ( α ( x ) a ) α α ( y ) + α α ( x ) ∗ A r ∗ B ( a ) α ( y ) − α α ( y ) ∗ A r ∗ B ( a ) α ( x ) , (3.10) l ∗ B ( β β ( a ))( α ( x ) ∗ A y ) = ( ρ B ( β ( a )) α ( x )) ∗ A α ( y ) − l ∗ B ( ρ A ( α ( x )) β ( a )) α ( y ) + α α ( x ) ∗ A ( l ∗ B ( β ( a )) y )+ r ∗ B ( r ∗ A ( y ) β ( a )) α α ( x ) , (3.11) where { α ( x ) , α ( y ) } A = α ( x ) ∗ A α ( y ) − α ( y ) ∗ A α ( x ) , ( ρ A ◦ α ) β = ( l ∗ A ◦ α ) β − ( r ∗ A ◦ α ) β , { β ( a ) , β ( b ) } B = β ( a ) ∗ B β ( b ) − β ( b ) ∗ B β ( a ) , ( ρ B ◦ β ) α = ( l ∗ B ◦ β ) α − ( r ∗ B ◦ β ) α . Then ( A, B, l ∗ A , r ∗ A , β , β , l ∗ B , r ∗ B , α , α ) is called a matched pair of BiHom-pre-Lie alge-bras. In this case, there exists a BiHom-pre-Lie algebra structure on the vector space A ⊕ B of the underlying vector spaces of A and B given by ( x + a ) ∗ ( y + b ) := x ∗ A y + ( l ∗ A ( x ) b + r ∗ A ( y ) a ) + a ∗ B b + ( l ∗ B ( a ) y + r ∗ B ( b ) x ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) . Proof.
The proof is obtained in a similar way as for Theorem 2.8.We denote this BiHom-pre-Lie algebra by
A ⊲⊳ l ∗ A ,r ∗ A ,β ,β l ∗ B ,r ∗ B ,α ,α B . Proposition 3.7.
Let ( A, B, l ∗ A , r ∗ A , β , β , l ∗ B , r ∗ B , α , α ) be a matched pair of regularBiHom-pre-Lie algebras ( A, ∗ A , α , α ) and ( B, ∗ B , β , β ) . Then, ( A, B, l ∗ A − ( r ∗ A ◦ α α − ) β − β ,β , β , l ∗ B − ( r ∗ B ◦ β β − ) α − α , α , α ) is a matched pair of the associated BiHom-Lie algebras ( A, {· , ·} A , α , α ) and ( B, {· , ·} B , β , β ) . Proof.
Let (
A, B, l ∗ A , r ∗ A , β , β , l ∗ B , r ∗ B , α , α ) be a matched pair of regular BiHom-pre-Liealgebras ( A, ∗ A , α , α ) and ( B, ∗ B , β , β ). In view of Proposition 3.5, the linear maps l ∗ A − ( r ∗ A ◦ α α − ) β − β : A → gl ( B ) and l ∗ B − ( r ∗ B ◦ α α − ) β − β : B → gl ( A ) are representationsof the underlying BiHom-Lie algebras ( A, {· , ·} A , α , α ) and ( B, {· , ·} B , β , β ), respectively.Therefore, (2.21) is equivalent to (3.8)-(3.9) and (2.22) is equivalent to (3.10)-(3.11).12ow, we recall the definition of BiHom-dendriform algebra [32] and their notions ofbimodule and matched pair given in [23]. Definition 3.8.
A BiHom-dendriform algebra is a quintuple ( A, ≺ , ≻ , α , α ) consisting ofa vector space A on which the operations ≺ , ≻ : A ⊗ A → A, and α , α : A → A are linearmaps such that α ◦ α = α ◦ α and for all x, y, z ∈ A the following equalities are satisfied: α ( x ≺ y ) = α ( x ) ≺ α ( y ) α ( x ≺ y ) = α ( x ) ≺ α ( y ) α ( x ≻ y ) = α ( x ) ≻ α ( y ) α ( x ≻ y ) = α ( x ) ≻ α ( y )( x ≺ y ) ≺ α ( z ) = α ( x ) ≺ ( y · z ) , ( x ≻ y ) ≺ α ( z ) = α ( x ) ≻ ( y ≺ z ) ,α ( x ) ≻ ( y ≻ z ) = ( x · y ) ≻ α ( z ) , where x · y = x ≺ y + x ≻ y. (3.12) Lemma 3.9. [32] Let ( A, ≺ , ≻ , α , α ) be a BiHom-dendriform algebra. Then, ( A, · := ≺ + ≻ , α , α ) is a BiHom-associative algebra. Definition 3.10.
Let ( A, ≺ , ≻ , α , α ) be a BiHom-dendriform algebra, and V be a vec-tor space. Let l ≺ , r ≺ , l ≻ , r ≻ : A → gl ( V ) , and β , β : V → V be six linear maps. Then,( l ≺ , r ≺ , l ≻ , r ≻ , β , β , V ) is called a bimodule of A if the following equations hold for any x, y ∈ A and v ∈ V : l ≺ ( x ≺ y ) β ( v ) = l ≺ ( α ( x )) l · ( y ) v, (3.13) r ≺ ( α ( x )) l ≺ ( y ) v = l ≺ ( α ( y )) r · ( x ) v, (3.14) r ≺ ( α ( y )) r ≺ ( y ) v = r ≺ ( x · y ) β ( v ) , (3.15) l ≺ ( x ≻ y ) β ( v ) = l ≻ ( α ( x )) l ≺ ( y ) v, (3.16) r ≺ ( α ( x )) l ≻ ( y ) v = l ≻ ( α ( y )) r ≺ ( x ) v, (3.17) r ≺ ( α ( x )) r ≻ ( y ) v = r ≻ ( y ≺ x ) β ( v ) , (3.18) l ≻ ( x · y ) β ( v ) = l ≻ ( α ( x )) l ≻ ( y ) v, (3.19) r ≻ ( α ( x )) l · ( y ) v = l ≻ ( α ( y )) r ≻ ( x ) v, (3.20) r ≻ ( α ( x )) r · ( y ) v = r ≻ ( y ≻ x ) β ( v ) , (3.21) β ( l ≺ ( x ) v ) = l ≺ ( α ( x )) β ( v ) , (3.22) β ( r ≺ ( x ) v ) = r ≺ ( α ( x )) β ( v ) , (3.23) β ( l ≺ ( x ) v ) = l ≺ ( α ( x )) β ( v ) , (3.24) β ( r ≺ ( x ) v ) = r ≺ ( α ( x )) β ( v ) (3.25) β ( l ≻ ( x ) v ) = l ≻ ( α ( x )) β ( v ) , (3.26) β ( r ≻ ( x ) v ) = r ≻ ( α ( x )) β ( v ) , (3.27) β ( l ≻ ( x ) v ) = l ≻ ( α ( x )) β ( v ) , (3.28) β ( r ≻ ( x ) v ) = r ≻ ( α ( x )) β ( v ) (3.29)where x · y = x ≺ y + x ≻ y, l · = l ≺ + l ≻ and r · = r ≺ + r ≻ . roposition 3.11. Let ( l ≺ , r ≺ , l ≻ , r ≻ , β , β , V ) be a bimodule of a BiHom-dendriform alge-bra ( A, ≺ , ≻ , α , α ) . Then, there exists a BiHom-dendriform algebra structure on the directsum A ⊕ V of the underlying vector spaces of A and V given by ( x + u ) ≺ ′ ( y + v ) := x ≺ y + l ≺ ( x ) v + r ≺ ( y ) u ( x + u ) ≻ ′ ( y + v ) := x ≻ y + l ≻ ( x ) v + r ≻ ( y ) u, ( α ⊕ β )( x + a ) := α ( x ) + β ( a )( α ⊕ β )( x + a ) := α ( x ) + β ( a ) for all x, y ∈ A, u, v ∈ V . We denote it by A × l ≺ ,r ≺ ,l ≻ ,r ≻ ,α ,α ,β ,β V . Proposition 3.12. [23] Let ( l ≺ , r ≺ , l ≻ , r ≻ , β , β , V ) be a bimodule of BiHom-dendriformalgebra ( A, ≺ , ≻ , α , α ) . Let ( A, · = ≺ + ≻ , α , α ) be the BiHom associative algebra. Then( l ≺ + l ≻ , r ≺ + r ≻ , β , β , V ) is a bimodule of ( A, · , α , α ) .Proof. We prove only the axiom (2.5). The others being proved similarly. For any x, y ∈ A and v ∈ V , we have( l ≺ + l ≻ )( x · y ) β ( v )=( l ≺ + l ≻ )( x ≺ y + x ≻ y ) β ( v )= l ≺ ( x ≺ y ) + l ≺ ( x ≻ y ) β ( v ) + l ≻ ( x · y ) β ( v )= l ≺ ( α ( x ))( l ≺ + l ≻ )( y ) v + l ≻ ( α ( x )) l ≺ ( y ) v + l ≻ ( α ( x )) l ≻ ( y ) v = l ≺ ( α ( x ))( l ≺ + l ≻ )( y ) v + l ≻ ( α ( x ))( l ≺ + l ≻ )( y ) v =( l ≺ + l ≻ )( α ( x ))( l ≺ + l ≻ )( y ) v. This finishes the proof.
