(Non-BiHom-Commutative) BiHom-Poisson algebras
aa r X i v : . [ m a t h . R A ] A ug (Non-BiHom-Commutative) BiHom-Poisson algebras Hadjer Adimi , Hanene Amri , Sami Mabrouk , Abdenacer Makhlouf
1. Universit´e Mohamed El Bachir El Ibrahimi de Bordj Bou Arr´eridj El-Anasser, 34030 - Algeria,E-mail: [email protected]. Universit´e Badji Mokhtar Annaba’ BP 12 d´epartement de math´ematiques, facult´e des sciences, AlgeriaE-mail: [email protected]. University of Gafsa, Faculty of Sciences Gafsa, 2112 Gafsa, TunisiaE-mail: [email protected]. IRIMAS - D´epartement de Math´ematiques, 6, rue des fr`eres Lumi`ere, F-68093 Mulhouse, FranceE-mail: [email protected]
Abstract
The aim of this paper is to introduce and study BiHom-Poisson algebras, in par-ticular Non-BiHom-Commutative BiHom-Poisson algebras. We discuss their repre-sentation theory and Semi-direct product. Furthermore, we characterize admissi-ble BiHom-Poisson algebras. Finally, we establish the classification of 2-dimensionalBiHom-Poisson algebras.
A vector space A is called a Poisson algebra provided that, beside addition, it has two K -bilinear operations which are related by derivation. First, with respect to multiplica-tion, A is a commutative associative algebra; denote the multiplication by µ ( a, b ) (or a · b or ab ), where a, b ∈ A . Second, A is a Lie algebra; traditionally here the Lie operation isdenoted by the Poisson brackets { a,b } , where a, b ∈ A . It is also assumed that these twooperations are connected by the Leibniz rule { a · b, c } = a ·{ b, c } + b ·{ a, c } , a , b , c ∈ A [6, 11].Poisson algebras are the key to recover Hamiltonian mechanics and are also central in thestudy of quantum groups. Manifolds with a Poisson algebra structure are known as Pois-son manifolds, of which the symplectic manifolds and the Poisson-Lie groups are a specialcase. Their generalization is known as Nambu algebras [15, 5, 2, 3], where the binarybracket is generalized to ternary or n -ary bracket. A Hom-algebra structure is a multipli-cation on a vector space where a usual structure is twisted by a homomorphism [12]. Thefirst motivation to study nonassociative Hom-algebras comes from quasi-deformations ofLie algebras of vector fields, in particular q -deformations of Witt and Virasoro algebras.The structure of (Non-Commutative)-Hom-Poisson algebras are twisted generalization of(Non-Commutative)-Poisson algebras [18]. A (Non-Commutative)-Hom-Poisson algebra A is defined by a linear self-map α , and two binary operations { , } (the Hom-Lie bracket)and µ (the Hom-associative product). The associativity, the Jacobi identity, and the Leib-niz identity in a (Non-Commutative)-Poisson algebra are replaced by their Hom-type (i.e. α -twisted) identities. Motivated by a categorical study of Hom-algebras and new type of1ategories, generalized algebraic structures endowed with two commuting multiplicativelinear maps, called BiHom-algebras including BiHom-associative algebras, BiHom-Lie al-gebras and BiHom-Bialgebras were introduced in [7]. Therefore, when the two linear mapsare the same, BiHom-algebras will be turn to Hom-algebras in some cases. Various studiesdeal with these new type of algebras, see [19, 8, 9] and references therein.The purpose of this paper is to study (Non-Commutative) BiHom-Poisson algebras.The paper is organized as follows. In Section 2, we review the definition of BiHom-associative and BiHom-Lie algebras and then generalize the Poisson algebra notion toBiHom case. This new structure is illustrated with some examples. In Section 3, westudy the concept of module of BiHom-Poisson algebra, which is based on BiHom-modulesof BiHom-associative and BiHom-Lie algebras. Then we define semi-direct product of(Non-Commutative) BiHom-Poisson algebras. In Section 4, we describe BiHom-Poissonalgebras using only one binary operation and the twisting maps via the polarization-depolarization process. We show that, admissible BiHom-Poisson algebras, and only theseBiHom-algebras, give rise to BiHom-Poisson algebras via polarization. In the last section,we give the classification of 2-dimensional BiHom-Poisson algebras. In this section, we recall some basic definitions about BiHom-associative and BiHom-Lie algebras [7] and then we generalize the Poisson algebras notion to BiHom case. Weassume that K will denote a commutative field of characteristic zero. Definition 2.1.
A BiHom-associative algebra is a quadruple ( A, µ, α, β ) consisting ofvector space A , a bilinear mapping µ : A × A → A and two homomorphisms α, β : A → A such that for x, y, z ∈ A we have αβ = βα,α ◦ µ = µ ◦ α ⊗ , β ◦ µ = µ ◦ β ⊗ ,µ ( α ( x ) , µ ( y, z )) = µ ( µ ( x, y ) , β ( z )) ( BiHom-associative condition ) , (2.1) where αβ = α ◦ β . We recall that the BiHom-commutative condition is µ ( β ( x ) , α ( y )) = µ ( β ( y ) , α ( x )), forall x, y ∈ A . Definition 2.2.
