Noncommutative Poisson bialgebras
aa r X i v : . [ m a t h . QA ] A p r NONCOMMUTATIVE POISSON BIALGEBRAS
JIEFENG LIU, CHENGMING BAI AND YUNHE SHENGA bstract . In this paper, we introduce the notion of a noncommutative Poisson bialgebra, andestablish the equivalence between matched pairs, Manin triples and noncommutative Poisson bial-gebras. Using quasi-representations and the corresponding cohomology theory of noncommutativePoisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the in-troduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxteroperators, more generally O -operators on noncommutative Poisson algebras, and noncommutativepre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Pois-son Yang-Baxter equation in some special noncommutative Poisson algebras obtained from thesestructures. C ontents
1. Introduction 11.1. Noncommutative Poisson algebras and pre-Poisson algebras 11.2. The bialgebra theory for noncommutative Poisson algebras 21.3. Noncommutative pre-Poisson algebras, Rota-Baxter operators and O -operators 31.4. Outline of the paper 42. Quasi-representations and cohomologies of noncommutative Poisson algebras 43. Matched pairs, Manin triples and (pseudo-)Poisson bialgebras 84. Coboundary (pseudo-)Poisson bialgebras 125. Noncommutative pre-Poisson algebras, Rota-Baxter operators and O -operators 175.1. Noncommutative pre-Poisson algebras 175.2. Rota-Baxter operators and O -operators on noncommutative Poisson algebras 19References 231. I ntroduction This paper aims to study the bialgebra theory for noncommutative Poisson algebras, in partic-ular coboundary ones. Skew-symmetric solutions of the Poisson Yang-Baxter equation in certainnoncommutative Poisson algebras are constructed using O -operators and noncommutative pre-Poisson algebras, which give coboundary noncommutative Poisson bialgebras.1.1. Noncommutative Poisson algebras and pre-Poisson algebras.
The notion of a noncom-mutative Poisson algebra was first given by Xu in [38], which is especially suitable for geometricsituations. Keywords: noncommutative Poisson algebra, noncommutative Poisson bialgebra, Poisson Yang-Baxter equation,noncommutative pre-Poisson algebra, Rota-Baxter operator MSC : 13D03, 16T10, 17B63
Definition 1.1. A noncommutative Poisson algebra is a triple ( P , · P , {− , −} P ) , where ( P , · P ) is anassociative algebra (not necessarily commutative) and ( P , {− , −} P ) is a Lie algebra, such that theLeibniz rule holds: { x , y · P z } P = { x , y } P · P z + y · P { x , z } P , ∀ x , y , z ∈ P . In [18], Flato, Gerstenhaber and Voronov introduced a more general notion of a Leibniz pairand study its cohomology and deformation theory. In particular, they gave the cohomology theoryof a noncommutative Poisson algebra associated to a representation using an innovative bicom-plex. Recently, Bao and Ye developed the cohomology theory of noncommutative Poisson al-gebras associated to quasi-representations through Yoneda-Ext groups and projective resolutionsin [9, 10]. Noncommutative Poisson algebras had been studied by many authors from di ff erentaspects [24, 25, 26, 39]. A Poisson algebra in the usual sense is the one where the associative mul-tiplication on P is commutative. Note that there is another noncommutative analogue of Poissonalgebras, namely double Poisson algebras ([36]), which will not be considered in this paper.Aguiar introduced the notion of a pre-Poisson algebra in [2] and constructed many examples.A pre-Poisson algebra contains a Zinbiel algebra and a pre-Lie algebra such that some compat-ibility conditions are satisfied. Zinbiel algebras, which are also called dual Leibniz algebras,were introduced by Loday in [29], and further studied in [28, 30]. Pre-Lie algebras are a classof nonassociative algebras coming from the study of convex homogeneous cones, a ffi ne mani-folds and a ffi ne structures on Lie groups, and cohomologies of associative algebras. They alsoappeared in many fields in mathematics and mathematical physics, such as complex and symplec-tic structures on Lie groups and Lie algebras, integrable systems, Poisson brackets and infinitedimensional Lie algebras, vertex algebras, quantum field theory and operads. See the survey [11]and the references therein for more details. A pre-Poisson algebra gives rise to a Poisson algebranaturally through the sub-adjacent commutative associative algebra of the Zinbiel algebra and thesub-adjacent Lie algebra of the pre-Lie algebra. Conversely, a Rota-Baxter operator action (moregenerally an O -operator action) on a Poisson algebra gives rise to a pre-Poisson algebra. We cansummarize these relations by the following diagram: Zinbiel algebra + pre-Lie algebra / / sub-adjacent (cid:15) (cid:15) Pre-Poisson algebrasub-adjacent (cid:15) (cid:15) comm associative algebra + Lie algebra / / Rota-Baxter action O O Poisson algebra.Rota-Baxter action O O The bialgebra theory for noncommutative Poisson algebras.
For a given algebraic struc-ture determined by a set of multiplications, a bialgebra structure on this algebra is obtained bya corresponding set of comultiplications together with a set of compatibility conditions betweenthe multiplications and comultiplications. For a finite dimensional vector space V with the givenalgebraic structure, this can be achieved by equipping the dual space V ∗ with the same algebraicstructure and a set of compatibility conditions between the structures on V and those on V ∗ .The great importance of the bialgebra theory and the noncommutative Poisson algebra servesas the main motivation for our interest in a suitable bialgebra theory for the noncommutativePoisson algebra in this paper.A good compatibility condition in a bialgebra is prescribed on the one hand by a strong motiva-tion and potential applications, and on the other hand by a rich structure theory and e ff ective con-structions. In the associative algebra context, an antisymmetric infinitesimal bialgebra [1, 3, 4, 6]has the same associative multiplications on A and A ∗ , and the comultiplication being a 1-cocycle ONCOMMUTATIVE POISSON BIALGEBRAS 3 on A with coe ffi cients in the tensor representation A ⊗ A . In the Lie algebra context, a Lie bial-gebra consists of a Lie algebra ( g , [ − , − ] g ) and a Lie coalgebra ( g , δ ), where δ : g → ⊗ g is a Liecomultiplication, such that the Lie comultiplication being a 1-cocycle on g with coe ffi cients in thetensor representation g ⊗ g . See [14, 17] for more details about Lie bialgebras and applications inmathematical physics. Thus, the representation theory and the cohomology theory usually playessential roles in the study of a bialgebra theory.In fact, there has been a bialgebra theory for the usual (commutative) Poisson algebras, theso-called Poisson bialgebras ([32]), in terms of the representation theory of Passion algebras.However, a direct generalization is not available for the noncommutative Poisson algebras. In thispaper, we apply quasi-representations instead of representations and the corresponding cohomol-ogy theory ([9, 39]) to study noncommutative Poisson bialgebras. Even though both Lie algebrasand associative algebras admit tensor representations as mentioned above, the tensor product oftwo representations of a noncommutative Poisson algebra is not a representation anymore, but aquasi-representation. This is the reason why quasi-representations, not representations, are themain ingredient in our study of noncommutative Poisson bialgebras. On the other hand, the dualof the regular representation ( L , R , ad) of a noncommutative Poisson algebra is also usually not arepresentation, but a quasi-representation. We introduce the concept of a coherent noncommuta-tive Poisson algebra to overcome this problem. Note that the usual Poisson algebras are coherent.Thus, the bialgebra theory for noncommutative Poisson algebras established in this paper containsall the results of Poisson bialgebras given in [32].Moreover, like the case of Poisson bialgebras, the study of coboundary noncommutative Pois-son bialgebras leads to the introduction of the Poisson Yang-Baxter equation in a coherent non-commutative Poisson algebra. A skew-symmetric solution of the Poisson Yang-Baxter equationnaturally gives a (coboundary) noncommutative Poisson bialgebra.In addition, coherent noncommutative Poisson algebras are closely related to compatible Liealgebras which play important roles in several fields in mathematics and mathematical physics([19, 20, 21, 31]). In fact, a coherent noncommutative Poisson algebra naturally gives a compat-ible Lie algebra. Consequently, a noncommutative Poisson bialgebra gives rise to a compatibleLie bialgebra ([37]), and there is also a similar relationship in the coboundary cases.1.3. Noncommutative pre-Poisson algebras, Rota-Baxter operators and O -operators. Thenotion of a dendriform algebra was introduced by Loday in [29] with motivation from periodicityof algebraic K-theory and operads.
Definition 1.2. A dendriform algebra is a vector space A with two bilinear maps ≻ : A ⊗ A −→ Aand ≺ : A ⊗ A −→ A such that for all x , y , z ∈ A, the following equalities hold: ( x ≺ y ) ≺ z = x ≺ ( y ≻ z + y ≺ z ) , ( x ≻ y ) ≺ z = x ≻ ( y ≺ z ) , x ≻ ( y ≻ z ) = ( x ≻ y + x ≺ y ) ≻ z . One can obtain an associative algebra as well as a pre-Lie algebra from a dendriform algebra.The relations among dendriform algebras, associative algebras, pre-Lie algebras and Lie algebrasare given as follows:dendriform algebra ( A , ≻ , ≺ ) x ∗ y = x ≻ y − y ≺ x / / x · y = x ≻ y + x ≺ y (cid:15) (cid:15) pre-Lie algebra ( A , ∗ ) [ x , y ] = x ∗ y − y ∗ x (cid:15) (cid:15) associative algebra ( A , · ) [ x , y ] = x · y − y · x / / Lie algebra ( A , [ − , − ]) . JIEFENG LIU, CHENGMING BAI AND YUNHE SHENG
A Zinbiel algebra can be viewed as a commutative dendriform algebra, namely x ≻ y = y ≺ x . In fact, from the operadic point of view, dendriform algebras and Zinbiel algebras can beviewed as the splitting of associative algebras and commutative associative algebras respectively([2, 7, 15, 33]).In this paper, we introduce the notion of a noncommutative pre-Poisson algebra, which con-sists of a dendriform algebra and a pre-Lie algebra, such that some compatibility conditionsare satisfied. Through the sub-adjacent associative algebra and the sub-adjacent Lie algebra, anoncommutative pre-Poisson algebra gives rise to a noncommutative Poisson algebra naturally.Thus, noncommutative pre-Poisson algebras can be viewed as the splitting of noncommutativePoisson algebras. We further introduce the notion of a Rota-Baxter operator (more generally an O -operator) on a noncommutative Poisson algebra, which is simultaneously a Rota-Baxter op-erator on the underlying associative algebra and a Rota-Baxter operator on the underlying Liealgebra. See [2, 5, 8, 13, 16, 22, 23, 34, 35] for more details on Rota-Baxter operators and O -operators. A noncommutative pre-Poisson algebra can be obtained through the action of aRota-Baxter operator (more generally an O -operator). The above relations can be summarizedinto the following commutative diagram: dendriform algebra + pre-Lie algebra / / sub-adjacent (cid:15) (cid:15) noncomm pre-Poisson algebrasub-adjacent (cid:15) (cid:15) associative algebra + Lie algebra / / Rota-Baxter action O O noncomm Poisson algebraRota-Baxter action O O We construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some specialnoncommutative Poisson algebras obtained from these structures.1.4.
Outline of the paper.