Theorem 3.13.
Let ( A, ≺ A , ≻ A , α , α ) and ( B, ≺ B , ≻ B , β , β ) be two BiHom-dendriformalgebras. Suppose that there are linear maps l ≺ A , r ≺ A , l ≻ A , r ≻ A : A → gl ( B ) and l ≺ B , r ≺ B , l ≻ B , r ≻ B : B → gl ( A ) such that for all x, y ∈ A, a, b ∈ B , the following equalities hold: r ≺ A ( α ( x ))( a ≺ B b ) = β ( a ) ≺ B ( r A ( x ) b ) + r ≺ A ( l B ( x ) β ( a )) , (3.30) l ≺ A ( l ≺ B ( x )) β ( b ) + ( r ≺ A ( x ) a ) ≺ B β ( b ) = β ( a ) ≺ B ( l ≺ A ( x ) b ) + r ≺ A ( r ≺ B ( b ) x ) β ( a ) , (3.31) l ≺ A ( α ( x ))( a ∗ B b ) = ( l ≺ A ( x ) a ) ∗ B β ( b ) + l ≺ A ( r ≺ A ( a ) x ) β ( b ) , (3.32) r ≺ A ( α ( x ))( a ≻ B b ) = r ≻ A ( l ≺ B ( b ) x ) β ( a ) + β ( a ) ≻ B ( r ≺ A ( x ) b ) , (3.33) l ≺ A ( l ≻ B ( a ) x ) β ( b ) +( r ≻ A ( x ) a ) ≺ B β ( b ) = β ( a ) ≻ B ( l ≺ A ( x ) b ) + r ≻ A ( r ≺ B ( b ) x ) β ( a ) (3.34) l ≻ A ( α ( x ))( a ≺ B b ) = ( l ≻ A ( x ) a ) ≺ B β ( b ) + l ≺ A ( r ≻ B ( a ) x ) β ( b ) , (3.35) r ≻ A ( α ( x ))( a ∗ B b ) = β ( a ) ≻ B ( r ≻ A ( x ) b ) + r ≻ A ( l ≻ B ( b ) x ) β ( a ) , (3.36) β ( a ) ≻ B ( l ≻ A ( x ) b ) + r ≻ A ( r ≻ B ( b ) x ) β ( a ) = l ≻ A ( l B ( a ) x ) β ( b ) + ( r A ( x ) a ) ≻ B β ( b ) , (3.37) l ≻ A ( α ( x ))( a ≻ B b ) = ( l A ( x ) a ) ≻ B β ( b ) + l ≻ A ( r B ( a ) x ) β ( b ) , (3.38)14 ≺ B ( β ( a ))( x ≺ A y ) = α ( x ) ≺ A ( r B ( a ) y ) + r ≺ B ( l A ( y ) a ) α ( x ) , (3.39) l ≺ B ( l ≺ A ( x ) a ) α ( y ) +( r ≺ B ( a ) x ) ≺ A α ( y ) = α ( x ) ≺ A ( l B ( a ) y ) + r ≺ B ( r A ( y ) a ) α ( x ) , (3.40) l ≺ B ( β ( a ))( x ∗ A y ) = ( l ≺ B ( a ) x ) ≺ A α ( y ) + l ≺ B ( r ≺ A ( x ) a ) α ( y ) , (3.41) r ≺ B ( β ( a ))( x ≻ A y ) = r ≻ B ( l ≺ B ( y ) a ) α ( x ) + α ( x ) ≻ A ( r ≺ B ( a ) y ) , (3.42) l ≺ B ( l ≻ A ( x ) a ) α ( y ) +( r ≻ B ( a ) x ) ≺ A α ( y ) = α ( x ) ≻ A ( l ≺ B ( a ) y ) + r ≻ B ( r ≺ A ( y ) a ) α ( x ) , (3.43) l ≻ B ( β ( a ))( x ≺ A y ) = ( l ≻ B ( a ) x ) ≺ A α ( y ) + l ≺ B ( r ≻ A ( x ) a ) α ( y ) , (3.44) r ≻ B ( β ( a ))( x ∗ A y ) = α ( x ) ≻ A ( r ≻ B ( a ) y ) + r ≻ B ( l ≻ A ( y ) a ) α ( x ) , (3.45) α ( x ) ≻ A ( l ≻ B ( a ) y ) + r ≻ B ( r ≻ A ( y ) a ) α ( x ) = l ≻ B ( l A ( x ) a ) α ( y ) + ( r B ( a ) x ) ≻ A α ( y ) , (3.46) l ≻ B ( β ( a ))( x ≻ A y ) = ( l B ( a ) x ) ≻ A α ( y ) + l ≻ B ( r A ( x ) a ) α ( y ) , (3.47) where x · A y = x ≺ A y + x ≻ A y, l · A = l ≺ A + l ≻ A , r · A = r ≺ A + r ≻ A ,a · B b = a ≺ B b + a ≻ B b, l · B = l ≺ B + l ≻ B , r · B = r ≺ B + r ≻ B . Then ( A, B, l ≺ A , r ≺ A , l ≻ A , r ≻ A , β , β , l ≺ B , r ≺ B , l ≻ B , r ≻ B , α , α ) is called a matched pair ofBiHom-dendriform algebras. In this case, there exists a BiHom-dendriform algebra struc-ture on the direct sum A ⊕ B of the underlying vector spaces of A and B given by ( x + a ) ≺ ( y + b ) := x ≺ A y + ( l ≺ A ( x ) b + r ≺ A ( y ) a ) + a ≺ B b + ( l ≺ B ( a ) y + r ≺ B ( b ) x ) , ( x + a ) ≻ ( y + b ) := x ≻ A y + ( l ≻ A ( x ) b + r ≻ A ( y ) a ) + a ≻ B b + ( l ≻ B ( a ) y + r ≻ B ( b ) x ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) . We denote this BiHom-dendriform algebra by
A ⊲⊳ l A ,r A ,β ,β l B ,r B ,α ,α B . Proposition 3.14. [23] Let ( A, B, l ≺ A , r ≺ A , l ≻ A , r ≻ A , β , β , l ≺ B , r ≺ B , l ≻ B , r ≻ B , α , α ) be amatched pair of a BiHom-dendriform algebras ( A, ≺ A , ≻ A , α , α ) and ( B, ≺ B , ≻ B , β , β ) .Then, ( A, B, l ≺ A + l ≻ A , r ≺ A + r ≻ A , β , β , l ≺ B + l ≻ B , r ≺ B + r ≻ B , α , α ) is a matched pair ofthe associated BiHom-associative algebras ( A, · A = ≺ A + ≻ A , α , α ) and ( B, · B = ≺ B + ≻ B , β , β ) .Proof. Let (
A, B, l ≺ A , r ≺ A , l ≻ A , r ≻ A , β , β , l ≺ B , r ≺ B , l ≻ B , r ≻ B , α , α ) be a matched pair of aBiHom-dendriform algebras ( A, ≺ A , ≻ A , α , α ) and ( B, ≺ B , ≻ B , β , β ). In view of Proposi-tion 3.12, the linear maps l ≺ A + l ≻ A , r ≺ A + r ≻ A : A → gl ( B ) and l ≺ B + l ≻ B , r ≺ B + r ≻ B : B → gl ( A ) are bimodules of the underlying BiHom-Lie algebras ( A, · A , α , α ) and ( B, · B , β , β ),respectively. Therefore, (2.12)-(2.14) are equivalents to (3.30)-(3.38) and (2.15)-(2.17) areequivalents to (3.39)-(3.47).Now, we introduce the definition of noncommutative BiHom-pre-Poisson algebra and wegive some results. 15 efinition 3.15. A noncommutative BiHom-pre-Poisson algebra is a 6-tuple ( A, ≺ , ≻ , ∗ , α , α )such that ( A, ≺ , ≻ , α , α ) is a BiHom-dendriform algebra and ( A, ∗ , α , α ) is a BiHom-pre-Lie algebra satisfying the following compatibility conditions:( α ( x ) ∗ α ( y ) − α ( y ) ∗ α ( x )) ≺ α ( z ) = α α ( x ) ∗ ( α ( y ) ≺ z ) − α α ( y ) ≺ ( α ( x ) ∗ z ) , (3.48) α ( x ) ≻ ( α α ( y ) ∗ α ( z ) − α ( z ) ∗ α ( y )) = α α ( y ) ∗ ( x ≻ α ( z )) − ( α ( y ) ∗ x ) ≻ α α ( z ) , (3.49)( α ( x ) ≺ α ( y ) + α ( x ) ≻ α ( y )) ∗ α ( z ) = ( α ( x ) ∗ α ( z )) ≻ α ( y ) + α α ( x ) ≺ ( α ( y ) ∗ z ) . (3.50) Theorem 3.16.