A BiHom-Lie algebra is a quadruple ( A, [ · , · ] , α, β ) consisting of vectorspace A , a bilinear mapping [ ., . ] : A × A → A and two homomorphisms α, β : A → A suchthat for x, y, z ∈ A we have αβ = βα,α ([ x, y ]) = [ α ( x ) , α ( y )] , β ([ x, y ]) = [ β ( x ) , β ( y )] , [ β ( x ) , α ( y )] = − [ β ( y ) , α ( x )] , ( BiHom-skew-symmetric ) (2.2) (cid:9) x,y,z [ β ( x ) , [ β ( y ) , α ( z )]] = 0 ( BiHom-Jacobi condition ) , (2.3) where (cid:9) x,y,z denotes summation over the cyclic permutation on x, y, z . α is a bijective morphism, then the identity (2.3) can be written[ β ( x ) , [ β ( y ) , α ( z )]] = [[ α − β ( x ) , β ( y )] , αβ ( z )] + [ β ( y ) , [ β ( x ) , α ( z )]] . (2.4) Definition 2.3.
A Poisson algebra is a triple ( A, {· , ·} , µ ) consisting of a vector space A and two bilinear maps {· , ·} , µ : A × A −→ A satisfying1. ( A, {· , ·} ) is a Lie algebra,2. ( A, µ ) is a commutative associative algebra,3. for all x, y ∈ A : { µ ( x, y ) , z } = µ ( { x, z } , y ) + µ ( x, { y, z } ) ( Compatibility identity ) . (2.5) If µ is non-commutative then ( A, {· , ·} , µ ) is a non-commutative Poisson algebra. Definition 2.4.
A BiHom-Poisson algebra is a 5-uple ( A, {· , ·} , µ, α, β ) consisting of avector space A , two bilinear maps {· , ·} , µ : A × A −→ A and two linear maps α, β : A −→ A satisfying1. ( A, {· , ·} , α, β ) is a BiHom-Lie algebra,2. ( A, µ, α, β ) is a BiHom-commutative BiHom-associative algebra,3. for all x, y ∈ A : { µ ( x, y ) , αβ ( z ) } = µ ( { x, β ( z ) } , α ( y )) + µ ( α ( x ) , { y, α ( z ) } ) . (2.6) If µ is non-BiHom-commutative then ( A, {· , ·} , µ, α, β ) is a non-BiHom-commutative BiHom-Poisson algebra. We are using here a right handed Leibniz rule, one may call such algebras right BiHom-Poisson algebras. We refer to [10] for left BiHom-Poisson algebras.
Remark 2.1.
Obviously, a BiHom-Poisson algebra ( A, {· , ·} , µ, α, β ) for which α = β and α injective is just a Hom-Poisson algebra ( A, {· , ·} , µ, α ) . Proposition 2.1.
Let ( A, µ, α, β ) be a BiHom-associative algebra where α and β are twobijective homomorphisms. Then the -uple ( A, {· , ·} , µ, α, β ) , where the bracket is definedby { x, y } = µ ( x, y ) − µ ( α − β ( y ) , αβ − ( x )) , for x, y ∈ A is a non-commutative BiHom-Poisson algebra.Proof. We show that α and β are compatible with the bracket {· , ·} . For all x, y ∈ A , wehave { α ( x ) , α ( y ) } = µ ( α ( x ) , α ( y )) − µ ( α − β ( α ( y )) , αβ − ( α ( x )))= µ ( α ( x ) , α ( y )) − µ ( β ( y ) , α β − ( x ))= α ( { x, y } ) . α is even and α ◦ β = β ◦ α . In the same way, we checkthat β ( { x, y } ) = { β ( x ) , β ( y ) } .The BiHom-skew-symmetry { β ( x ) , α ( y ) } = −{ β ( y ) , α ( x ) } is obvious.Therefore, it remains to prove the BiHom-Jacobi identity. For all x, y, z ∈ A , we have { β ( x ) , { β ( y ) , α ( z ) }} = µ ( β ( x ) , µ ( β ( y ) , α ( z ))) − µ ( µ ( α − β ( y ) , β ( z )) , αβ ( x )) − µ ( β ( x ) , µ ( β ( z ) , α ( y ))) + µ ( µ ( α − β ( z ) , β ( y )) , αβ ( x )) . And, we have { β ( y ) , { β ( z ) , α ( x ) }} = µ ( β ( y ) , µ ( β ( z ) , α ( x ))) − µ ( µ ( α − β ( z ) , β ( x )) , αβ ( y )) − µ ( β ( y ) , µ ( β ( x ) , α ( z ))) + µ ( µ ( α − β ( x ) , β ( z )) , αβ ( y )) . Similarly, { β ( z ) , { β ( x ) , α ( y ) }} = µ ( β ( z ) , µ ( β ( x ) , α ( y ))) − µ ( µ ( α − β ( x ) , β ( y )) , αβ ( z )) − µ ( β ( z ) , µ ( β ( y ) , α ( x ))) + µ ( µ ( α − β ( y ) , β ( x )) , αβ ( z )) . By BiHom-associativity, we find that (cid:9) x,y,z { β ( x ) , { β ( y ) , α ( z ) }} = 0 . Now, we show the compatibility condition, for x, y, z ∈ P , we have { µ ( x, y ) , αβ ( z ) } − µ ( { x, β ( z ) } , α ( y )) − µ ( α ( x ) { y, α ( z ) } )= µ ( µ ( x, y ) , αβ ( z )) − µ ( β ( z ) , µ ( αβ − ( x ) , αβ − ( y )) − µ ( µ ( x, β ( z )) , α ( y ))+ µ ( µ ( α − β ( z ) , αβ − ( x )) , α ( y )) − µ ( α ( x ) , µ ( y, α ( z ))) + µ ( α ( x ) , µ ( β ( z ) , αβ − ( y ))) = 0 . Definition 2.5.