In Section 2, we recall quasi-representations and the correspondingcohomology theory of noncommutative Poisson algebras, and introduce the notion of a coherentnoncommutative Poisson algebra for our later study of noncommutative Poisson bialgebras.In Section 3, we introduce the notions of matched pairs, Manin triples for noncommutativePoisson algebras and noncommutative (pseudo)-Poisson bialgebras. The equivalences betweenmatched pairs of coherent noncommutative Poisson algebras, Manin triples for noncommutativePoisson algebras and noncommutative Poisson bialgebras are established.In Section 4, we study coboundary noncommutative Poisson bialgebras with the help of quasi-representations of noncommutative Poisson algebras and the corresponding cohomology theory,which leads to the introduction of the Poisson Yang-Baxter equation in a coherent noncommuta-tive Poisson algebra.In Section 5, we introduce the notion of a noncommutative pre-Poisson algebra and a Rota-Baxter operator (more generally an O -operator) on a noncommutative Poisson algebra, by whichwe construct skew-symmetric solutions of the Poisson Yang-Baxter equation in certain specialnoncommutative Poisson algebras obtained from these structures.In this paper, all the vector spaces are over an algebraically closed field K of characteristic 0,and finite dimensional.2. Q uasi - representations and cohomologies of noncommutative P oisson algebras In this section, we recall (quasi)-representations of noncommutative Poisson algebras and thecorresponding cohomology theory.
ONCOMMUTATIVE POISSON BIALGEBRAS 5
Definition 2.1.
Let ( A , · A ) be an associative algebra and V a vector space. Let L , R : A −→ gl ( V ) be two linear maps with x → L x and x → R x respectively. The triple ( V ; L , R ) is called a representation of A if for all x , y ∈ A , we have L x · A y = L x ◦ L y , R x · A y v = R y ◦ R x , L x ◦ R y = R y ◦ L x . In fact, ( V ; L , R ) is a representation of an associative algebra A if and only if the direct sum A ⊕ V of vector spaces is an associative algebra (the semi-direct product) by defining the multi-plication on A ⊕ V by( x + v ) · ( L , R ) ( x + v ) = x · A x + L x v + R x v , ∀ x , x ∈ A , v , v ∈ V . We denote it by A ⋉ L , R V or simply by A ⋉ V . Lemma 2.2.
Let ( V ; L , R ) be a representation of an associative algebra ( A , · A ) . Define L ∗ : A −→ gl ( V ∗ ) and R ∗ : A −→ gl ( V ∗ ) by hL ∗ x α, v i = −h α, L x v i , hR ∗ x α, v i = −h α, R x v i , ∀ x ∈ A , α ∈ V ∗ , v ∈ V . Then ( V ∗ ; −R ∗ , −L ∗ ) is a representation of ( A , · A ) . Example 2.3.
Let ( A , · A ) be an associative algebra. Let L x and R x denote the left and right multi-plication operators, respectively, that is, L x y = x · A y , R y x = x · A y for all x , y ∈ A . Then ( A ; L , R )is a representation of ( A , · A ), called the regular representation . Furthermore, ( A ∗ ; − R ∗ , − L ∗ ) isalso a representation of ( A , · A ).Similarly, let ( g , [ − , − ] g ) be a Lie algebra and V a vector space. Let ρ : g → gl ( V ) be a linearmap. The pair ( V ; ρ ) is called a representation of g if for all x , y ∈ g , we have ρ ([ x , y ] g ) = [ ρ ( x ) , ρ ( y )] . In fact, ( V ; ρ ) is a representation of a Lie algebra g if and only if the direct sum g ⊕ V of vectorspaces is a Lie algebra (the semi-direct product) by defining the Lie bracket on g ⊕ V by[ x + v , x + v ] ρ = [ x , x ] g + ρ ( x )( v ) − ρ ( x )( v ) , ∀ x , x ∈ g , v , v ∈ V . We denote it by g ⋉ ρ V or simply by g ⋉ V . Moreover, let ( V ; ρ ) be a representation of a Lie algebra( g , [ − , − ] g ). Define ρ ∗ : g −→ gl ( V ∗ ) by h ρ ∗ ( x )( α ) , v i = −h α, ρ ( x )( v ) i , ∀ x ∈ g , α ∈ V ∗ , v ∈ V . Then ( V ∗ ; ρ ∗ ) is a representation of ( g , [ − , − ] g ). In particular, define ad : g → gl ( g ) by ad x y = [ x , y ] g for all x , y ∈ g . Then ( g ; ad) is a representation of ( g , [ − , − ] g ), called the adjoint represen-tation . Furthermore, ( g ∗ ; ad ∗ ) is also a representation of ( g , [ − , − ] g ). Definition 2.4. ([18, 39])
Let ( P , · P , {− , −} P ) be a noncommutative Poisson algebra. (i) A quasi-representation of P is a quadruple ( V ; L , R , ρ ) such that ( V ; L , R ) is a represen-tation of the associative algebra ( P , · P ) and ( V ; ρ ) is a representation of the Lie algebra ( P , {− , −} P ) satisfying L { x , y } P = ρ ( x ) L y − L y ρ ( x ) , (1) R { x , y } P = ρ ( x ) R y − R y ρ ( x ) , ∀ x , y ∈ P . (2) (ii) A quasi-representation ( V ; L , R , ρ ) is called a representation of P if we have (3) ρ ( x · P y ) = L x ρ ( y ) + R y ρ ( x ) . By a direct calculation, we have
JIEFENG LIU, CHENGMING BAI AND YUNHE SHENG
Proposition 2.5.
Let ( P , · P , {− , −} P ) be a noncommutative Poisson algebra. (i) If ( V ; L , R , ρ ) is a quasi-representation of P, then ( V ∗ ; −R ∗ , −L ∗ , ρ ∗ ) is also a quasi-representation of P; (ii) If ( V ; L , R , ρ ) is a quasi-representation and satisfies (4) ρ ( x · P y ) = ρ ( x ) L y + ρ ( y ) R x , then ( V ∗ ; −R ∗ , −L ∗ , ρ ∗ ) is a representation of P. Example 2.6.
Let ( P , · P , {− , −} P ) be a noncommutative Poisson algebra. Then ( P ; L , R , ad) is arepresentation of P , which is also called the regular representation of P . However, the dual( P ∗ ; − R ∗ , − L ∗ , ad ∗ ) is just a quasi-representation of P . Furthermore, it is straightforward to checkthat ( P ∗ ; − R ∗ , − L ∗ , ad ∗ ) is a representation of P if and only if the noncommutative Poisson algebra P satisfies(5) { x , y · P z } P + { y , z · P x } P + { z , x · P y } P = , ∀ x , y ∈ P . Definition 2.7.
A noncommutative Poisson algebra ( P , · P , {− , −} P ) is called coherent if it satisfies (5) . Proposition 2.8.
A noncommutative Poisson algebra ( P , · P , {− , −} P ) is coherent if and only if itsatisfies [ { x , y } P , z ] P + [ { z , x } P , y ] P + [ { y , z } P , x ] P = for all x , y , z ∈ P, where [ − , − ] P : P × P −→ P is the commutator Lie bracket defined by (6) [ x , y ] P = x · P y − y · P x . Proof.
By the Leibniz rule of the noncommutative Poisson algebra ( P , · P , {− , −} P ), we have { x , y · P z } P + { y , z · P x } P + { z , x · P y } P = { x , y } P · P z + y · P { x , z } P + { y , z } P · P x + z · P { y , x } P + { z , x } P · P y + x · P { z , y } P = { x , y } P · P z − z · P { x , y } P + { z , x } P · P y − y · P { z , x } P + { y , z } P · P x − x · P { y , z } P = [ { x , y } P , z ] P + [ { z , x } P , y ] P + [ { y , z } P , x ] P . Then the conclusion follows immediately.Similarly, a coherent noncommutative Poisson algebra ( P , · P , {− , −} P ) also satisfies { [ x , y ] P , z } P + { [ z , x ] P , y } P + { [ y , z ] P , x } P = . Corollary 2.9.
Any commutative Poisson algebra is coherent.
Coherent noncommutative Poisson algebras are closely related to compatible Lie algebras ([19,20, 21, 31]).
Definition 2.10. A compatible Lie algebra ( g , [ − , − ] , [ − , − ] ) consists of two Lie algebras ( g , [ − , − ] ) and ( g , [ − , − ] ) such that for any k , k ∈ K , the following bilinear operation (7) [ x , y ] = k [ x , y ] + k [ x , y ] , ∀ x , y ∈ g defines a Lie algebra structure on g . Proposition 2.11. ([19])
Let ( g , [ − , − ] ) and ( g , [ − , − ] ) be two Lie algebras. Then ( g , [ − , − ] , [ − , − ] ) is a compatible Lie algebra if and only if for any x , y , z ∈ g , the following equation holds (8) [[ x , y ] , z ] + [[ y , z ] , x ] + [[ z , x ] , y ] + [[ x , y ] , z ] + [[ y , z ] , x ] + [[ z , x ] , y ] = . ONCOMMUTATIVE POISSON BIALGEBRAS 7
Corollary 2.12.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra. Then we havea compatible Lie algebra ( P , {− , −} P , [ − , − ] P ) , where the bracket [ − , − ] P is given by (6) . Example 2.13.
Let ( A , · A ) be an associative algebra. Define a bracket {− , −} ~ as follows(9) { x , y } ~ = ~ ( x · A y − y · A x ) , where x , y ∈ A and ~ is a fixed number. Then ( A , · A , {− , −} ~ ) is a coherent noncommutative Poissonalgebra, which is called the standard noncommutative Poisson algebra .By a direct calculation, we have Proposition 2.14.
The standard noncommutative Poisson algebra ( P , · P , {− , −} ~ ) is compatiblewith any coherent noncommutative Poisson algebra ( P , · P , {− , −} P ) in the sense of that for anyk , k ∈ K , ( P , k · P + k · P , k {− , −} P + k {− , −} ~ ) is still a coherent noncommutative Poisson algebra. Example 2.15.
Let P be a 3-dimensional vector space with basis { e , e , e } . Define the non-zeromultiplication and the bracket operation on P by e · e = e , e · e = − e ; { e , e } = ae + be + ce , { e , e } = be , { e , e } = − ae , ∀ a , b , c ∈ K . Then ( P , · , {− , −} ) is a coherent noncommutative Poisson algebra. Example 2.16.
Let P be a 4-dimensional vector space with basis { e , e , e , e } . Define the non-zero multiplication and the non-zero bracket operation on P by e · e = e , e · e = e , e · e = e , e · e = − e ; { e , e } = ae , { e , e } = be , { e , e } = ce , ∀ a , b , c ∈ K . Then ( P , · , {− , −} ) is a coherent noncommutative Poisson algebra.It is straightforward to obtain the following conclusion. Proposition 2.17.