Let ( A, ≺ , ≻ , ∗ ) be a noncommutative pre-Poisson algebra [30] and α , α : A → A be two morphisms of A such that α α = α α . Then A α ,α := ( A, ≺ α ,α = ≺◦ ( α ⊗ α ) , ≻ α ,α = ≻ ◦ ( α ⊗ α ) , ∗ α ,α = ∗ ◦ ( α ⊗ α ) , α , α ) is a noncommutative BiHom-pre-Poisson algebra, called the Yau twist of A . Moreover, assume that ( A ′ , ≺ ′ , ≻ ′ , ∗ ′ ) is an-other noncommutative pre-Poisson algebra and α ′ , α ′ : A ′ → A ′ be a two commuting noncom-mutative pre-Poisson algebra morphisms. Let f : A → A ′ be a pre-Poisson algebra morphismsatisfying f ◦ α = α ′ ◦ f and f ◦ α = α ′ ◦ f . Then f : A α ,α → A ′ α ′ ,α ′ is a noncommutativeBiHom-pre-Poisson algebra morphism.Proof. We shall only prove relation (3.48) the others being proved analogously. Then, forany x, y, z ∈ A ,( α ( x ) ∗ α ,α α ( y ) − α ( y ) ∗ α ,α α ( x )) ≺ α ,α α ( z )=( α α ( x ) ∗ α α ( y ) − α α ( y ) ∗ α α ( x )) ≺ α ,α α ( z )=( α α ( x ) ∗ α α ( y ) − α α ( y ) ∗ α α ( x )) ≺ α ( z )= α α ( x ) ∗ ( α α ( y ) ≺ α ( z )) − α α ( y ) ≺ ( α α ( x ) ∗ α ( z ))= α α ( x ) ∗ α ,α ( α ( y ) ≺ α ,α z ) − α α ( y ) ≺ α ,α ( α ( x ) ∗ α ,α z ) . For the second assertion, we have f ( x ≺ α ,α y ) = f ( α ( x ) ≺ α ( y ))= f ( α ( x )) ≺ ′ f ( α ( y ))= α ′ f ( x ) ≺ ′ α ′ f ( y )= f ( x ) ≺ ′ α ,α f ( y ) . Similarly, we have f ( x ≻ α ,α y ) = f ( x ) ≻ ′ α ′ ,α ′ f ( y ) and f ( x ∗ α ,α y ) = f ( x ) ∗ ′ α ′ ,α ′ f ( y ). Thiscompletes the proof. Proposition 3.17.
More generally, let ( A, ≺ , ≻ , ∗ , α , α ) be a noncommutative BiHom-pre-Poisson algebra and α ′ , α ′ : A → A be a two noncommutative BiHom-pre-Poisson algebramorphisms such that any two of the maps α , α , α ′ , α ′ commute. Then ( A, ≺ α ′ ,α ′ , ≻ α ′ ,α ′ , ∗ α ′ ,α ′ , α ◦ α ′ , α ◦ α ′ ) is a noncommutative BiHom-pre-Poisson algebra. Corollary 3.18.
Let ( A, ≺ , ≻ , ∗ , α , α ) be a noncommutative BiHom-pre-Poisson algebraand n ∈ N ∗ . Then The nth derived noncommutative BiHom-pre-Poisson algebra of type of A is definedby A n = ( A, ≺ ( n ) = ≺ ◦ ( α n ⊗ α n ) , ≻ ( n ) = ≻ ◦ ( α n ⊗ α n ) , ∗ ( n ) = ∗ ◦ ( α n ⊗ α n ) , α n +11 , α n +12 ) . (ii) The nth derived noncommutative BiHom-pre-Poisson algebra of type of A is definedby A n = ( A, ≺ (2 n − = ≺ ◦ ( α n − ⊗ α n − ) , ≻ (2 n − = ≻ ◦ ( α n − ⊗ α n − ) , ∗ (2 n − = ∗ ◦ ( α n − ⊗ α n − ) , α n , α n ) . Proof.
Apply Theorem 3.17 with α ′ = α n , α ′ = α n and α ′ = α n − , α ′ = α n − respectively. Theorem 3.19.
Let ( A, ≺ , ≻ , ∗ , α , α ) be a regular noncommutative BiHom-pre-Poissonalgebra. Then ( A, · , {· , ·} , α , α ) is a noncommutative BiHom-Poisson algebra with x · y = x ≺ y + x ≻ y, and { x, y } = x ∗ y − α − α ( y ) ∗ α α − ( x ) , for any x, y ∈ A . We say that ( A, · , {· , ·} , α , α ) is the sub-adjacent noncommutative BiHom-Poisson algebra of ( A, ≺ , ≻ , ∗ , α , α ) and denoted by A c .Proof. By Lemma 3.2 and Lemma 3.9, we deduce that ( A, · , α , α ) is a BiHom-associativealgebra and ( A, {· , ·} , α , α ) is a BiHom-Lie algebra. Now, we show the BiHom-Leibnizidentity { α α ( x ) , y · z } − { α ( x ) , y } · α ( z ) − α ( y ) · { α ( x ) , z } = { α α ( x ) , y ≺ z + y ≻ z } − { α ( x ) , y } ≺ α ( z ) − { α ( x ) , y } ≻ α ( z ) − α ( y ) ≺ { α ( x ) , z } − α ( y ) ≻ { α ( x ) , z } = α α ( x ) ∗ ( y ≺ z ) + α α ( x ) ∗ ( y ≻ z ) − α − α ( y ≺ z ) ∗ α ( x ) − α − α ( y ≻ z ) ∗ α ( x ) − ( α ( x ) ∗ y ) ≺ α ( z ) + ( α − α ( y ) ∗ α ( x ))) ≺ α ( z ) − ( α ( x ) ∗ y ) ≻ α ( z ) − ( α − α ( y ) ∗ α ( x )) ≻ α ( z ) − α ( y ) ≺ ( α ( x ) ∗ z )+ α ( y ) ≺ ( α − α ( z ) ∗ α α − ( x )) − α ( y ) ≻ ( α ( x ) ∗ z ) + α ( y ) ≻ ( α − α ( z ) ∗ α α − ( x ))= (cid:16) α α ( x ) ∗ ( y ≺ z ) − α ( y ) ≺ ( α ( x ) ∗ z ) − ( α ( x ) ∗ y − α − α ( y ) ∗ α ( x )) ≺ ( z ) (cid:17) + (cid:16) α α ( x ) ∗ ( y ≻ z ) − ( α ( x ) ∗ y ) ≻ α ( z ) − α ( y ) ≻ ( α ( x ) ∗ z − α − α ( z ) ∗ α α − ( x ) (cid:17) + (cid:16) ( α − α ( y ) ∗ α ( x )) ≻ α ( z ) + α ( y ) ≺ ( α − α ( z ) ∗ α α − ( x )) − ( α − α ( y ) ≻ α − α ( z ) α − α ( y ) ≻ α − α ( z )) ∗ α ( x ) (cid:17) =0 + 0 + 0 = 0 ( by (3 . − (3 . A, · , {· , ·} , α , α ) is a noncommutative BiHom-Poisson algebra.17he relation existing between a noncommutative BiHom-Poisson algebra and noncom-mutative BiHom-pre-Poisson algebra, as illustrated by the following diagram:BiHom-dendriform alg+ BiHom-pre-Lie alg / / x ∗ y − α − α ( y ) α α − ( x ) x ≺ y + x ≻ y (cid:15) (cid:15) BiHom-pre-Poisson alg x ∗ y − α − α ( y ) α α − ( x ) x ≺ y + x ≻ y (cid:15) (cid:15) BiHom-associative alg+BiHom-Lie alg / / BiHom-Poisson alg.In the following we introduce the notions of bimodule and matched pair of noncommu-tative BiHom-pre-Poisson algebras and related relevant properties are also given
Definition 3.20.