Let ( A, µ, { ., . } , α, β ) and ( A ′ , µ ′ , { ., . } ′ , α ′ , β ′ ) be two BiHom-Poisson al-gebras. A linear map f : A → A ′ is a morphism of BiHom-Poisson algebras if it satisfiesfor all x , x ∈ A : f ( { x , x } ) = { f ( x ) , f ( x ) } ′ , (2.7) f ◦ µ = µ ′ ◦ f ⊗ , (2.8) f ◦ α = α ′ ◦ f. (2.9) f ◦ β = β ′ ◦ f. (2.10) It said to be a weak morphism if hold only the two first conditions.
Definition 2.6.
Let ( A, µ, { ., . } , α, β ) be a BiHom-Poisson algebra. It is said to be mul-tiplicative if α ( { x , x } ) = { α ( x ) , α ( x ) } ,β ( { x , x } ) = { β ( x ) , β ( x ) } ,α ◦ µ = µ ◦ α ⊗ .β ◦ µ = µ ◦ β ⊗ . It is said to be regular if α and β are bijective. roposition 2.2. Let ( A, {· , ·} , µ ) be an ordinary Poisson algebra over a field K and let α, β : A → A be two commuting morphisms. Define the two linear maps {· , ·} α,β , µ α,β : A ⊗ A −→ A by { x, y } α,β = { α ( x ) , β ( y ) } and µ α,β ( x, y ) = µ ( α ( x ) , β ( y )) , for all x, y ∈ A .Then A α,β := ( A, {· , ·} α,β , µ α,β , α, β ) is a BiHom-Poisson algebra.Proof. We already have (
A, µ, α, β ) is a BiHom-commutative BiHom-associative algebraand ( A, { , } α,β , α, β ) is a BiHom-Lie algebra. It remains to check the BiHom-Leibnizidentity. Let x, y, z ∈ A , we have { µ α,β ( x, y ) , αβ ( z ) } α,β − µ α,β ( { x, β ( z ) } α,β , α ( y )) − µ α,β ( α ( x ) , { y, α ( z ) } α,β )= { µ ( α ( x ) , αβ ( y )) , αβ ( z ) } − µ ( { α ( x ) , αβ ( z ) } , αβ ( y )) − µ ( α ( x ) , { αβ ( y ) , αβ ( z ) } )= { µ ( X, Y ) , Z ) } − µ ( { X, Z } , Y ) − µ ( X, { Y, Z } ) = 0 , where X = α ( x ) , Y = αβ ( y ) , Z = αβ ( z ) . Remark 2.2.
Let ( A, {· , ·} , µ, α, β ) be a BiHom-Poisson algebra and α ′ , β ′ : A → A twoBiHom-Poisson algebra morphisms such that any of the maps α, β, α ′ , β ′ commute. Definenew multiplications on A by: { x, y } ′ = { α ′ ( x ) , β ′ ( y ) } , µ ′ ( x, y ) = µ ( α ′ ( x ) , β ′ ( y )) . Then, ( A, {· , ·} ′ , µ ′ , α ′ α, β ′ β ) is a BiHom-Poisson algebra. Example 2.1.
Let { e , e , e } be a basis of a -dimensional vector space A over K .Consider the following multiplication µ , skew-symmetric bracket and linear map α on A − = K : µ ( e , e ) = e ,µ ( e , e ) = µ ( e , e ) = e , { e , e } = ae + be , { e , e } = ce + de ,α ( e ) = λ e + λ e , α ( e ) = λ e + λ e , α ( e ) = λ e + λ e , where a, b, c, d, λ , λ , λ , λ , λ , λ are parameters in K . Assume that β = Id , hence αβ = βα . Using Proposition 2.2, we construct the following multiplicative BiHom-Poissonalgebra defined by µ αβ ( e , e ) = λ e ,µ αβ ( e , e ) = λ e ,µ αβ ( e , e ) = λ e , { e , e } αβ = − ( λ a + λ c ) e − ( λ b + λ d ) e , { e , e } αβ = − ( λ a + λ c ) e − ( λ b + λ d ) e , { e , e } αβ = − ( λ a + λ c ) e − ( λ b + λ d ) e . Then we give an example of BiHom-Poisson algebra where α and β are arbitrary and { e , e , e } be a basis of a 3-dimensional vector space A over K .5 xample 2.2. α ( e ) = e , α ( e ) = e , α ( e ) = e .β ( e ) = e , β ( e ) = e .µ ( e , e ) = λ e ,µ ( e , e ) = λ e , { e , e } = ae , where a, λ are parameters in K . Another example of BiHom-Poisson algebras of dimension 3 with basis { e , e , e } isgiven where α and β are diagonal. Example 2.3. α ( e ) = ae , α ( e ) = be .β ( e ) = ce , β ( e ) = de .µ ( e , e ) = λ e , { e , e } = λ e , where a, b, c, d, λ , λ are parameters in K . In the sequel we define a direct sum and tensor product of a BiHom-Poisson algebraand a BiHom-associative symmetric algebra.
Theorem 2.1.
Let ( A , µ , { ., . } , α , β ) and ( A , µ , { ., . } , α , β ) be two (non-BiHom-commutative) BiHom-Poisson algebras. Let µ A ⊕ A be a bilinear map on A ⊕ A definedfor x , y ∈ A and x , y ∈ A by µ ( x + x , y + y ) = µ ( x , y ) + µ ( x , y ) , { ., . } A ⊕ A a bilinear map defined by { x + x , y + y } A ⊕ A = { x , y } + { x , y } and α A ⊕ A a linear map defined by α A ⊕ A ( x + x ) = α ( x ) + α ( x ) , and β A ⊕ A a linear map defined by β A ⊕ A ( x + x ) = β ( x ) + β ( x ) . Then ( A ⊕ A , µ A ⊕ A , { ., . } A ⊕ A , α A ⊕ A , β A ⊕ A ) is a BiHom-Poisson algebra. Theorem 2.2.