Let ( P , · P , {− , −} P ) be a noncommutative Poisson algebra and ( V ; L , R , ρ ) arepresentation. Then ( P ⊕ V , · ( L , R ) , {− , −} ρ ) is a noncommutative Poisson algebra, where ( P ⊕ V , · ( L , R ) ) is the semi-direct product associative algebra P ⋉ ( L , R ) V and ( P ⊕ V , {− , −} ρ ) is the semi-direct product Lie algebra P ⋉ ρ V.Furthermore, if the noncommutative Poisson algebra P is coherent and the representation ( V ; L , R , ρ ) satisfies (4) , then ( P ⊕ V , · ( L , R ) , {− , −} ρ ) is a coherent noncommutative Poisson algebra. We denote a semi-direct product noncommutative Poisson algebra by P ⋉ ( L , R ,ρ ) V .The cohomology complex for a noncommutative Poisson algebra ( P , · P , {− , −} P ) associated toa quasi-representation ( V ; L , R , ρ ) is given as follows ([9]). Denote C i , j ( P , V ) = Hom(( ⊗ i P ) ⊗ ( ∧ j P ) , V ) and C n ( P , V ) = X i + j = n C i , j ( P , V ) . The coboundary operator δ n : C n ( P , V ) −→ C n + ( P , V ) is given by δ n = X i + j = n (ˆ δ i , j + ( − i ¯ δ i , j ) , JIEFENG LIU, CHENGMING BAI AND YUNHE SHENG where ˆ δ i , j : C i , j ( P , V ) −→ C i + , j ( P , V ) is given byˆ δ i , j ϕ (cid:0) ( a ⊗ · · · ⊗ a i + ) ⊗ ( x ∧ · · · ∧ x j ) (cid:1) = L a ϕ (( a ⊗ · · · ⊗ a i + ) ⊗ ( x ∧ · · · ∧ x j )) + i X k = ( − k ϕ (cid:0) ( a ⊗ · · · ⊗ a k · P a k + ⊗ · · · ⊗ a i + ) ⊗ ( x ∧ · · · ∧ x j ) (cid:1) + ( − i + R a i + ϕ (cid:0) ( a ⊗ · · · ⊗ a i ) ⊗ ( x ∧ · · · ∧ x j ) (cid:1) and ¯ δ i , j : C i , j ( P , V ) −→ C i , j + ( P , V ) is given by¯ δ i , j ϕ (cid:0) ( a ⊗ · · · ⊗ a i ) ⊗ ( x ∧ · · · ∧ x j + ) (cid:1) = j + X l = ( − l + (cid:16) ρ ( x l ) ϕ (cid:0) ( a ⊗ · · · ⊗ a i ) ⊗ ( x ∧ · · · ˆ x l · · · ∧ x j + ) (cid:1) − i X k = ϕ (cid:0) ( a ⊗ · · · ⊗ { x l , a k } P ⊗ · · · ⊗ a i ) ⊗ ( x ∧ · · · ˆ x l · · · ∧ x j + ) (cid:1)(cid:17) + X ≤ p ≤ q ≤ j + ( − p + q ϕ (cid:0) ( a ⊗ · · · ⊗ a i ) ⊗ ( { x p , x q } P ∧ x ∧ · · · ˆ x p · · · ˆ x q · · · ∧ x j + ) (cid:1) for all ϕ ∈ C i , j ( P , V ) and a , · · · , a i + , x , · · · , x j + ∈ P .In particular, a 1-cochain ϕ + ψ , where ϕ ∈ C , ( P , V ) and ψ ∈ C , ( P , V ), is a 1-cocycle on P with coe ffi cients in the quasi-representation ( V ; L , R , ρ ) means that δ ( ϕ + ψ ) =
0, i.e.ˆ δ , ϕ ( a ⊗ b ) = , (cid:0) ˆ δ , ψ − ¯ δ , ϕ (cid:1) ( a ⊗ x ) = , ¯ δ , ψ ( x ⊗ y ) = , ∀ a , b , x , y ∈ P . More precisely, ϕ ( a · P b ) = L a ϕ ( b ) + R b ϕ ( a );(10) ϕ ( { x , a } P ) = ρ ( x ) ϕ ( a ) − L a ψ ( x ) + R a ψ ( x );(11) ψ ( { x , y } P ) = ρ ( x ) ψ ( y ) − ρ ( y ) ψ ( x ) . (12)A 1-cochain ϕ + ψ ∈ C ( P , V ) is a 1-coboundary on P with coe ffi cients in the quasi-representation( V ; L , R , ρ ) if and only if there exists an element u ∈ V such that(13) L a u − R a u = ϕ ( a ) and ρ ( x ) u = ψ ( x ) , ∀ a , x ∈ P .
3. M atched pairs , M anin triples and ( pseudo -)P oisson bialgebras A matched pair of Lie algebras is a pair of Lie algebras ( g , [ − , − ] g ) and ( g , [ − , − ] g ) togetherwith two representations ρ : g −→ gl ( g ) and ̺ : g −→ gl ( g ) satisfying ̺ ( α )[ x , y ] g = [ ̺ ( α ) x , y ] g + [ x , ̺ ( α ) y ] g − ̺ ( ρ ( x ) α ) y + ̺ ( ρ ( y ) α ) x ,ρ ( x )[ α, β ] g = [ ρ ( x ) α, β ] g + [ α, ρ ( x ) β ] g − ρ ( ̺ ( α ) x ) β + ρ ( ̺ ( β ) x ) α for all x , y ∈ g and α, β ∈ g . In this case, there exists a Lie algebra structure on the vector space d = g ⊕ g given by(14) [ x + α, y + β ] d = [ x , y ] g + ̺ ( α ) y − ̺ ( β ) x + [ α, β ] g + ρ ( x ) β − ρ ( y ) α. It is denoted by g ⊲⊳ ̺ρ g or simply by g ⊲⊳ g , and called the double of the matched pair. More-over, every Lie algebra which is a direct sum of the underlying vector spaces of two subalgebrasis the double of a matched pair of Lie algebras. ONCOMMUTATIVE POISSON BIALGEBRAS 9 A matched pair of associative algebras is a pair of associative algebras ( A , · ) and ( A , · )together with two representations ( L , R ) : A −→ gl ( A ) and ( ˜ L , ˜ R ) : A −→ gl ( A ) satisfying L x ( α · β ) = L ˜ R α x β + ( L x α ) · β, R x ( α · β ) = R ˜ L β x α + α · R x β, ˜ L α ( x · y ) = ˜ L R x α y + ( ˜ L α x ) · y , ˜ R α ( x · y ) = ˜ R L y α x + x · ˜ R α y , L ˜ L α x β + ( R x α ) · β − R ˜ R β x α − α · L x β = , ˜ L L x α y + ( ˜ R α x ) · y − ˜ R R y α x − x · ˜ L α y = x , y ∈ A and α, β ∈ A . In this case, there exists an associative algebra structure on thevector space A = A ⊕ A given by(15) ( x + α ) · A ( y + β ) = x · y + ˜ L α y + ˜ R β x + α · β + L x β + R y α. It is denoted by A ⊲⊳ ( ˜ L , ˜ R )( L , R ) A or simply by A ⊲⊳ A , and called the double of the matched pair.Moreover, every associative algebra which is a direct sum of the underlying vector spaces of twosubalgebras is the double of a matched pair of associative algebras. Definition 3.1.
Let ( P , · , {− , −} P ) and ( P , · , {− , −} P ) be two noncommutative Poisson alge-bras. A matched pair of noncommutative Poisson algebras is a pair of noncommutative Pois-son algebras ( P , P ) together with two representations ( L , R , ρ ) : P −→ gl ( P ) and ( ˜ L , ˜ R , ̺ ) : P −→ gl ( P ) such that P ⊲⊳ ̺ρ P is a matched pair of Lie algebras, P ⊲⊳ ( ˜ L , ˜ R )( L , R ) P is a matchedpair of associative algebras and for all x , y ∈ P and α, β ∈ P , the following equalities hold: ρ ( x )( α · β ) = ( ρ ( x ) α ) · β + α · ρ ( x ) β − L ̺ ( α ) x β − R ̺ ( β ) x α, (16) L x { α, β } P = { α, L x β } P − ρ ( ˜ R β x ) α − L ̺ ( α ) x β + ( ρ ( x ) α ) · β, (17) ̺ ( α )( x · y ) = ( ̺ ( α ) x ) · y + x · ̺ ( α ) y − ˜ L ρ ( x ) α y − ˜ R ρ ( y ) α x , (18) ˜ L α { x , y } P = { x , ˜ L α y } P − ̺ ( R y α ) x − ˜ L ρ ( x ) α y + ( ̺ ( α ) x ) · y . (19) Proposition 3.2.
Let ( P , · , {− , −} P ) and ( P , · , {− , −} P ) be two noncommutative Poisson al-gebras. If ( P , P ) is a matched pair of noncommutative Poisson algebras with representations ( L , R , ρ ) : P −→ gl ( P ) and ( ˜ L , ˜ R , ̺ ) : P −→ gl ( P ) , then there exists a noncommutativePoisson algebra structure on P = P ⊕ P given by { x + α, y + β } P = { x , y } P + ̺ ( α ) y − ̺ ( β ) x + { α, β } P + ρ ( x ) β − ρ ( y ) α, (20) ( x + α ) · P ( y + β ) = x · y + ˜ L α y + ˜ R β x + α · β + L x β + R y α (21) for all x , y ∈ P and α, β ∈ P . Proof.
The verification is a routine calculation and thus omitted.We denote this noncommutative Poisson algebra by P ⊲⊳ ( ˜ L , ˜ R ,̺ )( L , R ,ρ ) P or simply by P ⊲⊳ P , andcall it the double of the matched pair. Moreover, every noncommutative Poisson algebra whichis a direct sum of the underlying vector spaces of two subalgebras is the double of a matched pairof noncommutative Poisson algebras.A quadratic noncommutative Poisson algebra is a noncommutative Poisson algebra P equippedwith a nondegenerate symmetric bilinear form B ( − , − ) on P which is invariant in the sense that(22) B ( { a , b } P , c ) = B ( a , { b , c } P ) , B ( a · P b , c ) = B ( a , b · P c ) , ∀ a , b , c ∈ P . Definition 3.3. A Manin triple of noncommutative Poisson algebras is a triple (( P , B ) , P , P ) ,where ( P , B ) is a quadratic noncommutative Poisson algebra, ( P , · , {− , −} P ) and ( P , · , {− , −} P ) are noncommutative Poisson subalgebras of P, such that (i) P = P ⊕ P as vector spaces; (ii) P and P are isotropic with respect to B ( − , − ) . Remark 3.4.
It is obvious that a Manin triple of noncommutative Poisson algebras is simultane-ously a Manin triple of Lie algebras ( [12] ) and an analogue of a Manin triple in the context ofassociative algebras which is called a double construction of Frobenius algebras in [6] . Proposition 3.5.
Let ( P , P , P ) be a Manin triple of noncommutative Poisson algebras. Then thenoncommutative Poisson algebras P, P and P must be coherent. Proof.