Let ( A, ≺ , ≻ , ∗ , α , α ) be a noncommutative BiHom-pre-Poisson algebra.A bimodule of A is a 9-tuple ( l ≺ , r ≺ , l ≻ , r ≻ , l ∗ , r ∗ , β , β , V ) such that ( l ∗ , r ∗ , β , β , V ) is a bi-module of the BiHom-pre-Lie algebra ( A, ∗ , α , α ) and ( l ≺ , r ≺ , l ≻ , r ≻ , β , β , V ) is a bimoduleof the BiHom-dendriform algebra ( A, ≺ , ≻ , α , α ) satisfying for all x, y ∈ A and v ∈ V : l ≺ ( { α ( x ) , α ( y ) } ) β ( v ) = l ∗ ( α α ( x )) l ≺ ( α ( y )) v − l ≺ ( α α ( y )) l ∗ ( α ( x )) v, (3.51) r ≺ ( α ( x )) ρ ( α ( y )) β ( v ) = l ∗ ( α α ( y )) r ≺ ( x ) β ( v ) − r ≺ ( α ( y ) ∗ x ) β β ( v ) , (3.52) − r ≺ ( α ( x )) ρ ( α ( y )) β ( v ) = r ∗ ( α ( y ) ≺ x ) β β ( v ) − l ≺ ( α α ( y )) r ∗ ( x ) β ( v ) , (3.53) l ≻ ( α ( x )) ρ ( α α ( y )) β ( v ) = l ∗ ( α α ( y )) l ≻ ( x ) β ( v ) − l ≻ ( α ( y ) ∗ z ) β β ( v ) , (3.54) r ≻ ( { α α ( x ) , α ( y ) } ) β ( v ) = l ∗ ( α α ( x )) r ≻ ( α ( y )) v − r ≻ ( α α ( y )) l ∗ ( α ( y )) v, (3.55) − l ≻ ( α ( x )) ρ ( α ( y )) β ( v ) = r ∗ ( x ≻ α ( y )) β β ( v ) − r ≻ ( α α ( y )) r ∗ ( x ) β ( v ) , (3.56) l ∗ ( α ( x ) · α ( y )) β ( v ) = r ≻ ( a ( y )) l ∗ ( α ( x )) β ( v ) + l ≺ ( α α ( x )) l ∗ ( α ( y )) v, (3.57) r ∗ ( α ( x )) l · ( α ( y )) β ( v ) = l ≻ ( α ( y ) ∗ α ( x )) β ( v ) + l ≺ ( α α ( y )) r ∗ ( x ) β ( v ) , (3.58) r ∗ ( α ( x )) r · ( α ( y )) β ( v ) = r ≻ ( α ( y )) r ∗ ( α ( x )) β ( v ) + r ≺ ( α ( y ) ∗ x ) β β ( v ) , (3.59)where x · y = x ≺ y + x ≻ y, l · = l ≺ + l ≻ , r · = r ≺ + r ≻ , { α ( x ) , α ( y ) } = α ( x ) ∗ α ( y ) − α ( y ) ∗ α ( x ) , ( ρ ◦ α ) β = ( l ∗ ◦ α ) β − ( r ∗ ◦ α ) β . Proposition 3.21.
Let ( l ≺ , r ≺ , l ≻ , r ≻ , l ∗ , r ∗ , β , β , V ) be a bimodule of a noncommutativeBiHom-pre-Poisson algebra ( A, ≺ , ≻ , ∗ , α , α ) . Then, there exists a noncommutative BiHom-pre-Poisson algebra structure on the direct sum A ⊕ V of the underlying vector spaces of A and V given by ( x + u ) ≺ ′ ( y + v ) := x ≺ y + l ≺ ( x ) v + r ≺ ( y ) u, ( x + u ) ≻ ′ ( y + v ) := x ≻ y + l ≻ ( x ) v + r ≻ ( y ) u, ( x + u ) ∗ ′ ( y + v ) := x ∗ y + l ∗ ( x ) v + r ∗ ( y ) u, ( α ⊕ β )( x + u ) := α ( x ) + β ( u ) , ( α ⊕ β )( x + u ) := α ( x ) + β ( u ) , for all x, y ∈ A, u, v ∈ V . We denote it by A × l ≺ ,r ≺ ,l ≻ ,r ≻ ,l ∗ ,r ∗ ,α ,α ,β ,β V . roof. We prove only the axiom (3.48). The axioms (3.49), (3.50) being proved similarly.For any x , x , x ∈ A and v , v , v ∈ V , we have(( α + β )( x + v ) ∗ ′ ( α + β )( x + v )) ≺ ′ ( α + β )( x + v ) − (( α + β )( x + v ) ∗ ′ ( α + β )( x + v )) ≺ ′ ( α + β )( x + v )=(( α ( x ) ∗ α ( x )) + l ∗ ( α ( x )) β ( v ) + r ∗ ( α ( x )) β ( v )) ≺ ′ ( α + β )( x + v ) − ( α ( x ) ∗ α ( x ) + l ∗ ( α ( x ) β ( v ) + r ∗ ( α ( x )) β ( v )) ≺ ′ ( α + β )( x + v )=( α ( x ) ∗ α ( x )) ≺ α ( x ) + l ≺ ( α ( x ) ∗ α ( x )) β ( v ) + r ≺ ( α ( x )) l ∗ ( α ( x )) β ( v ) − r ≺ ( α ( x )) r ∗ ( α ( x )) β ( v ) − ( α ( x ) ∗ α ( x )) ≺ α ( x ) − l ≺ ( α ( x ) ∗ α ( x )) β ( v )+ r ≺ ( α ( x )) l ∗ ( α ( x )) β ( v ) − r ≺ ( α ( x )) r ∗ ( α ( x )) β ( v )= { α ( x ) , α ( x ) } + l ≺ ( { α ( x ) , α ( x ) } ) β ( v )+ r ≺ ( α ( x )) ρ ( α ( x )) β ( v ) + r ≺ ( α ( x )) ρ ( α ( x )) β ( v ) . On the other hand ,( α + β )( α + β )( x + v ) ∗ ′ ( α ( x + v ) ≺ ( x + v )) − ( α + β )( α + β )( x + v ) ≺ ′ (( α + β )( x + v ) ∗ ′ ( x + v ))=( α α ( x ) + β β ( v )) ∗ ′ ( α ( x ) ≺ x + l ≺ ( α ( x )) v + r ≺ ( x ) β ( v ) − ( α α ( x ) + β β ( v )) ≺ ′ ( α ( x ) ∗ x + l ∗ ( α ( x )) v + r ∗ ( x ) α ( v ))= α α ( x ) ∗ α ( x ) ≺ x ) + l ∗ ( α α ( x )) l ≺ ( α ( x )) v + l ∗ ( α α ( x )) r ≺ ( x ) β ( v )+ r ∗ ( α ( x ) ≺ x ) β β ( v ) − α α ( x ) ≺ ( α ( x ) ∗ x ) − l ≺ ( α α ( x )) l ∗ ( α ( x )) v − l ≺ ( α α ( x )) r ∗ ( x ) α ( v ) − r ≺ ( α ( x ) ∗ x ) β β ( v ) . By equations (3.48), (3.51)-(3.53) we deduce that(( α + β )( x + v ) ∗ ′ ( α + β )( x + v )) ≺ ′ ( α + β )( x + v ) − (( α + β )( x + v ) ∗ ′ ( α + β )( x + v )) ≺ ′ ( α + β )( x + v )=( α + β )( α + β )( x + v ) ∗ ′ ( α ( x + v ) ≺ ( x + v )) − ( α + β )( α + β )( x + v ) ≺ ′ (( α + β )( x + v ) ∗ ( x + v ))There is an example of bimodule of noncommutative BiHom-pre-Poisson algebra Example . Let ( A, ≺ , ≻ , ∗ , α , α ) be a noncommutative BiHom-pre-Poisson algebra.Then ( L ≺ , R ≺ , L ≻ , R ≻ , L ∗ , R ∗ , α , α , A ) is called a regular bimodule of A , where L ≺ ( x ) y = x ≺ y, R ≺ ( x ) y = y ≺ x, L ≻ ( x ) y = x ≻ y, R ≻ ( x ) y = y ≻ x and L ∗ ( x ) y = x ∗ y, R ∗ ( x ) y = y ∗ x , for all x, y ∈ A . Proposition 3.23. If f : ( A, ≺ , ≻ , ∗ , α , α ) −→ ( A ′ , ≺ , ≻ , ∗ , β , β ) is a morphismof noncommutative BiHom-pre-Poisson algebra, then ( l ≺ , r ≺ , l ≻ , r ≻ , l ∗ , r ∗ , β , β , A ′ ) be-comes a bimodule of A via f , i.e, l ≺ ( x ) y = f ( x ) ≺ y, r ≺ ( x ) y = y ≺ f ( x ) , l ≻ ( x ) y = f ( x ) ≻ y, r ≻ ( x ) y = y ≻ f ( x ) and l ∗ ( x ) y = f ( x ) ∗ y, r ∗ ( x ) y = y ∗ f ( x ) for all ( x, y ) ∈ A × A ′ . roof. We prove only the axiom (3.57). The others being proved similarly. For any x, y ∈ A and z ∈ A ′ , we have l ∗ ( α ( x ) · α ( y )) β ( z )= f ( α ( x ) · α ( y )) ∗ β ( z ))=( β f ( x ) · β f ( y )) ∗ β ( z )=( β f ( x ) ∗ β ( z )) ≻ β f ( y ) + β β f ( x ) ≺ ( β f ( y ) ∗ z ) ( by (3 . f ( α ( x )) ∗ β ( z )) ≻ f ( α ( y )) + f ( α α ( x )) ≺ ( f ( α ( y )) ∗ z )= r ≻ ( α ( y ))( f ( α ( x )) ∗ β ( z )) + l ≺ ( α α ( x ))( f ( α ( y )) ∗ z )= r ≻ ( α ( y )) l ∗ ( α ( x )) β ( z ) + l ≺ ( α α ( x )) l ∗ ( α ( y )) z. This finishes the proof.