Let ( A, µ, { ., . } , α, β ) be a BiHom-Poisson algebra and ( B, µ ′ , α ′ , β ′ ) be aBiHom-associative symmetric algebra, then ( A ⊗ B, { ., . } A ⊗ B , µ ⊗ µ ′ , α ⊗ α ′ , β ⊗ β ′ ) , is a BiHom-Poisson algebra, where { ., . } A ⊗ B = { ., . } ⊗ µ ′ . roof. Since µ and µ ′ are both BiHom-associative multiplication whence a tensor prod-uct µ ⊗ µ ′ is BiHom-associative. Also the BiHom-commutativity of µ ⊗ µ ′ , the BiHom-skewsymmetry of { ., . } and the BiHom-commutativity of µ imply the BiHom-skewsymmetryof { ., . } A ⊗ B . Same, since the BiHom-Jacobi identity of { ., . } and the BiHom-associative of µ ′ are satisfy then { ., . } A ⊗ B is a BiHom-Lie bracket on A ⊗ B . Therefore, it remains tocheck the BiHom-Leibniz identity. We have LHS = { µ ⊗ µ ′ ( a ⊗ b , a ⊗ b ) , αβ ⊗ α ′ β ′ ( a ⊗ b ) } A ⊗ B = { µ ( a , b ) ⊗ µ ′ ( a , b ) , αβ ( a ) ⊗ α ′ β ′ ( b ) } A ⊗ B = { µ ( a , b ) , αβ ( a ) } A | {z } a ′ ⊗ µ ′ ( µ ′ ( a , b ) , α ′ β ′ ( b )) | {z } b ′ and RHS = µ ⊗ µ ′ ( a ⊗ b , { β ( a ) ⊗ β ′ ( b ) , α ( a ) ⊗ α ′ ( b ) } A ⊗ B )+ µ ⊗ µ ′ ( { α ( a ) ⊗ α ′ ( b ) , a ⊗ b } A ⊗ B , α ⊗ β ′ ( a ⊗ b ))= µ ⊗ µ ′ ( α ( a ) ⊗ α ′ ( b ) , { a , α ( a ) } ⊗ µ ′ ( b , α ′ ( b )))+ µ ⊗ µ ′ ( { a , β ( a ) } ⊗ µ ′ ( b , β ′ ( b )) , α ( a ) ⊗ α ′ ( b ))= µ ( α ( a ) , { a , α ( a ) } ) | {z } c ′ ⊗ µ ′ ( α ′ ( b ) , µ ′ ( b , α ′ ( b ))) | {z } d ′ + µ ( { a , β ( a ) } , α ( a )) | {z } e ′ ⊗ µ ′ ( µ ′ ( b , β ′ ( b )) , α ′ ( b ) | {z } f ′ . With BiHom-Leibniz identity we have a ′ = c ′ + e ′ , and using the BiHom-associativitycondition we have b ′ = d ′ = f ′ . Therefore the left hand side is equal to the right hand sideand the BiHom-Leibniz identity is proved. Then( A ⊗ B, µ ⊗ µ ′ , { ., . } A ⊗ B , ( α ⊗ α ′ , β ⊗ β ′ ))is a BiHom-Poisson algebra. In this section we introduce a representation theory of BiHom-Poisson algebras andprovide a semi-direct product construction.
Definition 3.1.
A representation of a BiHom-Lie algebra ( A, {· , ·} , α, β ) on a vector space V with respect to two commuting maps γ, ν ∈ End ( V ) is a linear map ρ {· , ·} : A −→ End ( V ) , such that for any x, y ∈ A , the following equalities are satisfied: ρ {· , ·} ( α ( x )) ◦ γ = γ ◦ ρ {· , ·} ( x ) , (3.11) ρ {· , ·} ( β ( x )) ◦ ν = ν ◦ ρ {· , ·} ( x ) , (3.12) ρ {· , ·} ( { β ( x ) , y } ) ◦ ν = ( ρ {· , ·} ( αβ ( x )) ◦ ρ {· , ·} ( y ) − ρ {· , ·} ( β ( y )) ◦ ρ {· , ·} ( α ( x )) . (3.13)7 roposition 3.1. Let ( A, {· , ·} ) be a Lie algebra and ρ : A → End ( V ) be a representationof the Lie algebra on V . Let α, β : A → A be two commuting morphisms and let γ, ν : V → V be two commuting linear maps such that ρ {· , ·} ( α ( x )) ◦ γ = γ ◦ ρ {· , ·} ( x ) , ρ {· , ·} ( β ( x )) ◦ ν = ν ◦ ρ {· , ·} ( x ) and ρ {· , ·} ( α ( x )) ◦ ν = − ρ {· , ·} ( β ( x )) ◦ γ . Define e ρ {· , ·} = ρ {· , ·} ( α ( x )) ◦ γ. Then ( V, e ρ, γ, ν ) is a representation of the BiHom-Lie algebra A .Proof. Let x, y ∈ A , e ρ {· , ·} ( { β ( x ) , y } α,β ) ◦ ν − e ρ {· , ·} ( αβ ( x )) ◦ e ρ {· , ·} ( y ) e ρ {· , ·} ( β ( y )) ◦ e ρ {· , ·} ( α ( x ))= e ρ {· , ·} ( αβ ( x ) , β ( y )) ◦ ν − e ρ {· , ·} ( αβ ( x )) ◦ ρ {· , ·} ( α ( y )) ◦ ν + e ρ {· , ·} ( β ( y )) ◦ ρ {· , ·} ( α ( x )) ◦ ν = ρ {· , ·} ( α β ( x ) , αβ ( y )) ◦ ν − ρ {· , ·} ( α β ( x )) ◦ ρ {· , ·} ( αβ ( y )) ◦ ν + ρ {· , ·} ( αβ ( y )) ◦ ρ {· , ·} ( α β ( x )) ◦ ν = (cid:0) ρ {· , ·} ( α β ( x ) , αβ ( y )) − ρ {· , ·} ( α β ( x )) ◦ ρ {· , ·} ( αβ ( y )) + ρ {· , ·} ( αβ ( y )) ◦ ρ {· , ·} ( α β ( x )) (cid:1) ◦ ν = 0 . Let ( A, {· , ·} , α, β ) be a BiHom-Lie algebra. Let ρ {· , ·} : A → End ( V ) be a represen-tation of the BiHom-Lie algebra on V with respect to γ and ν . Assume that the maps α and ν are bijective. On the direct sum of the underlying vector spaces A ⊕ V , define e α, e β : A ⊕ V −→ A ⊕ V by e α ( x + a ) = α ( x ) + γ ( a ) , e β ( x + a ) = β ( x ) + ν ( a ) , and define a skewsymmetric bilinear map [ · , · ] A ⊕ V : A ⊕ V × A ⊕ V −→ A ⊕ V by[( x + a ) , ( y + b )] A ⊕ V = { x, y } + ρ {· , ·} ( x )( b ) − ρ {· , ·} ( α − β ( y ))( γν − ( a )) . (3.14) Theorem 3.1. [19] With the above notations, ( A ⊕ V, [ · , · ] A ⊕ V , e α, e β ) is a BiHom-Liealgebra. Definition 3.2.
