Related papers: Poisson algebras of block-upper-triangular bilinea…

In the algebra Sym(gl(m)) we consider Poisson pencils generated by the linear Poisson-Lie bracket {,}_{gl(m)} and that corresponding to the so-called Reflection Equation Algebra. Each bracket of such a pencil has the Poisson center…

In this paper we study associative algebras with a Poisson algebra structure on the center acting by derivations on the rest of the algebra. These structures, which we call Poisson fibred algebras, appear in the study of quantum groups at…

This article presents a differential groupoid with ``coaction'' of the groupoid underlying the Quantum Euclidean Group (i.e. its $C^*$-algebra is the $C^*$-algebra of this quantum group). The dual of the Lie algebroid is a Poisson manifold…

If g is a quasitriangular Lie bialgebra, one can asks what is the geometrical meaning of its r-matrix. A first answer was given in a paper by Weinstein and Xu, using purely geometrical means: roughly, one has that the formal Poisson group…

The standard Poisson structure on the rectangular matrix variety M_{m,n}(C) is investigated, via the orbits of symplectic leaves under the action of the maximal torus T of GL_{m+n}(C). These orbits, finite in number, are shown to be smooth…

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 consider a skew-symmetric $n$-ary bracket on the polynomial algebra $K[x_1,\ldots,x_n,x_{n+1}]$ ($n\geq 2$) over a field $K$ of characteristic zero defined by $\{a_1,\ldots,a_n\}=J(a_1,\ldots,a_n,C)$, where $C$ is a fixed element of…

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…

It is shown that every $2$-shifted Poisson structure on a finitely generated semi-free commutative differential graded algebra $A$ defines a very explicit infinitesimal $2$-braiding on the homotopy $2$-category of the symmetric monoidal…

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…

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 construct explicitly a class of coboundary Poisson-Lie structures on the group of formal diffeomorphisms of ${\Bbb R}^n$. Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra $W_n$…

The correspondence between Poisson structures and symplectic groupoids, analogous to the one of Lie algebras and Lie groups, plays an important role in Poisson geometry; it offers, in particular, a unifying framework for the study of…

Let g be a quasitriangular Lie bialgebra over a field k of characteristic zero, and let g^* be its dual Lie bialgebra. We prove that the formal Poisson group F[[g^*]] is a braided Hopf algebra. More generally, we prove that if (U_h,R) is…

Given a manifold M with an action of a quadratic Lie algebra d, such that all stabilizer algebras are co-isotropic in d, we show that the product M\times d becomes a Courant algebroid over M. If the bilinear form on d is split, the choice…

Wilcox has considered a twisted semigroup algebra structure on the partition algebra $\mathbb{C}A_k(n)$, but it appears that there has not previously been any known basis that gives $\mathbb{C}A_k(n)$ the structure of a "non-twisted"…

Motivated by a search for Lie group structures on groups of Poisson diffeomorphisms [24], we investigate linearizability of Poisson structures of Poisson groupoids around the unit section. After extending the Lagrangian neighbourhood…

We describe transposed Poisson structures on the upper triangular matrix Lie algebra $T_n(F)$, $n>1$, over a field $F$ of characteristic zero. We prove that, for $n>2$, any such structure is either of Poisson type or the orthogonal sum of a…

We review nonabelian Poisson structures on affine and projective spaces over $\mathbb{C}$. We also construct a class of examples of nonabelian Poisson structures on $\mathbb{C} P^{n-1}$ for $n>2$. These nonabelian Poisson structures depend…

We construct quadratic finite-dimensional Poisson algebras and their quantum versions related to rank N and degree one vector bundles over elliptic curves with n marked points. The algebras are parameterized by the moduli of curves. For N=2…