Corollary 3.24.
Let ( l ≺ , r ≺ , l ≻ , r ≻ , l ∗ , r ∗ , β , β , V ) be a bimodule of a regular noncommuta-tive BiHom-pre-Poisson algebra ( A, ≺ , ≻ , ∗ , α , α ) such that β is bijective. Let ( A, · , {· , ·} , α , α ) be the subadjacent of ( A, ≺ , ≻ , ∗ , α , α ) . Then ( l ≺ + l ≻ , r ≺ + r ≻ , l ∗ − ( r ∗ ◦ α α − ) β − β , β , β , V )is a representation of ( A, · , {· , ·} , α , α ) .Proof. It follows from the relation between the noncommutative BiHom-pre-Poisson algebraand the associated noncommutative BiHom-Poisson algebra. More precisely by Poposition3.5 and Poposition 3.12, we deduce that ( l ∗ − ( r ∗ ◦ α α − ) β − β , β , β , V ) is a representationof ( A, {· , ·} , α , α ) and ( l ≺ + l ≻ , r ≺ + r ≻ , β , β , V ) is a bimodule of ( A, · , α , α ). Now, therest, it is easy ( in a similar way as for Poposition 3.5 and Poposition 3.12) to verify theaxioms (2.24)-(2.26) Corollary 3.25.
Let ( l ≺ , r ≺ , l ≻ , r ≻ , l ∗ , r ∗ , β , β , V ) be a bimodule of a regular noncommuta-tive BiHom-pre-Poisson algebra ( A, ≺ , ≻ , ∗ , α , α ) such that β is bijective. Let ( A, · , {· , ·} , α , α ) be the subadjacent of ( A, ≺ , ≻ , ∗ , α , α ) . Then
1) ( l ≺ , r ≻ , l ∗ − ( r ∗ ◦ α α − ) β − β , β , β , V ) is bimodule of ( A, · , {· , ·} , α , α );2) ( l ≺ + l ≻ , , , r ≺ + r ≻ , l ∗ , r ∗ , β , β , V ) and ( l ≺ , , , r ≻ , l ∗ , r ∗ , β , β , V ) are bimodules of ( A, ≺ , ≻ , ∗ , α , α );3) the noncommutative BiHom-pre-Poisson algebras A × l ≺ ,r ≺ ,l ≻ ,r ≻ ,l ∗ ,r ∗ ,α ,α ,β ,β V and A × l ≺ + l ≻ , , ,r ≺ + r ≻ ,l ∗ ,r ∗ ,α ,α ,β ,β V have the same associated noncommutative BiHom-Poisson algebra A × l ≺ + l ≻ ,r ≺ + r ≻ ,l ∗ − ( r ∗ ◦ α α − ) β − β ,α ,α ,β ,β V. Proof.
It results from a direct computation.The following result gives a construction of a bimodule of a BiHom-pre-Poisson algebraby means of the Yau twist procedure 20 heorem 3.26.
Let ( A, ≺ , ≻ , ∗ , α , α ) be a noncommutative BiHom-pre-Poisson algebra, ( l ≺ , r ≺ , l ≻ , r ≻ , l ∗ , r ∗ , β , β , V ) be a bimodule of A . Let α ′ , α ′ be two endomorphisms of A suchthat any two of the maps α , α ′ , α , α ′ commute and β ′ , β ′ be linear maps of V such thatany two of the maps β , β ′ , β , β ′ commute. Suppose furthermore that β ′ ◦ l ≺ = ( l ≺ ◦ α ′ ) β ′ , β ′ ◦ l ≺ = ( l ≺ ◦ α ′ ) β ′ ,β ′ ◦ l ≻ = ( l ≻ ◦ α ′ ) β ′ , β ′ ◦ l ≻ = ( l ≻ ◦ α ′ ) β ′ ,β ′ ◦ l ∗ = ( l ∗ ◦ α ′ ) β ′ , β ′ ◦ l ∗ = ( l ∗ ◦ α ′ ) β ′ , and β ′ ◦ r ≺ = ( r ≺ ◦ α ′ ) β ′ , β ′ ◦ r ≺ = ( r ≺ ◦ α ′ ) β ′ ,β ′ ◦ r ≻ = ( r ≻ ◦ α ′ ) β ′ , β ′ ◦ r ≻ = ( r ≻ ◦ α ′ ) β ′ ,β ′ ◦ r ∗ = ( r ∗ ◦ α ′ ) β ′ , β ′ ◦ r ∗ = ( r ∗ ◦ α ′ ) β ′ , and write A α ′ ,α ′ for the noncommutative BiHom-pre-Poisson algebra ( A, ≺ α ′ ,α ′ , ≻ α ′ ,α ′ , ∗ α ′ ,α ′ , α α ′ , α α ′ ) and V β ′ ,β ′ = ( e l ≺ , e r ≺ , e l ≻ , e r ≻ , e l ∗ , e r ∗ , β β ′ , β β ′ , V ) , where e l ≺ = ( l ≺ ◦ α ′ ) β ′ , e r ≺ = ( r ≺ ◦ α ′ ) β ′ , e l ≻ = ( l ≻ ◦ α ′ ) β ′ , e r ≻ = ( r ≻ ◦ α ′ ) β ′ , e l ∗ = ( l ∗ ◦ α ′ ) β ′ , e r ∗ = ( r ∗ ◦ α ′ ) β ′ . Then V β ′ ,β ′ is a bimodule of A α ′ ,α ′ .Proof. We prove only one axiom. The others being proved similarly. For any x, y ∈ A and v ∈ V , we have e l ≺ ( { α α ′ ( x ) , α α ′ ( y ) } α ′ ,α ′ β β ′ ( v )= e l ≺ ( { α α ′ α ′ ( x ) , α α ′ α ′ ( y ) } β β ′ ( v )= l ≺ ( { α α ′ α ′ ( x ) , α α ′ α ′ ( y ) } β ′ ( v )= l ∗ ( α α α ′ α ′ ( x )) l ≺ ( α α ′ α ′ ( y )) β ′ ( v ) − l ≺ ( α α α ′ α ′ ( y )) l ∗ ( α α ′ α ′ ( x )) β ′ ( v ) ( by (3 . e l ∗ ( α α ′ α α ′ ( x )) e l ≺ ( α α ′ ( y )) v − e l ≺ ( α α ′ α α ′ ( y )) l ∗ ( α α ′ ( x )) v Taking α ′ = α p , α ′ = α p and β ′ = β q , β ′ = β q leads to the following statement: Corollary 3.27.