First we show that the noncommutative Poisson algebra P is coherent. Since P is asubalgebra of P , we only need to prove that for any x , y , z ∈ P , the following equality holds: { x · P y , z } P = { x , y · P z } P + { y , z · P x } P . (23)By the invariance of B and the Leibniz rule, for α ∈ P , we have B (cid:0) { x · P y , z } P − { x , y · P z } P − { y , z · P x } P , α (cid:1) = B (cid:0) y · P { z , α } P − { y · P z , α } P + { y , α } P · P z , x (cid:1) = . Then by the nondegeneracy of B , (23) follows immediately.Also by the invariance of B , for all x , y , z ∈ P and α, β ∈ P , we have B (cid:0) { x · P y , α } P − { x , y · P α } P − { y , α · P x } P , z (cid:1) = B (cid:0) { z , x · P y } P − { z , x } P · P y − x · P { z , y } P , α (cid:1) = , and B (cid:0) { x · P y , α } P − { x , y · P α } P − { y , α · P x } P , β (cid:1) = − B (cid:0) { x · P y , β } P − { x , β } P · P y − x · P { y , β } P , α (cid:1) = , which imply that { x · P y , α } P = { x , y · P α } P + { y , α · P x } P . Similarly, for x ∈ P and α, β, γ ∈ P , we also have { α · P β, γ } P = { α, β · P γ } P + { β, γ · P α } P , { α · P β, x } P = { α, β · P x } P + { β, x · P α } P . Thus the noncommutative Poisson algebras P and P are coherent.We recall the notions of Lie bialgebras and antisymmetric infinitesimal bialgebras before wegive the notion of a noncommutative Poisson bialgebra.A Lie bialgebra is a pair (( g , [ − , − ] g ) , δ ), where ( g , [ − , − ] g ) is a Lie algebra and δ : g −→ ∧ g is a linear map such that ( g , δ ) is a Lie coalgebra and δ is a 1-cocycle on g with coe ffi cients in therepresentation ( g ⊗ g ; ad ⊗ + ⊗ ad), i.e.(24) δ ([ x , y ] g ) = (ad x ⊗ + ⊗ ad x ) δ ( y ) − (ad y ⊗ + ⊗ ad y ) δ ( x ) , ∀ x , y ∈ g . In particular, a Lie bialgebra ( g , δ ) is called coboundary if δ is a 1-coboundary, that is, thereexists an r ∈ g ⊗ g such that(25) δ ( x ) = (ad x ⊗ + ⊗ ad x ) r , ∀ x , y ∈ g . A coboundary Lie bialgebra is usually denoted by ( g , r ).An antisymmetric infinitesimal bialgebra is a pair (( A , · A ) , ∆ ), where ( A , · A ) is an associativealgebra and ∆ : A −→ A ⊗ A is a linear map such that ( A , ∆ ) is a coassociative coalgebra and ∆ isa 1-cocycle on A with coe ffi cients in the representation ( A ⊗ A ; 1 ⊗ L , R ⊗ ∆ ( x · A y ) = (1 ⊗ L x ) ∆ ( y ) + ( R y ⊗ ∆ ( x ) , ∀ x , y ∈ A (26) ONCOMMUTATIVE POISSON BIALGEBRAS 11 satisfying(27) ( L y ⊗ − ⊗ R y ) ∆ ( x ) + τ (cid:16) ( L x ⊗ − ⊗ R x ) ∆ ( y ) (cid:17) = , where τ : A ⊗ A −→ A ⊗ A is the exchange operator defined by(28) τ ( x ⊗ y ) = y ⊗ x , ∀ x , y ∈ A . In particular, an antisymmetric infinitesimal bialgebra ( A , ∆ ) is called coboundary if ∆ is a 1-coboundary, that is, there exists an r ∈ A ⊗ A such that(29) ∆ ( x ) = (1 ⊗ L x − R x ⊗ r , ∀ x , y ∈ A . A coboundary antisymmetric infinitesimal bialgebra is usually denoted by ( A , r ). Definition 3.6. A noncommutative pseudo-Poisson bialgebra is a triple (( P , · P , {− , −} P ) , ∆ , δ ) ,where ( P , · P , {− , −} P ) is a coherent noncommutative Poisson algebra, ∆ : P −→ P ⊗ P and δ : P −→ ∧ P are linear maps such that (a) ( P ∗ , ∆ ∗ , δ ∗ ) is a noncommutative Poisson algebra, where δ ∗ : ∧ P ∗ −→ P ∗ and ∆ ∗ : P ∗ ⊗ P ∗ −→ P ∗ are the dual maps of ∆ and δ , defined by h ∆ ∗ ( α, β ) , x i = h ∆ ( x ) , α ⊗ β i , h δ ∗ ( α, β ) , x i = h δ ( x ) , α ∧ β i , ∀ x ∈ P , α, β ∈ P ∗ ;(30)(b) δ + ∆ ∈ C , ( P , P ⊗ P ) ⊕ C , ( P , P ⊗ P ) is a -cocycle on P associated to the quasi-representation ( P ⊗ P ; 1 ⊗ L , R ⊗ , ad ⊗ + ⊗ ad) , i.e. ∆ ( x · P y ) = (1 ⊗ L x ) ∆ ( y ) + ( R y ⊗ ∆ ( x ) , (31) ∆ ( { x , y } P ) = (ad x ⊗ + ⊗ ad x ) ∆ ( y ) − (1 ⊗ L y ) δ ( x ) + ( R y ⊗ δ ( x ) , (32) δ ( { x , y } P ) = (ad x ⊗ + ⊗ ad x ) δ ( y ) − (ad y ⊗ + ⊗ ad y ) δ ( x );(33)(c) △ and δ satisfy ( L y ⊗ − ⊗ R y ) ∆ ( x ) + τ (cid:16) ( L x ⊗ − ⊗ R x ) ∆ ( y ) (cid:17) = , (34) ( L x ⊗ δ ( y ) + ( R y ⊗ δ ( x ) + (1 ⊗ ad x ) ∆ ( y ) + (1 ⊗ ad y ) τ (cid:0) ∆ ( x ) (cid:1) = δ ( x · P y ) . (35) A noncommutative pseudo-Poisson bialgebra (( P , · P , {− , −} P ) , ∆ , δ ) is called a noncommutativePoisson bialgebra if P ∗ is also a coherent noncommutative Poisson algebra. Remark 3.7.
Even though ( P ; L , R , ad) is a representation of a noncommutative Poisson algebra ( P , · P , {− , −} P ) , ( P ⊗ P ; 1 ⊗ L , R ⊗ , ad ⊗ + ⊗ ad) is just a quasi-representation. This is the reasonwhy we need to use quasi-representations to deal with the bialgebra theory of noncommutativePoisson algebras. Remark 3.8.
Let (( P , · P , {− , −} P ) , ∆ , δ ) be a noncommutative Poisson bialgebra. Then by the factthat ( P , δ ) is a Lie coalgebra and (33) , (( P , {− , −} P ) , δ ) is a Lie bialgebra. By the fact that ( P , ∆ ) isa coassociative coalgebra, (31) and (34) , (( P , · P ) , ∆ ) is an antisymmetric infinitesimal bialgebra. Theorem 3.9.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra with two co-multiplications ∆ : P −→ ⊗ P and δ : P −→ ∧ P. Suppose that δ ∗ and ∆ ∗ induce a coherentnoncommutative Poisson algebra structure on P ∗ . Set { α, β } P ∗ = δ ∗ ( α, β ) and α · P ∗ β = ∆ ∗ ( α, β ) .Then the following statements are equivalent: (i) (( P , · P , {− , −} P ) , ∆ , δ ) is a noncommutative Poisson bialgebra. (ii) ( P , P ∗ ; − R ∗ , − L ∗ , ad ∗ , − R ∗ , − L ∗ , ad ∗ ) is a matched pair of coherent noncommutative Pois-son algebras, where R ∗ , L ∗ and ad ∗ are given by (36) h R ∗ α x , β i = −h x , β · P ∗ α i , h L ∗ α x , β i = −h x , α · P ∗ β i , h ad ∗ α x , β i = −h x , { α, β } P ∗ i for all x ∈ P , α, β ∈ P ∗ . (iii) (( P ⊕ P ∗ , B ) , P , P ∗ ) is a Manin triple of noncommutative Poisson algebras, where theinvariant symmetric bilinear form B on P ⊕ P ∗ is given by (37) B ( x + α, y + β ) = h x , β i + h α, y i , ∀ x , y ∈ P , α, β ∈ P ∗ . Proof.
First, we show that (i) and (ii) are equivalent. It is known that (( P , {− , −} P ) , δ P ) is a Liebialgebra if and only if ( P , P ∗ ; ad ∗ , ad ∗ ) is a matched pair of Lie algebras and (( P , · P ) , ∆ P ) is anantisymmetric infinitesimal bialgebra if and only if ( P , P ∗ ; − R ∗ , − L ∗ , − R ∗ , − L ∗ ) is a matched pairof associative algebras. By a straightforward calculation, we have(32) ⇐⇒ (16) ⇐⇒ (19) and (35) ⇐⇒ (17) ⇐⇒ (18) , in which L = L ∗ , R = R ∗ , ρ = ad ∗ and ˜ L = L ∗ , ˜ R = R ∗ , ̺ = ad ∗ .Next, we show that (ii) and (iii) are equivalent. It is known ([12]) that ( P , P ∗ ; ad ∗ , ad ∗ ) is amatched pair of Lie algebras if and only if ( P ⊕ P ∗ , P , P ∗ ; B ) is a Manin triple of Lie algebras.Similarly, ( P , P ∗ ; − R ∗ , − L ∗ , − R ∗ , − L ∗ ) is a matched pair of associative algebras if and only if ( P ⊕ P ∗ , P , P ∗ ; B ) is a double construction of Frobenius algebra ([6], also see Remark 3.4). The restconditions in a matched pair of coherent noncommutative Poisson algebras are equivalent to theLeibniz rule that the noncommutative Poisson algebra structure on P ⊕ P ∗ satisfies. Thus, (ii) and(iii) are equivalent. Remark 3.10.
Recall ( [37] ) that a compatible Lie bialgebra structure on a compatible Liealgebra ( g , [ − , − ] , [ − , − ] ) is a pair of linear maps α, β : g → g ⊗ g such that for any k , k ∈ K , k α + k β is a Lie bialgebra structure on the Lie algebra ( g , k [ − , − ] + k [ − , − ] ) . A compatibleLie bialgebra is equivalent to a Manin triple of compatible Lie algebras in the following sense:assume that ( g , [ − , − ] , [ − , − ] ) and ( g ∗ , {− , −} , {− , −} ) are compatible Lie algebras, there is acompatible Lie algebra structure ( ~ − , − (cid:127) , ~ − , − (cid:127) ) on the direct sum of the underlying vectorspaces of g and g ∗ such that the bilinear form given by (37) is invariant in the sense B ( ~ u , v (cid:127) , w ) = B ( u , ~ v , w (cid:127) ) , B ( ~ u , v (cid:127) , w ) = B ( u , ~ v , w (cid:127) ) , ∀ u , v , w ∈ g ⊕ g ∗ . It is obvious that a Manin triple of noncommutative Poisson algebras naturally gives a Manintriple of compatible Lie algebras and hence a noncommutative Poisson bialgebra naturally givesa compatible Lie bialgebra.
4. C oboundary ( pseudo -)P oisson bialgebras To begin with, we recall some important results of coboundary Lie bialgebras and coboundaryantisymmetric infinitesimal bialgebras.Let ( g , [ − , − ] g ) be a Lie algebra and r ∈ g ⊗ g . Then the linear map δ defined by (25) makes( g , δ ) into a coboundary Lie bialgebra if and only if the following conditions are satisfied(i) (ad x ⊗ + ⊗ ad x )( r + τ ( r )) = x ⊗ ⊗ + ⊗ ad x ⊗ + ⊗ ⊗ ad x )([ r , r ] + [ r , r ] + [ r , r ]) = C ( r ) = [ r , r ] + [ r , r ] + [ r , r ] = classical Yang-Baxter equation ( CYBE ) . ONCOMMUTATIVE POISSON BIALGEBRAS 13
Let ( A , · ) be an associative algebra and r ∈ g ⊗ g . Then the linear map ∆ defined by (29)makes ( A , ∆ ) into a coboundary antisymmetric infinitesimal bialgebra if and only if the followingconditions are satisfied(i) ( L x ⊗ − ⊗ R x )(1 ⊗ L y − R y ⊗ r + τ ( r )) = ⊗ ⊗ L x − R x ⊗ ⊗ r · r + r · r − r · r ) = A ( r ) = r · r + r · r − r · r = associative Yang-Baxter equation ( AYBE ) . See [1, 3, 6, 8] for more details.Next, we introduce the notion of a coboundary noncommutative (pseudo-)Poisson bialgebra. Definition 4.1.
