Noncommutative Poisson bialgebras

Abstract

In this paper, we introduce the notion of a noncommutative Poisson bialgebra, and establish the equivalence between matched pairs, Manin triples and noncommutative Poisson bialgebras. Using quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, we study coboundary noncommutative Poisson bialgebras which leads to the introduction of the Poisson Yang-Baxter equation. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra. Rota-Baxter operators, more generally $\mathcal{O}$-operators on noncommutative Poisson algebras, and noncommutative pre-Poisson algebras are introduced, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.

Introduction

This paper aims to study the bialgebra theory for noncommutative Poisson algebras, in particular coboundary ones. Skew-symmetric solutions of the Poisson Yang-Baxter equation in certain noncommutative Poisson algebras are constructed using $\mathcal{O}$-operators and noncommutative pre-Poisson algebras, which give coboundary noncommutative Poisson bialgebras.

The notion of a noncommutative Poisson algebra was first given by Xu in [38], which is especially suitable for geometric situations.

Definition 1.1

A noncommutative Poisson algebra is a triple $(P,{\cdot }_P,{\{-,-\}}_P)$, where $(P,{\cdot }_P)$ is an associative algebra (not necessarily commutative) and $(P,{\{-,-\}}_P)$ is a Lie algebra, such that the Leibniz rule holds:

$${\{x,y{\cdot }_{P}z\}}_P={\{x,y\}}_P{\cdot }_{P}z+y{\cdot }_P{\{x,z\}}_P,\forall x,y,z\in P.$$

In [18], Flato, Gerstenhaber and Voronov introduced a more general notion of a Leibniz pair and study its cohomology and deformation theory. In particular, they gave the cohomology theory of a noncommutative Poisson algebra associated to a representation using an innovative bicomplex. Recently, Bao and Ye developed the cohomology theory of noncommutative Poisson algebras associated to quasi-representations through Yoneda-Ext groups and projective resolutions in [9], [10]. Noncommutative Poisson algebras had been studied by many authors from different aspects [24], [25], [26], [39]. A Poisson algebra in the usual sense is the one where the associative multiplication on P is commutative. Note that there is another noncommutative analogue of Poisson algebras, 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 compatibility 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 class of nonassociative algebras coming from the study of convex homogeneous cones, affine manifolds and affine structures on Lie groups, and cohomologies of associative algebras. They also appeared in many fields in mathematics and mathematical physics, such as complex and symplectic structures on Lie groups and Lie algebras, integrable systems, Poisson brackets and infinite dimensional 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 algebra naturally through the sub-adjacent commutative associative algebra of the Zinbiel algebra and the sub-adjacent Lie algebra of the pre-Lie algebra. Conversely, a Rota-Baxter operator action (more generally an $\mathcal{O}$-operator action) on a Poisson algebra gives rise to a pre-Poisson algebra. We can summarize these relations by the following diagram:

For a given algebraic structure determined by a set of multiplications, a bialgebra structure on this algebra is obtained by a corresponding set of comultiplications together with a set of compatibility conditions between the multiplications and comultiplications. For a finite dimensional vector space V with the given algebraic structure, this can be achieved by equipping the dual space $V^{⁎}$ with the same algebraic structure 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 serves as the main motivation for our interest in a suitable bialgebra theory for the noncommutative Poisson algebra in this paper.

A good compatibility condition in a bialgebra is prescribed on the one hand by a strong motivation and potential applications, and on the other hand by a rich structure theory and effective constructions. 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 on A with coefficients in the tensor representation $A\otimes A$. In the Lie algebra context, a Lie bialgebra consists of a Lie algebra $(\mathfrak{g},{[-,-]}_{\mathfrak{g}}$) and a Lie coalgebra $(\mathfrak{g},\delta )$, where $\delta :\mathfrak{g}\to {\otimes }^2\mathfrak{g}$ is a Lie comultiplication, such that the Lie comultiplication being a 1-cocycle on $\mathfrak{g}$ with coefficients in the tensor representation $\mathfrak{g}\otimes \mathfrak{g}$. See [14], [17] for more details about Lie bialgebras and applications in mathematical physics. Thus, the representation theory and the cohomology theory usually play essential roles in the study of a bialgebra theory.