Let ( A, ≺ , ≻ , ∗ , α , α ) be a noncommutative BiHom-pre-Poisson algebra ( l ≺ , r ≺ , l ≻ , r ≻ , l ∗ , r ∗ , β , β , V ) a bimodule of A . Then V β q ,β q is a bimodule of A α p ,α p for anynonnegative integers p , p , q and q . Let ( l ≺ , r ≺ , l ≻ , r ≻ , l ⋄ , r ⋄ , β , β , V ) be a bimodule of a noncommutative BiHom-pre-Poissonalgebra ( A, ≺ , ≻ , ⋄ , α , α ) and let l ∗≺ , r ∗≺ , l ∗≻ , r ∗≻ , l ∗⋄ , r ∗⋄ : A → gl ( V ∗ ) , furthermore α ∗ , α ∗ : A ∗ → A ∗ , β ∗ , β ∗ : V ∗ → V ∗ be the dual maps of respectively α , α , β and β such that h l ∗≺ ( x ) u ∗ , v i = h u ∗ , l ≺ ( x ) v i , h r ∗≺ ( x ) u ∗ , v i = h u ∗ , r ≺ ( x ) v ih l ∗≻ ( x ) u ∗ , v i = h u ∗ , l ≻ ( x ) v i , h r ∗≻ ( x ) u ∗ , v i = h u ∗ , r ≻ ( x ) v ih l ∗⋄ ( x ) u ∗ , v i = h u ∗ , l ⋄ ( x ) v i , h r ∗⋄ ( x ) u ∗ , v i = h u ∗ , r ⋄ ( x ) v i α ∗ ( x ∗ ( y )) = x ∗ ( α ( y )) , α ∗ ( x ∗ ( y )) = x ∗ ( α ( y )) β ∗ ( u ∗ ( v )) = u ∗ ( β ( v )) , β ∗ ( u ∗ ( v )) = u ∗ ( β ( v ))21 roposition 3.28. Let ( l ≺ , r ≺ , l ≻ , r ≻ , l ⋄ , r ⋄ , β , β , V ) be a bimodule of a noncommutativeBiHom-pre-Poisson algebra ( A, ≺ , ≻ , ⋄ , α , α ) . Then ( l ∗≺ , r ∗≺ , l ∗≻ , r ∗≻ , l ∗⋄ , r ∗⋄ , β ∗ , β ∗ , V ∗ ) is a bi-module of ( A, ≺ , ≻ , ⋄ , α , α ) provided that β ( l ⋄ ( α ( x ) · α ( y )) u = β l ⋄ ( α ( x )) r ≻ ( a ( y )) u + l ⋄ ( α ( y )) l ≺ ( α α ( x )) u, (3.60) β l · ( α ( y )) r ⋄ ( α ( x )) u = β ( l ≻ ( α ( y ) ⋄ α ( x ))) u + β r ⋄ ( x ) l ≺ ( α α ( y )) u, (3.61) β r · ( α ( y )) r ⋄ ( α ( x )) u = β r ⋄ ( α ( x )) r ≻ ( α ( y )) u + β β ( r ≺ ( α ( y ) ⋄ x )) u, (3.62) β ρ ( α α ( y )) l ≻ ( α ( x )) u = β l ≻ ( x ) l ⋄ ( α α ( y )) u − β β ( l ≻ ( α ( y ) ⋄ z )) u, (3.63) β ( r ≻ ( { α α ( x ) , α ( y ) } )) u = r ≻ ( α ( y )) l ⋄ ( α α ( x )) u − l ⋄ ( α ( y )) r ≻ ( α α ( y )) u, (3.64) − β ρ ( α ( y )) l ≻ ( α ( x )) u = β β ( r ⋄ ( x ≻ α ( y ))) u − β r ⋄ ( x ) r ≻ ( α α ( y )) u, (3.65) β ( l ≺ ( { α ( x ) , α ( y ) } ) u = l ≺ ( α ( y )) l ⋄ ( α α ( x )) u − l ⋄ ( α ( x )) l ≺ ( α α ( y )) u, (3.66) β ρ ( α ( y )) r ≺ ( α ( x )) u = β r ≺ ( x ) l ⋄ ( α α ( y )) u − β β ( r ≺ ( α ( y ) ⋄ x )) u, (3.67) − β ρ ( α ( y )) r ≺ ( α ( x )) u = β β ( r ⋄ ( α ( y ) ≺ x )) u − β r ⋄ ( x ) l ≺ ( α α ( y ))( v ) , (3.68) for all x, y ∈ A and u ∈ V .Proof. Straightforward.
Theorem 3.29.
Let ( A, ≺ A , ≻ A , ∗ A , α , α ) and ( B, ≺ B , ≻ B , ∗ B , β , β ) be two noncommuta-tive BiHom-pre-Poisson algebra. Suppose that there are linear maps l ≺ A , r ≺ A , l ≻ A , r ≻ A , l ∗ A , r ∗ A : A → gl ( B ) , and l ≺ B , r ≺ B , l ≻ B , r ≻ B , l ∗ B , r ∗ B : B → gl ( A ) such that A ⊲⊳ l ∗ A ,r ∗ A ,β ,β l ∗ B ,r ∗ B ,α ,α B is amatched pair of BiHom-pre-Lie algebras and A ⊲⊳ l ≺ A ,r ≺ A ,l ≻ A ,r ≻ A ,β ,β l ≺ B ,r ≺ B ,l ≻ B ,r ≻ B ,α ,α B is a matched pair ofBiHom-dendriform algebra and for all x, y ∈ A, a, b ∈ B , the following equalities hold: − l ≺ A ( ρ B ( β ( a )) α ( x )) β ( b ) + ρ A ( α ( x )) β ( a ) ≺ B β ( b )= l ∗ A ( α α ( x ))( β ( a ) ≺ B b ) − β β ( a ) ≺ B ( l ∗ A ( α ( x )) b ) − r ≺ A ( r ∗ B ( b ) α ( x )) β β ( a ) , (3.69) l ≺ A ( ρ B ( β ( a )) α ( x )) β ( b ) − ( ρ A ( α ( x )) β ( a )) ≺ B β ( b )= β β ( a ) ∗ B ρ ( α ( x )) b + r ∗ A ( r ≺ B ( b ) α ( x )) β β ( a ) − l ≺ A ( α α ( x ))( β ( a ) ∗ B b ) , (3.70) r ≺ A ( α ( x ))( { β ( a ) , β ( b ) } B ) = β β ( a ) ∗ B ( r ≺ A ( x ) β ( b ))+ r ∗ A ( l ≺ B ( β ( b )) x ) β β ( a ) − l ≺ A ( α α ( x ))( β ( a ) ∗ B b ) , (3.71) l ≻ A ( α ( x )) { β β ( a ) , β ( b ) } = β β ( a ) ∗ B ( l ≺ A ( x ) β ( b ))+ r ∗ A ( r ≺ B ( β ( b )) x ) β β ( a ) − ( r ∗ A ( x ) β ( a )) ≺ B β β ( b ) − l ≺ A ( l ∗ A ( β ( a )) x ) β β ( b ) , (3.72) β ( a ) ≺ B ( ρ A ( α α ( x ))) β ( b ) − r ≺ A ( ρ B ( β ( b )) α ( x )) β ( a )= l ∗ A ( α α ( x ))( a ≻ B β ( b )) − ( l ∗ A ( α ( x )) a ) ≺ A β β ( b ) − l ≻ A ( r ∗ B ( a ) α ( x )) β β ( b ) , (3.73) − β ( a ) ≻ B ( ρ ( α ( x )) α ( b )) + r ≻ A ( ρ ( β β ( b )) α ( x )) β ( a )= β β ( b ) ∗ B ( r ≻ A ( α ( x )) b ) + r ∗ A ( l ≻ B ( a ) α ( x )) − r ≻ A ( α α ( x ))( β ( b ) ∗ B a ) , (3.74)22 l · B ( α ( x )) β ( a )) ∗ B β ( b ) + l ∗ A ( r · B ( β ( a )) α ( x )) β ( b )= ( l ∗ A ( α ( x )) β ( b )) ≻ B β ( a ) + l ≻ A ( r ∗ B ( β ( b )) α ( x )) β ( a )+ l ≺ A ( α α ( x ))( β ( a ) ∗ B b ) , (3.75) l ∗ A ( r · A ( α ( x )) β ( a )) β ( b ) + ( r · A ( α ( x )) β ( a )) ∗ B β ( b )= r ≻ A ( α ( x ))( β ( a ) ∗ B β ( b )) + β β ( a ) ≺ B ( l ∗ A ( α ( x )) b )+ r ≺ A ( r ∗ B ( b ) α ( x )) β β ( a ) , (3.76) r ∗ A ( α ( x ))( β ( a ) · B β ( b )) = ( r ∗ A ( α ( x )) β ( a )) ≻ B β ( b )+ l ≻ A ( l ∗ B ( β ( a )) α ( x )) β ( b ) + β β ( a ) ≺ B ( r ∗ A ( x ) β ( b ))+ r ≺ A ( l ∗ B ( β ( a )) α ( x )) β ( b ) + β β ( a ) ≺ B ( r ∗ A ( x ) β ( b ))+ r ≺ A ( l ∗ B ( β ( b )) x ) β β ( x ) , (3.77) − l ≺ B ( ρ A ( α ( x )) β ( a )) α ( y ) + ρ B ( β ( a )) α ( x ) ≺ A α ( y )= l ∗ B ( β β ( a ))( α ( x ) ≺ A y ) − α α ( x ) ≺ A ( l ∗ B ( β ( a )) y ) − r ≺ B ( r ∗ A ( y ) β ( a )) α α ( x ) , (3.78) l ≺ B ( ρ A ( α ( x )) β ( a )) α ( y ) − ( ρ B ( β ( a )) α ( x )) ≺ A α ( y )= α α ( x ) ∗ A ρ ( β ( a )) y + r ∗ B ( r ≺ A ( y ) β ( a )) α α ( x ) − l ≺ B ( β β ( a ))( α ( x ) ∗ A y ) , (3.79) r ≺ B ( β ( a ))( { α ( x ) , α ( y ) } A ) = α α ( x ) ∗ A ( r ≺ B ( a ) α ( y ))+ r ∗ B ( l ≺ A ( α ( y )) a ) α α ( x ) − l ≺ B ( β β ( a ))( α ( x ) ∗ A y ) , (3.80) l ≻ B ( β ( a )) { α α ( x ) , α ( y ) } = α α ( x ) ∗ A ( l ≺ B ( x ) α ( y ))+ r ∗ B ( r ≺ A ( α ( y )) a ) α α ( x ) − ( r ∗ B ( a ) α ( x )) ≺ A α α ( y ) − l ≺ B ( l ∗ B ( α ( x )) a ) α α ( y ) , (3.81) α ( x ) ≺ A ( ρ B ( β β ( a ))) α ( y ) − r ≺ B ( ρ A ( α ( y )) β ( a )) α ( x )= l ∗ B ( β β ( x ))( a ≻ B α ( b )) − ( l ∗ A ( β ( a )) x ) ≺ B α α ( y ) − l ≻ B ( r ∗ A ( x ) β ( a )) α α ( y ) , (3.82) − α ( x ) ≻ A ( ρ ( β ( a )) β ( y )) + r ≻ B ( ρ ( α α ( y )) β ( a )) α ( x )= α α ( y ) ∗ A ( r ≻ B ( β ( a )) y ) + r ∗ B ( l ≻ A ( x ) β ( a )) − r ≻ B ( β β ( a ))( α ( y ) ∗ A x ) , (3.83)( l · A ( β ( a )) α ( x )) ∗ A α ( y ) + l ∗ B ( r · A ( α ( x )) β ( a )) α ( y )= ( l ∗ B ( β ( a )) α ( y )) ≻ A α ( x ) + l ≻ B ( r ∗ A ( α ( y )) β ( a )) α ( x )+ l ≺ B ( β β ( a ))( α ( x ) ∗ A y ) , (3.84) l ∗ B ( r · B ( β ( a )) α ( x )) α ( y ) + ( r · B ( β ( a )) α ( x )) ∗ A α ( y )= r ≻ B ( β ( a ))( α ( x ) ∗ A α ( y )) + α α ( x ) ≺ A ( l ∗ B ( β ( a )) y )+ r ≺ B ( r ∗ A ( y ) β ( a )) α α ( x ) , (3.85) r ∗ B ( β ( a ))( α ( x ) · A α ( y )) = ( r ∗ B ( β ( a )) α ( x )) ≻ A α ( y )+ l ≻ B ( l ∗ A ( α ( x )) β ( a )) α ( y ) + α α ( x ) ≺ A ( r ∗ B ( a ) α ( y ))+ r ≺ B ( l ∗ A ( α ( x )) β ( a )) α ( y ) + α α ( x ) ≺ A ( r ∗ B ( a ) α ( y ))+ r ≺ B ( l ∗ A ( α ( y )) a ) α α ( a ) , (3.86)23 here x · A y = x ≺ A y + x ≻ A y, l · A = l ≺ A + l ≻ A , r · A = r ≺ A + r ≻ A ,a · B b = a ≺ B b + a ≻ B b, l · B = l ≺ B + l ≻ B , r · B = r ≺ B + r ≻ B , { α ( x ) , α ( y ) } A = α ( x ) ∗ A α ( y ) − α ( y ) ∗ A α ( x ) , { β ( a ) , β ( b ) } B = β ( a ) ∗ B β ( b ) − β ( b ) ∗ B β ( a ) , ( ρ A ◦ α ) β = ( l ∗ A ◦ α ) β − ( r ∗ A ◦ α ) β , ( ρ B ◦ β ) α = ( l ∗ B ◦ β ) α − ( r ∗ B ◦ β ) α . Then ( A, B, l ≺ A , r ≺ A , l ≻ A , r ≻ A , l ∗ A , r ∗ A , β , β , l ≺ B , r ≺ B , l ≻ B , r ≻ B , l ∗ B , r ∗ B , α , α ) is called a matchedpair of noncommutative BiHom-pre-Poisson algebras. In this case, there exists a noncommu-tative BiHom-pre-Poisson algebra structure on the direct sum A ⊕ B of the underlying vectorspaces of A and B given by ( x + a ) ≺ ( y + b ) := ( x ≺ A y + r ≺ B ( b ) x + l ≺ B ( a ) y ) + ( l ≺ A ( x ) b + r ≺ A ( y ) a + a ≺ B b ) , ( x + a ) ≻ ( y + b ) := ( x ≻ A y + r ≻ B ( b ) x + l ≻ B ( a ) y ) + ( l ≻ A ( x ) b + r ≻ A ( y ) a + a ≻ B b )( x + a ) ∗ ( y + b ) := ( x ∗ A y + r ∗ B ( b ) x + l ∗ B ( a ) y ) + ( l ∗ A ( x ) b + r ∗ A ( y ) a + a ∗ B b ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) , ( α ⊕ β )( x + a ) := α ( x ) + β ( a ) , for any x, y ∈ A, a, b ∈ B .Proof. It is obtained in a similar way as for Theorem 2.8.Let
A ⊲⊳ l ≺ A ,r ≺ A ,l ≻ A ,r ≻ A ,l ∗ A ,r ∗ A ,β ,β l ≺ B ,r ≺ B ,l ≻ B ,r ≻ B ,l ∗ B ,r ∗ B ,α ,α B denote this noncommutative BiHom-pre-Poisson al-gebra. Corollary 3.30.
Let ( A, B, l ≺ A , r ≺ A , l ≻ A , r ≻ A , l ∗ A , r ∗ A , β , β , l ≺ B , r ≺ B , l ≻ B , r ≻ B , l ∗ B , r ∗ B , α , α ) be a matched pair of regular noncommutative BiHom-pre-Poisson algebras ( A, ≺ A , ≻ A , ∗ A , α , α ) and ( B, ≺ B , ≻ B , ∗ B , β , β ) . Then, ( A, B, l ≺ A + l ≻ A , r ≺ A + r ≻ A , l ∗ A − ( r ∗ A ◦ α α − ) β − β , β , β , l ≺ B + l ≻ B , r ≺ B + r ≻ B , l ∗ B − ( r ∗ B ◦ β β − ) α − α , α , α ) is a matched pair of the associated noncom-mutative BiHom-Poisson algebras ( A, · A , {· , ·} A , α , α ) and ( B, · B , {· , ·} B , β , β ) .Proof. Let (
A, B, l ≺ A , r ≺ A , l ≻ A , r ≻ A , l ∗ A , r ∗ A , β , β , l ≺ B , r ≺ B , l ≻ B , r ≻ B , l ∗ B , r ∗ B , α , α ) be a matchedpair of regular noncommutative BiHom-pre-Poisson algebras ( A, ≺ A , ≻ A , ∗ A , α , α ) and ( B, ≺ B , ≻ B , ∗ B , β , β ). Then by Proposition 3.7 and Proposition 3.14, ( A, B, l ≺ A + l ≻ A , r ≺ A + r ≻ A , β , β , l ≺ B + l ≻ B , r ≺ B + r ≻ B , α , α ) is a matched pair of the associated BiHom-associativealgebras ( A, · A , α , α ) and ( B, · B , β , β ) and ( A, B, l ∗ A − ( r ∗ A ◦ α α − ) β − β , β , β , l ∗ B − ( r ∗ B ◦ β β − ) α − α , α , α ) is a matched pair of the associated BiHom-Lie algebras ( A, {· , ·} A , α , α )and ( B, {· , ·} B , β , β ). Besides, in view of Corollary 3.24, the linear maps l ≺ A + l ≻ A , r ≺ A + r ≻ A , l ∗ A − ( r ∗ A ◦ α α − ) β − β : A → gl ( B ) and l ≺ B + l ≻ B , r ≺ B + r ≻ B , l ∗ B − ( r ∗ B ◦ β β − ) α − α : B → gl ( A ) are a representations of the underlying noncommutative BiHom-Poisson alge-bras ( A, · A , {· , ·} A , α , α ) and ( B, · B , {· , ·} B , β , β ), respectively. Therefore, (2.27)-(2.28) areequivalents to (3.69)-(3.77) and (2.29)-(2.30) are equivalents to (3.78)-(3.86).24 O -operators of noncommutative BiHom-Poisson al-gebras In this section we introduce the notions of an O -operator of noncommutative BiHom- Poissonalgebras and we give some related properties. Definition 4.1.