A noncommutative (pseudo-)Poisson bialgebra ( P , ∆ , δ ) is called coboundary ifthere exists an r ∈ P ⊗ P such that (40) δ ( x ) = (1 ⊗ ad x + ad x ⊗ r , ∆ ( x ) = (1 ⊗ L x − R x ⊗ r , ∀ x ∈ P . We denote a coboundary noncommutative (pseudo-)Poisson bialgebra by ( P , r ). Theorem 4.2.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra and r ∈ P ⊗ P.Define ∆ : P −→ P ⊗ P and δ : P −→ ∧ P by (40) . Then ( P , ∆ , δ ) is a noncommutative pseudo-Poisson bialgebra if and only if the following conditions are satisfied: (i) ( L x ⊗ − ⊗ R x )(1 ⊗ L y − R y ⊗ r + τ ( r )) = ; (ii) (ad x ⊗ + ⊗ ad x )( r + τ ( r )) = x ⊗ ⊗ L y − R y ⊗ r + τ ( r )) = ; (iv) (1 ⊗ ⊗ L x − R x ⊗ ⊗ A ( r ) = ; (v) (ad x ⊗ ⊗ + ⊗ ad x ⊗ + ⊗ ⊗ ad x ) C ( r ) = ; (vi) for r = P j a j ⊗ b j ∈ P ⊗ P, (ad x ⊗ ⊗ A ( r ) + (1 ⊗ ⊗ L x − ⊗ R x ⊗ C ( r ) + X j τ (cid:16) (1 ⊗ ad a j )(1 ⊗ L x − R x ⊗ r + τ ( r )) (cid:17) ⊗ b j = . Proof.
First by Conditions (i) and (iv), the dual map ∆ ∗ : ⊗ P ∗ −→ P ∗ defines an associativealgebra structure on P ∗ . By Conditions (ii) and (v), the dual map δ ∗ : ∧ P ∗ −→ P ∗ defines aLie algebra structure on P ∗ . By the fact that the noncommutative Poisson algebra P is coherentand (vi), we can deduce that the Leibniz rule is satisfied. Thus, ( P ∗ , ∆ ∗ , δ ∗ ) is a noncommutativePoisson algebra, which implies that Condition (a) in Definition 3.6 holds.Then since ∆ + δ is a coboundary, it is naturally closed. This implies that Condition (b) inDefinition 3.6 holds.Finally, by the definition of δ given by (40), it is straightforward to deduce that (34) holds. Bya direct calculation, we can obtain that (35) is equivalent to (cid:0) ⊗ (ad x · P y − ad x L y ) (cid:1) r + (1 ⊗ ad y R x ) τ ( r ) − (cid:0) L x ⊗ ad y (cid:1) ( r + τ ( r )) = . Since the noncommutative Poisson algebra ( P , · P , {− , −} P ) is coherent, we havead x · P y − ad x L y = ad y R x . Thus (35) is equivalent to(1 ⊗ ad y R x )( r + τ ( r )) − (cid:0) L x ⊗ ad y (cid:1) ( r + τ ( r )) = , which is equivalent to Condition (iii). Thus Condition (c) in Definition 3.6 holds. The converse can be proved similarly. We omit the details.
Definition 4.3.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra and r ∈ P ⊗ P.The equation (41) A ( r ) = C ( r ) = is called the Poisson Yang-Baxter equation (
PYBE ) in P. Let ( g , [ − , − ] g ) be a Lie algebra and r ∈ ∧ g a solution of the CYBE. Then the Lie algebrastructure [ − , − ] g ∗ : g ∗ × g ∗ −→ g ∗ induced by r is given by(42) [ α, β ] g ∗ = ad ∗ r ♯ ( α ) β − ad ∗ r ♯ ( β ) α, ∀ α, β ∈ g ∗ , where r ♯ : g ∗ −→ g is defined by(43) h r ♯ ( α ) , β i = r ( α, β ) . Furthermore, r ♯ is a Lie algebra homomorphism from the Lie algebra ( g ∗ , [ − , − ] g ∗ ) to ( g , [ − , − ] g ).Let ( A , · A ) be an associative algebra and r ∈ ∧ A a solution of the AYBE. Then the associativealgebra structure · A ∗ : A ∗ ⊗ A ∗ −→ A ∗ induced by r is given by(44) α · A ∗ β = − R ∗ r ♯ ( α ) β − L ∗ r ♯ ( β ) α, ∀ α, β ∈ A ∗ . Furthermore, r ♯ is an associative algebra homomorphism from the associative algebra ( A ∗ , · A ∗ ) to( A , · A ).The following theorem shows that a skew-symmetric solution of the PYBE in a coherent non-commutative Poisson algebra gives rise to a noncommutative Poisson bialgebra. Theorem 4.4.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra and r ∈ ∧ P asolution of the
PYBE . Then the maps · P ∗ : = ∆ ∗ : ⊗ P ∗ −→ P ∗ and {− , −} P ∗ : = δ ∗ : ∧ P ∗ −→ P ∗ ,where ∆ and δ are given by (40) , induce a coherent noncommutative Poisson algebra structureon P ∗ such that ( P , P ∗ ) is a noncommutative Poisson bialgebra. Proof.
By the fact that r is skew-symmetric, we deduce that (i)-(iii) in Theorem 4.2 hold. By thefact that r is a skew-symmetric solution of the PYBE, we deduce that (iv)-(vi) in Theorem 4.2hold. Thus, ( P , P ∗ ) is a noncommutative pseudo-Poisson bialgebra.Then by (42) and (44), we have { α · P ∗ β, γ } P ∗ − { α, β · P ∗ γ } P ∗ − { β, γ · P ∗ α } P ∗ = ad ∗ r ♯ ( α · P ∗ β ) γ − ad ∗ r ♯ ( γ ) ( α · P ∗ β ) + ad ∗ r ♯ ( β · P ∗ γ ) α − ad ∗ r ♯ ( α ) ( β · P ∗ γ ) + ad ∗ r ♯ ( γ · P ∗ α ) β − ad ∗ r ♯ ( β ) ( γ · P ∗ α ) = ad ∗ r ♯ ( α ) · P r ♯ ( β ) γ + ad ∗ r ♯ ( γ ) R ∗ r ♯ ( α ) β + ad ∗ r ♯ ( γ ) L ∗ r ♯ ( β ) α + ad ∗ r ♯ ( β ) · P r ♯ ( γ ) α + ad ∗ r ♯ ( α ) R ∗ r ♯ ( β ) γ + ad ∗ r ♯ ( α ) L ∗ r ♯ ( γ ) β + ad ∗ r ♯ ( γ ) · P r ♯ ( α ) β + ad ∗ r ♯ ( β ) R ∗ r ♯ ( γ ) α + ad ∗ r ♯ ( β ) L ∗ r ♯ ( α ) γ = (cid:0) ad ∗ r ♯ ( α ) · P r ♯ ( β ) + ad ∗ r ♯ ( α ) R ∗ r ♯ ( β ) + ad ∗ r ♯ ( β ) L ∗ r ♯ ( α ) (cid:1) γ + (cid:0) ad ∗ r ♯ ( γ ) · P r ♯ ( α ) + ad ∗ r ♯ ( γ ) R ∗ r ♯ ( α ) + ad ∗ r ♯ ( α ) L ∗ r ♯ ( γ ) (cid:1) β + (cid:0) ad ∗ r ♯ ( β ) · P r ♯ ( γ ) + ad ∗ r ♯ ( β ) R ∗ r ♯ ( γ ) + ad ∗ r ♯ ( γ ) L ∗ r ♯ ( β ) (cid:1) α = . The last equality holds because h ad ∗ x · P y α + ad ∗ x R ∗ y α + ad ∗ y L ∗ x α, z i = h x · P { y , z } P + { x , z } P · P y − { x · P y , z } P , α i = , ONCOMMUTATIVE POISSON BIALGEBRAS 15 which follows from the Leibniz rule of the noncommutative Poisson algebra P . We deduce that( P ∗ , · P ∗ , {− , −} P ∗ ) is a coherent noncommutative Poisson algebra. Thus, ( P , P ∗ ) is a noncommuta-tive Poisson bialgebra. Remark 4.5.
The study of coboundary compatible Lie bialgebras and the classical Yang-Baxterequation in compatible Lie algebras was also given in [37] . It is straightforward to see thata skew-symmetric solution of the Poisson Yang-Baxter equation in a coherent noncommutativePoisson algebra ( P , · P , {− , −} P ) is a skew-symmetric solution of the classical Yang-Baxter equationin the compatible Lie algebra ( P , {− , −} P , [ − , − ] P ) and hence gives a (coboundary) compatible Liebialgebra. Corollary 4.6.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra and r ∈ ∧ P asolution of the
PYBE . Then r ♯ : P ∗ −→ P is a noncommutative Poisson algebra homomorphismfrom the coherent noncommutative Poisson algebra ( P ∗ , · P ∗ , {− , −} P ∗ ) to ( P , · P , {− , −} P ) . Example 4.7.
Let ( A , · A , {− , −} ~ ) be the standard noncommutative Poisson algebra given by Ex-ample 2.13. If r ∈ ∧ A is a solution of the AYBE in the associative algebra ( A , · A ), then r is asolution of the PYBE in the standard noncommutative Poisson algebra A . Furthermore, the dualnoncommutative Poisson algebra ( A ∗ , · A ∗ , {− , −} P ∗ ) induced by r is also standard, where α · A ∗ β = − R ∗ r ♯ ( α ) β − L ∗ r ♯ ( β ) α, { α, β } P ∗ = ~ ( α · A ∗ β − β · A ∗ α ) , ∀ α, β ∈ A ∗ . Example 4.8.
Consider the coherent noncommutative Poisson algebra P given by Example 2.15,then r given by r = κ e ∧ e + κ e ∧ e is a solution of the PYBE in P , where κ and κ are constants. Example 4.9.
Consider the coherent noncommutative Poisson algebra P given by Example 2.16,then r given by r = κ e ∧ e + κ e ∧ e + κ e ∧ e + κ e ∧ e − κ e ∧ e and r = κ e ∧ e + κ e ∧ e + ( κ + κ ) e ∧ e are solutions of the PYBE in P , where κ , κ and κ are constants.Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra and r ∈ ∧ P . Assume that r is nondegenerate, i.e. r ♯ : P ∗ −→ P is invertible. Define ω ∈ ∧ P ∗ by ω ( x , y ) = h ( r ♯ ) − ( x ) , y i , ∀ x , y ∈ P . Then we have
Proposition 4.10.
With the above notations, r ∈ ∧ P is a solution of the
PYBE in a coherentnoncommutative Poisson algebra ( P , · P , {− , −} P ) if and only if ω is both a Connes cocycle on theassociative ( P , · P ) and a symplectic structure on the Lie algebra ( P , {− , −} P ) , i.e., ω ( x · P y , z ) + ω ( y · P z , x ) + ω ( z · P x , y ) = ,ω ( { x , y } P , z ) + ω ( { y , z } P , x ) + ω ( { z , x } P , y ) = , ∀ x , y , z ∈ P . Proof.