In fact, there has been a bialgebra theory for the usual (commutative) Poisson algebras, the so-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 this paper, we apply quasi-representations instead of representations and the corresponding cohomology theory ([9], [39]) to study noncommutative Poisson bialgebras. Even though both Lie algebras and associative algebras admit tensor representations as mentioned above, the tensor product of two representations of a noncommutative Poisson algebra is not a representation anymore, but a quasi-representation. This is the reason why quasi-representations, not representations, are the main ingredient in our study of noncommutative Poisson bialgebras. On the other hand, the dual of the regular representation $(L,R,\mathrm{ad})$ of a noncommutative Poisson algebra is also usually not a representation, but a quasi-representation. We introduce the concept of a coherent noncommutative 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 contains all the results of Poisson bialgebras given in [32].

Moreover, like the case of Poisson bialgebras, the study of coboundary noncommutative Poisson bialgebras leads to the introduction of the Poisson Yang-Baxter equation in a coherent noncommutative Poisson algebra. A skew-symmetric solution of the Poisson Yang-Baxter equation naturally gives a (coboundary) noncommutative Poisson bialgebra.

In addition, coherent noncommutative Poisson algebras are closely related to compatible Lie algebras 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 compatible Lie algebra. Consequently, a noncommutative Poisson bialgebra gives rise to a compatible Lie bialgebra ([37]), and there is also a similar relationship in the coboundary cases.

The notion of a dendriform algebra was introduced by Loday in [29] with motivation from periodicity of algebraic K-theory and operads.

Definition 1.2

A dendriform algebra is a vector space A with two bilinear maps $\succ :A\otimes A⟶A$ and $\prec :A\otimes A⟶A$ such that for all $x,y,z\in A$, the following equalities hold:

$$(x\prec y)\prec z=x\prec (y\succ z+y\prec z),(x\succ y)\prec z=x\succ (y\prec z),x\succ (y\succ z)=(x\succ y+x\prec y)\succ z.$$

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 algebras are given as follows:

A Zinbiel algebra can be viewed as a commutative dendriform algebra, namely $x\succ y=y\prec x$. In fact, from the operadic point of view, dendriform algebras and Zinbiel algebras can be viewed 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 consists of a dendriform algebra and a pre-Lie algebra, such that some compatibility conditions are satisfied. Through the sub-adjacent associative algebra and the sub-adjacent Lie algebra, a noncommutative pre-Poisson algebra gives rise to a noncommutative Poisson algebra naturally. Thus, noncommutative pre-Poisson algebras can be viewed as the splitting of noncommutative Poisson algebras. We further introduce the notion of a Rota-Baxter operator (more generally an $\mathcal{O}$-operator) on a noncommutative Poisson algebra, which is simultaneously a Rota-Baxter operator on the underlying associative algebra and a Rota-Baxter operator on the underlying Lie algebra. See [2], [5], [8], [13], [16], [22], [23], [34], [35] for more details on Rota-Baxter operators and $\mathcal{O}$-operators. A noncommutative pre-Poisson algebra can be obtained through the action of a Rota-Baxter operator (more generally an $\mathcal{O}$-operator). The above relations can be summarized into the following commutative diagram:

We construct skew-symmetric solutions of the Poisson Yang-Baxter equation in some special noncommutative Poisson algebras obtained from these structures.

In Section 2, we recall quasi-representations and the corresponding cohomology theory of noncommutative Poisson algebras, and introduce the notion of a coherent noncommutative Poisson algebra for our later study of noncommutative Poisson bialgebras.

In Section 3, we introduce the notions of matched pairs, Manin triples for noncommutative Poisson algebras and noncommutative (pseudo)-Poisson bialgebras. The equivalences between matched pairs of coherent noncommutative Poisson algebras, Manin triples for noncommutative Poisson 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 noncommutative Poisson algebra.

In Section 5, we introduce the notion of a noncommutative pre-Poisson algebra and a Rota-Baxter operator (more generally an $\mathcal{O}$-operator) on a noncommutative Poisson algebra, by which we construct skew-symmetric solutions of the Poisson Yang-Baxter equation in certain special noncommutative Poisson algebras obtained from these structures.

In this paper, all the vector spaces are over an algebraically closed field $\mathbb{K}$ of characteristic 0, and finite dimensional.