Related papers: Quantization of Poisson algebraic groups and Poiss…
The dual Lie bialgebra of a certain ``quasitriangular'' Lie bialgebra structure on the Heisenberg Lie algebra determines a (non-compact) Poisson--Lie group G. The compatible Poisson bracket on G is non-linear, but it can still be realized…
We compute the Poisson cohomology of the linear Poisson structure dual to the n-dimensional "book" Lie algebra, defined by [e_0,e_i]=e_i, [e_i,e_j]=0, for i,j=1,...,n-1.
Following a recent group theoretical quantization of the symplectic space S={(phi in R mod 2pi, p>0)} in terms of irreducible unitary representations of the group SO(1,2) the present paper proposes an application of those results to the old…
In this work, we find the Poisson superalgebras related to schemes of quantization. Initially, we consider the Dirac superbracket in the context of the quantization of constrained systems. Next, we show the existence of a Poisson…
We describe the Poisson ideals and attached symplectic geometry of a cluster algebra with compatible Poisson structure. We apply these results to determine the spectrum of a quantum cluster algebra. As an application, we describe the…
We prove that the group of automorphisms of the Lie algebra $\Der_K (P_n)$ of derivations of a polynomial algebra $P_n=K[x_1,..., x_n]$ over a field of characteristic zero is canonically isomorphic to the the group of automorphisms of the…
It is known that the computation of the Poisson cohomology is closely related to the classification of singularities of Poisson structures. In this paper, we will first look for the normal forms of germs at (0,0) of Poisson structures on…
We classify the centers of the quantized Weyl algebras that are PI and derive explicit formulas for the discriminants of these algebras over a general class of polynomial central subalgebras. Two different approaches to these formulas are…
We classify real Poisson structures on complex toric manifolds of type $(1,1)$ and initiate an investigation of their Poisson cohomology. For smooth toric varieties, such structures are necessarily algebraic and are homogeneous quadratic in…
Let k be an algebraically closed field of odd characteristic. We describe derivations of a large class of quantizations of affine normal Poisson varieties over k.
In this article we discuss the geometric quantization on a certain type of infinite dimensional super-disc. Such systems are quite natural when we analyze coupled bosons and fermions. The large-N limit of a system like that corresponds to a…
A Poisson algebra is a Lie algebra endowed with a commutative associative product in such a way that the Lie and associative products are compatible via a Leibniz rule. If we part from a Lie color algebra, instead of a Lie algebra, a…
We construct a first order local model for Poisson manifolds around a large class of Poisson submanifolds and we give conditions under which this model is a local normal form. The resulting linearization theorem includes as special cases…
By considering suitable Poisson groupoids, we develop an approach to obtain Lie group structures on (subgroups of) the Poisson diffeomorphism groups of various classes of Poisson manifolds. As applications, we show that the Poisson…
This paper is concerned with the Poisson transform of differential forms on the hyperbolic space $H^n(\mathbb R)$. Consider an integer $p$ such that $1\leqslant p\leqslant n$ and let $q$ be either $p-1$ or $p$. For $1<r<\infty$, we prove…
All factorizable Lie bialgebra structures on complex reductive Lie algebras were described by Belavin and Drinfeld. We classify the symplectic leaves of the full class of corresponding connected Poisson-Lie groups. A formula for their…
We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet--Deny theorem holds for compact quantum groups; also, the result of…
The notion of Poisson manifold with compatible pseudo-metric was introduced by the author in [1]. In this paper, we introduce a new class of Lie algebras which we call a pseudo-Rieamannian Lie algebras. The two notions are strongly related:…
There is a simple and natural quantization of differential forms on odd Poisson supermanifolds, given by the relation [f,dg]={f,g} for any two functions f and g. We notice that this non-commutative differential algebra has a geometrical…
We propose an algebraic viewpoint of the problem of deformation quantization of the so called almost Poisson algebras, which are algebras with a commutative associative product and an antisymmetric bracket which is a bi-derivation but does…