Related papers: Poisson brackets on some skew PBW extensions

We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the…

We examine PBW deformations of finite group extensions of skew polynomial rings, in particular the quantum Drinfeld orbifold algebras defined by the first author. We give a homological interpretation, in terms of Gerstenhaber brackets, of…

In this paper we prove the universal property of skew $PBW$ extensions generalizing this way the well known universal property of skew polynomial rings. For this, we will show first a result about the existence of this class of…

Let $B$ be a Poisson algebra $\Bbb C[x_1,\ldots, x_k]$ with Poisson bracket such that $$\{x_j,x_i\}=c_{ji}x_ix_j+p_{ji}$$ for all $j>i$, where $c_{ji}\in\Bbb C$ and $p_{ji}\in\Bbb C[x_1,\ldots,x_i]$. Here we obtain an iterated skew…

In this work we study the automorphisms of skew $PBW$ extensions and skew quantum polynomials. We use Artamonov's works as reference for getting the principal results about automorphisms for generic skew $PBW$ extensions and generic skew…

Poisson brackets on the polynomial algebra C[x,y,z] are studied. A description of all such brackets is given and, for a significant class of Poisson brackets, the Poisson prime ideals and Poisson primitive ideals are determined. The results…

We give a physicist oriented survey of Poisson-Lie symmetries of classical systems. We consider finite dimensional geometric actions and the chiral WZNW model as examples for the general construction. An essential point is that quadratic…

As a generalization of the linear Poisson bracket on the dual space of a Lie algebra, we introduce certain non-linear Poisson brackets which are ``cocycle perturbations'' of the linear Poisson bracket. We show that these special Poisson…

We suggest two explicit descriptions of the Poisson q-W algebras which are Poisson algebras of regular functions on certain algebraic group analogues of the Slodowy transversal slices to adjoint orbits in a complex semisimple Lie algebra g.…

The classical commutative coding theory has been recently extended to noncommutative rings of polynomial type. There are many interesting works in coding theory over single Ore extensions. In this review article we present the most relevant…

We present a class of Poisson structures on trivial extension algebras which generalize some known structures induced by Poisson modules. We show that there exists a one-to-one correspondence between such a class of Poisson structures and…

In this paper we study skew Poincar\'e-Birkhoff-Witt extensions over weak symmetric and $(\Sigma,\Delta)$-weak symmetry rings. Since these extensions generalize Ore extensions of injective type and another noncommutative rings of polynomial…

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…

Recently Pascal Baseilhac and Stefan Kolb obtained a PBW basis for the $q$-Onsager algebra $\mathcal O_q$. They defined the PBW basis elements recursively, and it is obscure how to express them in closed form. To mitigate the difficulty, we…

We introduce a bracket on 1-forms defined on ${\cal J}^{\infty}(S^1, \mathbb{R}^n)$, the infinite jet extension of the space of loops and prove that it satisfies the standard properties of a Poisson bracket. Using this bracket, we show that…

We prove that if R is a left Noetherian and left regular ring then the same is true for any bijective skew PBW extension A of R. From this we get Serre's Theorem for such extensions. We show that skew PBW extensions and its localizations…

Quadratic Poisson brackets on associative algebras are studied. Such a bracket compatible with the multiplication is related to a differentiation in tensor square of the underlying algebra. Jacobi identity means that this differentiation…

In this paper we extend some results obtained by Artamonov and Sabitov for quantum polynomials to skew quantum polynomials and quasi-commutative bijective skew PBW extensions. Moreover, we find a counterexample to the conjecture proposed in…

In this paper, we define and study the universal enveloping algebra of a Poisson superalgebra. In particular, a new PBW Theorem for Lie-Rinehart superalgebras is proved leading to a PBW Theorem for Poisson superalgebras, we show the…

In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived using the so called Ostrogradsky transformation.