Related papers: Double quantization of $\cp$ type orbits by genera…
Given a simple Lie algebra $\gggg$, we consider the orbits in $\gggg^*$ which are of R-matrix type, i.e., which possess a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the so-called R-matrix bracket. We call an…
In this paper we describe a multiparameter deformation of the function algebra of a semisimple coadjoint orbit. In the first section we use the representation of the Lie algebra on a generalized Verma module to quantize the Kirillov bracket…
Two types of Poisson pencils connected to classical R-matrices and their quantum counterparts are considered. A representation theory of the quantum algebras related to some symmetric orbits in $sl(n)^*$ is constructed. A twisted version of…
We describe two types of Poisson pencils generated by a linear bracket and a quadratic one arising from a classical R-matrix. A quantization scheme is discussed for each. The quantum algebras are represented as the enveloping algebras of…
We study one and two parameter quantizations of the function algebra on a semisimple orbit in the coadjoint representation of a simple Lie group subject to the condition that the multiplication on the quantized algebra is invariant under…
We check The Vaisman condition of geometric quantization for R-matrix type Poisson pencil on a coadjoint orbit of a compact Lie group. It is shown that this condition isn't satisfied.
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…
A version of Kirillov's orbit method states that the primitive spectrum of a generic quantisation $A$ of a Poisson algebra $Z$ should correspond bijectively to the symplectic leaves of $\operatorname{Spec}(Z)$. In this article we consider a…
A version of quantum orbit method is presented for real forms of equal rank of quantum complex simple groups. A quantum moment map is constructed, based on the canonical isomorphism between a quantum Heisenberg algebra and an algebra of…
Let $G$ be the complex general linear group and $g$ its Lie algebra equipped with a factorizable Lie bialgebra structure; let $U_h$ be the corresponding quantum group. We construct explicit $U_h$-equivariant quantization of Poisson orbit…
We discuss a framework for quantizing a Poisson manifold via the quantization of its symplectic groupoid, that combines the tools of geometric quantization with the results of Renault's theory of groupoid C*-algebras. This setting allows…
In our previous paper, we constructed an explicit GL(n)-equivariant quantization of the Kirillov--Kostant-Souriau bracket on a semisimple coadjoint orbit. In the present paper, we realize that quantization as a subalgebra of endomorphisms…
We present an explicit $\U_h\bigl(gl(n,\C)\bigr)$-equivariant quantization on coadjoint orbits of $GL(n,\C)$. It forms a two-parameter family quantizing the Poisson pair of the reflection equation and Kirillov-Kostant-Souriau brackets.
We classify filtered quantizations of conical symplectic singularities and use this to show that all filtered quantizations of symplectic quotient singularities are spherical Symplectic reflection algebras of Etingof and Ginzburg. We…
Let $O$ be a closed Poisson conjugacy class of the complex algebraic Poisson group GL(n) relative to the Drinfeld-Jimbo factorizable classical r-matrix. Denote by $T$ the maximal torus of diagonal matrices in GL(n). With every $a\in O\cap…
Star products on the classical double group of a simple Lie group and on corresponding symplectic grupoids are given so that the quantum double and the "quantized tangent bundle" are obtained in the deformation description. "Complex"…
In this letter, we revisit the quantisation problem for a fundamental model of classical mechanics - the Zhukovsky-Volterra top. We have discovered a four-parametric pencil of compatible Poisson brackets, comprising two quadratic and two…
Generalized flag manifolds endowed with the Bruhat-Poisson bracket are compact Poisson homogeneous spaces, whose decompositions in symplectic leaves coincide with their stratifications in Schubert cells. In this note it is proved that the…
The orbit method of Kirillov is used to derive the p-mechanical brackets [math-ph/0007030, quant-ph/0212101]. They generate the quantum (Moyal) and classic (Poisson) brackets on respective orbits corresponding to representations of the…
Let M be a coadjoint semisimple orbit of a simple Lie group G. Let $U_h(\g)$ be a quantum group corresponding to G. We construct a universal family of $U_h(\g)$ invariant quantizations of the sheaf of functions on M and describe all such…