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J. Donin

Publications and source records attributed to J. Donin.

At least 19 recordsLinked to original sources
arXiv2004

Quantum groupoids and dynamical categories

In this paper we realize the dynamical categories introduced in our previous paper as categories of modules over bialgebroids; we study the bialgebroids arising in this way. We define quasitriangular structure on bialgebroids and present examples of quasitriangular bialgebroids related to the dynamical categories. We show that dynamical twists over an arbitrary base give rise to bialgebroid twists. We prove that the classical dynamical r-matrices over an arbitrary base manifold are in one-to-one correspondence with a special class of coboundary Lie bialgebroids.

math.QA↗
arXiv2003

Dynamical Yang-Baxter equation and quantum vector bundles

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, {\em etc}. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and their Levi subalgebra. Using this twist, we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive Lie groups.

math.QA↗
arXiv2002

$\U_q(sl(n))$-invariant quantization of symmetric coadjoint orbits via reflection equation algebra

We study relations between the two-parameter $\U_q(sl(n))$-invariant deformation quantization on $sl^*(n)$ and the reflection equation algebra. The latter is described by a quantum permutation on $\End(\C^n)$ given explicitly. The reflection equation algebra is used for constructing the one-parameter quantization on coadjoint orbits, including symmetric and certain bisymmetric and nilpotent ones. Our approach is based on embedding the quantized function algebras on the orbits into the algebra of functions on the quantum group $SL_q(n)$ via reflection equation algebra characters.

math.QA↗
arXiv2002

Method of quantum characters in equivariant quantization

Let $G$ be a reductive Lie group, $\g$ its Lie algebra, and $M$ a $G$-manifold. Suppose $\A_h(M)$ is a $\U_h(\g)$-equivariant quantization of the function algebra $\A(M)$ on $M$. We develop a method of building $\U_h(\g)$-equivariant quantization on $G$-orbits in $M$ as quotients of $\A_h(M)$. We are concerned with those quantizations that may be simultaneously represented as subalgebras in $\U^*_h(\g)$ and quotients of $\A_h(M)$. It turns out that they are in one-to-one correspondence with characters of the algebra $\A_h(M)$. We specialize our approach to the situation $\g=gl(n,\C)$, $M=\End(\C^n)$, and $\A_h(M)$ the so-called reflection equation algebra associated with the representation of $\U_h(\g)$ on $\C^n$. For this particular case, we present in an explicit form all possible quantizations of this type; they cover symmetric and bisymmetric orbits. We build a two-parameter deformation family and obtain, as a limit case, the $\U(\g)$-equivariant quantization of the Kirillov-Kostant-Souriau bracket on symmetric orbits.

math.QA↗
arXiv2002

Quantum coadjoint orbits of GL(n) and generalized Verma modules

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 of a generalized Verma module. As a corollary, we obtain an explicit description of the annihilators of generalized Verma modules over U(gl(n)). As an application, we construct real forms of the quantum orbits and classify finite dimensional representations. We compute the non-commutative Connes index for basic homogenous vector bundles over the quantum orbits.

math.QA↗
arXiv2002

On universal solution to reflection equation

For a given quasitriangular Hopf algebra $\Ha$ we study relations between the braided group $\tilde \Ha^*$ and Drinfeld's twist. We show that the braided bialgebra structure of $\tilde \Ha^*$ is naturally described by means of twisted tensor powers of $\Ha$ and their module algebras. We introduce universal solution to the reflection equation (RE) and deduce a fusion prescription for RE-matrices

math.QA↗
arXiv2002

Classification of polarized deformation quantizations

We give a classification of polarized deformation quantizations on a symplectic manifold with a (complex) polarization. Also, we establish a formula which relates the characteristic class of a polarized deformation quantization to its Fedosov class and the Chern class of the polarization.

math.QA↗
arXiv2002

Reflection Equation, Twist, and Equivariant Quantization

We prove that the reflection equation (RE) algebra $\La_R$ associated with a finite dimensional representation of a quasitriangular Hopf algebra $\Ha$ is twist-equivalent to the corresponding Faddeev-Reshetikhin-Takhtajan (FRT) algebra. We show that $\La_R$ is a module algebra over the twisted tensor square \twist{$\Ha$}{$\Ha$} and the double $\D(\Ha)$. We define FRT- and RE-type algebras and apply them to the problem of equivariant quantization on Lie groups and matrix spaces.

math.QA↗
arXiv2001

Quantum G-manifolds

Let $G$ be a Lie group, $\g$ its Lie algebra, and $U_h(\g)$ the corresponding quantum group. We consider some examples of $U_h(\g)$-invariant one and two parameter quantizations on $G$-manifolds.

math.QA↗
arXiv2000

Polarized deformation quantization

Let $A$ be a star product on a symplectic manifold $(M,ω_0)$, $\frac{1}{t}[ω]$ its Fedosov class, where $ω$ is a deformation of $ω_0$. We prove that for a complex polarization of $ω$ there exists a commutative subalgebra, $O$, in $A$ that is isomorphic to the algebra of functions constant along the polarization. Let $F(A)$ consists of elements of $A$ whose commutator with $O$ belongs to $O$. Then, $F(A)$ is a Lie algebra which is an $O$-extension of the Lie algebra of derivations of $O$. We prove a formula which relates the class of this extension, the Fedosov class, and the Chern class of $P$.

