Related papers: Double Multiplicative Poisson Vertex Algebras
A noncommutative (NC) version of Poisson geometry was initiated by Van den Bergh by introducing at the level of associative algebras the formalism of double Poisson brackets. Their key property is to induce (standard) Poisson brackets under…
Double Poisson structures (a la Van den Bergh) on commutative algebras are studied; the main result shows that there are no non-trivial such structures on polynomial algebras of Krull dimension greater than one. For a general commutative…
A new Lax representation for the Bogoyavlensky lattice is found, its $r$--matrix interpretation is elaborated. The $r$--matrix structure turns out to be related to a highly nonlocal quadratic Poisson structure on a direct sum of associative…
The notions of vertex Lie algebra and vertex Poisson algebra are presented and connections among vertex Lie algebras, vertex Poisson algebras and vertex algebras are discussed.
Logarithmic vertex algebras were introduced in our previous paper, motivated by logarithmic conformal field theory. Non-local Poisson vertex algebras were introduced by De Sole and Kac, motivated by the theory of integrable systems. We…
Quadratic Poisson brackets on a vector space equipped with a bilinear multiplication are studied. A notion of a bracket compatible with the multiplication is introduced and an effective criterion of such compatibility is given. Among…
We construct a method to obtain the algebraic classification of Poisson algebras defined on a commutative associative algebra, and we apply it to obtain the classification of the $3$-dimensional Poisson algebras. In addition, we study the…
Poisson algebra is usually defined to be a commutative algebra together with a Lie bracket, and these operations are required to satisfy the Leibniz rule. We describe Poisson structures in terms of a single bilinear operation. This enables…
The aim of this paper is to introduce the notion of (noncommutative) transposed Poisson conformal algebras, which serve as the conformal analogues of transposed Poisson algebras and admit a rich class of identities. We show that the tensor…
The paper is devoted to the Poisson brackets compatible with multiplication in associative algebras. These brackets are shown to be quadratic and their relations with the classical Yang--Baxter equation are revealed. The paper also contains…
The relations between integrable Poisson algebras with three generators and two-dimensional manifolds are investigated. Poisson algebraic maps are also discussed.
A class of Poisson algebras considered as a Poisson version of the multiparameter quantized coordinate rings of symplectic and Euclidean $2n$-spaces is constructed and the prime Poisson ideals and the symplectic ideals of these Poisson…
This book offers an introduction to vertex algebra based on a new approach. The new approach says that a vertex algebra is an associative algebra such that the underlying Lie algebra is a vertex Lie algebra. In particular, vertex algebras…
Let $A=F[x,y]$ be the polynomial algebra on two variables $x,y$ over an algebraically closed field $F$ of characteristic zero. Under the Poisson bracket, $A$ is equipped with a natural Lie algebra structure. It is proven that the maximal…
The notion of Poisson dialgebras was introduced by Loday. In this article, we propose a new definition with some modifications that is supported by several canonical examples coming from Poisson algebra modules, averaging operators on…
We introduce coupled double Poisson brackets on an associative algebra $A$ as pairs consisting of a generalized Van den Bergh's double Poisson bracket and a generalized Fairon--McCulloch's right double Poisson bracket subject to a…
Poisson superpair is a pair of Poisson superalgebra structures on a super commutative associative algebra, whose any linear combination is also a Poisson superalgebra structure. In this paper, we first construct certain linear and quadratic…
We develop a theory of noncommutative Poisson extensions. For an augmented dg algebra \(A\), we show that any shifted double Poisson bracket on \(A\) induces a graded Lie algebra structure on the reduced cyclic homology. Under the…
We study a family of algebras defined using a locally-finite endomorphism called a braiding map. When the braiding map is semi-simple, the algebra is a generalized vertex algebra, while when the braiding map is locally-nilpotent we have a…
We describe a conjectural classification of Poisson vertex algebras of CFT type and of Poisson vertex algebras in one differential variable (= scalar Hamiltonian operators).