It follows from the fact that r is a solution of the AYBE if and only if ω is a Connes cocycleon the associative ( P , · P , ) ([6]) and r is a solution of the CYBE if and only if ω is a symplecticstructure on the Lie algebra ( P , {− , −} P ) ([14]).Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra. An element s ∈ P ⊗ P iscalled ( L , R , ad) -invariant if(45) (1 ⊗ L x − R x ⊗ s = , (ad x ⊗ + ⊗ ad x ) s = , ∀ x ∈ P . Proposition 4.11.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra. Let r = a + s ∈ P ⊗ P with skew-symmetric part a and symmetric part s . If the symmetric part s of r is ( L , R , ad) -invariant and r is a solution of the PYBE , then ( P , r ) gives a coboundary noncommuta-tive pseudo-Poisson bialgebra. Furthermore, this coboundary noncommutative pseudo-Poissonbialgebra is a noncommutative Poisson bialgebra if and only if the symmetric part s of r satisfies (46) ad ∗ s ♯ ( α ) · P s ♯ ( β ) γ + ad ∗ s ♯ ( γ ) · P s ♯ ( α ) β + ad ∗ s ♯ ( β ) · P s ♯ ( γ ) α = , where s ♯ : P ∗ −→ P is defined by (47) h s ♯ ( α ) , β i = s ( α, β ) , ∀ α, β ∈ P ∗ . Proof.
Since the symmetric part s of r is ( L , R , ad)-invariant and r is a solution of the PYBE, byTheorem 4.2, ( P , r ) gives a coboundary noncommutative pseudo-Poisson bialgebra.By the invariance of s and A ( r ) =
0, we have a ♯ ( α ) · P a ♯ ( β ) − a ♯ ( α · P ∗ β ) = s ♯ ( α ) · P s ♯ ( β ) , where α · P ∗ β = − R ∗ a ♯ ( α ) β − L ∗ a ♯ ( β ) α. By this fact, with a similar proof of Theorem 4.4, the non-commutative Poisson algebra ( P ∗ , · P ∗ , {− , −} P ∗ ) is coherent with { α, β } P ∗ = ad ∗ a ♯ ( α ) β − ad ∗ a ♯ ( β ) α if andonly if the symmetric part s of r satisfies (46). We omit the details.At the end of this section, we establish the Drinfeld double theory for noncommutative Poissonbialgebras. Let (( P , · P , {− , −} P ) , ∆ , δ ) be a noncommutative Poisson bialgebra. By Theorem 3.5and Theorem 3.9, there is a coherent noncommutative Poisson algebra structure on D = P ⊕ P ∗ with the associative multiplication ∗ D : D × D −→ D and the Lie bracket {− , −} D : D × D −→ D given by ( x + α ) ∗ D ( y + β ) = x · P y − R ∗ α y − L ∗ β x + α · P ∗ β − R ∗ x β − L ∗ y α, (48) { x + α, y + β } D = { x , y } P + ad ∗ α y − ad ∗ β x + { α, β } P ∗ + ad ∗ x β − ad ∗ y α (49)for all x , y ∈ P , α, β ∈ P ∗ , where R ∗ , L ∗ and ad ∗ are given by (36).Let { e , · · · , e n } be a basis of P and { e ∗ , · · · , e ∗ n } its dual basis. Let r = P i e i ⊗ e ∗ i ∈ D ⊗ D . Proposition 4.12.
With the above notations, r = P i e i ⊗ e ∗ i ∈ D ⊗ D is a solution of the PYBE inthe coherent noncommutative Poisson algebra D such that ( D , D ∗ ) is a noncommutative Poissonbialgebra. Proof.
Obviously the symmetric part of r is s = P i ( e i ⊗ e ∗ i + e ∗ i ⊗ e i ) and the skew-symmetric partof r is a = P i ( e i ⊗ e ∗ i − e ∗ i ⊗ e i ). By the Drinfeld double theory of Lie algebras and the associativedouble theory of associative algebras (see [6] for more details on the associative double theory),we know that s is ( L , R , ad)-invariant and r satisfies the classical and the associative Yang-Baxterequations. Thus r is a solution of the PYBE, and by Proposition 4.11, ( D , r ) is a noncommutative ONCOMMUTATIVE POISSON BIALGEBRAS 17 pseudo-Poisson bialgebra. By a straightforward calculation, the noncommutative Poisson algebrastructure on D ∗ is given by ( x + α ) ∗ D ∗ ( y + β ) = x · P y − α · P ∗ β, { x + α, y + β } D ∗ = { x , y } P − { α, β } P ∗ . It is easy to check that the noncommutative Poisson algebra structure on D ∗ is coherent. Thus( D , D ∗ ) is a noncommutative Poisson bialgebra.5. N oncommutative pre -P oisson algebras , R ota -B axter operators and O - operators In this section, we introduce the notions of noncommutative pre-Poisson algebras and Rota-Baxter operators (more generally O -operators) on a noncommutative Poisson algebra. We showthat on the one hand, an O -operator on a noncommutative Poisson algebra gives a noncommuta-tive pre-Poisson algebra, and on the other hand, a noncommutative pre-Poisson algebra naturallygives an O -operator on the sub-adjacent noncommutative Poisson algebra. We use O -operatorsand noncommutative pre-Poisson algebras to construct some skew-symmetric solutions of thePoisson Yang-Baxter equation.5.1. Noncommutative pre-Poisson algebras.
Recall that a pre-Lie algebra is a pair ( A , ∗ ),where A is a vector space, and ∗ : A ⊗ A −→ A is a bilinear multiplication satisfying that forall x , y , z ∈ A , the associator ( x , y , z ) = ( x ∗ y ) ∗ z − x ∗ ( y ∗ z ) is symmetric in x , y , i.e.,( x , y , z ) = ( y , x , z ) , or equivalently , ( x ∗ y ) ∗ z − x ∗ ( y ∗ z ) = ( y ∗ x ) ∗ z − y ∗ ( x ∗ z ) . Lemma 5.1. ([11])
Let ( A , ∗ ) be a pre-Lie algebra. The commutator [ x , y ] A = x ∗ y − y ∗ x definesa Lie algebra structure on A, which is called the sub-adjacent Lie algebra of ( A , ∗ ) and denotedby A c . Furthermore, L : A → gl ( A ) defined by (50) L x y = x ∗ y , ∀ x , y ∈ Agives a representation of A c on A. There is a similar relationship between dendriform algebras and associative algebras.
Lemma 5.2. ([30])
Let ( A , ≻ , ≺ ) be a dendriform algebra. Then ( A , · ) is an associative algebra,where x · y = x ≻ y + x ≺ y. Moreover, for x ∈ A, define L ≻ x , R ≺ x : A −→ gl ( A ) by (51) L ≻ x ( y ) = x ≻ y , R ≺ x ( y ) = y ≺ x , ∀ y ∈ A . Then ( A ; L ≻ , R ≺ ) is a representation of the associative algebra ( A , · ) . Now we are ready to give the main notion in this subsection.
Definition 5.3. A noncommutative pre-Poisson algebra is a quadruple ( A , ≻ , ≺ , ∗ ) such that ( A , ≻ , ≺ ) is a dendriform algebra and ( A , ∗ ) is a pre-Lie algebra satisfying the following compat-ibility conditions: ( x ∗ y − y ∗ x ) ≻ z = x ∗ ( y ≻ z ) − y ≻ ( x ∗ z ) , (52) x ≺ ( y ∗ z − z ∗ y ) = y ∗ ( x ≺ z ) − ( y ∗ x ) ≺ z , (53) ( x ≻ y + x ≺ y ) ∗ z = ( x ∗ z ) ≺ y + x ≻ ( y ∗ z ) . (54) A noncommutative pre-Poisson algebra ( A , ≻ , ≺ , ∗ ) is called coherent if it also satisfies (55) ( x ≻ y + x ≺ y ) ∗ z = x ∗ ( y ≻ z ) + y ∗ ( z ≺ x ) . Remark 5.4.
Aguiar introduced the notion of pre-Poisson algebras in [2] , as the splitting ofPoisson algebras. If the dendriform algebra in a noncommutative pre-Poisson algebra reduces toa Zinbiel algebra, then we obtain a pre-Poisson algebra. Noncommutative pre-Poisson algebrascan be viewed as the splitting of noncommutative Poisson algebras. See [7, 33] for more detailsof splitting of operads.
Proposition 5.5.
A noncommutative pre-Poisson algebra ( A , ≻ , ≺ , ∗ ) is coherent if and only if itsatisfies ( x ∗ y ) ◦ z − x ◦ ( y ∗ z ) = ( y ∗ x ) ◦ z − y ◦ ( x ∗ z ) for all x , y , z ∈ A, where x ◦ y = x ≻ y − y ≺ x. Proof.
By (52)-(54) in the definition of noncommutative pre-Poisson algebra , we have x ∗ ( y ≻ z ) + y ∗ ( z ≺ x ) − ( x ≻ y + x ≺ y ) ∗ z = ( x ∗ y − y ∗ x ) ≻ z + y ≻ ( x ∗ z ) + z ≺ ( y ∗ x − x ∗ y ) + ( y ∗ z ) ≺ x − ( x ∗ z ) ≺ y − x ≻ ( y ∗ z ) = (cid:0) ( x ∗ y ) ≻ z − z ≺ ( x ∗ y ) (cid:1) − (cid:0) x ≻ ( y ∗ z ) − ( y ∗ z ) ≺ x (cid:1) − (cid:0) ( y ∗ x ) ≻ z − z ≺ ( y ∗ x ) (cid:1) + (cid:0) y ≻ ( x ∗ z ) − ( x ∗ z ) ≺ y (cid:1) = ( x ∗ y ) ◦ z − x ◦ ( y ∗ z ) − ( y ∗ x ) ◦ z + y ◦ ( x ∗ z ) . Then the conclusion follows immediately.Similarly, a coherent noncommutative pre-Poisson algebra ( A , ≻ , ≺ , ∗ ) also satisfies( x ◦ y ) ∗ z − x ∗ ( y ◦ z ) = ( y ◦ x ) ∗ z − y ∗ ( x ◦ z ) . Definition 5.6. A compatible pre-Lie algebra ( A , ◦ , ∗ ) consists of two pre-Lie algebras ( A , ◦ ) and ( A , ∗ ) such that for any k , k ∈ K , the following bilinear operation (56) x ⋆ y = k x ◦ y + k x ∗ y , ∀ x , y ∈ A , defines a pre-Lie algebra structure on A. It is straightforward to obtain the following result.
Proposition 5.7.
Let ( A , ◦ ) and ( A , ∗ ) be two pre-Lie algebras. Then ( A , ◦ , ∗ ) is a compatiblepre-Lie algebra if and only if for all x , y , z ∈ A, (57) ( x ◦ y ) ∗ z − x ∗ ( y ◦ z ) + ( x ∗ y ) ◦ z − x ◦ ( y ∗ z ) = ( y ◦ x ) ∗ z − y ∗ ( x ◦ z ) + ( y ∗ x ) ◦ z − y ◦ ( x ∗ z ) . Corollary 5.8.
Let ( A , ≻ , ≺ , ∗ ) be a coherent noncommutative pre-Poisson algebra. Then ( A , ◦ , ∗ ) is a compatible pre-Lie algebra, where x ◦ y = x ≻ y − y ≺ x for all x , y ∈ A. Example 5.9.
Let ( A , ≻ , ≺ ) be a dendriform algebra. Then ( A , ≻ , ≺ , ∗ ~ ) is a coherent noncommu-tative pre-Poisson algebra, where ~ is a fixed number and the pre-Lie algebra structure ∗ ~ is givenby x ∗ ~ y = ~ ( x ≻ y − y ≺ x ) , ∀ x , y ∈ A . A noncommutative pre-Poisson algebra gives rise to a noncommutative Poisson algebra and arepresentation on itself naturally.