Let ( A, µ, α, β ) be a commutative BiHom-associative algebra. A repre-sentation (or a BiHom-module) on a vector space V with respect to γ, ν ∈ End ( V ) is alinear map ρ µ : A −→ End ( V ) , such that for any x, y ∈ A , the following equalities aresatisfied: ρ µ ( α ( x )) ◦ γ = γ ◦ ρ µ ( x ) , ρ µ ( β ( x )) ◦ ν = ν ◦ ρ µ ( x ) , (3.15) ρ µ ( µ ( x, y )) ◦ ν = ρ µ ( α ( x )) ρ µ ( y ) . (3.16)Let ( A, µ, α, β ) be a commutative BiHom-associative algebra and (
V, ρ µ , γ, ν ) be arepresentation of A . On the direct sum of the underlying vector spaces A ⊕ V , define e α, e β : A ⊕ V −→ A ⊕ V by e α ( x + a ) = α ( x ) + γ ( a ) and e β ( x + a ) = β ( x ) + ν ( a )and define a bilinear map µ A ⊕ V : A ⊕ V × A ⊕ V −→ A ⊕ V by µ A ⊕ V ( x + a, y + b ) = µ ( x, y ) + ρ µ ( x )( b ) + ρ µ ( α − β ( y ))( γν − ( a )) . (3.17) Theorem 3.2.
With the above notations, ( A ⊕ V, µ A ⊕ V , e α, e β ) is a commutative BiHom-associative algebra. roof. By the fact that α, β are algebra homomorphisms with respect to µ , for x, y ∈ A, a, b ∈ V , we have e α ( µ A⊕ V ( x + a, y + b )) = e α ( µ ( x, y ) + ρ µ ( x )( b ) + ρ µ ( α − β ( y ))( γν − ( a )))= α ( µ ( x, y )) + γ ( ρ µ ( x )( b )) + γ ( ρ µ ( α − β ( y ))( γν − ( a )))) µ ( α ( x ) , α ( y ))) + ρ µ ( α ( x ))( γ ( b ))) + ρ µ ( α − β ( α ( y )))( γν − ( γ ( a )))))= µ A⊕ V ( α ( x ) + γ ( a ) , α ( y ) + γ ( b ))= µ A⊕ V ( e α ( x + a ) , e α ( y + b )) . If ( A , µ, α, β ) is a commutative BiHom-associative algebra, µ A ⊕ V ( e α ( x + a ) , µ A ⊕ V ( ( y + b, z + c ) ) ) = µ A ⊕ V ( µ A ⊕ V ( x + a, y + b ) ) , e β ( z + c )) , (3.18)for x, y, z ∈ A and a, b, c ∈ V . Developing (3.18), we have µ A ⊕ V ( e α ( x + a ) , µ A ⊕ V ( ( y + b, z + c ) ) )= µ A ⊕ V ( e α ( x + a ) , µ ( y, z ) + ρ µ ( y )( c ) + ρ µ ( α − β ( z ))( γν − ( b )))= µ ( α ( x ) , µ ( y, z )) + ρ µ ( α ( x )) ◦ ρ µ ( y )( c )) + ρ µ ( α ( x )) ◦ ρ µ ( α − β ( z ))( γν − ( b )))+ ρ µ ( µ ( α − β ( y ) , α − β ( z )))( γ ν − ( a )) . Similarly µ A ⊕ V ( µ A ⊕ V ( x + a, y + b ) , e β ( z + c ) ) = µ A ⊕ V ( µ ( x, y ) + ρ µ ( x )( b ) + ρ µ ( α − β ( y ))( γν − ( a )) , e β ( z + c ) ) = µ ( µ ( x, y ) , β ( z )) + ρ µ ( µ ( x, y )) ◦ ν ( c ) + ρ µ ( α − β ( z )) ◦ ρ µ ( αβ − ( x ))( γν − ( b ))+ ρ µ ( α − β ( z )) ◦ ρ µ ( y )( γ ν − ( a )) . Definition 3.3.