Let ( A, · , α , α ) be a BiHom-associative algebra and ( l, r, β , β , V ) bea bimodule of A . Then, a linear map T : V → A is called an O -operator associated to( l, r, β , β , V ), if T satisfies α T = T β , α T = T β and T ( u ) · T ( v ) = T ( l ( T ( u )) v + r ( T ( v )) u ) for all u, v ∈ V. Lemma 4.2. [23] Let ( A, · , α , α ) be a BiHom-associative algebra, and let ( l, r, β , β , V ) be a bimodule. Let T : V → A be an O -operator associated to ( l, r, β , β , V ) . Then, thereexists a BiHom-dendriform algebra structure on V given by u ≻ v = l ( T ( u )) v, u ≺ v = r ( T ( v )) u for all u, v ∈ V . Now we recall the definition of an O -operator on a BiHom-Lie algebra associated toa given representation, which generalize the Rota-Baxter operator of weight 0 introducedin [31]. Definition 4.3.
Let ( A, {· , ·} , α , α ) be a BiHom-Lie algebra, and let ( ρ, β , β , V ) be arepresentation of A . Then, a linear map T : V → A is called an O -operator associated to( ρ, β , β , V ), if T satisfies α T = T β , α T = T β and { T ( u ) , T ( v ) } = T ( ρ ( T ( u )) v − ρ ( T ( β − β ( v ))) β β − ( u )) for all u, v ∈ V. Example . An O -operator on a BiHom-Lie algebra ( A, {· , ·} , α , α ) with respect to theadjoint representation is called a Rota-Baxter operator on A . Lemma 4.5.
Let T : V → A be an O -operator on a BiHom-Lie algebra ( A, {· , ·} , α , α ) with respect to a representation ( ρ, β , β , V ) . Define a multiplication ∗ on V by u ∗ v = ρ ( T ( u )) v, ∀ u, v ∈ V. (4.1) Then ( V, ∗ , α , α ) is a BiHom-pre-Lie algebra. Definition 4.6.
Let ( A, · , {· , ·} , α , α ) be a noncommutative BiHom-Poisson algebra, andlet ( l, r, ρ, β , β , V ) be a representation of A . A linear operator T : V → A is called an O -operator on A if T is both an O -operator on the BiHom-associative algebra ( A, · , α , α )and an O -operator on the BiHom-Lie algebra ( A, {· , ·} , α , α ). Example . An O -operator on a noncommutative BiHom-Poisson algebra ( A, · , {· , ·} , α , α )with respect the regular representation is called a Rota-Baxter operator on A .25 heorem 4.8. Let ( A, · , {· , ·} , α , α ) be a noncommutative BiHom-Poisson algebra and T : V → A an O -operator on A with respect to the representation ( l, r, ρ, β , β , V ) . Define newoperations ≺ , ≻ and ∗ on V by u ≺ v = l ( T ( u )) v, u ≻ v = r ( T ( v )) u, u ∗ v = ρ ( T ( u )) v. (4.2) Then ( V, ≺ , ≻ , ∗ , α , α ) is a noncommutative BiHom-pre-Poisson algebra. Moreover, T ( V ) = { T ( v ); v ∈ V } ⊂ A is a subalgebra of A and there is an induced noncommutative BiHom-pre-Poisson algebra structure on T ( V ) given by T ( u ) ≺ T ( v ) = T ( u ≺ v ) , T ( u ) ≻ T ( v ) = T ( u ≻ v ) , T ( u ) ∗ T ( v ) = T ( u ∗ v ) , (4.3) for all u, v ∈ V .Proof. By Lemma 4.2 and Lemma 4.5, we deduce that ( A, ≺ , ≻ , α , α ) is a BiHom-dendriformalgebra and ( A, ∗ , α , α ) is a BiHom-pre-Lie algebra. Now, we prove only the axiom (3.48).The other being proved similarly, for any x, y, z ∈ V we have( β ( x ) ∗ β ( y ) − β ( y ) ∗ β ( x )) ≺ β ( z ) − β β ( x ) ∗ ( β ( y ) ≺ z ) + β β ( y ) ≺ ( β ( x ) ∗ z )=( ρ ( T ( β ( x )) β ( y ) − ρ ( T ( β ( y )) β ( x )) ≺ β ( z ) − ρ ( T ( β β ( x )))( β ( y ) ≺ z ) + l ( T ( β β ( y )))( β ( x ) ∗ z )= l ( T ( ρ ( T ( β ( x )) β ( y ) − ρ ( T ( β ( y ))) β ( x )) β ( z ) − ρ ( T ( β β ( x ))) l ( β ( y )) z + l ( T ( β β ( y ))) ρ ( β ( x )) z = l ( { T ( β ( x ) , T ( β ( y )) } ) β ( z ) − ρ ( T ( β β ( x ))) l ( β ( y )) z + l ( T ( β β ( y ))) ρ ( β ( x )) z = 0 ( by (2 . . Therefore, ( V, ≺ , ≻ , ∗ , α , α ) is a BiHom-pre-Posson algebra.The rest is straightforward. Corollary 4.9.
Let ( A, · , {· , ·} , α , α ) be a noncommutative BiHom-Poisson algebra. Thenthere is a noncommutative BiHom-pre-Poisson algebra structure on A such that its sub-adjacent noncommutative BiHom-Poisson algebra is exactly ( A, · , {· , ·} , α , α ) if and only ifthere exists an invertible O -operator on ( A, · , {· , ·} , α , α ) . Proof.
Suppose that there exists an invertible O -operator T : V → A associated to therepresentation ( l, r, ρ, β , β , V ), then the compatible noncommutative BiHom-pre-Poissonalgebra structure on A , for all x, y ∈ A is given by x ≺ y = T ( l ( x ) T − ( y )) , x ≻ y = T ( r ( y ) T − ( x )) , x ∗ y = T ( ρ ( x ) T − ( y )) ∀ x, y ∈ A. Conversely, let ( A, ≺ , ≻ , ∗ , α , α ) be a noncommutative BiHHom-pre-Poisson algebra and( A, · , {· , ·} , α , α ) the sub-adjacent noncommutative BiHom-Poisson algebra. Then the iden-tity map id is an O -operator on A with respect to the regular representation ( L ≺ , R ≻ , ad, α , α , A ).26 xample . Let ( A, · , {· , ·} , α , α ) be a noncommutative BiHom-Poisson algebra and R : A −→ A a Rota-Baxter operator. Define new operations on A by x ≺ y = R ( x ) · y, x ≻ y = x · R ( y ) , x ∗ y = { R ( x ) , y } . Then ( A, ≺ , ≻ , ∗ , α , α ) is a noncommutative BiHom-pre-Poisson algebra and R is a homo-morphism from the sub-adjacent noncommutative BiHom-Poisson algebra ( A, · ′ , {· , ·} ′ , α , α )to ( A, · , {· , ·} , α , α ), where x · ′ y = x ≺ y + x ≻ y and { x, y } ′ = x ∗ y − α − α ( y ) ∗ α α − ( x ).The inverse relation existing between a noncommutative BiHom-pre-Poisson algebra andnoncommutative BiHom-Poisson algebra, as illustrated by the following diagram:BiHom-associative alg+BiHom-Lie alg / / { R ( x ) ,y } R ( x ) · y, x · R ( y ) (cid:15) (cid:15) BiHom-Poisson alg { R ( x ) ,y } R ( x ) · y, x · R ( y ) (cid:15) (cid:15) BiHom-dendriform alg+ BiHom-pre-Lie alg / / BiHom-pre-Poisson alg.
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