Theorem 5.10.
Let ( A , ≻ , ≺ , ∗ ) be a noncommutative pre-Poisson algebra. ONCOMMUTATIVE POISSON BIALGEBRAS 19 (i)
Define x · y = x ≻ y + x ≺ y and { x , y } = x ∗ y − y ∗ x , ∀ x , y ∈ A . Then ( A , · , {− , −} ) is a noncommutative Poisson algebra, which is called the sub-adjacentnoncommutative Poisson algebra of ( A , ≻ , ≺ , ∗ ) and denoted by A c . (ii) If the noncommutative pre-Poisson algebra ( A , ≻ , ≺ , ∗ ) is coherent, then A c is also coher-ent. (iii) ( A ; L ≻ , R ≺ , L ) is a representation of the sub-adjacent noncommutative Poisson algebra A c ,where L ≻ and R ≺ are given by (51) and L is given by (50) . Proof. (i) By Lemma 5.1 and 5.2, we deduce that ( A , · ) is an associative algebra and ( A , {− , −} ) isa Lie algebra. By (52)-(54), we have { x , y · z } − { x , y } · z − y · { x , z } = (cid:16) x ∗ ( y ≻ z ) − y ≻ ( x ∗ z ) − ( x ∗ y − y ∗ x ) ≻ z (cid:17) + (cid:16) x ∗ ( y ≺ z ) − ( x ∗ y ) ≺ z − y ≺ ( x ∗ z − z ∗ x ) (cid:17) + (cid:16) ( y ∗ x ) ≺ z + y ≻ ( z ∗ x ) − ( y ≻ z + y ≺ z ) ∗ x (cid:17) = , which implies that ( A , · , {− , −} ) is a noncommutative Poisson algebra.(ii) If the noncommutative pre-Poisson algebra ( A , ≻ , ≺ , ∗ ) is coherent, by (55), we have { x · y , z } − { x , y · z } − { y , z · x } = ( x · y ) ∗ z − z ∗ ( x ≻ y ) − z ∗ ( x ≺ y ) + ( y · z ) ∗ x − x ∗ ( y ≻ z ) − x ∗ ( y ≺ z ) + ( z · x ) ∗ y − y ∗ ( z ≻ x ) − y ∗ ( z ≺ x ) = (cid:0) ( x · y ) ∗ z − x ∗ ( y ≻ z ) − y ∗ ( z ≺ x ) (cid:1) + (cid:0) ( z · x ) ∗ y − z ∗ ( x ≻ y ) − x ∗ ( y ≺ z ) (cid:1) + (cid:0) ( y · z ) ∗ x − y ∗ ( z ≻ x ) − z ∗ ( x ≺ y ) (cid:1) = , which implies that (5) holds. Thus, ( A , · , {− , −} ) is coherent.(iii) By Lemma 5.1, ( A ; L ) is a representation of the sub-adjacent Lie algebra A c . By Lemma5.2, ( A ; L ≻ , R ≺ ) is a representation of the associative algebra ( A , · ). Moreover, (52) implies that(1) holds, (53) implies that (2) holds and (54) implies that (3) holds. Thus ( A ; L ≻ , R ≺ , L ) is arepresentation of the noncommutative Poisson algebra A c . Corollary 5.11.
Let ( A , ≻ , ≺ , ∗ ) be a coherent noncommutative pre-Poisson algebra. Then thedual ( A ∗ ; − R ∗≺ , − L ∗≻ , L ∗ ) is also a representation of A c , where L ∗≻ , R ∗≺ : A −→ gl ( V ∗ ) are given by h L ∗≻ x α, y i = −h α, x ≻ y i , h R ∗≺ x α, y i = −h α, y ≺ x i , ∀ x , y ∈ A , α ∈ A ∗ . Rota-Baxter operators and O -operators on noncommutative Poisson algebras. A linearmap T : V −→ A is called an O -operator on an associative algebra ( A , · ) with respect to arepresentation ( V ; L , R ) if T satisfies(58) T ( u ) · T ( v ) = T ( L T ( u ) v + R T ( v ) u ) , ∀ u , v ∈ V . In particular, an O -operator on an associative algebra ( A , · ) with respect to the regular representa-tion is called a Rota-Baxter operator on A . Lemma 5.12. ([8])
Let ( A , · ) be an associative algebra and ( V ; L , R ) a representation. Let T : V → A be an O -operator on ( A , · ) with respect to ( V ; L , R ) . Then there exists a dendriformalgebra structure on V given byu ≻ v = L T ( u ) v , u ≺ v = R T ( v ) u , ∀ u , v ∈ V . A linear map T : V −→ g is called an O -operator ([27]) on a Lie algebra ( g , [ − , − ] g ) withrespect to a representation ( V ; ρ ) if T satisfies(59) [ T ( u ) , T ( v )] g = T (cid:16) ρ ( T ( u ))( v ) − ρ ( T ( v ))( u ) (cid:17) , ∀ u , v ∈ V . In particular, an O -operator on a Lie algebra ( g , [ − , − ] g ) with respect to the adjoint representationis called a Rota-Baxter operator on g . Lemma 5.13. ([5])
Let T : V → g be an O -operator on a Lie algebra ( g , [ − , − ] g ) with respect toa representation ( V ; ρ ) . Define a multiplication ∗ on V by (60) u ∗ v = ρ ( T u )( v ) , ∀ u , v ∈ V . Then ( V , ∗ ) is a pre-Lie algebra. Let ( V ; L , R , ρ ) be a representation of a noncommutative Poisson algebra ( P , · P , {− , −} P ). Definition 5.14. (i)
A linear operator T : V −→ P is called an O -operator on P if T is bothan O -operator on the associative algebra ( P , · P ) and an O -operator on the Lie algebra ( P , {− , −} P ) ; (ii) A linear operator B : P −→ P is called a
Rota-Baxter operator on
P if B is both aRota-Baxter operator on the associative algebra ( P , · P ) and a Rota-Baxter operator onthe Lie algebra ( P , {− , −} P ) . When ( P , · P , {− , −} P ) is a usual Poisson algebra, i.e. · P is commutative, we recover the notionof a Rota-Baxter operator on a Poisson algebra introduced by Aguiar in [2].It is obvious that B : P −→ P is a Rota-Baxter operator on P if and only if B is an O -operatoron P with respect to the representation ( P ; L , R , ad). Example 5.15.
Let B be a Rota-Baxter operator on an associative algebra ( A , · ). Then B is aRota-Baxter operator on the coherent noncommutative Poisson algebra ( A , · , {− , −} ~ ) given byExample 2.13. Example 5.16.
Let ( A , ≻ , ≺ , ∗ ) be a noncommutative pre-Poisson algebra. Then the identity mapid is an O -operator on A c with respect to the representation ( A ; L ≻ , R ≺ , L ).Obviously, we have Proposition 5.17.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra and r ∈∧ P a solution of the
PYBE . Then r ♯ : P ∗ −→ P is an O -operator on P with respect to therepresentation ( P ∗ ; − R ∗ , − L ∗ , ad ∗ ) . An O -operator on a noncommutative Poisson algebra gives a noncommutative pre-Poissonalgebra. Theorem 5.18.
Let ( P , · P , {− , −} P ) be a noncommutative Poisson algebra and T : V −→ P an O -operator on P with respect to the representation ( V ; L , R , ρ ) . Define new operations ≻ , ≺ and ∗ on V by u ≻ v = L T ( u ) v , u ≺ v = R T ( v ) u , u ∗ v = ρ ( T ( u )) v . Then ( V , ≻ , ≺ , ∗ ) is a noncommutative pre-Poisson algebra and T is a homomorphism from V c to ( P , · P , {− , −} P ) .Furthermore, if the representation satisfies (4) , then ( V , ≻ , ≺ , ∗ ) is a coherent noncommutativepre-Poisson algebra. ONCOMMUTATIVE POISSON BIALGEBRAS 21
Proof.
First by the fact that T is an O -operator on the associative algebra ( P , · P ) as well as an O -operator on the Lie algebra ( P , {− , −} P ) with respect to the representations ( V ; L , R ) and ( V ; ρ )respectively, we deduce that ( V , ≻ , ≺ ) is a dendriform algebra and ( V , ∗ ) is a pre-Lie algebra.Denote by { u , v } T : = u ∗ v − v ∗ u , then by the fact T ([ u , v ] T ) = { T ( u ) , T ( v ) } P and (1),( u ∗ v − v ∗ u ) ≻ w − u ∗ ( v ≻ w ) + v ≻ ( u ∗ w ) = { u , v } T ≻ w − u ∗ ( v ≻ w ) + v ≻ ( u ∗ w ) = L T { u , v } T w − ρ ( T ( u )) L T ( v ) w + L T ( v ) ρ ( T ( u )) w = L { T ( u ) , T ( v ) } P w − ρ ( T ( u )) L T ( v ) w + L T ( v ) ρ ( T ( u )) w = , which implies that (52) holds. Similarly, by (2), we can show that (53) also holds.Denote by u · T v : = u ≻ v + v ≺ u , then by the fact T ( u · T v ) = T ( u ) · P T ( v ) and (3),( u ≻ v + u ≺ v ) ∗ w − ( u ∗ w ) ≺ v − u ≻ ( v ∗ w ) = ( u · T v ) ∗ w − ( u ∗ w ) ≺ v − u ≻ ( v ∗ w ) = ρ ( T ( u · T v )) w − R T ( v ) ρ ( T ( u )) w − L T ( u ) ρ ( T ( v )) w = ρ ( T ( u ) · P T ( v )) w − R T ( v ) ρ ( T ( u )) w − L T ( u ) ρ ( T ( v )) w = , which implies that (54) holds. Thus, ( V , ≻ , ≺ , ∗ ) is a noncommutative pre-Poisson algebra. It isobvious that T is a homomorphism from V c to ( P , · P , {− , −} P ).If the representation satisfies (4), then we have( u ≻ v + v ≺ u ) ∗ w − u ∗ ( v ≻ w ) − v ∗ ( w ≺ u ) = ρ ( T ( L T ( u ) v + R T ( v ) u )) w − ρ ( T ( u ))( L T ( v ) w ) − ρ ( T ( v ))( L T ( u ) w ) = ρ ( T ( u ) · P T ( v )) w − ρ ( T ( u ))( L T ( v ) w ) − ρ ( T ( v ))( L T ( u ) w ) = , which implies that ( V , ≻ , ≺ , ∗ ) is coherent. Corollary 5.19.
Let ( P , · P , {− , −} P ) be a noncommutative Poisson algebra and T : V −→ P an O -operator on P with respect to the representation ( V ; L , R , ρ ) . Then T ( V ) = { T ( v ) | v ∈ V } ⊂ Pis a subalgebra of P and there is an induced noncommutative pre-Poisson algebra structure onT ( V ) given byT ( u ) ≻ T ( v ) = T ( u ≻ v ) , T ( u ) ≺ T ( v ) = T ( u ≺ v ) , T ( u ) ∗ T ( v ) = T ( u ∗ v ) for all u , v ∈ V. Corollary 5.20.