Let ( A, {· , ·} , µ, α, β ) be a BiHom-Poisson algebra, V be a vector space and ρ {· , ·} , ρ µ : A −→ End ( V ) be two linear maps and also γ, ν : V −→ V be two linear maps.Then ( V, ρ {· , ·} , ρ µ , γ, ν ) is called a representation of A if ( V, ρ {· , ·} , γ, ν ) is a representationof ( A, {· , ·} , α, β ) and ( V, ρ µ , γ, ν ) is a representation of ( A, µ, α, β ) and they are compatiblein the sense that for any x, y ∈ Aρ {· , ·} ( µ ( x, y )) ν = ρ µ ( β ( y )) ρ {· , ·} ( x ) + ρ µ ( α ( x )) ρ {· , ·} ( y ) , (3.19) ρ µ ( { β ( x ) , y } ) ν = − ρ µ ( αβ ( x )) ρ {· , ·} ( y ) − ρ {· , ·} ( β ( y )) ρ µ ( α ( x )) . (3.20) Theorem 3.3.
Let ( A, {· , ·} , µ, α, β ) be a BiHom-Poisson algebra and ( V, ρ {· , ·} , ρ µ , γ, ν ) be a representation of A . Then ( A ⊕ V, µ A ⊕ V , {· , ·} A ⊕ V , e α, e β ) is a commutative BiHom-Poisson algebra, where the maps µ A ⊕ V , {· , ·} A ⊕ V , e α and e β are defined in Theorem 3.2 andTheorem 3.1. roof. We need only to show that the Leibniz identity is satisfied. Let x, y, z ∈ A and a, b, c ∈ V , we have { µ A ⊕ V ( x + a, y + b ) , e α e β ( z + c ) } A ⊕ V − µ A ⊕ V ( { x + a, e β ( z + c ) } A ⊕ V , e α ( y + b )) − µ A ⊕ V ( e α ( x + a ) , { y + b, e α ( z + c ) } A ⊕ V )= { µ ( x, y ) + ρ µ ( x )( b ) + ρ µ ( α − β ( y )) γν − ( a ) , e α e β ( z + c ) } A ⊕ V − µ A ⊕ V ( { x, β ( z ) } + ρ {· , ·} ( α − β ( x )) ν ( c ) − ρ {· , ·} ( α − β ( z )) γν − ( a ) , e α ( y + b )) − µ A ⊕ V ( e α ( x + a ) , { y, α ( z ) } + ρ {· , ·} ( y ) γ ( c ) − ρ {· , ·} ( β ( z )) γν − ( b ))= { µ ( x, y ) , αβ ( z ) } + ρ {· , ·} ( µ ( x, y )) γν ( c ) − ρ {· , ·} ( β ( z )) γν − ( ρ µ ( x )( b ) + ρ µ ( α − β ( y )) γν − ( a )) − µ ( { x, β ( z ) } , α ( y )) − ρ µ ( { x, β ( z ) } ) γ ( b ) − ρ µ ( β ( y )) γν − ( ρ {· , ·} ( α − β ( x )) ν ( c ) − ρ {· , ·} ( α − β ( z )) γν − ( a )) − µ ( α ( x ) , { y, α ( z ) } ) − ρ µ ( α ( x ))( ρ {· , ·} ( y ) γ ( c ) − ρ {· , ·} ( α ( z ))( b )) − ρ µ ( { α − β ( y ) , β ( z ) } ) γ ν − ( a )= { µ ( x, y ) , αβ ( z ) } + ρ {· , ·} ( µ ( x, y )) γν ( c ) − ρ {· , ·} ( β ( z ))( ρ µ ( αβ − ( x )) γν − ( b )) − ρ {· , ·} ( β ( z ))( ρ µ ( y ) γ ν − ( a )) − µ ( { x, β ( z ) } , α ( y )) − ρ µ ( { x, β ( z ) } ) γ ( b ) − ρ µ ( β ( y ))( ρ {· , ·} ( x ) γ ( c )) − ρ µ ( β ( y ))( ρ {· , ·} ( β ( z )) γ ν − ( a )) − µ ( α ( x ) , { y, α ( z ) } ) − ρ µ ( α ( x ))( ρ {· , ·} ( y ) γ ( c )) + ρ µ ( α ( x ))( ρ {· , ·} ( α ( z ))( b )) − ρ µ ( { α − β ( y ) , β ( z ) } ) γ ν − ( a ) = 0 . A Poisson algebra has two binary operations, the Lie bracket and the commutativeassociative product. In this section we describe BiHom-Poisson algebra using only onebinary operation and the twisting maps via the polarization-depolarization procedure.
Definition 4.1.
Let ( A, µ, α, β ) be a BiHom-algebra. Then A is called an admissibleBiHom-Poisson algebra if it satisfies as α,β ( β ( x ) , α ( y ) , α ( z )) = 13 { µ ( µ ( β ( x ) , αβ ( z )) , α ( y )) − µ ( µ ( β ( z ) , α ( x )) , α ( y ))+ µ ( µ ( β ( y ) , αβ ( z )) , α ( x )) − µ ( µ ( β ( y ) , α ( x )) α β ( z )) } , (4.21) for all x, y, z ∈ A , where as α,β is the BiHom-associator of A defined by as α,β ( x, y, z ) = µ ( µ ( x, z ) , β ( y )) − µ ( α ( x ) , µ ( x, z )) (4.22)If the BiHom-algebra ( A, µ, α, β ) is regular then the identity (4.21) is equivalent to as α,β ( x, y, z ) = 13 { µ ( µ ( x, α − β ( z )) , α ( y )) − µ ( µ ( α − β ( z ) , αβ − ( x )) , α ( y ))+ µ ( µ ( α − β ( y ) , α − β ( z )) , α β − ( x )) − µ ( µ ( α − β ( y ) , αβ − ( x )) β ( z )) } . (4.23) Proposition 4.1.
