Related papers: Poisson spectra in polynomial algebras

We consider ideals in the ring $\mathbb{Z}_2[x_1,\ldots, x_n]$ that contain the polynomials $x_i^2 - x_i$ for $i = 1, \ldots, n$ and give various results related to the one-to-one correspondence between these ideals and the subsets of…

Developing ideas based on combinatorial formulas for characteristic classes we introduce the algebra modeling secondary characteristic classes associated to $N$ connections. Certain elements of the algebra correspond to the ordinary and…

Using a Poisson bracket representation, in 3D, of the Lie algebra $\mathfrak{sl}(2)$, we first use highest weight representations to embed this into larger Lie algebras. These are then interpreted as symmetry and conformal symmetry algebras…

We present the classical Poisson-Lichnerowicz cohomology for the Poisson algebra of polynomials $\mathbb{C}[X_{1},..., X_{n}]$ using exterior calculus. After presenting some non homogeneous Poisson brackets on this algebra, we compute…

Let $\Bbbk$ be a field of characteristic zero. For any positive integer $n$ and any scalar $a\in\Bbbk$, we construct a family of Artin-Schelter regular algebras $R(n,a)$, which are quantisations of Poisson structures on…

Let $P$ be a Poisson algebra with a Lie bracket $\{, \}$ over a field $\F$ of characteristic $p\geq 0$. In this paper, the Lie structure of $P$ is investigated. In particular, if $P$ is solvable with respect to its Lie bracket, then we…

In this paper we introduce the class of graded Poisson color algebras as the natural generalization of graded Poisson algebras and graded Poisson superalgebras. For $\Lambda$ an arbitrary abelian group, we show that any of such…

The present paper is devoted to the complete classification of $4$-dimensional complex Poisson algebras, taking into account a classification, up to isomorphism, of the complex commutative associative algebras of dimension $4$, as well as…

The degenerations of Poisson-type algebras are studied in the following varieties in dimension two: Leibniz--Poisson algebras, transposed Leibniz--Poisson algebras, Novikov--Poisson algebras, commutative pre-Lie algebras, anti-pre-Lie…

The algebraic and geometric classification of all complex $3$-dimensional transposed Poisson algebras is obtained. Also, we discuss strong special $3$-dimensional transposed Poisson algebras.

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…

Let F,G in C[x_1,...,x_n] be two polynomials in n variables x_1,...,x_n over the complex numbers field C. In this paper, we prove that if the degree of the Poisson bracket [F,G] is small enough then there are strict constraints for…

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 describe double Poisson brackets in the sense of M. Van den Bergh on certain finite-dimensional algebras. In particular we prove that all possible double Poisson brackets on matrix algebras are "inner", i.e. given by some…

We introduce the notion of a multiplicative Poisson $\lambda$-bracket, which plays the same role in the theory of Hamiltonian differential-difference equations as the usual Poisson $\lambda$-bracket plays in the theory of Hamiltonian PDE.…

We develop the theory of double multiplicative Poisson vertex algebras. These structures, defined at the level of associative algebras, are shown to be such that they induce a classical structure of multiplicative Poisson vertex algebra on…

In a Hamiltonian system with first class constraints observables can be defined as elements of a quotient Poisson bracket algebra. In the gauge fixing method observables form a quotient Dirac bracket algebra. We show that these two algebras…

We study the algebraic structure of the Poisson algebra P(O) of polynomials on a coadjoint orbit O of a semisimple Lie algebra. We prove that P(O) splits into a direct sum of its center and its derived ideal. We also show that P(O) is…

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…

Various coordinate rings of varieties appearing in the theory of Poisson Lie groups and Poisson homogeneous spaces belong to the large, axiomatically defined class of symmetric Poisson nilpotent algebras, e.g. coordinate rings of Schubert…