Let ( P , · P , {− , −} P ) be a noncommutative Poisson algebra. There is a noncommu-tative pre-Poisson algebra structure on P such that its sub-adjacent noncommutative Poisson al-gebra is exactly ( P , · P , {− , −} P ) if and only if there exists an invertible O -operator on ( P , · P , {− , −} P ) . Proof. If T : V −→ P is an invertible O -operator on P with respect to the representation( V ; L , R , ρ ), then the compatible noncommutative pre-Poisson algebra structure on P is givenby x ≻ y = T ( L x T − ( y )) , x ≺ y = T ( R y T − ( x )) , x ∗ y = T ( ρ ( x )( T − ( y )))for all x , y ∈ P .Conversely, let ( P , ≻ , ≺ , ∗ ) be a noncommutative pre-Poisson algebra and ( P , · P , {− , −} P ) thesub-adjacent noncommutative Poisson algebra. Then the identity map id is an O -operator on P with respect to the representation ( P ; L ≻ , R ≺ , L ). Example 5.21.
Let ( P , · P , {− , −} P ) be a noncommutative Poisson algebra and B : P −→ P aRota-Baxter operator. Define new operations on P by x ≻ y = B ( x ) · P y , x ≺ y = x · P B ( y ) , x ∗ y = {B ( x ) , y } P . Then ( P , ≻ , ≺ , ∗ ) is a noncommutative pre-Poisson algebra and B is a homomorphism from thesub-adjacent noncommutative Poisson algebra ( P , · B , {− , −} B ) to ( P , · P , {− , −} P ), where x · B y = x ≻ y + x ≺ y and { x , y } B = x ∗ y − y ∗ x . Example 5.22.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra and r ∈ ∧ P a solution of the PYBE. Then ( P ∗ , ≻ , ≺ , ∗ ) is a coherent noncommutative pre-Poisson algebra,where α ≻ β = − R ∗ r ♯ ( α ) β, α ≺ β = − L ∗ r ♯ ( β ) α, α ∗ β = ad ∗ r ♯ ( α ) β, ∀ α, β ∈ P ∗ . By Proposition 4.10, we have
Example 5.23.
Let ( P , · P , {− , −} P ) be a coherent noncommutative Poisson algebra and ω ∈ ∧ P ∗ non-degenerate. If ω is both a Connes cocycle on the associative algebra ( P , · P ) and a symplecticstructure on the Lie algebra ( P , {− , −} P ), then ( P , ≻ , ≺ , ∗ ) is a noncommutative pre-Poisson algebra,where ≻ , ≺ and ∗ are determined by ω ( x ≻ y , z ) = ω ( y , z · P x ) , ω ( x ≺ y , z ) = ω ( x , y · P z ) , ω ( x ∗ y , z ) = − ω ( y , { x , z } P ) , ∀ x , y , z ∈ P . Theorem 5.24.
Let ( V ; L , R , ρ ) be a representation of a coherent noncommutative Poisson al-gebra ( P , · P , {− , −} P ) satisfying (4) . Let T : V −→ P be a linear map which is identified withan element in ( P ⋉ ( −R ∗ , −L ∗ ,ρ ∗ ) V ∗ ) ⊗ ( P ⋉ ( −R ∗ , −L ∗ ,ρ ∗ ) V ∗ ) . Then T = T − τ ( T ) is a skew-symmetricsolution of the PYBE in P ⋉ ( −R ∗ , −L ∗ ,ρ ∗ ) V ∗ if and only if T is an O -operator on P with respect tothe representation ( V ; L , R , ρ ) , where τ is the exchange operator given by (28) . Proof.
By Proposition 2.17 and the fact that ( V ; L , R , ρ ) is a representation satisfying (4), it fol-lows that P ⋉ ( −R ∗ , −L ∗ ,ρ ∗ ) V ∗ is a coherent noncommutative Poisson algebra.By Corollary 3.10 in [8], T = T − τ ( T ) is a skew-symmetric solution of the associative Yang-Baxter equation in P ⋉ ( −R ∗ , −L ∗ ) V ∗ if and only if T is an O -operator on ( P , · P ) with respect to therepresentation ( L , R ). By the conclusion in [5], T = T − τ ( T ) is a skew-symmetric solution ofthe classical Yang-Baxter equation in P ⋉ ρ ∗ V ∗ if and only if T is an O -operator on ( P , {· , ·} P )with respect to the representation ρ . Thus, T = T − τ ( T ) is a skew-symmetric solution of thePoisson Yang-Baxter equation if and only if T = T − τ ( T ) is both a solution of the associativeYang-Baxter equation in P ⋉ ( −R ∗ , −L ∗ ) V ∗ and the classical Yang-Baxter equation in P ⋉ ρ ∗ V ∗ , whichimplies that T = T − τ ( T ) is a skew-symmetric solution of the Poisson Yang-Baxter equation in P ⋉ ( −R ∗ , −L ∗ ,ρ ∗ ) V ∗ if and only if T is an O -operator associated to the representation ( V ; L , R , ρ ).By Example 5.16 and Theorem 5.24, we get Corollary 5.25.
Let ( A , ≻ , ≺ , ∗ ) be a coherent noncommutative pre-Poisson algebra. Then r = P i ( e i ⊗ e ∗ i − e ∗ i ⊗ e i ) is a skew-symmetric solution of the PYBE in the coherent noncommutativePoisson algebra A c ⋉ ( − R ∗≺ , − L ∗≻ , L ∗ ) A ∗ , where { e , · · · , e n } is a basis of A and { e ∗ , · · · , e ∗ n } is the dualbasis. Example 5.26.
Let ( A , ≻ , ≺ ) be a dendriform algebra. By Example 5.9, ( A , ≻ , ≺ , ∗ ~ ) is a coherentnoncommutative pre-Poisson algebra, where x ∗ ~ y = ~ ( x ≻ y − y ≺ x ) for all x , y ∈ A . Thus r = P i ( e i ⊗ e ∗ i − e ∗ i ⊗ e i ) is a skew-symmetric solution of the PYBE in the coherent noncommutative ONCOMMUTATIVE POISSON BIALGEBRAS 23
Poisson algebra A c ⋉ ( − R ∗≺ , − L ∗≻ , L ∗ ) A ∗ , where { e , · · · , e n } is a basis of A and { e ∗ , · · · , e ∗ n } is the dualbasis. Acknowledgements.
This research is supported by NSFC (11922110, 11901501, 11931009).C. Bai is also supported by the Fundamental Research Funds for the Central Universities andNankai ZhiDe Foundation. R eferences [1] M. Aguiar, Infinitesimal Hopf algebras. In New trends in Hopf algebra theory (La Falda, 1999),
Contemp.Math.
Lett. Math. Phys.
54 (2000), 263-277. 2, 4, 18, 20[3] M. Aguiar, On the associative analog of Lie bialgebras.
J. Algebra
244 (2001), 492-532. 2, 13[4] M. Aguiar, Infinitesimal bialgebras, pre-Lie and dendriform algebras. In Hopf algebras,
Lecture Notes in Pureand Appl. Math.
J. Phys. A: Math. Theo.
40 (2007),11073-11082. 4, 20, 22[6] C. Bai, Double constructions of Frobenius algebras, Connes cocycles and their duality.
J. Noncommut. Geom.
Int.Math. Res. Not. O -operators on associative algebras and associative Yang-Baxter equations. PacificJ. Math.
256 (2012), 257-289. 4, 13, 19, 22[9] Y. Bao and Y. Ye, Cohomology structure for a Poisson algebra: I.
J. Algebra Appl.
15 (2016), no. 2, 1650034,17 pp. 2, 3, 7[10] Y. Bao and Y. Ye, Cohomology structure for a Poisson algebra: II.
Sci. China Math. (2019).https: // doi.org / / s11425-019-1591-6. 2[11] D. Burde, Left-symmetric algebras and pre-Lie algebras in geometry and physics. Cent. Eur. J. Math.
Comm. Math. Phys.
210 (2000), 249-273. 4[14] V. Drinfeld, Hamiltonian structure on the Lie groups, Lie bialgebras and the geometric sense of the classicalYang-Baxter equations.
Soviet Math. Dokl.
27 (1983), 68-71. 3, 16[15] K. Ebrahimi-Fard, Loday-type algebras and the Rota-Baxter relation.
Lett. Math. Phys.
61 (2002), no. 2, 139-147. 4[16] K. Ebrahimi-Fard, L. Guo and D. Kreimer, Integrable renormalization II: the general case.
Ann. Henri Poincare
Lectures in Mathematical Physics , Inter-national Press, Somerville, MA, 2002. 3[18] M. Flato, M. Gerstenhaber and A. A. Voronov, Cohomology and deformation of Leibniz pairs.
Lett. Math.Phys.
34 (1995), 77-90. 2, 5[19] I. Z. Golubchik and V. V. Sokolov, Compatible Lie brackets and integrable equations of the principal chiralmodel type.
Funct. Anal. Appl.
36 (2002), 172-181. 3, 6[20] I. Z. Golubchik and V. V. Sokolov, Compatible Lie brackets and the Yang-Baxter equation.
Theor. Math. Phys.
146 (2006), 159-169. 3, 6[21] I. Z. Golubchik and V. V. Sokolov, Factorization of the loop algebras and compatible Lie brackets.
J. NonlinearMath. Phys.
12 (2005), 343-350. 3, 6[22] L. Guo, An introduction to Rota-Baxter algebra. Surveys of Modern Mathematics, 4. International Press,Somerville, MA; Higher Education Press, Beijing, 2012. xii +
226 pp. 4[23] L. Guo and W. Keigher, Baxter algebras and shu ffl e products. Adv. Math.
150 (2000), 117-149. 4[24] F. Kubo, Finite-dimensional non-commutative Poisson algebras.
J. Pure Appl. Algebra
113 (1996), 307-314.2 [25] F. Kubo, Finite-dimensional simple Leibniz pairs and simple Poisson modules.
Lett. Math. Phys.
43 (1998),21-29. 2[26] F. Kubo, Non-commutative Poisson algebra structures on a ffi ne Kac-Moody aglebras. J. Pure Appl. Algebra
126 (1998), 267-286. 2[27] B. A. Kupershmidt, What a classical r -matrix really is. J. Nonlinear Math. Phys.
Manuscripta Math.
96 (1998), 295-315. 2[29] J.-L. Loday, Cup product for Leibniz cohomology and dual Leibniz algebras.
Math. Scand.
77, Univ. LouisPasteur, Strasbourg, 1995, pp. 189-196. 2, 3[30] J.-L. Loday, Dialgebras. In Dialgebras and related operads.
Lecture Notes in Math.
J. Math. Phys.
47 (2006),013506. 3, 6[32] X. Ni and C. Bai, Poisson bialgebras.
J. Math. Phys.
54 (2013), 023515. 3[33] J. Pei, C. Bai and L. Guo, Splitting of operads and Rota-Baxter operators on operads.
Appl. Cate. Stru.
Bull. Amer. Math. Soc.
75 (1969), 325-329,330-334. 4[35] K. Uchino, Twisting on associative algebras and Rota-Baxter type operators.
J. Noncommut. Geom.
Trans. Amer. Math. Soc.
360 (2008), 5711-5769. 2[37] M. Wu and C. Bai, Compatible Lie bialgebras.
Comm. Theor. Phys.
63 (2015), 653-664. 3, 12, 15[38] P. Xu, Noncommutative Poisson algebras.
Amer. J. Math.
116 (1994), 101-125. 1[39] Y. Yang, Y. Yao and Y. Ye, (Quasi-)Poisson enveloping algebras.
Acta Math. Sin. (Engl. Ser.)
29 (2013), 105-118. 2, 3, 529 (2013), 105-118. 2, 3, 5