Let ( A, µ ) be an admissible Poisson algebra and α, β : A → A twocommuting Poisson algebra morphisms. Then ( A, µ α,β = µ ◦ ( α ⊗ β ) , α, β ) is an admissibleBiHom-Poisson algebra. roof. Let x, y, z ∈ A µ α,β ( µ α,β ( β ( x ) , α ( y )) , α β ( z )) − µ α,β ( αβ ( x ) , µ α,β ( α ( y ) , α ( z ))) − { µ α,β ( µ α,β ( β ( x ) , αβ ( z )) , α ( y )) − µ α,β ( µ α,β ( β ( z ) , α ( x )) , α ( y )) + µ α,β ( µ α,β ( β ( y ) , αβ ( z )) , α ( x )) − µ α,β ( µ α,β ( β ( y ) , α ( x )) α β ( z )) } = µ ( µ ( α β ( x ) , α β ( y )) , α β ( z )) − µ ( α β ( x ) , µ ( α β ( y ) , α β ( z ))) − { µ ( µ ( α β ( x ) , α β ( z )) , α β ( y )) − µ ( µ ( α β ( z ) , α β ( x )) , α β ( y )) + µ ( µ ( α β ( y ) , α β ( z )) , α β ( x )) − µ ( µ ( α β ( y ) , α β ( x )) α β ( z )) } = 0 As usual in (4.21) the product µ is denoted by juxtapositions of elements in A . Anadmissible BiHom-Poisson algebra with α = β = Id is exactly an admissible Poissonalgebra as defined in [6].To compare BiHom-Poisson algebras and admissible BiHom-Poisson algebras, we needthe following function, which generalizes a similar function in [14]. Definition 4.2.
Let ( A, µ, α, β ) be a regular BiHom-algebra. Define the quadruple P ( A ) = ( A, {· , ·} , • , α, β ) , (4.24) where { x, y } = ( µ ( x, y ) − µ ( α − β ( y ) αβ − ( x ))) and x • y = ( µ ( x, y )+ µ ( α − β ( y ) αβ − ( x ))) called the polarization of A . We call P the polarization function . The following result says that admissible BiHom-Poisson algebras, and only theseBiHom-algebras, give rise to BiHom-Poisson algebras via polarization.
Theorem 4.1.
Let ( A, µ, α, β ) be a regular BiHom-algebra. Then the polarization P ( A ) isa regular BiHom-Poisson algebra if and only if A is an admissible BiHom-Poisson algebra.Proof. For any x, y, z ∈ A , we will check that the ( A, {· , ·} , α, β ) is a BiHom-Lie algebra.Indeed, we have { β ( x ) , α ( y ) } = β ( x ) α ( y ) − β ( y ) α ( x ) = −{ β ( y ) , α ( x ) } , so the antisymmetry of {· , ·} is satisfied. Now, we verify the BiHom-Jacobi identity { β ( x ) , { β ( y ) , α ( z ) }} + { β ( y ) , { β ( z ) , α ( x ) }} + { β ( z ) , { β ( x ) , α ( y ) }} = { β ( x ) ,
12 ( β ( y ) · α ( z ) − β ( z ) · α ( y )) } + { β ( y ) ,
12 ( β ( z ) · α ( x ) − β ( x ) · α ( z )) } + { β ( z ) ,
12 ( β ( x ) · α ( y ) − β ( y ) · α ( x )) } = 14 (cid:16) − as αβ ( β ( x ) , β ( y ) , α ( z )) + as αβ ( α − β ( x ) , β ( z ) , α ( y )) − as αβ ( α − β ( y ) , β ( z ) , α ( x )) + as αβ ( α − β ( z ) , β ( y ) , α ( x ))+ as αβ ( α − β ( y ) , β ( x ) , α ( z )) − as αβ ( α − β ( z ) , β ( x ) , α ( y )) (cid:17) = 0 . A, • , α, β ) is a BiHom-commutative BiHom-associative algebra. In-deed, for any x, y, z ∈ A , the prove of BiHom-commutativity of µ is similar to the BiHom-skewsymmetry of {· , ·} checked before.( x • y ) • β ( z ) − α ( x ) • ( y • z ) = 12 ( µ ( x, y ) − µ ( α − β ( y ) , αβ − ( x ))) • β ( z ) − α ( x ) •
12 ( µ ( y, z ) − µ ( α − β ( z ) , αβ − ( y ))) = 14 (cid:16) as αβ ( x, y, z ) − as αβ ( α − β ( z ) , y, α β − ( x ))+ as αβ ( α − β ( z ) , αβ − ( x ) , αβ − ( y )) + µ ( µ ( α − β ( z ) , αβ − ( x )) , α ( y )) − µ ( µ ( α − β ( y ) , αβ − ( x )) , β ( z )) + µ ( µ ( α − β ( y ) , α − β ( z )) , α β − ( x )) − as αβ ( x, α − β ( y ) , αβ − ( z )) − µ ( µ ( x, α − β ( y )) , α ( z )) = 0 . Finally, we check the condition : { x • y, αβ ( z ) } − { x, β ( z ) } • α ( y ) − α ( x ) • { y, α ( z ) } . Indeed, we have { x • y, αβ ( z ) } − { x, β ( z ) } • α ( y ) − α ( x ) • { y, α ( z ) } = 14 (cid:16) as αβ ( x, y, α ( z )) − as αβ ( x, β ( z ) , αβ − ( y )) − as αβ ( α − β ( y ) , β ( z ) , α β − ( x )) + as ( α − β ( z ) , y, α β − ( x ))+ as αβ ( α − β ( z ) , αβ − ( x ) , αβ − ( y )) + as αβ ( α − β ( y ) , αβ − ( x ) , α ( z )) (cid:17) = 0 . The proof is finished.The following result says that there is a bijective correspondence between admissibleBiHom-Poisson algebras and BiHom-Poisson algebras via polarization and depolarization.