math.QA↗
arXiv2000

$\Uh$ invariant Quantization of Coadjoint Orbits and Vector Bundles over them

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 quantizations. We also describe all two parameter $U_h(\g)$ invariant quantizations on M, which can be considered as $U_h(\g)$ invariant quantizations of the Kirillov-Kostant-Souriau (KKS) Poisson bracket on M. We also consider how those quantizations relate to the natural polarizations of M with respect to the KKS bracket. Using polarizations, we quantize the sheaves of sections of vector bundles on M as one- and two-sided $U_h(\g)$ invariant modules over a quantized function sheaf.

math.QA↗
arXiv1999

Double quantization on coadjoint representations of simple Lie groups and its orbits

Let M be a manifold with an action of a Lie group G, $\A$ the function algebra on M. The first problem we consider is to construct a $U_h(\g)$ invariant quantization, $\A_h$, of $\A$, where $U_h(\g)$ is a quantum group corresponding to G. Let s be a G invariant Poisson bracket on M. The second problem we consider is to construct a $U_h(\g)$ invariant two parameter (double) quantization, $\A_{t,h}$, of $\A$ such that $\A_{t,0}$ is a G invariant quantization of $s$. We call $\A_{t,h}$ a $U_h(\g)$ invariant quantization of the Poisson bracket s. In the paper we study the cases when G is a simple Lie group and $M$ is the coadjoint representation $\g^*$ of G or a semisimple orbit in this representation. The paper is founded on the papers: J.Donin, Double quantization on the coadjoint representation of $sl(n)^*$, Czechoslovak J. of Physics, 47 (1997), no 11, 1115-1122, q-alg/9707031, and J.Donin, D.Gurevich, and S.Shnider, Double Quantization on Some Orbits in the Coadjoint Representations of Simple Lie Groups, Com. Math. Phys., 204 (1999), no. 1, 39-60, math/9807159, and contains some additional results.

math.QA↗
arXiv1998

Double quantization of $\cp$ type orbits by generalized Verma modules

It is known that symmetric orbits in ${\bf g}^*$ for any simple Lie algebra ${\bf g}$ are equiped with a Poisson pencil generated by the Kirillov-Kostant-Souriau bracket and the reduced Sklyanin bracket associated to the "canonical" R-matrix. We realize quantization of this Poisson pencil on $\cp$ type orbits (i.e. orbits in $sl(n+1)^*$ whose real compact form is $ CP^n$) by means of q-deformed Verma modules.

math.QA↗
arXiv1997

Double quantization on the coadjoint representation of sl(n)

For $\g=sl(n)$ we construct a two parametric $U_h(\g)$-invariant family of algebras, $(S\g)_{t,h}$, which defines a quantization of the function algebra $S\g$ on the coadjoint representation and in the parameter $t$ gives a quantization of the Lie bracket. The family induces a two parametric deformation of the function algebra of any maximal orbit which is a quantization of the Kirillov-Kostant-Souriau bracket in the parameter $t$. In addition we construct a quantum de Rham complex on $\g^*$.

q-alg↗
arXiv1997

Two types of Poisson pencils and related quantum objects

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 quantum mechanics is discussed.

q-alg↗
arXiv1995

Quantum hyperboloid and braided modules

When a quantum hyperboloid is realized, as a three - parameter algebra $\ahqc$, in the usual manner, the following problem arises: what is a ``representation theory'' of this algebra? We construct the series of all spin representations of $\ahqc$, and we discuss a braided version of the orbit method, i.e. a correspondence between orbits in $\gggg^*$ and $\gggg$-modules. A braided trace and a braided involution are discussed as well.

q-alg↗
arXiv1995

Cohomological construction of quantized universal enveloping algebras

Given an associative algebra $A$, and the category, $\cC$, of its finite dimensional modules, additional structures on the algebra $A$ induce corresponding ones on the category $\cC$. Thus, the structure of a rigid quasi-tensor (braided monoidal) category on $Rep_A$ is induced by an algebra homomorphism $A\to A\otimes A$ (comultiplication), coassociative up to conjugation by $Φ\in A^{\otimes 3}$ (associativity constraint) and cocommutative up to conjugation by $\cR\in A^{\otimes 2}$ (commutativity constraint), together with an antiautomorphism (antipode), $S$, of $A$ satisfying the certain compatibility conditions. A morphism of quasi-tensor structures is given by an element $F\in A^{\otimes 2}$ with suitable induced actions on $Φ$, $\cR$ and $S$. Drinfeld defined such a structure on $A=U(\cG)[[h]]$ for any semisimple Lie algebra $\cG$ with the usual comultiplication and antipode but nontrivial $\cR$ and $Φ$ and proved that the corresponding quasi-tensor category is isomomorphic to the category of representations of the Drinfeld-Jimbo (DJ) quantum universal enveloping algebra (QUE), $U_h(\cG)$. In the paper we give a direct cohomological construction of the $F$ which reduces $Φ$ to the trivial associativity constraint, without any assumption on the prior existence of a strictly coassociative QUE. Thus we get a new approach to the DJ quantization. We prove that $F$ can be chosen to satisfy some additional invariance conditions under (anti)automorphisms of $U(\cG)[[h]]$, in particular, $F$ gives an isomorphism of rigid quasi-tensor categories. Moreover, we prove that for pure imaginary values of the deformation parameter, the elements $F$, $R$ and $Φ$ can be chosen to be

q-alg↗