Corollary 4.1.
Let ( A, {· , ·} , • , α, β ) be a BiHom-Poisson algebra. Define the BiHom-algebra P − ( A ) = ( A, µ = {· , ·} + • , α, β ) , (4.25) then P − ( A ) is an admissible BiHom-Poisson algebra called the depolarization of A . Wecall P − the depolarization function .Proof. If (
A, µ, α ) is a regular admissible BiHom-Poisson algebra, then P ( A ) is a BiHom-Poisson algebra by Theorem 4.1. We have P − ( P ( A )) = A because µ ( x, y ) = 12 ( µ ( x, y ) − µ ( α − β ( y ) , αβ − ( x )))+ 12 ( µ ( x, y )+ µ ( α − β ( y ) , αβ − ( x )) , ∀ x, y ∈ A. Let ( A, {· , ·} , µ, α, β ) be a BiHom-Poisson algebra, in this section, we provide a list of2-dimensional BiHom-Poisson algebras, where the morphisms α and β are diagonal.12lgebras Multiplications Brackets Morphisms Alg µ ( e , e ) = c e , µ ( e , e ) = c e , { e , e } = d e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , { e , e } = d e , { e , e } = e , α = (cid:18) a (cid:19) ,β = (cid:18) b (cid:19) .Alg µ ( e , e ) = c e , µ ( e , e ) = c e , { e , e } = d e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , { e , e } = e , { e , e } = d e , α = (cid:18) a
00 0 (cid:19) ,β = (cid:18) b
00 0 (cid:19) .Alg µ ( e , e ) = c e , { e , e } = e , { e , e } = d e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , µ ( e , e ) = c b e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) b
00 1 (cid:19) .Alg µ ( e , e ) = c e , { e , e } = d e , { e , e } = d e , { e , e } = e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) b (cid:19) .Alg µ ( e , e ) = c e , µ ( e , e ) = c b e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) b (cid:19) .Alg µ ( e , e ) = c e , { e , e } = d e , α = (cid:18) a (cid:19) ,β = (cid:18) (cid:19) . lg µ ( e , e ) = c e , µ ( e , e ) = c a e , { e , e } = d e , α = (cid:18) a (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , { e , e } = d e , α = (cid:18) a
00 1 (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , µ ( e , e ) = c a e , { e , e } = d e , α = (cid:18) a
00 1 (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , { e , e } = e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , µ ( e , e ) = c e , { e , e } = e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) b
00 1 (cid:19) .Alg µ ( e , e ) = c e , { e , e } = d e , { e , e } = e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , µ ( e , e ) = c e , { e , e } = d e , { e , e } = e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , { e , e } = e , { e , e } = d e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) .Alg µ ( e , e ) = c e , { e , e } = d e , { e , e } = e , α = (cid:18) (cid:19) ,β = (cid:18) (cid:19) . eferences [1] H. Amri, A. Makhlouf, Non-commutative ternary Nambu-Poisson algebras andternary Hom-Nambu-Poisson algebras. Journal of Generalized Lie Theory and Appli-cations, 9 :221 (2015).[2] G. Dito, M. Flato, D. Sternheimer, and L. Takhtajan, Deformation quantization andNambu mechanics. Commun. Math. Phys., , 1–22, (1997).[3] G. Dito, M. Flato, and D. Sternheimer. Nambu mechanics, n-ary operations andtheir quantization.In Deformation Theory and Symplectic Geometry. Math. Phys.Stud. , Kluwer Academic Publishers, Dordrecht, Boston, 43–66, (1997).[4] V.T. Filippov, n -Lie algebras,(Russian), Sibirsk. Mat. Zh. , no. 6, 126–140 (1985).(English translation: Siberian Math. J. , no. 6, 879–891, (1985))[5] V.T. Filippov, n -Lie algebras,(Russian), Sibirsk. Mat. Zh. , no. 6, 126–140 (1985).(English translation: Siberian Math. J. , no. 6, 879–891, (1985))[6] M. Goze and E. Remm, Poisson algebras in terms of non-associative algebras, J. Alg.320 (2008) 294–317.[7] G. Graziani, A. Makhlouf, C. Menini, F. Panaite, BiHom-associative algebras,BiHom-Lie algebras and BiHom-bialgebras. Symmetry, Integrability and Geometry :Methods and Applications SIGMA 11 (2015), 086, 34 pages[8] L. Liu, A. Makhlouf, C. Menini, F. Panaite, BiHom-pre-Lie algebras, BiHom-Leibnizalgebras and Rota-Baxter operators on BiHom-Lie algebras, arXiv:1706.00474.(2017).[9] L. Liu, A. Makhlouf, C. Menini, F. Panaite, Rota-Baxter operators on BiHom-associative algebras and related structures. Colloq. Math., Vol. 161 (2020), 263–294.[10] L. Liu, A. Makhlouf, C. Menini, F. Panaite, Tensor products and perturbations ofBiHom-Novikov-Poisson algebras. Preprint 2020.[11] F. Kubo, Finite-dimensional non-commutative Poisson algebras, J. Pure Appl. Alg.113 (1996) 307–314.[12] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. 2(2008) 51–64.[13] A. Makhlouf and S. Silvestrov, Hom-algebras and Hom-coalgebras, J. Alg. Appl. 9(2010) 1–37.[14] M. Markl and E. Remm, Algebras with one operation including Poisson and otherLie-admissible algebras, J. Alg. 299 (2006) 171–189.[15] Y. Nambu, Generalized Hamiltonian mechanics, Phys. Rev. D (3),7